diff --git a/doc/pub/week40/html/._week40-bs000.html b/doc/pub/week40/html/._week40-bs000.html
index 07f00c933..5bc847095 100644
--- a/doc/pub/week40/html/._week40-bs000.html
+++ b/doc/pub/week40/html/._week40-bs000.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -231,7 +321,7 @@ MathJax.Hub.Config({
 <!-- ------------------- main content ---------------------- -->
 <div class="jumbotron">
 <center>
-<h1>Week 40: Neural networks</h1>
+<h1>Week 40: Gradient descent methods (continued) and start Neural networks</h1>
 </center>  <!-- document title -->
 
 <!-- author(s): Morten Hjorth-Jensen -->
@@ -247,7 +337,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Oct 8, 2022</h4>
+<h4>October 2-6, 2023</h4>
 </center> <!-- date -->
 <br>
 
@@ -272,7 +362,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs008.html">9</a></li>
   <li><a href="._week40-bs009.html">10</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs001.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
@@ -286,7 +376,7 @@ MathJax.Hub.Config({
 </footer>
 -->
 <center style="font-size:80%">
-<!-- copyright --> &copy; 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+<!-- copyright --> &copy; 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 </center>
 </body>
 </html>
diff --git a/doc/pub/week40/html/._week40-bs001.html b/doc/pub/week40/html/._week40-bs001.html
index f53794cc3..59d315e61 100644
--- a/doc/pub/week40/html/._week40-bs001.html
+++ b/doc/pub/week40/html/._week40-bs001.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,22 +319,42 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0001"></a>
 <!-- !split -->
-<h2 id="plan-for-week-40" class="anchor">Plan for week 40 </h2>
+<h2 id="plans-for-week-40" class="anchor">Plans for week 40 </h2>
 
+<div class="panel panel-default">
+<div class="panel-body">
+<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 <ul>
-<li> Thursday: Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.</li>
-<ul>
-  <li> <a href="https://youtu.be/sCHiGSoG2lE" target="_self">Video of lecture</a></li>
+  <li> Work on project 1 and discussions on how to structure your report</li>
+  <li> No weekly exercises for week 40, project work only</li>
+  <li> <a href="https://youtu.be/tVW1ZDmZnwM" target="_self">Video on how to write scientific reports recorded during one of the lab sessions</a></li>
+  <li> A general guideline can be found at <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md" target="_self"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</tt></a>.</li>
 </ul>
-<li> Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.</li>
+</div>
+</div>
+  
+
+<div class="panel panel-default">
+<div class="panel-body">
+<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
 <ul>
-  <li> <a href="https://youtu.be/hDTtA7PRRfI" target="_self">Video of lecture</a></li>
+  <li> Stochastic Gradient descent with examples and automatic differentiation</li>
+  <li> Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.</li>  
+  <li> Readings and Videos:</li>
+<ul>
+    <li> These lecture notes</li>
+    <li> For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.</li>
+    <li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_self">Aurelien Geron's chapter 4 on stochastic gradient descent</a></li>
+    <li> For neural networks we recommend Goodfellow et al chapter 6.</li>
+    <li> <a href="https://www.youtube.com/watch?v=sDv4f4s2SB8" target="_self">Video on gradient descent</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI" target="_self">Video on stochastic gradient descent</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_self">Neural Networks demystified</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_self">Building Neural Networks from scratch</a></li>
 </ul>
 </ul>
-<p>Reading suggestions for both days: <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_self">Aurelien Geron's chapter 10</a> and Hastie et al chapter 11.
-For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
-For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
-</p>
+</div>
+</div>
+
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -262,7 +372,7 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.
   <li><a href="._week40-bs009.html">10</a></li>
   <li><a href="._week40-bs010.html">11</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs002.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs002.html b/doc/pub/week40/html/._week40-bs002.html
index 25cccf96e..5b9bfb8a4 100644
--- a/doc/pub/week40/html/._week40-bs002.html
+++ b/doc/pub/week40/html/._week40-bs002.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,10 +319,16 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0002"></a>
 <!-- !split -->
-<h2 id="overview-video-on-stochastic-gradient-descent" class="anchor">Overview video on Stochastic Gradient Descent </h2>
-
-<a href="https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" target="_self">What is Stochastic Gradient Descent</a>
+<h2 id="summary-from-last-week-using-gradient-descent-methods-limitations" class="anchor">Summary from last week, using gradient descent methods, limitations </h2>
 
+<ul>
+<li> <b>Gradient descent (GD) finds local minima of our function</b>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</li>
+<li> <b>GD is sensitive to initial conditions</b>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</li>
+<li> <b>Gradients are computationally expensive to calculate for large datasets</b>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called &quot;mini batches&quot;. This has the added benefit of introducing stochasticity into our algorithm.</li>
+<li> <b>GD is very sensitive to choices of learning rates</b>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</li>
+<li> <b>GD treats all directions in parameter space uniformly.</b> Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</li> 
+<li> GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</li>
+</ul>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -250,7 +346,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs010.html">11</a></li>
   <li><a href="._week40-bs011.html">12</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs003.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs003.html b/doc/pub/week40/html/._week40-bs003.html
index b474ffd27..93afec08b 100644
--- a/doc/pub/week40/html/._week40-bs003.html
+++ b/doc/pub/week40/html/._week40-bs003.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,15 +319,9 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0003"></a>
 <!-- !split -->
-<h2 id="practical-tips" class="anchor">Practical tips </h2>
+<h2 id="overview-video-on-stochastic-gradient-descent" class="anchor">Overview video on Stochastic Gradient Descent </h2>
 
-<ul>
-<li> <b>Randomize the data when making mini-batches</b>. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.</li>
-<li> <b>Transform your inputs</b>. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.</li>
-<li> <b>Monitor the out-of-sample performance.</b> Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This <em>early stopping</em> significantly improves performance in many settings.</li>
-<li> <b>Adaptive optimization methods don't always have good generalization.</b> Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.</li>
-</ul>
-<p>Geron's text, see chapter 11, has several interesting discussions.</p>
+<a href="https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" target="_self">What is Stochastic Gradient Descent</a>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -257,7 +341,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs011.html">12</a></li>
   <li><a href="._week40-bs012.html">13</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs004.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs004.html b/doc/pub/week40/html/._week40-bs004.html
index 5e0858c7a..12cea2617 100644
--- a/doc/pub/week40/html/._week40-bs004.html
+++ b/doc/pub/week40/html/._week40-bs004.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,33 +319,18 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0004"></a>
 <!-- !split -->
-<h2 id="automatic-differentiation" class="anchor">Automatic differentiation </h2>
+<h2 id="batches-and-mini-batches" class="anchor">Batches and mini-batches </h2>
 
-<p><a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_self">Automatic differentiation (AD)</a>, 
-also called algorithmic
-differentiation or computational differentiation,is a set of
-techniques to numerically evaluate the derivative of a function
-specified by a computer program. AD exploits the fact that every
-computer program, no matter how complicated, executes a sequence of
-elementary arithmetic operations (addition, subtraction,
-multiplication, division, etc.) and elementary functions (exp, log,
-sin, cos, etc.). By applying the chain rule repeatedly to these
-operations, derivatives of arbitrary order can be computed
-automatically, accurately to working precision, and using at most a
-small constant factor more arithmetic operations than the original
-program.
-</p>
+<p>In gradient descent we compute the cost function and its gradient for all data points we have.</p>
 
-<p>Automatic differentiation is neither:</p>
-
-<ul>
-<li> Symbolic differentiation, nor</li>
-<li> Numerical differentiation (the method of finite differences).</li>
-</ul>
-<p>Symbolic differentiation can lead to inefficient code and faces the
-difficulty of converting a computer program into a single expression,
-while numerical differentiation can introduce round-off errors in the
-discretization process and cancellation
+<p>In large-scale applications such as the <a href="https://www.image-net.org/challenges/LSVRC/" target="_self">ILSVRC challenge</a>, the
+training data can have on order of millions of examples. Hence, it
+seems wasteful to compute the full cost function over the entire
+training set in order to perform only a single parameter update. A
+very common approach to addressing this challenge is to compute the
+gradient over batches of the training data. For example, a typical batch could contain some thousand  examples from
+an  entire training set of several millions. This batch is then used to
+perform a parameter update.
 </p>
 
 <p>
@@ -277,7 +352,7 @@ discretization process and cancellation
   <li><a href="._week40-bs012.html">13</a></li>
   <li><a href="._week40-bs013.html">14</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs005.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs005.html b/doc/pub/week40/html/._week40-bs005.html
index 233203cff..b1fba64b4 100644
--- a/doc/pub/week40/html/._week40-bs005.html
+++ b/doc/pub/week40/html/._week40-bs005.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
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         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,11 +319,31 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0005"></a>
 <!-- !split -->
-<h2 id="videos-on-neural-networks" class="anchor">Videos on Neural Networks </h2>
+<h2 id="stochastic-gradient-descent-sgd" class="anchor">Stochastic Gradient Descent (SGD) </h2>
 
-<a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_self">Neural Networks demystified</a>
+<p>In stochastic gradient descent, the extreme case is the case where we
+have only one batch, that is we include the whole data set.
+</p>
 
-<a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_self">Building Neural Networks from scratch</a>
+<p>This process is called Stochastic Gradient
+Descent (SGD) (or also sometimes on-line gradient descent). This is
+relatively less common to see because in practice due to vectorized
+code optimizations it can be computationally much more efficient to
+evaluate the gradient for 100 examples, than the gradient for one
+example 100 times. Even though SGD technically refers to using a
+single example at a time to evaluate the gradient, you will hear
+people use the term SGD even when referring to mini-batch gradient
+descent (i.e. mentions of MGD for &#8220;Minibatch Gradient Descent&#8221;, or BGD
+for &#8220;Batch gradient descent&#8221; are rare to see), where it is usually
+assumed that mini-batches are used. The size of the mini-batch is a
+hyperparameter but it is not very common to cross-validate or bootstrap it. It is
+usually based on memory constraints (if any), or set to some value,
+e.g. 32, 64 or 128. We use powers of 2 in practice because many
+vectorized operation implementations work faster when their inputs are
+sized in powers of 2.
+</p>
+
+<p>In our notes with  SGD we mean stochastic gradient descent with mini-batches.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -255,7 +365,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs013.html">14</a></li>
   <li><a href="._week40-bs014.html">15</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs006.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs006.html b/doc/pub/week40/html/._week40-bs006.html
index a05002ee7..8a678db09 100644
--- a/doc/pub/week40/html/._week40-bs006.html
+++ b/doc/pub/week40/html/._week40-bs006.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,17 +319,22 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0006"></a>
 <!-- !split -->
-<h2 id="neural-networks" class="anchor">Neural networks </h2>
+<h2 id="stochastic-gradient-descent" class="anchor">Stochastic Gradient Descent </h2>
 
-<p>Artificial neural networks are computational systems that can learn to
-perform tasks by considering examples, generally without being
-programmed with any task-specific rules. It is supposed to mimic a
-biological system, wherein neurons interact by sending signals in the
-form of mathematical functions between layers. All layers can contain
-an arbitrary number of neurons, and each connection is represented by
-a weight variable.
+<p>Stochastic gradient descent (SGD) and variants thereof address some of
+the shortcomings of the Gradient descent method discussed above.
 </p>
 
+<p>The underlying idea of SGD comes from the observation that the cost
+function, which we want to minimize, can almost always be written as a
+sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \),
+</p>
+$$
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}). 
+$$
+
+
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -261,7 +356,7 @@ a weight variable.
   <li><a href="._week40-bs014.html">15</a></li>
   <li><a href="._week40-bs015.html">16</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs007.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs007.html b/doc/pub/week40/html/._week40-bs007.html
index 6c010bff9..777c770ff 100644
--- a/doc/pub/week40/html/._week40-bs007.html
+++ b/doc/pub/week40/html/._week40-bs007.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,63 +319,21 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0007"></a>
 <!-- !split -->
-<h2 id="artificial-neurons" class="anchor">Artificial neurons  </h2>
+<h2 id="computation-of-gradients" class="anchor">Computation of gradients </h2>
 
-<p>The field of artificial neural networks has a long history of
-development, and is closely connected with the advancement of computer
-science and computers in general. A model of artificial neurons was
-first developed by McCulloch and Pitts in 1943 to study signal
-processing in the brain and has later been refined by others. The
-general idea is to mimic neural networks in the human brain, which is
-composed of billions of neurons that communicate with each other by
-sending electrical signals.  Each neuron accumulates its incoming
-signals, which must exceed an activation threshold to yield an
-output. If the threshold is not overcome, the neuron remains inactive,
-i.e. has zero output.
+<p>This in turn means that the gradient can be
+computed as a sum over \( i \)-gradients 
 </p>
-
-<p>This behaviour has inspired a simple mathematical model for an artificial neuron.</p>
-
 $$
-\begin{equation}
- y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
-\tag{1}
-\end{equation}
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
 $$
 
-<p>Here, the output \( y \) of the neuron is the value of its activation function, which have as input
-a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
-</p>
-
-<p>Conceptually, it is helpful to divide neural networks into four
-categories:
-</p>
-<ol>
-<li> general purpose neural networks for supervised learning,</li>
-<li> neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),</li>
-<li> neural networks for sequential data such as Recurrent Neural Networks (RNNs), and</li>
-<li> neural networks for unsupervised learning such as Deep Boltzmann Machines.</li>
-</ol>
-<p>In natural science, DNNs and CNNs have already found numerous
-applications. In statistical physics, they have been applied to detect
-phase transitions in 2D Ising and Potts models, lattice gauge
-theories, and different phases of polymers, or solving the
-Navier-Stokes equation in weather forecasting.  Deep learning has also
-found interesting applications in quantum physics. Various quantum
-phase transitions can be detected and studied using DNNs and CNNs,
-topological phases, and even non-equilibrium many-body
-localization. Representing quantum states as DNNs quantum state
-tomography are among some of the impressive achievements to reveal the
-potential of DNNs to facilitate the study of quantum systems.
-</p>
-
-<p>In quantum information theory, it has been shown that one can perform
-gate decompositions with the help of neural. 
-</p>
-
-<p>The applications are not limited to the natural sciences. There is a
-plethora of applications in essentially all disciplines, from the
-humanities to life science and medicine.
+<p>Stochasticity/randomness is introduced by only taking the
+gradient on a subset of the data called minibatches.  If there are \( n \)
+data points and the size of each minibatch is \( M \), there will be \( n/M \)
+minibatches. We denote these minibatches by \( B_k \) where
+\( k=1,\cdots,n/M \).
 </p>
 
 <p>
@@ -310,7 +358,7 @@ humanities to life science and medicine.
   <li><a href="._week40-bs015.html">16</a></li>
   <li><a href="._week40-bs016.html">17</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs008.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs008.html b/doc/pub/week40/html/._week40-bs008.html
index 8c0ab4f99..b9f8ef17b 100644
--- a/doc/pub/week40/html/._week40-bs008.html
+++ b/doc/pub/week40/html/._week40-bs008.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,29 +319,28 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0008"></a>
 <!-- !split -->
-<h2 id="neural-network-types" class="anchor">Neural network types </h2>
-
-<p>An artificial neural network (ANN), is a computational model that
-consists of layers of connected neurons, or nodes or units.  We will
-refer to these interchangeably as units or nodes, and sometimes as
-neurons.
+<h2 id="sgd-example" class="anchor">SGD example </h2>
+<p>As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
+and we choose to have \( M=5 \) minibathces,
+then each minibatch contains two data points. In particular we have
+\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
+(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you
+have only a single batch with all data points and on the other extreme,
+you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e
+\( B_k = \mathbf{x}_k \).
 </p>
 
-<p>It is supposed to mimic a biological nervous system by letting each
-neuron interact with other neurons by sending signals in the form of
-mathematical functions between layers.  A wide variety of different
-ANNs have been developed, but most of them consist of an input layer,
-an output layer and eventual layers in-between, called <em>hidden
-layers</em>. All layers can contain an arbitrary number of nodes, and each
-connection between two nodes is associated with a weight variable.
+<p>The idea is now to approximate the gradient by replacing the sum over
+all data points with a sum over the data points in one the minibatches
+picked at random in each gradient descent step 
 </p>
+$$
+\nabla_{\beta}
+C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
+c_i(\mathbf{x}_i, \mathbf{\beta}).
+$$
 
-<p>Neural networks (also called neural nets) are neural-inspired
-nonlinear models for supervised learning.  As we will see, neural nets
-can be viewed as natural, more powerful extensions of supervised
-learning methods such as linear and logistic regression and soft-max
-methods we discussed earlier.
-</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -276,7 +365,7 @@ methods we discussed earlier.
   <li><a href="._week40-bs016.html">17</a></li>
   <li><a href="._week40-bs017.html">18</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs009.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs009.html b/doc/pub/week40/html/._week40-bs009.html
index 5ae6ad5a5..f4d23cae6 100644
--- a/doc/pub/week40/html/._week40-bs009.html
+++ b/doc/pub/week40/html/._week40-bs009.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,19 +319,19 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0009"></a>
 <!-- !split -->
-<h2 id="feed-forward-neural-networks" class="anchor">Feed-forward neural networks </h2>
+<h2 id="the-gradient-step" class="anchor">The gradient step </h2>
 
-<p>The feed-forward neural network (FFNN) was the first and simplest type
-of ANNs that were devised. In this network, the information moves in
-only one direction: forward through the layers.
-</p>
+<p>Thus a gradient descent step now looks like </p>
+$$
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
+$$
 
-<p>Nodes are represented by circles, while the arrows display the
-connections between the nodes, including the direction of information
-flow. Additionally, each arrow corresponds to a weight variable
-(figure to come).  We observe that each node in a layer is connected
-to <em>all</em> nodes in the subsequent layer, making this a so-called
-<em>fully-connected</em> FFNN.
+<p>where \( k \) is picked at random with equal
+probability from \( [1,n/M] \). An iteration over the number of
+minibathces (n/M) is commonly referred to as an epoch. Thus it is
+typical to choose a number of epochs and for each epoch iterate over
+the number of minibatches, as exemplified in the code below.
 </p>
 
 <p>
@@ -268,7 +358,7 @@ to <em>all</em> nodes in the subsequent layer, making this a so-called
   <li><a href="._week40-bs017.html">18</a></li>
   <li><a href="._week40-bs018.html">19</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs010.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs010.html b/doc/pub/week40/html/._week40-bs010.html
index f26dba49f..6771de428 100644
--- a/doc/pub/week40/html/._week40-bs010.html
+++ b/doc/pub/week40/html/._week40-bs010.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,27 +319,51 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0010"></a>
 <!-- !split -->
-<h2 id="convolutional-neural-network" class="anchor">Convolutional Neural Network </h2>
+<h2 id="simple-example-code" class="anchor">Simple example code </h2>
 
-<p>A different variant of FFNNs are <em>convolutional neural networks</em>
-(CNNs), which have a connectivity pattern inspired by the animal
-visual cortex. Individual neurons in the visual cortex only respond to
-stimuli from small sub-regions of the visual field, called a receptive
-field. This makes the neurons well-suited to exploit the strong
-spatially local correlation present in natural images. The response of
-each neuron can be approximated mathematically as a convolution
-operation.  (figure to come)
-</p>
 
-<p>Convolutional neural networks emulate the behaviour of neurons in the
-visual cortex by enforcing a <em>local</em> connectivity pattern between
-nodes of adjacent layers: Each node in a convolutional layer is
-connected only to a subset of the nodes in the previous layer, in
-contrast to the fully-connected FFNN.  Often, CNNs consist of several
-convolutional layers that learn local features of the input, with a
-fully-connected layer at the end, which gathers all the local data and
-produces the outputs. They have wide applications in image and video
-recognition.
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span> 
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">10</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
+
+j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
+        <span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #408080; font-style: italic">#Compute new suggestion for </span>
+        j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Taking the gradient only on a subset of the data has two important
+benefits. First, it introduces randomness which decreases the chance
+that our opmization scheme gets stuck in a local minima. Second, if
+the size of the minibatches are small relative to the number of
+datapoints (\( M <  n \)), the computation of the gradient is much
+cheaper since we sum over the datapoints in the \( k-th \) minibatch and not
+all \( n \) datapoints.
 </p>
 
 <p>
@@ -277,7 +391,7 @@ recognition.
   <li><a href="._week40-bs018.html">19</a></li>
   <li><a href="._week40-bs019.html">20</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs011.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs011.html b/doc/pub/week40/html/._week40-bs011.html
index fc02ecf58..9f1d0deef 100644
--- a/doc/pub/week40/html/._week40-bs011.html
+++ b/doc/pub/week40/html/._week40-bs011.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,18 +319,18 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0011"></a>
 <!-- !split -->
-<h2 id="recurrent-neural-networks" class="anchor">Recurrent neural networks </h2>
+<h2 id="when-do-we-stop" class="anchor">When do we stop? </h2>
 
-<p>So far we have only mentioned ANNs where information flows in one
-direction: forward. <em>Recurrent neural networks</em> on the other hand,
-have connections between nodes that form directed <em>cycles</em>. This
-creates a form of internal memory which are able to capture
-information on what has been calculated before; the output is
-dependent on the previous computations. Recurrent NNs make use of
-sequential information by performing the same task for every element
-in a sequence, where each element depends on previous elements. An
-example of such information is sentences, making recurrent NNs
-especially well-suited for handwriting and speech recognition.
+<p>A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the \( \beta \) that
+gave the lowest value.
 </p>
 
 <p>
@@ -268,7 +358,7 @@ especially well-suited for handwriting and speech recognition.
   <li><a href="._week40-bs019.html">20</a></li>
   <li><a href="._week40-bs020.html">21</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs012.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs012.html b/doc/pub/week40/html/._week40-bs012.html
index 60a3046c3..6db89d90c 100644
--- a/doc/pub/week40/html/._week40-bs012.html
+++ b/doc/pub/week40/html/._week40-bs012.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,18 +319,17 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0012"></a>
 <!-- !split -->
-<h2 id="other-types-of-networks" class="anchor">Other types of networks </h2>
+<h2 id="slightly-different-approach" class="anchor">Slightly different approach </h2>
 
-<p>There are many other kinds of ANNs that have been developed. One type
-that is specifically designed for interpolation in multidimensional
-space is the radial basis function (RBF) network. RBFs are typically
-made up of three layers: an input layer, a hidden layer with
-non-linear radial symmetric activation functions and a linear output
-layer (''linear'' here means that each node in the output layer has a
-linear activation function). The layers are normally fully-connected
-and there are no cycles, thus RBFs can be viewed as a type of
-fully-connected FFNN. They are however usually treated as a separate
-type of NN due the unusual activation functions.
+<p>Another approach is to let the step length \( \gamma_j \) depend on the
+number of epochs in such a way that it becomes very small after a
+reasonable time such that we do not move at all. Such approaches are
+also called scaling. There are many such ways to <a href="https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1" target="_self">scale the learning
+rate</a>
+and <a href="https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf" target="_self">discussions here</a>. See
+also
+<a href="https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1" target="_self"><tt>https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1</tt></a>
+for a discussion of different scaling functions for the learning rate.
 </p>
 
 <p>
@@ -268,7 +357,7 @@ type of NN due the unusual activation functions.
   <li><a href="._week40-bs020.html">21</a></li>
   <li><a href="._week40-bs021.html">22</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs013.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs013.html b/doc/pub/week40/html/._week40-bs013.html
index 1b0603e17..cd7a53a73 100644
--- a/doc/pub/week40/html/._week40-bs013.html
+++ b/doc/pub/week40/html/._week40-bs013.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,15 +319,63 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0013"></a>
 <!-- !split -->
-<h2 id="multilayer-perceptrons" class="anchor">Multilayer perceptrons  </h2>
+<h2 id="time-decay-rate" class="anchor">Time decay rate </h2>
 
-<p>One uses often so-called fully-connected feed-forward neural networks
-with three or more layers (an input layer, one or more hidden layers
-and an output layer) consisting of neurons that have non-linear
-activation functions.
+<p>As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in <em>time</em> \( t \).</p>
+
+<p>In this way we can fix the number of epochs, compute \( \beta \) and
+evaluate the cost function at the end. Repeating the computation will
+give a different result since the scheme is random by design. Then we
+pick the final \( \beta \) that gives the lowest value of the cost
+function.
 </p>
 
-<p>Such networks are often called <em>multilayer perceptrons</em> (MLPs).</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span> 
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">step_length</span>(t,t0,t1):
+    <span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">500</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
+t0 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+t1 <span style="color: #666666">=</span> <span style="color: #666666">10</span>
+
+gamma_j <span style="color: #666666">=</span> t0<span style="color: #666666">/</span>t1
+j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
+        <span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #408080; font-style: italic">#Compute new suggestion for beta</span>
+        t <span style="color: #666666">=</span> epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i
+        gamma_j <span style="color: #666666">=</span> step_length(t,t0,t1)
+        j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;gamma_j after </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> epochs: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (n_epochs,gamma_j))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -264,7 +402,7 @@ activation functions.
   <li><a href="._week40-bs021.html">22</a></li>
   <li><a href="._week40-bs022.html">23</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs014.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs014.html b/doc/pub/week40/html/._week40-bs014.html
index 9ba98008b..833ff1fed 100644
--- a/doc/pub/week40/html/._week40-bs014.html
+++ b/doc/pub/week40/html/._week40-bs014.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,20 +319,97 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0014"></a>
 <!-- !split -->
-<h2 id="why-multilayer-perceptrons" class="anchor">Why multilayer perceptrons?  </h2>
+<h2 id="code-with-a-number-of-minibatches-which-varies" class="anchor">Code with a Number of Minibatches which varies </h2>
 
-<p>According to the <em>Universal approximation theorem</em>, a feed-forward
-neural network with just a single hidden layer containing a finite
-number of neurons can approximate a continuous multidimensional
-function to arbitrary accuracy, assuming the activation function for
-the hidden layer is a <b>non-constant, bounded and
-monotonically-increasing continuous function</b>.
-</p>
+<p>In the code here we vary the number of mini-batches.</p>
+
+<!-- code=text (!bc pycode) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"># Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
+print(&quot;Own inversion&quot;)
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f&quot;Eigenvalues of Hessian Matrix:{EigValues}&quot;)
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+
+for iter in range(Niterations):
+    gradients = 2.0/n*X.T @ ((X @ theta)-y)
+    theta -= eta*gradients
+print(&quot;theta from own gd&quot;)
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print(&quot;theta from own sdg&quot;)
+print(theta)
+
+plt.plot(xnew, ypredict, &quot;r-&quot;)
+plt.plot(xnew, ypredict2, &quot;b-&quot;)
+plt.plot(x, y ,&#39;ro&#39;)
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r&#39;$x$&#39;)
+plt.ylabel(r&#39;$y$&#39;)
+plt.title(r&#39;Random numbers &#39;)
+plt.show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>Note that the requirements on the activation function only applies to
-the hidden layer, the output nodes are always assumed to be linear, so
-as to not restrict the range of output values.
-</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -269,7 +436,7 @@ as to not restrict the range of output values.
   <li><a href="._week40-bs022.html">23</a></li>
   <li><a href="._week40-bs023.html">24</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs015.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
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index 5447cafc6..f10af4e22 100644
--- a/doc/pub/week40/html/._week40-bs015.html
+++ b/doc/pub/week40/html/._week40-bs015.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,15 +319,13 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0015"></a>
 <!-- !split -->
-<h2 id="illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" class="anchor">Illustration of a single perceptropn model and a multi-perceptron model </h2>
+<h2 id="replace-or-not" class="anchor">Replace or not </h2>
 
-<center>  <!-- FIGURE -->
-<hr class="figure">
-<center>
-<p class="caption">Figure 1:  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer. </p>
-</center>
-<p><img src="figures/nns.png" width="600" align="bottom"></p>
-</center>
+<p>In the above code, we have use replacement in setting up the
+mini-batches. The discussion
+<a href="https://sebastianraschka.com/faq/docs/sgd-methods.html" target="_self">here</a> may be
+useful.  
+</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -264,7 +352,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs023.html">24</a></li>
   <li><a href="._week40-bs024.html">25</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs016.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs016.html b/doc/pub/week40/html/._week40-bs016.html
index 46672f801..15db620ae 100644
--- a/doc/pub/week40/html/._week40-bs016.html
+++ b/doc/pub/week40/html/._week40-bs016.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,68 +319,39 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0016"></a>
 <!-- !split -->
-<h2 id="examples-of-xor-or-and-and-gates" class="anchor">Examples of XOR, OR and AND gates </h2>
+<h2 id="momentum-based-gd" class="anchor">Momentum based GD </h2>
 
-<p>Let us first try to fit various gates using standard linear
-regression. The gates we are thinking of are the classical XOR, OR and
-AND gates, well-known elements in computer science. The tables here
-show how we can set up the inputs \( x_1 \) and \( x_2 \) in order to yield a
-specific target \( y_i \).
+<p>The stochastic gradient descent (SGD) is almost always used with a
+<em>momentum</em> or inertia term that serves as a memory of the direction we
+are moving in parameter space.  This is typically implemented as
+follows
 </p>
 
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\tag{1}
+\end{align}
+$$
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
-<span style="color: #BA2121; font-style: italic">Simple code that tests XOR, OR and AND gates with linear regression</span>
-<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
+<p>where we have introduced a momentum parameter \( \gamma \), with
+\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to
+indicate the gradient is to be taken over a different mini-batch at
+each step. We call this algorithm gradient descent with momentum
+(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a
+running average of recently encountered gradients and
+\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory
+used in the averaging procedure. Consistent with this, when
+\( \gamma=0 \), this just reduces down to ordinary SGD as discussed
+earlier. An equivalent way of writing the updates is
+</p>
 
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #408080; font-style: italic"># Design matrix</span>
-X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The invers of X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+$$
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+$$
 
-<span style="color: #408080; font-style: italic"># The XOR gate </span>
-yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
-ThetaXOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-
-
-<span style="color: #408080; font-style: italic"># The OR gate </span>
-yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
-ThetaOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-
-
-<span style="color: #408080; font-style: italic"># The OR gate </span>
-yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
-ThetaAND  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
-<p>What is happening here?</p>
+<p>where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -317,7 +378,7 @@ ThetaAND  <span style="color: #666666">=</span> Xinv <span style="color: #666666
   <li><a href="._week40-bs024.html">25</a></li>
   <li><a href="._week40-bs025.html">26</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs017.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs017.html b/doc/pub/week40/html/._week40-bs017.html
index f0fd22c78..be2351601 100644
--- a/doc/pub/week40/html/._week40-bs017.html
+++ b/doc/pub/week40/html/._week40-bs017.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,80 +319,31 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0017"></a>
 <!-- !split -->
-<h2 id="does-logistic-regression-do-a-better-job" class="anchor">Does Logistic Regression do a better Job? </h2>
+<h2 id="more-on-momentum-based-approaches" class="anchor">More on momentum based approaches </h2>
 
+<p>Let us try to get more intuition from these equations. It is helpful
+to consider a simple physical analogy with a particle of mass \( m \)
+moving in a viscous medium with drag coefficient \( \mu \) and potential
+\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \),
+then its motion is described by
+</p>
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
-<span style="color: #BA2121; font-style: italic">Simple code that tests XOR and OR gates with linear regression</span>
-<span style="color: #BA2121; font-style: italic">and logistic regression</span>
-<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
+$$
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+$$
 
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<p>We can discretize this equation in the usual way to get</p>
 
-<span style="color: #408080; font-style: italic"># Design matrix</span>
-X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The invers of X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
 
-<span style="color: #408080; font-style: italic"># The XOR gate </span>
-yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
-ThetaXOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<p>Rearranging this equation, we can rewrite this as</p>
 
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
 
-<span style="color: #408080; font-style: italic"># The OR gate </span>
-yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
-ThetaOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-
-
-<span style="color: #408080; font-style: italic"># The OR gate </span>
-yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
-ThetaAND  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-
-<span style="color: #408080; font-style: italic"># Now we change to logistic regression</span>
-
-
-<span style="color: #408080; font-style: italic"># Logistic Regression</span>
-logreg <span style="color: #666666">=</span> LogisticRegression()
-logreg<span style="color: #666666">.</span>fit(X, yOR)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for OR gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yOR)))
-
-logreg<span style="color: #666666">.</span>fit(X, yXOR)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for XOR gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yXOR)))
-
-
-logreg<span style="color: #666666">.</span>fit(X, yAND)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for AND gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yAND)))
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
-<p>Not exactly impressive, but somewhat better.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -329,7 +370,7 @@ logreg<span style="color: #666666">.</span>fit(X, yAND)
   <li><a href="._week40-bs025.html">26</a></li>
   <li><a href="._week40-bs026.html">27</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs018.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs018.html b/doc/pub/week40/html/._week40-bs018.html
index 55a8107dc..d53a8fe4c 100644
--- a/doc/pub/week40/html/._week40-bs018.html
+++ b/doc/pub/week40/html/._week40-bs018.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,38 +319,57 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0018"></a>
 <!-- !split -->
-<h2 id="adding-neural-networks" class="anchor">Adding Neural Networks </h2>
+<h2 id="momentum-parameter" class="anchor">Momentum parameter </h2>
 
+<p>Notice that this equation is identical to previous one if we identify
+the position of the particle, \( \mathbf{w} \), with the parameters
+\( \boldsymbol{\theta} \). This allows us to identify the momentum
+parameter and learning rate with the mass of the particle and the
+viscous drag as:
+</p>
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># and now neural networks with Scikit-Learn and the XOR</span>
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
 
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPClassifier
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_classification
-X, yXOR <span style="color: #666666">=</span> make_classification(n_samples<span style="color: #666666">=100</span>, random_state<span style="color: #666666">=1</span>)
-FFNN <span style="color: #666666">=</span> MLPClassifier(random_state<span style="color: #666666">=1</span>, max_iter<span style="color: #666666">=300</span>)<span style="color: #666666">.</span>fit(X, yXOR)
-FFNN<span style="color: #666666">.</span>predict_proba(X)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Test set accuracy with Feed Forward Neural Network  for XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>FFNN<span style="color: #666666">.</span>score(X, yXOR)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
+<p>Thus, as the name suggests, the momentum parameter is proportional to
+the mass of the particle and effectively provides inertia.
+Furthermore, in the large viscosity/small learning rate limit, our
+memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+</p>
 
+<p>Why is momentum useful? SGD momentum helps the gradient descent
+algorithm gain speed in directions with persistent but small gradients
+even in the presence of stochasticity, while suppressing oscillations
+in high-curvature directions. This becomes especially important in
+situations where the landscape is shallow and flat in some directions
+and narrow and steep in others. It has been argued that first-order
+methods (with appropriate initial conditions) can perform comparable
+to more expensive second order methods, especially in the context of
+complex deep learning models.
+</p>
+
+<p>These beneficial properties of momentum can sometimes become even more
+pronounced by using a slight modification of the classical momentum
+algorithm called Nesterov Accelerated Gradient (NAG).
+</p>
+
+<p>In the NAG algorithm, rather than calculating the gradient at the
+current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one
+calculates the gradient at the expected value of the parameters given
+our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma
+\mathbf{v}_{t-1}) \). This yields the NAG update rule
+</p>
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\tag{2}
+\end{align}
+$$
+
+<p>One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -287,7 +396,7 @@ FFNN<span style="color: #666666">.</span>predict_proba(X)
   <li><a href="._week40-bs026.html">27</a></li>
   <li><a href="._week40-bs027.html">28</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs019.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs019.html b/doc/pub/week40/html/._week40-bs019.html
index 9f99e2657..0b84fc505 100644
--- a/doc/pub/week40/html/._week40-bs019.html
+++ b/doc/pub/week40/html/._week40-bs019.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,19 +319,29 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0019"></a>
 <!-- !split -->
-<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
+<h2 id="second-moment-of-the-gradient" class="anchor">Second moment of the gradient </h2>
 
-<p>The output \( y \) is produced via the activation function \( f \)</p>
-$$
- y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
-$$
+<p>In stochastic gradient descent, with and without momentum, we still
+have to specify a schedule for tuning the learning rates \( \eta_t \)
+as a function of time.  As discussed in the context of Newton's
+method, this presents a number of dilemmas. The learning rate is
+limited by the steepest direction which can change depending on the
+current position in the landscape. To circumvent this problem, ideally
+our algorithm would keep track of curvature and take large steps in
+shallow, flat directions and small steps in steep, narrow directions.
+Second-order methods accomplish this by calculating or approximating
+the Hessian and normalizing the learning rate by the
+curvature. However, this is very computationally expensive for
+extremely large models. Ideally, we would like to be able to
+adaptively change the step size to match the landscape without paying
+the steep computational price of calculating or approximating
+Hessians.
+</p>
 
-<p>This function receives \( x_i \) as inputs.
-Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \). 
-In an FFNN of such neurons, the <em>inputs</em> \( x_i \) are the <em>outputs</em> of
-the neurons in the preceding layer. Furthermore, an MLP is
-fully-connected, which means that each neuron receives a weighted sum
-of the outputs of <em>all</em> neurons in the previous layer.
+<p>Recently, a number of methods have been introduced that accomplish
+this by tracking not only the gradient, but also the second moment of
+the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
+<a href="https://arxiv.org/abs/1412.6980" target="_self">ADAM</a>.
 </p>
 
 <p>
@@ -269,7 +369,7 @@ of the outputs of <em>all</em> neurons in the previous layer.
   <li><a href="._week40-bs027.html">28</a></li>
   <li><a href="._week40-bs028.html">29</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs020.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs020.html b/doc/pub/week40/html/._week40-bs020.html
index 9f34ba368..89aebe3b3 100644
--- a/doc/pub/week40/html/._week40-bs020.html
+++ b/doc/pub/week40/html/._week40-bs020.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,47 +319,32 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0020"></a>
 <!-- !split -->
-<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
+<h2 id="rms-prop" class="anchor">RMS prop </h2>
 
-<p>First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),</p>
-
-$$
-\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
-\tag{2}
-\end{equation}
-$$
-
-<p>Here \( b_i \) is the so-called bias which is normally needed in
-case of zero activation weights or inputs. How to fix the biases and
-the weights will be discussed below.  The value of \( z_i^1 \) is the
-argument to the activation function \( f_i \) of each node \( i \), The
-variable \( M \) stands for all possible inputs to a given node \( i \) in the
-first layer.  We define  the output \( y_i^1 \) of all neurons in layer 1 as
+<p>In RMS prop, in addition to keeping a running average of the first
+moment of the gradient, we also keep track of the second moment
+denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule
+for RMS prop is given by
 </p>
 
 $$
-\begin{equation}
- y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\right)
-\tag{3}
-\end{equation}
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\tag{3}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
 $$
 
-<p>where we assume that all nodes in the same layer have identical
-activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
-In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
-</p>
-
-$$
-\begin{equation}
- y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
-\tag{4}
-\end{equation}
-$$
-
-<p>where \( N_l \) is the number of nodes in layer \( l \). When the output of
-all the nodes in the first hidden layer are computed, the values of
-the subsequent layer can be calculated and so forth until the output
-is obtained.
+<p>where \( \beta \) controls the averaging time of the second moment and is
+typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate
+typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8}  \) is a
+small regularization constant to prevent divergences. Multiplication
+and division by vectors is understood as an element-wise operation. It
+is clear from this formula that the learning rate is reduced in
+directions where the norm of the gradient is consistently large. This
+greatly speeds up the convergence by allowing us to use a larger
+learning rate for flat directions.
 </p>
 
 <p>
@@ -297,7 +372,7 @@ is obtained.
   <li><a href="._week40-bs028.html">29</a></li>
   <li><a href="._week40-bs029.html">30</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs021.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs021.html b/doc/pub/week40/html/._week40-bs021.html
index 75ab68b2d..58a80c032 100644
--- a/doc/pub/week40/html/._week40-bs021.html
+++ b/doc/pub/week40/html/._week40-bs021.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,29 +319,57 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0021"></a>
 <!-- !split -->
-<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
+<h2 id="adam-optimizer-https-arxiv-org-abs-1412-6980" class="anchor"><a href="https://arxiv.org/abs/1412.6980" target="_self">ADAM optimizer</a> </h2>
 
-<p>The output of neuron \( i \) in layer 2 is thus,</p>
+<p>A related algorithm is the ADAM optimizer. In
+<a href="https://arxiv.org/abs/1412.6980" target="_self">ADAM</a>, we keep a running average of
+both the first and second moment of the gradient and use this
+information to adaptively change the learning rate for different
+parameters.  The method isefficient when working with large
+problems involving lots data and/or parameters.  It is a combination of the
+gradient descent with momentum algorithm and the RMSprop algorithm
+discussed above.
+</p>
+
+<p>In addition to keeping a running average of the first and
+second moments of the gradient
+(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and
+\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM
+performs an additional bias correction to account for the fact that we
+are estimating the first two moments of the gradient using a running
+average (denoted by the hats in the update rule below). The update
+rule for ADAM is given by (where multiplication and division are once
+again understood to be element-wise operations below)
+</p>
 
 $$
 \begin{align}
- y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
-\tag{5}\\
- &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
-\tag{6}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\tag{4}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\tag{5}
 \end{align}
 $$
 
-<p>where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads</p>
+<p>where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and
+second moment and are typically taken to be \( 0.9 \) and \( 0.99 \)
+respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.
+</p>
+
+<p>Like in RMSprop, the effective step size of a parameter depends on the
+magnitude of its gradient squared.  To understand this better, let us
+rewrite this expression in terms of the variance
+\( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t -
+(\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The
+update rule for this parameter is given by
+</p>
 
 $$
-\begin{align}
- y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
-\tag{7}\\
- &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
-  + b_1^3\right]
-\tag{8}
-\end{align}
+\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 +  m_t^2 }+\epsilon}.
 $$
 
 
@@ -280,7 +398,7 @@ $$
   <li><a href="._week40-bs029.html">30</a></li>
   <li><a href="._week40-bs030.html">31</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs022.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs022.html b/doc/pub/week40/html/._week40-bs022.html
index e81ba6f23..45d096092 100644
--- a/doc/pub/week40/html/._week40-bs022.html
+++ b/doc/pub/week40/html/._week40-bs022.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,22 +319,11 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0022"></a>
 <!-- !split -->
-<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
+<h2 id="algorithms-and-codes-for-adagrad-rmsprop-and-adam" class="anchor">Algorithms and codes for Adagrad, RMSprop and Adam </h2>
 
-<p>We can generalize this expression to an MLP with \( l \) hidden
-layers. The complete functional form is,
-</p>
+<p>The algorithms we have implemented are well described in the text by <a href="https://www.deeplearningbook.org/contents/optimization.html" target="_self">Goodfellow, Bengio and Courville, chapter 8</a>.</p>
 
-$$
-\begin{align}
-&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
-\tag{9}
-\end{align}
-$$
-
-<p>which illustrates a basic property of MLPs: The only independent
-variables are the input values \( x_n \).
-</p>
+<p>The codes which implement these algorithms are discussed after our presentation of automatic differentiation.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -271,7 +350,7 @@ variables are the input values \( x_n \).
   <li><a href="._week40-bs030.html">31</a></li>
   <li><a href="._week40-bs031.html">32</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs023.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs023.html b/doc/pub/week40/html/._week40-bs023.html
index 014cf2c2e..b11f3de28 100644
--- a/doc/pub/week40/html/._week40-bs023.html
+++ b/doc/pub/week40/html/._week40-bs023.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,31 +319,15 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0023"></a>
 <!-- !split -->
-<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
+<h2 id="practical-tips" class="anchor">Practical tips </h2>
 
-<p>This confirms that an MLP, despite its quite convoluted mathematical
-form, is nothing more than an analytic function, specifically a
-mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
-\hat{y} \in \mathbb{R}^m \).
-</p>
-
-<p>Furthermore, the flexibility and universality of an MLP can be
-illustrated by realizing that the expression is essentially a nested
-sum of scaled activation functions of the form
-</p>
-
-$$
-\begin{equation}
- f(x) = c_1 f(c_2 x + c_3) + c_4
-\tag{10}
-\end{equation}
-$$
-
-<p>where the parameters \( c_i \) are weights and biases. By adjusting these
-parameters, the activation functions can be shifted up and down or
-left and right, change slope or be rescaled which is the key to the
-flexibility of a neural network.
-</p>
+<ul>
+<li> <b>Randomize the data when making mini-batches</b>. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.</li>
+<li> <b>Transform your inputs</b>. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.</li>
+<li> <b>Monitor the out-of-sample performance.</b> Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This <em>early stopping</em> significantly improves performance in many settings.</li>
+<li> <b>Adaptive optimization methods don't always have good generalization.</b> Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.</li>
+</ul>
+<p>Geron's text, see chapter 11, has several interesting discussions.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -280,7 +354,7 @@ flexibility of a neural network.
   <li><a href="._week40-bs031.html">32</a></li>
   <li><a href="._week40-bs032.html">33</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs024.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs024.html b/doc/pub/week40/html/._week40-bs024.html
index b9258cdb1..6810e7037 100644
--- a/doc/pub/week40/html/._week40-bs024.html
+++ b/doc/pub/week40/html/._week40-bs024.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,41 +319,105 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0024"></a>
 <!-- !split -->
-<h3 id="matrix-vector-notation" class="anchor">Matrix-vector notation </h3>
+<h2 id="automatic-differentiation" class="anchor">Automatic differentiation </h2>
 
-<p>We can introduce a more convenient notation for the activations in an A NN. </p>
-
-<p>Additionally, we can represent the biases and activations
-as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector 
-is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively. 
+<p><a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_self">Automatic differentiation (AD)</a>, 
+also called algorithmic
+differentiation or computational differentiation,is a set of
+techniques to numerically evaluate the derivative of a function
+specified by a computer program. AD exploits the fact that every
+computer program, no matter how complicated, executes a sequence of
+elementary arithmetic operations (addition, subtraction,
+multiplication, division, etc.) and elementary functions (exp, log,
+sin, cos, etc.). By applying the chain rule repeatedly to these
+operations, derivatives of arbitrary order can be computed
+automatically, accurately to working precision, and using at most a
+small constant factor more arithmetic operations than the original
+program.
 </p>
 
-<p>We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors. 
-With this notation, the sum becomes a matrix-vector multiplication, and we can write
-the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
+<p>Automatic differentiation is neither:</p>
+
+<ul>
+<li> Symbolic differentiation, nor</li>
+<li> Numerical differentiation (the method of finite differences).</li>
+</ul>
+<p>Symbolic differentiation can lead to inefficient code and faces the
+difficulty of converting a computer program into a single expression,
+while numerical differentiation can introduce round-off errors in the
+discretization process and cancellation
+</p>
+
+<p>Python has tools for so-called <b>automatic differentiation</b>.
+Consider the following example
 </p>
 $$
-\begin{equation}
- \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = 
- f_2\left(\left[\begin{array}{ccc}
-    w^2_{11} &w^2_{12} &w^2_{13} \\
-    w^2_{21} &w^2_{22} &w^2_{23} \\
-    w^2_{31} &w^2_{32} &w^2_{33} \\
-    \end{array} \right] \cdot
-    \left[\begin{array}{c}
-           y^1_1 \\
-           y^1_2 \\
-           y^1_3 \\
-          \end{array}\right] + 
-    \left[\begin{array}{c}
-           b^2_1 \\
-           b^2_2 \\
-           b^2_3 \\
-          \end{array}\right]\right).
-\tag{11}
-\end{equation}
+f(x) = \sin\left(2\pi x + x^2\right)
 $$
 
+<p>which has the following derivative</p>
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) 
+$$
+
+<p>Using <b>autograd</b> we have</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+
+<span style="color: #408080; font-style: italic"># To do elementwise differentiation:</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> elementwise_grad <span style="color: #008000; font-weight: bold">as</span> egrad 
+
+<span style="color: #408080; font-style: italic"># To plot:</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span> 
+
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sin(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f_grad_analytic</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi <span style="color: #666666">+</span> <span style="color: #666666">2*</span>x)
+
+<span style="color: #408080; font-style: italic"># Do the comparison:</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">1</span>,<span style="color: #666666">1000</span>)
+
+f_grad <span style="color: #666666">=</span> egrad(f)
+
+computed <span style="color: #666666">=</span> f_grad(x)
+analytic <span style="color: #666666">=</span> f_grad_analytic(x)
+
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&#39;Derivative computed from Autograd compared with the analytical derivative&#39;</span>)
+plt<span style="color: #666666">.</span>plot(x,computed,label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;autograd&#39;</span>)
+plt<span style="color: #666666">.</span>plot(x,analytic,label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;analytic&#39;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;x&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;y&#39;</span>)
+plt<span style="color: #666666">.</span>legend()
+
+plt<span style="color: #666666">.</span>show()
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The max absolute difference is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(computed <span style="color: #666666">-</span> analytic))))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -290,7 +444,7 @@ $$
   <li><a href="._week40-bs032.html">33</a></li>
   <li><a href="._week40-bs033.html">34</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs025.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs025.html b/doc/pub/week40/html/._week40-bs025.html
index 519e1152e..1bfa65577 100644
--- a/doc/pub/week40/html/._week40-bs025.html
+++ b/doc/pub/week40/html/._week40-bs025.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -228,26 +318,56 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0025"></a>
-<!-- !split -->
-<h3 id="matrix-vector-notation-and-activation" class="anchor">Matrix-vector notation  and activation </h3>
+<!-- !split  -->
+<h2 id="using-autograd" class="anchor">Using autograd </h2>
 
-<p>The activation of node \( i \) in layer 2 is</p>
-
-$$
-\begin{equation}
- y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
- f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\tag{12}
-\end{equation}
-$$
-
-<p>This is not just a convenient and compact notation, but also a useful
-and intuitive way to think about MLPs: The output is calculated by a
-series of matrix-vector multiplications and vector additions that are
-used as input to the activation functions. For each operation
-\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
+<p>Here we
+experiment with what kind of functions Autograd is capable
+of finding the gradient of. The following Python functions are just
+meant to illustrate what Autograd can do, but please feel free to
+experiment with other, possibly more complicated, functions as well.
 </p>
 
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f1</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">**3</span> <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+
+f1_grad <span style="color: #666666">=</span> grad(f1)
+
+<span style="color: #408080; font-style: italic"># Remember to send in float as argument to the computed gradient from Autograd!</span>
+a <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+
+<span style="color: #408080; font-style: italic"># See the evaluated gradient at a using autograd:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The gradient of f1 evaluated at a = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> using autograd is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(a,f1_grad(a)))
+
+<span style="color: #408080; font-style: italic"># Compare with the analytical derivative, that is f1&#39;(x) = 3*x**2 </span>
+grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">3*</span>a<span style="color: #666666">**2</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The gradient of f1 evaluated at a = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> by finding the analytic expression is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(a,grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -273,7 +393,7 @@ used as input to the activation functions. For each operation
   <li><a href="._week40-bs033.html">34</a></li>
   <li><a href="._week40-bs034.html">35</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs026.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs026.html b/doc/pub/week40/html/._week40-bs026.html
index f0ee706ff..f62f1327c 100644
--- a/doc/pub/week40/html/._week40-bs026.html
+++ b/doc/pub/week40/html/._week40-bs026.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,20 +319,70 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0026"></a>
 <!-- !split -->
-<h3 id="activation-functions" class="anchor">Activation functions  </h3>
+<h2 id="autograd-with-more-complicated-functions" class="anchor">Autograd with more complicated functions </h2>
 
-<p>A property that characterizes a neural network, other than its
-connectivity, is the choice of activation function(s).  As described
-in, the following restrictions are imposed on an activation function
-for a FFNN to fulfill the universal approximation theorem
+<p>To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
 </p>
 
-<ul>
-  <li> Non-constant</li>
-  <li> Bounded</li>
-  <li> Monotonically-increasing</li>
-  <li> Continuous</li>
-</ul>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f2</span>(x1,x2):
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">3*</span>x1<span style="color: #666666">**3</span> <span style="color: #666666">+</span> x2<span style="color: #666666">*</span>(x1 <span style="color: #666666">-</span> <span style="color: #666666">5</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+
+<span style="color: #408080; font-style: italic"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
+f2_grad_x1 <span style="color: #666666">=</span> grad(f2,<span style="color: #666666">0</span>)
+
+<span style="color: #408080; font-style: italic"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
+f2_grad_x2 <span style="color: #666666">=</span> grad(f2,<span style="color: #666666">1</span>)
+
+x1 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+x2 <span style="color: #666666">=</span> <span style="color: #666666">3.0</span> 
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Evaluating at x1 = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">, x2 = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x1,x2))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;-&quot;</span><span style="color: #666666">*30</span>)
+
+<span style="color: #408080; font-style: italic"># Compare with the analytical derivatives:</span>
+
+<span style="color: #408080; font-style: italic"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
+f2_grad_x1_analytical <span style="color: #666666">=</span> <span style="color: #666666">9*</span>x1<span style="color: #666666">**2</span> <span style="color: #666666">+</span> x2
+
+<span style="color: #408080; font-style: italic"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
+f2_grad_x2_analytical <span style="color: #666666">=</span> x1 <span style="color: #666666">-</span> <span style="color: #666666">5</span>
+
+<span style="color: #408080; font-style: italic"># See the evaluated derivations:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f2 w.r.t x1: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x1(x1,x2) ))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f2 w.r.t x1: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x1(x1,x2) ))
+
+<span style="color: #008000">print</span>()
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f2 w.r.t x2: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x2(x1,x2) ))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f2 w.r.t x2: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x2(x1,x2) ))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.</p>
+
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -268,7 +408,7 @@ for a FFNN to fulfill the universal approximation theorem
   <li><a href="._week40-bs034.html">35</a></li>
   <li><a href="._week40-bs035.html">36</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs027.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs027.html b/doc/pub/week40/html/._week40-bs027.html
index f2f9acb2a..641ced72a 100644
--- a/doc/pub/week40/html/._week40-bs027.html
+++ b/doc/pub/week40/html/._week40-bs027.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,29 +319,55 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0027"></a>
 <!-- !split -->
-<h3 id="activation-functions-logistic-and-hyperbolic-ones" class="anchor">Activation functions, Logistic and Hyperbolic ones  </h3>
+<h2 id="more-complicated-functions-using-the-elements-of-their-arguments-directly" class="anchor">More complicated functions using the elements of their arguments directly </h2>
 
-<p>The second requirement excludes all linear functions. Furthermore, in
-a MLP with only linear activation functions, each layer simply
-performs a linear transformation of its inputs.
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f3</span>(x): <span style="color: #408080; font-style: italic"># Assumes x is an array of length 5 or higher</span>
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">2*</span>x[<span style="color: #666666">0</span>] <span style="color: #666666">+</span> <span style="color: #666666">3*</span>x[<span style="color: #666666">1</span>] <span style="color: #666666">+</span> <span style="color: #666666">5*</span>x[<span style="color: #666666">2</span>] <span style="color: #666666">+</span> <span style="color: #666666">7*</span>x[<span style="color: #666666">3</span>] <span style="color: #666666">+</span> <span style="color: #666666">11*</span>x[<span style="color: #666666">4</span>]<span style="color: #666666">**2</span>
+
+f3_grad <span style="color: #666666">=</span> grad(f3)
+
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">4</span>,<span style="color: #666666">5</span>)
+
+<span style="color: #408080; font-style: italic"># Print the computed gradient:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed gradient of f3 is: &quot;</span>, f3_grad(x))
+
+<span style="color: #408080; font-style: italic"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
+f3_grad_analytical <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">2</span>, <span style="color: #666666">3</span>, <span style="color: #666666">5</span>, <span style="color: #666666">7</span>, <span style="color: #666666">22*</span>x[<span style="color: #666666">4</span>]])
+
+<span style="color: #408080; font-style: italic"># Print the analytical gradient:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical gradient of f3 is: &quot;</span>, f3_grad_analytical)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
 </p>
 
-<p>Regardless of the number of layers, the output of the NN will be
-nothing but a linear function of the inputs. Thus we need to introduce
-some kind of non-linearity to the NN to be able to fit non-linear
-functions Typical examples are the logistic <em>Sigmoid</em>
-</p>
-
-$$
- f(x) = \frac{1}{1 + e^{-x}},
-$$
-
-<p>and the <em>hyperbolic tangent</em> function</p>
-$$
- f(x) = \tanh(x)
-$$
-
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -277,7 +393,7 @@ $$
   <li><a href="._week40-bs035.html">36</a></li>
   <li><a href="._week40-bs036.html">37</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs028.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs028.html b/doc/pub/week40/html/._week40-bs028.html
index a4ea2de8b..ce18f2ae7 100644
--- a/doc/pub/week40/html/._week40-bs028.html
+++ b/doc/pub/week40/html/._week40-bs028.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -228,15 +318,8 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0028"></a>
-<!-- !split -->
-<h3 id="relevance" class="anchor">Relevance </h3>
-
-<p>The <em>sigmoid</em> function are more biologically plausible because the
-output of inactive neurons are zero. Such activation function are
-called <em>one-sided</em>. However, it has been shown that the hyperbolic
-tangent performs better than the sigmoid for training MLPs.  has
-become the most popular for <em>deep neural networks</em>
-</p>
+<!-- !split  -->
+<h2 id="functions-using-mathematical-functions-from-numpy" class="anchor">Functions using mathematical functions from Numpy </h2>
 
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -245,77 +328,23 @@ become the most popular for <em>deep neural networks</em>
     <div class="inner_cell">
       <div class="input_area">
         <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a </span>
-<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
-<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
-<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f4</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sqrt(<span style="color: #666666">1+</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>exp(x) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>sin(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)
 
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mt</span>
+f4_grad <span style="color: #666666">=</span> grad(f4)
 
-z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.1</span>)
-sigma_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1/</span>(<span style="color: #666666">1+</span>numpy<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z)))
-sigma <span style="color: #666666">=</span> sigma_fn(z)
+x <span style="color: #666666">=</span> <span style="color: #666666">2.7</span>
 
-fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
-ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
-ax<span style="color: #666666">.</span>plot(z, sigma)
-ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
-ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
-ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sigmoid function&#39;</span>)
+<span style="color: #408080; font-style: italic"># Print the computed derivative:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f4 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f4_grad(x)))
 
-plt<span style="color: #666666">.</span>show()
+<span style="color: #408080; font-style: italic"># The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi</span>
+f4_grad_analytical <span style="color: #666666">=</span> x<span style="color: #666666">/</span>np<span style="color: #666666">.</span>sqrt(<span style="color: #666666">1</span> <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>exp(x) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)<span style="color: #666666">*2*</span>np<span style="color: #666666">.</span>pi
 
-<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
-z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.02</span>)
-step_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1.0</span> <span style="color: #008000; font-weight: bold">if</span> z <span style="color: #666666">&gt;=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
-step <span style="color: #666666">=</span> step_fn(z)
-
-fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
-ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
-ax<span style="color: #666666">.</span>plot(z, step)
-ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
-ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
-ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;step function&#39;</span>)
-
-plt<span style="color: #666666">.</span>show()
-
-<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
-z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">0.1</span>)
-t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>sin(z)
-
-fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
-ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
-ax<span style="color: #666666">.</span>plot(z, t)
-ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
-ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi,<span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi])
-ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sine function&#39;</span>)
-
-plt<span style="color: #666666">.</span>show()
-
-<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
-<span style="color: #BA2121; font-style: italic">unit&quot;&quot;&quot;</span>
-z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>, <span style="color: #666666">.1</span>)
-zero <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>zeros(<span style="color: #008000">len</span>(z))
-y <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>max([zero, z], axis<span style="color: #666666">=0</span>)
-
-fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
-ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
-ax<span style="color: #666666">.</span>plot(z, y)
-ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
-ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
-ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Rectified linear unit&#39;</span>)
-
-plt<span style="color: #666666">.</span>show()
+<span style="color: #408080; font-style: italic"># Print the analytical gradient:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical gradient of f4 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f4_grad_analytical))
 </pre>
 </div>
       </div>
@@ -357,7 +386,7 @@ plt<span style="color: #666666">.</span>show()
   <li><a href="._week40-bs036.html">37</a></li>
   <li><a href="._week40-bs037.html">38</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs029.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs029.html b/doc/pub/week40/html/._week40-bs029.html
index affa73644..0bf2d94f8 100644
--- a/doc/pub/week40/html/._week40-bs029.html
+++ b/doc/pub/week40/html/._week40-bs029.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,38 +319,44 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0029"></a>
 <!-- !split -->
-<h2 id="the-multilayer-perceptron-mlp" class="anchor">The multilayer  perceptron (MLP) </h2>
+<h2 id="more-autograd" class="anchor">More autograd </h2>
 
-<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
-<ol>
-<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
-<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
-<li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
-</ol>
-<p>As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
-units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-</p>
 
-<p>For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
-</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f5</span>(x):
+    <span style="color: #008000; font-weight: bold">if</span> x <span style="color: #666666">&gt;=</span> <span style="color: #666666">0</span>:
+        <span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">**2</span>
+    <span style="color: #008000; font-weight: bold">else</span>:
+        <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">-3*</span>x <span style="color: #666666">+</span> <span style="color: #666666">1</span>
 
-<p>The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
-</p>
+f5_grad <span style="color: #666666">=</span> grad(f5)
+
+x <span style="color: #666666">=</span> <span style="color: #666666">2.7</span>
+
+<span style="color: #408080; font-style: italic"># Print the computed derivative:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f5 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f5_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation \( \hat{z} \), we can think of this as a basis
-expansion of the original inputs \( \hat{x} \). The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in  an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
-</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -287,7 +383,7 @@ like logistic regression or linear regression and their modifications on the oth
   <li><a href="._week40-bs037.html">38</a></li>
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs030.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs030.html b/doc/pub/week40/html/._week40-bs030.html
index 1f389a48c..29767aa57 100644
--- a/doc/pub/week40/html/._week40-bs030.html
+++ b/doc/pub/week40/html/._week40-bs030.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,30 +319,85 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0030"></a>
 <!-- !split -->
-<h2 id="from-one-to-many-layers-the-universal-approximation-theorem" class="anchor">From one to many layers, the universal approximation theorem </h2>
+<h2 id="and-with-loops" class="anchor">And  with loops </h2>
 
-<p>A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
-</p>
 
-<p>As stated earlier, 
-an important theorem in studies of neural networks, restated without
-proof here, is the <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_self">universal approximation
-theorem</a>.
-</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f6_for</span>(x):
+    val <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">10</span>):
+        val <span style="color: #666666">=</span> val <span style="color: #666666">+</span> x<span style="color: #666666">**</span>i
+    <span style="color: #008000; font-weight: bold">return</span> val
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f6_while</span>(x):
+    val <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    <span style="color: #008000; font-weight: bold">while</span> i <span style="color: #666666">&lt;</span> <span style="color: #666666">10</span>:
+        val <span style="color: #666666">=</span> val <span style="color: #666666">+</span> x<span style="color: #666666">**</span>i
+        i <span style="color: #666666">=</span> i <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">return</span> val
+
+f6_for_grad <span style="color: #666666">=</span> grad(f6_for)
+f6_while_grad <span style="color: #666666">=</span> grad(f6_while)
+
+x <span style="color: #666666">=</span> <span style="color: #666666">0.5</span>
+
+<span style="color: #408080; font-style: italic"># Print the computed derivaties of f6_for and f6_while</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f6_for at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f6_for_grad(x)))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f6_while at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f6_while_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #408080; font-style: italic"># Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9</span>
+<span style="color: #408080; font-style: italic"># The analytical derivative is: sum(i*x**(i-1)) </span>
+f6_grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">10</span>):
+    f6_grad_analytical <span style="color: #666666">+=</span> i<span style="color: #666666">*</span>x<span style="color: #666666">**</span>(i<span style="color: #666666">-1</span>)
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f6 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f6_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
-</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -279,7 +424,7 @@ the potential of being universal approximators.
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs031.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs031.html b/doc/pub/week40/html/._week40-bs031.html
index dcf807242..c0d5990f8 100644
--- a/doc/pub/week40/html/._week40-bs031.html
+++ b/doc/pub/week40/html/._week40-bs031.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,33 +319,57 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0031"></a>
 <!-- !split -->
-<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" class="anchor">Deriving the back propagation code for a multilayer perceptron model </h2>
+<h2 id="using-recursion" class="anchor">Using recursion </h2>
 
-<p>As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_self">back propagation algorithm</a>.
-</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<p>The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
-</p>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f7</span>(n): <span style="color: #408080; font-style: italic"># Assume that n is an integer</span>
+    <span style="color: #008000; font-weight: bold">if</span> n <span style="color: #666666">==</span> <span style="color: #666666">1</span> <span style="color: #AA22FF; font-weight: bold">or</span> n <span style="color: #666666">==</span> <span style="color: #666666">0</span>:
+        <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">else</span>:
+        <span style="color: #008000; font-weight: bold">return</span> n<span style="color: #666666">*</span>f7(n<span style="color: #666666">-1</span>)
 
-<p>To derive these equations let us start with a plain regression problem
-and define our cost function as
-</p>
+f7_grad <span style="color: #666666">=</span> grad(f7)
 
-$$
-{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
-$$
+n <span style="color: #666666">=</span> <span style="color: #666666">2.0</span>
 
-<p>where the $t_i$s are our \( n \) targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs \( \hat{x} \) are given by \( y_i \).  Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with \( K \)
-classes.
-</p>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f7 at n = </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(n,f7_grad(n)))
+
+<span style="color: #408080; font-style: italic"># The function f7 is an implementation of the factorial of n.</span>
+<span style="color: #408080; font-style: italic"># By using the product rule, one can find that the derivative is:</span>
+
+f7_grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">int</span>(n)<span style="color: #666666">-1</span>):
+    tmp <span style="color: #666666">=</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">for</span> k <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">int</span>(n)<span style="color: #666666">-1</span>):
+        <span style="color: #008000; font-weight: bold">if</span> k <span style="color: #666666">!=</span> i:
+            tmp <span style="color: #666666">*=</span> (n <span style="color: #666666">-</span> k)
+    f7_grad_analytical <span style="color: #666666">+=</span> tmp
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f7 at n = </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(n,f7_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -281,6 +395,8 @@ classes.
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs032.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs032.html b/doc/pub/week40/html/._week40-bs032.html
index 225fc0f06..620551e5f 100644
--- a/doc/pub/week40/html/._week40-bs032.html
+++ b/doc/pub/week40/html/._week40-bs032.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,41 +319,44 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0032"></a>
 <!-- !split -->
-<h2 id="definitions" class="anchor">Definitions </h2>
+<h2 id="unsupported-functions" class="anchor">Unsupported functions </h2>
+<p>Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.</p>
 
-<p>With our definition of the targets \( \hat{t} \), the outputs of the
-network \( \hat{y} \) and the inputs \( \hat{x} \) we
-define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
-\( l \)-th layer as a function of the bias, the weights which add up from
-the previous layer \( l-1 \) and the forward passes/outputs
-\( \hat{a}^{l-1} \) from the previous layer as
-</p>
+<p>Assigning a value to the variable being differentiated with respect to</p>
 
-$$
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
-$$
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f8</span>(x): <span style="color: #408080; font-style: italic"># Assume x is an array</span>
+    x[<span style="color: #666666">2</span>] <span style="color: #666666">=</span> <span style="color: #666666">3</span>
+    <span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">*2</span>
 
-<p>where \( b_k^l \) are the biases from layer \( l \).  Here \( M_{l-1} \)
-represents the total number of nodes/neurons/units of layer \( l-1 \). The
-figure here illustrates this equation.  We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-</p>
+f8_grad <span style="color: #666666">=</span> grad(f8)
 
-$$
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-$$
+x <span style="color: #666666">=</span> <span style="color: #666666">8.4</span>
 
-<p>With the activation values \( \hat{z}^l \) we can in turn define the
-output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
-and their nodes.  It means we have
-</p>
-
-$$
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-$$
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f8 is:&quot;</span>,f8_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
+<p>Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -288,6 +381,9 @@ $$
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs033.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs033.html b/doc/pub/week40/html/._week40-bs033.html
index d147d28b9..2022e3031 100644
--- a/doc/pub/week40/html/._week40-bs033.html
+++ b/doc/pub/week40/html/._week40-bs033.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,22 +319,80 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0033"></a>
 <!-- !split -->
-<h2 id="derivatives-and-the-chain-rule" class="anchor">Derivatives and the chain rule </h2>
+<h2 id="the-syntax-a-dot-b-when-finding-the-dot-product" class="anchor">The syntax a.dot(b) when finding the dot product </h2>
 
-<p>From the definition of the activation \( z_j^l \) we have</p>
-$$
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-$$
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f9</span>(a): <span style="color: #408080; font-style: italic"># Assume a is an array with 2 elements</span>
+    b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1.0</span>,<span style="color: #666666">2.0</span>])
+    <span style="color: #008000; font-weight: bold">return</span> a<span style="color: #666666">.</span>dot(b)
 
-<p>and</p>
-$$
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
-$$
+f9_grad <span style="color: #666666">=</span> grad(f9)
 
-<p>With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))</p>
-$$
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
-$$
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1.0</span>,<span style="color: #666666">0.0</span>])
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f9 is:&quot;</span>,f9_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f9_alternative</span>(x): <span style="color: #408080; font-style: italic"># Assume a is an array with 2 elements</span>
+    b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1.0</span>,<span style="color: #666666">2.0</span>])
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>dot(x,b) <span style="color: #408080; font-style: italic"># The same as x_1*b_1 + x_2*b_2</span>
+
+f9_alternative_grad <span style="color: #666666">=</span> grad(f9_alternative)
+
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">3.0</span>,<span style="color: #666666">0.0</span>])
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The gradient of f9 is:&quot;</span>,f9_alternative_grad(x))
+
+<span style="color: #408080; font-style: italic"># The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively</span>
+<span style="color: #408080; font-style: italic"># w.r.t x is (b_1, b_2).</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
 
 <p>
@@ -269,6 +417,10 @@ $$
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs034.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs034.html b/doc/pub/week40/html/._week40-bs034.html
index 34ff1e2d1..9babf2712 100644
--- a/doc/pub/week40/html/._week40-bs034.html
+++ b/doc/pub/week40/html/._week40-bs034.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,25 +319,33 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0034"></a>
 <!-- !split -->
-<h2 id="derivative-of-the-cost-function" class="anchor">Derivative of the cost function </h2>
+<h2 id="recommended-to-avoid" class="anchor">Recommended to avoid </h2>
+<p>The documentation recommends to avoid inplace operations such as</p>
 
-<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
-
-<p>Let us specialize to the output layer \( l=L \). Our cost function is</p>
-$$
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-$$
-
-<p>The derivative of this function with respect to the weights is</p>
-
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
-$$
-
-<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
-$$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
-$$
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;">a <span style="color: #666666">+=</span> b
+a <span style="color: #666666">-=</span> b
+a<span style="color: #666666">*=</span> b
+a <span style="color: #666666">/=</span>b
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
 
 <p>
@@ -271,6 +369,11 @@ $$
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs035.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs035.html b/doc/pub/week40/html/._week40-bs035.html
index 81736a06d..5d926ac4d 100644
--- a/doc/pub/week40/html/._week40-bs035.html
+++ b/doc/pub/week40/html/._week40-bs035.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,49 +319,83 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0035"></a>
 <!-- !split -->
-<h2 id="bringing-it-together-first-back-propagation-equation" class="anchor">Bringing it together, first back propagation equation </h2>
+<h2 id="using-autograd-with-ols" class="anchor">Using Autograd with OLS </h2>
 
-<p>We have thus</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
-$$
-
-<p>Defining</p>
-$$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-$$
-
-<p>and using the Hadamard product of two vectors we can write this as</p>
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-
-<p>This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
+<p>We conclude the part on optmization by showing how we can make codes
+for linear regression and logistic regression using <b>autograd</b>. The
+first example shows results with ordinary leats squares.
 </p>
 
-<p>Notice that everything in the above equations is easily computed.  In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \).  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-</p>
 
-$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-$$
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<p>With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
-$$
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(beta):
+    <span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> beta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+<span style="color: #408080; font-style: italic"># define the gradient</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> training_gradient(theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
+Xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
+ypredict <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta)
+ypredict2 <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta_linreg)
+
+plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">&quot;b-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Random numbers &#39;</span>)
+plt<span style="color: #666666">.</span>show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
 
 <p>
@@ -294,6 +418,12 @@ $$
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="._week40-bs044.html">45</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs036.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs036.html b/doc/pub/week40/html/._week40-bs036.html
index c6ce91aef..fa409740a 100644
--- a/doc/pub/week40/html/._week40-bs036.html
+++ b/doc/pub/week40/html/._week40-bs036.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,20 +319,83 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0036"></a>
 <!-- !split -->
-<h2 id="derivatives-in-terms-of-z-j-l" class="anchor">Derivatives in terms of \( z_j^L \) </h2>
+<h2 id="same-code-but-now-with-momentum-gradient-descent" class="anchor">Same code but now with momentum gradient descent </h2>
 
-<p>It is also easy to see that our previous equation can be written as</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-$$
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
-$$
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(beta):
+    <span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> beta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #408080; font-style: italic">#+np.random.randn(n,1)</span>
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">30</span>
+
+<span style="color: #408080; font-style: italic"># define the gradient</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> training_gradient(theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+    <span style="color: #008000">print</span>(<span style="color: #008000">iter</span>,gradients[<span style="color: #666666">0</span>],gradients[<span style="color: #666666">1</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+<span style="color: #408080; font-style: italic"># Now improve with momentum gradient descent</span>
+change <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+delta_momentum <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    <span style="color: #408080; font-style: italic"># calculate gradient</span>
+    gradients <span style="color: #666666">=</span> training_gradient(theta)
+    <span style="color: #408080; font-style: italic"># calculate update</span>
+    new_change <span style="color: #666666">=</span> eta<span style="color: #666666">*</span>gradients<span style="color: #666666">+</span>delta_momentum<span style="color: #666666">*</span>change
+    <span style="color: #408080; font-style: italic"># take a step</span>
+    theta <span style="color: #666666">-=</span> new_change
+    <span style="color: #408080; font-style: italic"># save the change</span>
+    change <span style="color: #666666">=</span> new_change
+    <span style="color: #008000">print</span>(<span style="color: #008000">iter</span>,gradients[<span style="color: #666666">0</span>],gradients[<span style="color: #666666">1</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd wth momentum&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely</p>
-$$
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
-$$
 
-<p>That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. </p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -262,6 +415,13 @@ $$
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="._week40-bs044.html">45</a></li>
+  <li><a href="._week40-bs045.html">46</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
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 </ul>
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diff --git a/doc/pub/week40/html/._week40-bs037.html b/doc/pub/week40/html/._week40-bs037.html
index e9a3e93aa..f649a9615 100644
--- a/doc/pub/week40/html/._week40-bs037.html
+++ b/doc/pub/week40/html/._week40-bs037.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,63 +319,69 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0037"></a>
 <!-- !split -->
-<h2 id="bringing-it-together" class="anchor">Bringing it together </h2>
+<h2 id="but-noen-of-these-can-compete-with-newton-s-method" class="anchor">But noen of these can compete with Newton's method </h2>
 
-<p>We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are</p>
 
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Newton&#39;s method</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-$$
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
-\tag{13}
-\end{equation}
-$$
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(beta):
+    <span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> beta)<span style="color: #666666">**2</span>)
 
-<p>and</p>
-$$
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\tag{14}
-\end{equation}
-$$
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
 
-<p>and</p>
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+beta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(beta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+<span style="color: #408080; font-style: italic"># Note that here the Hessian does not depend on the parameters beta</span>
+invH <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(H)
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-$$
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\tag{15}
-\end{equation}
-$$
+beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">5</span>
+
+<span style="color: #408080; font-style: italic"># define the gradient</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> training_gradient(beta)
+    beta <span style="color: #666666">-=</span> invH <span style="color: #666666">@</span> gradients
+    <span style="color: #008000">print</span>(<span style="color: #008000">iter</span>,gradients[<span style="color: #666666">0</span>],gradients[<span style="color: #666666">1</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;beta from own Newton code&quot;</span>)
+<span style="color: #008000">print</span>(beta)
+</pre>
 </div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
 </div>
 
 
-<p>An interesting consequence of the above equations is that when the
-activation \( a_k^{L-1} \) is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
-</p>
-
-<p>Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
-</p>
-
-<p>We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one \( L-1 \) in terms of the errors in the final output layer.
-</p>
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -304,6 +400,14 @@ one \( L-1 \) in terms of the errors in the final output layer.
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="._week40-bs044.html">45</a></li>
+  <li><a href="._week40-bs045.html">46</a></li>
+  <li><a href="._week40-bs046.html">47</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs038.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs038.html b/doc/pub/week40/html/._week40-bs038.html
index 242b4c780..7db2d6054 100644
--- a/doc/pub/week40/html/._week40-bs038.html
+++ b/doc/pub/week40/html/._week40-bs038.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,32 +319,104 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0038"></a>
 <!-- !split -->
-<h2 id="final-back-propagating-equation" class="anchor">Final back propagating equation </h2>
+<h2 id="including-stochastic-gradient-descent-with-autograd" class="anchor">Including Stochastic Gradient Descent with Autograd </h2>
+<p>In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using <b>autograd</b>.</p>
 
-<p>We have that (replacing \( L \) with a general layer \( l \))</p>
-$$
-\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
-$$
 
-<p>We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-$$
-\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
-$$
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
 
-<p>and recalling that</p>
-$$
-z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
-$$
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
 
-<p>with \( M_l \) being the number of nodes in layer \( l \), we obtain</p>
-$$
-\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
-$$
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-<p>This is our final equation.</p>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>training_gradient(y, X, theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
+Xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
+ypredict <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta)
+ypredict2 <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta_linreg)
+
+plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">&quot;b-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Random numbers &#39;</span>)
+plt<span style="color: #666666">.</span>show()
+
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+t0, t1 <span style="color: #666666">=</span> <span style="color: #666666">5</span>, <span style="color: #666666">50</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">learning_schedule</span>(t):
+    <span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+<span style="color: #408080; font-style: italic"># Can you figure out a better way of setting up the contributions to each batch?</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        eta <span style="color: #666666">=</span> learning_schedule(epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i)
+        theta <span style="color: #666666">=</span> theta <span style="color: #666666">-</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own sdg&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>We are now ready to set up the algorithm for back propagation and learning the weights and biases.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -273,6 +435,15 @@ $$
   <li class="active"><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="._week40-bs044.html">45</a></li>
+  <li><a href="._week40-bs045.html">46</a></li>
+  <li><a href="._week40-bs046.html">47</a></li>
+  <li><a href="._week40-bs047.html">48</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs039.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs039.html b/doc/pub/week40/html/._week40-bs039.html
index 8e9bc3615..f2019bd8e 100644
--- a/doc/pub/week40/html/._week40-bs039.html
+++ b/doc/pub/week40/html/._week40-bs039.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,75 +319,97 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0039"></a>
 <!-- !split -->
-<h2 id="setting-up-the-back-propagation-algorithm" class="anchor">Setting up the Back propagation algorithm </h2>
+<h2 id="same-code-but-now-with-momentum-gradient-descent" class="anchor">Same code but now with momentum gradient descent </h2>
 
-<p>The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>First, we set up the input data \( \hat{x} \) and the activations
-\( \hat{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \hat{a}^1 \).
-</p>
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>training_gradient(y, X, theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+t0, t1 <span style="color: #666666">=</span> <span style="color: #666666">5</span>, <span style="color: #666666">50</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">learning_schedule</span>(t):
+    <span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+
+change <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+delta_momentum <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
+
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        eta <span style="color: #666666">=</span> learning_schedule(epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i)
+        <span style="color: #408080; font-style: italic"># calculate update</span>
+        new_change <span style="color: #666666">=</span> eta<span style="color: #666666">*</span>gradients<span style="color: #666666">+</span>delta_momentum<span style="color: #666666">*</span>change
+        <span style="color: #408080; font-style: italic"># take a step</span>
+        theta <span style="color: #666666">-=</span> new_change
+        <span style="color: #408080; font-style: italic"># save the change</span>
+        change <span style="color: #666666">=</span> new_change
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own sdg with momentum&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
 </div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
 </div>
 
 
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \hat{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \hat{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-</div>
-
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all</p>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-</div>
-</div>
-
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-</div>
-</div>
-
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-</div>
-</div>
-
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -314,6 +426,16 @@ Here it is convenient to use stochastic gradient descent (see the examples below
   <li><a href="._week40-bs038.html">39</a></li>
   <li class="active"><a href="._week40-bs039.html">40</a></li>
   <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="._week40-bs044.html">45</a></li>
+  <li><a href="._week40-bs045.html">46</a></li>
+  <li><a href="._week40-bs046.html">47</a></li>
+  <li><a href="._week40-bs047.html">48</a></li>
+  <li><a href="._week40-bs048.html">49</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs040.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs040.html b/doc/pub/week40/html/._week40-bs040.html
index 99f9f7bee..559edf959 100644
--- a/doc/pub/week40/html/._week40-bs040.html
+++ b/doc/pub/week40/html/._week40-bs040.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -229,74 +319,78 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0040"></a>
 <!-- !split -->
-<h2 id="setting-up-the-back-propagation-algorithm" class="anchor">Setting up the Back propagation algorithm </h2>
+<h2 id="similar-second-order-function-now-problem-but-now-with-adagrad" class="anchor">Similar (second order function now) problem but now with AdaGrad </h2>
 
-<p>The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>First, we set up the input data \( \boldsymbol{x} \) and the activations
-\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \boldsymbol{a}^1 \).
-</p>
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+3*</span>x <span style="color: #666666">+4*</span>x<span style="color: #666666">*</span>x
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">*</span>x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+<span style="color: #408080; font-style: italic"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+<span style="color: #408080; font-style: italic"># Guess for unknown parameters theta</span>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">3</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #408080; font-style: italic"># Value for learning rate</span>
+eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
+<span style="color: #408080; font-style: italic"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  <span style="color: #666666">=</span> <span style="color: #666666">1e-8</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    Giter <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        Giter <span style="color: #666666">+=</span> gradients<span style="color: #666666">*</span>gradients
+        update <span style="color: #666666">=</span> gradients<span style="color: #666666">*</span>eta<span style="color: #666666">/</span>(delta<span style="color: #666666">+</span>np<span style="color: #666666">.</span>sqrt(Giter))
+        theta <span style="color: #666666">-=</span> update
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own AdaGrad&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
 </div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
 </div>
 
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \boldsymbol{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-</div>
-
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all</p>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-</div>
-</div>
-
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-</div>
-</div>
-
-
-<div class="panel panel-default">
-<div class="panel-body">
-<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-</div>
-</div>
-
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
+<p>Running this code we note an almost perfect agreement with the results from matrix inversion.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -313,6 +407,18 @@ Here it is convenient to use stochastic gradient descent (see the examples below
   <li><a href="._week40-bs038.html">39</a></li>
   <li><a href="._week40-bs039.html">40</a></li>
   <li class="active"><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs041.html">42</a></li>
+  <li><a href="._week40-bs042.html">43</a></li>
+  <li><a href="._week40-bs043.html">44</a></li>
+  <li><a href="._week40-bs044.html">45</a></li>
+  <li><a href="._week40-bs045.html">46</a></li>
+  <li><a href="._week40-bs046.html">47</a></li>
+  <li><a href="._week40-bs047.html">48</a></li>
+  <li><a href="._week40-bs048.html">49</a></li>
+  <li><a href="._week40-bs049.html">50</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
+  <li><a href="._week40-bs041.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
 </div>  <!-- end container -->
diff --git a/doc/pub/week40/html/._week40-bs041.html b/doc/pub/week40/html/._week40-bs041.html
index 3e98bab2b..885d97403 100644
--- a/doc/pub/week40/html/._week40-bs041.html
+++ b/doc/pub/week40/html/._week40-bs041.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
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         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -308,41 +318,83 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0041"></a>
-<!-- !split  -->
-<h2 id="setting-up-a-multi-layer-perceptron-model-for-classification" class="anchor">Setting up a Multi-layer perceptron model for classification  </h2>
+<!-- !split -->
+<h2 id="rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" class="anchor">RMSprop for adaptive learning rate with Stochastic Gradient Descent </h2>
 
-<p>We are now gong to develop an example based on the MNIST data
-base. This is a classification problem and we need to use our
-cross-entropy function we discussed in connection with logistic
-regression. The cross-entropy defines our cost function for the
-classificaton problems with neural networks.
-</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<p>In binary classification with two classes \( (0, 1) \) we define the
-logistic/sigmoid function as the probability that a particular input
-is in class \( 0 \) or \( 1 \).  This is possible because the logistic
-function takes any input from the real numbers and inputs a number
-between 0 and 1, and can therefore be interpreted as a probability. It
-also has other nice properties, such as a derivative that is simple to
-calculate.
-</p>
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
 
-<p>For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \)
-is in class 0 or 1 is just. We let \( \theta \) represent the unknown weights and biases to be adjusted by our equations). The variable \( x \)
-represents our activation values \( z \). We have
-</p>
-$$
-P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp{(- \boldsymbol{x}})} ,
-$$
+n <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+3*</span>x <span style="color: #666666">+4*</span>x<span style="color: #666666">*</span>x<span style="color: #408080; font-style: italic"># +np.random.randn(n,1)</span>
 
-<p>and</p>
-$$
-P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,
-$$
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">*</span>x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
 
-<p>where \( y \in \{0, 1\} \)  and \( \boldsymbol{\theta} \) represents the weights and biases
-of our network.
-</p>
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+<span style="color: #408080; font-style: italic"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+<span style="color: #408080; font-style: italic"># Guess for unknown parameters theta</span>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">3</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #408080; font-style: italic"># Value for learning rate</span>
+eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
+<span style="color: #408080; font-style: italic"># Value for parameter rho</span>
+rho <span style="color: #666666">=</span> <span style="color: #666666">0.99</span>
+<span style="color: #408080; font-style: italic"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  <span style="color: #666666">=</span> <span style="color: #666666">1e-8</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    Giter <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+	<span style="color: #408080; font-style: italic"># Accumulated gradient</span>
+	<span style="color: #408080; font-style: italic"># Scaling with rho the new and the previous results</span>
+        Giter <span style="color: #666666">=</span> (rho<span style="color: #666666">*</span>Giter<span style="color: #666666">+</span>(<span style="color: #666666">1-</span>rho)<span style="color: #666666">*</span>gradients<span style="color: #666666">*</span>gradients)
+	<span style="color: #408080; font-style: italic"># Taking the diagonal only and inverting</span>
+        update <span style="color: #666666">=</span> gradients<span style="color: #666666">*</span>eta<span style="color: #666666">/</span>(delta<span style="color: #666666">+</span>np<span style="color: #666666">.</span>sqrt(Giter))
+	<span style="color: #408080; font-style: italic"># Hadamard product</span>
+        theta <span style="color: #666666">-=</span> update
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own RMSprop&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
 
 <p>
@@ -370,7 +422,7 @@ of our network.
   <li><a href="._week40-bs049.html">50</a></li>
   <li><a href="._week40-bs050.html">51</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs042.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs042.html b/doc/pub/week40/html/._week40-bs042.html
index 3d2d024f9..7db9300b0 100644
--- a/doc/pub/week40/html/._week40-bs042.html
+++ b/doc/pub/week40/html/._week40-bs042.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,56 +319,89 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0042"></a>
 <!-- !split -->
-<h2 id="defining-the-cost-function" class="anchor">Defining the cost function </h2>
+<h2 id="and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" class="anchor">And finally <a href="https://arxiv.org/pdf/1412.6980.pdf" target="_self">ADAM</a> </h2>
 
-<p>Our cost function is given as (see the Logistic regression lectures)</p>
-$$
-\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
-y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .
-$$
 
-<p>This last equality means that we can interpret our <em>cost</em> function as a sum over the <em>loss</em> function
-for each point in the dataset \( \mathcal{L}_i(\boldsymbol{\theta}) \).  
-The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
-than maximizing a negative number.  
-</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:  </p>
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
 
-<p>\( y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and</p>
+n <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+3*</span>x <span style="color: #666666">+4*</span>x<span style="color: #666666">*</span>x<span style="color: #408080; font-style: italic"># +np.random.randn(n,1)</span>
 
-\( y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \)
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">*</span>x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
 
-<p>i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \))..  </p>
 
-<p>If \( \boldsymbol{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th
-output vector \( \boldsymbol{y}_i \).  
-The probability of \( \boldsymbol{x}_i \) being in class \( c \) will be given by the softmax function:  
-</p>
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+<span style="color: #408080; font-style: italic"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+<span style="color: #408080; font-style: italic"># Guess for unknown parameters theta</span>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">3</span>,<span style="color: #666666">1</span>)
 
-$$
-P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
-{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,
-$$
+<span style="color: #408080; font-style: italic"># Value for learning rate</span>
+eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
+<span style="color: #408080; font-style: italic"># Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980</span>
+beta1 <span style="color: #666666">=</span> <span style="color: #666666">0.9</span>
+beta2 <span style="color: #666666">=</span> <span style="color: #666666">0.999</span>
+<span style="color: #408080; font-style: italic"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  <span style="color: #666666">=</span> <span style="color: #666666">1e-7</span>
+<span style="color: #008000">iter</span> <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    first_moment <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    second_moment <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    <span style="color: #008000">iter</span> <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        <span style="color: #408080; font-style: italic"># Computing moments first</span>
+        first_moment <span style="color: #666666">=</span> beta1<span style="color: #666666">*</span>first_moment <span style="color: #666666">+</span> (<span style="color: #666666">1-</span>beta1)<span style="color: #666666">*</span>gradients
+        second_moment <span style="color: #666666">=</span> beta2<span style="color: #666666">*</span>second_moment<span style="color: #666666">+</span>(<span style="color: #666666">1-</span>beta2)<span style="color: #666666">*</span>gradients<span style="color: #666666">*</span>gradients
+        first_term <span style="color: #666666">=</span> first_moment<span style="color: #666666">/</span>(<span style="color: #666666">1.0-</span>beta1<span style="color: #666666">**</span><span style="color: #008000">iter</span>)
+        second_term <span style="color: #666666">=</span> second_moment<span style="color: #666666">/</span>(<span style="color: #666666">1.0-</span>beta2<span style="color: #666666">**</span><span style="color: #008000">iter</span>)
+	<span style="color: #408080; font-style: italic"># Scaling with rho the new and the previous results</span>
+        update <span style="color: #666666">=</span> eta<span style="color: #666666">*</span>first_term<span style="color: #666666">/</span>(np<span style="color: #666666">.</span>sqrt(second_term)<span style="color: #666666">+</span>delta)
+        theta <span style="color: #666666">-=</span> update
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own ADAM&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>which reduces to the logistic function in the binary case.  
-The likelihood of this \( C \)-class classifier
-is now given as:  
-</p>
-
-$$
-P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
-$$
-
-<p>Again we take the negative log-likelihood to define our cost function:  </p>
-
-$$
-\mathcal{C}(\boldsymbol{\theta}) = - \log{P(\mathcal{D} \mid \boldsymbol{\theta})}.
-$$
-
-<p>See the logistic regression lectures for a full definition of the cost function.</p>
-
-<p>The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -385,7 +428,7 @@ $$
   <li><a href="._week40-bs050.html">51</a></li>
   <li><a href="._week40-bs051.html">52</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs043.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs043.html b/doc/pub/week40/html/._week40-bs043.html
index 359d6dbaa..2acb627a0 100644
--- a/doc/pub/week40/html/._week40-bs043.html
+++ b/doc/pub/week40/html/._week40-bs043.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,50 +319,63 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0043"></a>
 <!-- !split -->
-<h2 id="example-binary-classification-problem" class="anchor">Example: binary classification problem </h2>
+<h2 id="and-logistic-regression" class="anchor">And Logistic Regression </h2>
 
-<p>As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \( \beta \) as</p>
-$$
-\mathcal{C}(\boldsymbol{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\boldsymbol{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\boldsymbol{\beta})}\right),
-$$
 
-<p>where we had defined the logistic (sigmoid) function</p>
-$$
-p(y_i =1\vert x_i,\boldsymbol{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}},
-$$
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
 
-<p>and</p>
-$$
-p(y_i =0\vert x_i,\boldsymbol{\beta})=1-p(y_i =1\vert x_i,\boldsymbol{\beta}).
-$$
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sigmoid</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">0.5</span> <span style="color: #666666">*</span> (np<span style="color: #666666">.</span>tanh(x <span style="color: #666666">/</span> <span style="color: #666666">2.</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
 
-<p>The parameters \( \boldsymbol{\beta} \) were defined using a minimization method like gradient descent or Newton-Raphson's method. </p>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">logistic_predictions</span>(weights, inputs):
+    <span style="color: #408080; font-style: italic"># Outputs probability of a label being true according to logistic model.</span>
+    <span style="color: #008000; font-weight: bold">return</span> sigmoid(np<span style="color: #666666">.</span>dot(inputs, weights))
 
-<p>Now we replace \( x_i \) with the activation \( z_i^l \) for a given layer \( l \) and the outputs as \( y_i=a_i^l=f(z_i^l) \), with \( z_i^l \) now being a function of the weights \( w_{ij}^l \) and biases \( b_i^l \). 
-We have then
-</p>
-$$
-a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}},
-$$
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">training_loss</span>(weights):
+    <span style="color: #408080; font-style: italic"># Training loss is the negative log-likelihood of the training labels.</span>
+    preds <span style="color: #666666">=</span> logistic_predictions(weights, inputs)
+    label_probabilities <span style="color: #666666">=</span> preds <span style="color: #666666">*</span> targets <span style="color: #666666">+</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> preds) <span style="color: #666666">*</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> targets)
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">-</span>np<span style="color: #666666">.</span>sum(np<span style="color: #666666">.</span>log(label_probabilities))
 
-<p>with </p>
-$$
-z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,
-$$
+<span style="color: #408080; font-style: italic"># Build a toy dataset.</span>
+inputs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0.52</span>, <span style="color: #666666">1.12</span>,  <span style="color: #666666">0.77</span>],
+                   [<span style="color: #666666">0.88</span>, <span style="color: #666666">-1.08</span>, <span style="color: #666666">0.15</span>],
+                   [<span style="color: #666666">0.52</span>, <span style="color: #666666">0.06</span>, <span style="color: #666666">-1.30</span>],
+                   [<span style="color: #666666">0.74</span>, <span style="color: #666666">-2.49</span>, <span style="color: #666666">1.39</span>]])
+targets <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #008000; font-weight: bold">True</span>, <span style="color: #008000; font-weight: bold">True</span>, <span style="color: #008000; font-weight: bold">False</span>, <span style="color: #008000; font-weight: bold">True</span>])
 
-<p>where the superscript \( l-1 \) indicates that these are the outputs from layer \( l-1 \).
-Our cost function at the final layer \( l=L \) is now
-</p>
-$$
-\mathcal{C}(\boldsymbol{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right),
-$$
+<span style="color: #408080; font-style: italic"># Define a function that returns gradients of training loss using Autograd.</span>
+training_gradient_fun <span style="color: #666666">=</span> grad(training_loss)
 
-<p>where we have defined the targets \( t_i \). The derivatives of the cost function with respect to the output \( a_i^L \) are then easily calculated and we get</p>
-$$
-\frac{\partial \mathcal{C}(\boldsymbol{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. 
-$$
+<span style="color: #408080; font-style: italic"># Optimize weights using gradient descent.</span>
+weights <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0.0</span>, <span style="color: #666666">0.0</span>, <span style="color: #666666">0.0</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Initial loss:&quot;</span>, training_loss(weights))
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">100</span>):
+    weights <span style="color: #666666">-=</span> training_gradient_fun(weights) <span style="color: #666666">*</span> <span style="color: #666666">0.01</span>
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Trained loss:&quot;</span>, training_loss(weights))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>In case we use another activation function than the logistic one, we need to evaluate other derivatives. </p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -379,7 +402,7 @@ $$
   <li><a href="._week40-bs051.html">52</a></li>
   <li><a href="._week40-bs052.html">53</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs044.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs044.html b/doc/pub/week40/html/._week40-bs044.html
index d2dfe412a..205a16cbd 100644
--- a/doc/pub/week40/html/._week40-bs044.html
+++ b/doc/pub/week40/html/._week40-bs044.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,24 +319,48 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0044"></a>
 <!-- !split -->
-<h2 id="the-softmax-function" class="anchor">The Softmax function </h2>
-<p>In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need</p>
-$$
-\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
-\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}.
-$$
+<h2 id="introducing-jax-https-jax-readthedocs-io-en-latest" class="anchor">Introducing <a href="https://jax.readthedocs.io/en/latest/" target="_self">JAX</a> </h2>
 
-<p>For the Softmax function we have</p>
-$$
-f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.
-$$
+<p>Presently, instead of using <b>autograd</b>, we recommend using <a href="https://jax.readthedocs.io/en/latest/" target="_self">JAX</a></p>
 
-<p>Its derivative with respect to \( z_j^l \) gives </p>
-$$
-\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), 
-$$
+<p><b>JAX</b> is Autograd and <a href="https://www.tensorflow.org/xla" target="_self">XLA (Accelerated Linear Algebra))</a>,
+brought together for high-performance numerical computing and machine learning research.
+It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.
+</p>
+
+<p>Here's a simple example on how you can use <b>JAX</b> to compute the derivate of the logistic function.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">jax.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">jnp</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">jax</span> <span style="color: #008000; font-weight: bold">import</span> grad, jit, vmap
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sum_logistic</span>(x):
+  <span style="color: #008000; font-weight: bold">return</span> jnp<span style="color: #666666">.</span>sum(<span style="color: #666666">1.0</span> <span style="color: #666666">/</span> (<span style="color: #666666">1.0</span> <span style="color: #666666">+</span> jnp<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x)))
+
+x_small <span style="color: #666666">=</span> jnp<span style="color: #666666">.</span>arange(<span style="color: #666666">3.</span>)
+derivative_fn <span style="color: #666666">=</span> grad(sum_logistic)
+<span style="color: #008000">print</span>(derivative_fn(x_small))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>which in case of the simply binary model reduces to  having \( i=j \). </p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -353,7 +387,7 @@ $$
   <li><a href="._week40-bs052.html">53</a></li>
   <li><a href="._week40-bs053.html">54</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs045.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs045.html b/doc/pub/week40/html/._week40-bs045.html
index 21c778586..d00fd1a9d 100644
--- a/doc/pub/week40/html/._week40-bs045.html
+++ b/doc/pub/week40/html/._week40-bs045.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -308,19 +318,18 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0045"></a>
-<!-- !split  -->
-<h2 id="developing-a-code-for-doing-neural-networks-with-back-propagation" class="anchor">Developing a code for doing neural networks with back propagation </h2>
+<!-- !split -->
+<h2 id="introduction-to-neural-networks" class="anchor">Introduction to Neural networks </h2>
 
-<p>One can identify a set of key steps when using neural networks to solve supervised learning problems:  </p>
+<p>Artificial neural networks are computational systems that can learn to
+perform tasks by considering examples, generally without being
+programmed with any task-specific rules. It is supposed to mimic a
+biological system, wherein neurons interact by sending signals in the
+form of mathematical functions between layers. All layers can contain
+an arbitrary number of neurons, and each connection is represented by
+a weight variable.
+</p>
 
-<ol>
-<li> Collect and pre-process data</li>  
-<li> Define model and architecture</li>  
-<li> Choose cost function and optimizer</li>  
-<li> Train the model</li>  
-<li> Evaluate model performance on test data</li>  
-<li> Adjust hyperparameters (if necessary, network architecture)</li>
-</ol>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -346,7 +355,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs053.html">54</a></li>
   <li><a href="._week40-bs054.html">55</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs046.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs046.html b/doc/pub/week40/html/._week40-bs046.html
index 5897043d6..f21db8d22 100644
--- a/doc/pub/week40/html/._week40-bs046.html
+++ b/doc/pub/week40/html/._week40-bs046.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,117 +319,65 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0046"></a>
 <!-- !split -->
-<h2 id="collect-and-pre-process-data" class="anchor">Collect and pre-process data </h2>
+<h2 id="artificial-neurons" class="anchor">Artificial neurons  </h2>
 
-<p>Here we will be using the MNIST dataset, which is readily available through the <b>scikit-learn</b>
-package. You may also find it for example <a href="http://yann.lecun.com/exdb/mnist/" target="_self">here</a>.  
-The <em>MNIST</em> (Modified National Institute of Standards and Technology) database is a large database
-of handwritten digits that is commonly used for training various image processing systems.  
-The MNIST dataset consists of 70 000 images of size \( 28\times 28 \) pixels, each labeled from 0 to 9.  
-The scikit-learn dataset we will use consists of a selection of 1797 images of size \( 8\times 8 \) collected and processed from this database.  
+<p>The field of artificial neural networks has a long history of
+development, and is closely connected with the advancement of computer
+science and computers in general. A model of artificial neurons was
+first developed by McCulloch and Pitts in 1943 to study signal
+processing in the brain and has later been refined by others. The
+general idea is to mimic neural networks in the human brain, which is
+composed of billions of neurons that communicate with each other by
+sending electrical signals.  Each neuron accumulates its incoming
+signals, which must exceed an activation threshold to yield an
+output. If the threshold is not overcome, the neuron remains inactive,
+i.e. has zero output.
 </p>
 
-<p>To feed data into a feed-forward neural network we need to represent
-the inputs as a design/feature matrix \( X = (n_{inputs}, n_{features}) \).  Each
-row represents an <em>input</em>, in this case a handwritten digit, and
-each column represents a <em>feature</em>, in this case a pixel.  The
-correct answers, also known as <em>labels</em> or <em>targets</em> are
-represented as a 1D array of integers 
-\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \).
+<p>This behaviour has inspired a simple mathematical model for an artificial neuron.</p>
+
+$$
+\begin{equation}
+ y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
+\tag{6}
+\end{equation}
+$$
+
+<p>Here, the output \( y \) of the neuron is the value of its activation function, which have as input
+a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
 </p>
 
-<p>As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from
-measurements of height (in m)  
-and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:  
+<p>Conceptually, it is helpful to divide neural networks into four
+categories:
+</p>
+<ol>
+<li> general purpose neural networks for supervised learning,</li>
+<li> neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),</li>
+<li> neural networks for sequential data such as Recurrent Neural Networks (RNNs), and</li>
+<li> neural networks for unsupervised learning such as Deep Boltzmann Machines.</li>
+</ol>
+<p>In natural science, DNNs and CNNs have already found numerous
+applications. In statistical physics, they have been applied to detect
+phase transitions in 2D Ising and Potts models, lattice gauge
+theories, and different phases of polymers, or solving the
+Navier-Stokes equation in weather forecasting.  Deep learning has also
+found interesting applications in quantum physics. Various quantum
+phase transitions can be detected and studied using DNNs and CNNs,
+topological phases, and even non-equilibrium many-body
+localization. Representing quantum states as DNNs quantum state
+tomography are among some of the impressive achievements to reveal the
+potential of DNNs to facilitate the study of quantum systems.
 </p>
 
-<p>$$ X = \begin{bmatrix}
-1.85 &amp; 81\\
-1.71 &amp; 65\\
-1.95 &amp; 103\\
-1.55 &amp; 42\\
-1.63 &amp; 56
-\end{bmatrix} ,$$  
+<p>In quantum information theory, it has been shown that one can perform
+gate decompositions with the help of neural. 
 </p>
 
-<p>and the targets would be:  </p>
-
-<p>$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$  </p>
-
-<p>Since each input image is a 2D matrix, we need to flatten the image
-(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a
-design/feature matrix. This means we lose all spatial information in the
-image, such as locality and translational invariance. More complicated
-architectures such as Convolutional Neural Networks can take advantage
-of such information, and are most commonly applied when analyzing
-images.
+<p>The applications are not limited to the natural sciences. There is a
+plethora of applications in essentially all disciplines, from the
+humanities to life science and medicine.
 </p>
 
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># import necessary packages</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
-
-
-<span style="color: #408080; font-style: italic"># ensure the same random numbers appear every time</span>
-np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
-
-<span style="color: #408080; font-style: italic"># display images in notebook</span>
-<span style="color: #666666">%</span>matplotlib inline
-plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">&#39;figure.figsize&#39;</span>] <span style="color: #666666">=</span> (<span style="color: #666666">12</span>,<span style="color: #666666">12</span>)
-
-
-<span style="color: #408080; font-style: italic"># download MNIST dataset</span>
-digits <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>load_digits()
-
-<span style="color: #408080; font-style: italic"># define inputs and labels</span>
-inputs <span style="color: #666666">=</span> digits<span style="color: #666666">.</span>images
-labels <span style="color: #666666">=</span> digits<span style="color: #666666">.</span>target
-
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;inputs = (n_inputs, pixel_width, pixel_height) = &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(inputs<span style="color: #666666">.</span>shape))
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;labels = (n_inputs) = &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(labels<span style="color: #666666">.</span>shape))
-
-
-<span style="color: #408080; font-style: italic"># flatten the image</span>
-<span style="color: #408080; font-style: italic"># the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64</span>
-n_inputs <span style="color: #666666">=</span> <span style="color: #008000">len</span>(inputs)
-inputs <span style="color: #666666">=</span> inputs<span style="color: #666666">.</span>reshape(n_inputs, <span style="color: #666666">-1</span>)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;X = (n_inputs, n_features) = &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(inputs<span style="color: #666666">.</span>shape))
-
-
-<span style="color: #408080; font-style: italic"># choose some random images to display</span>
-indices <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(n_inputs)
-random_indices <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice(indices, size<span style="color: #666666">=5</span>)
-
-<span style="color: #008000; font-weight: bold">for</span> i, image <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(digits<span style="color: #666666">.</span>images[random_indices]):
-    plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #666666">5</span>, i<span style="color: #666666">+1</span>)
-    plt<span style="color: #666666">.</span>axis(<span style="color: #BA2121">&#39;off&#39;</span>)
-    plt<span style="color: #666666">.</span>imshow(image, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>gray_r, interpolation<span style="color: #666666">=</span><span style="color: #BA2121">&#39;nearest&#39;</span>)
-    plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Label: </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> digits<span style="color: #666666">.</span>target[random_indices[i]])
-plt<span style="color: #666666">.</span>show()
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -445,7 +403,7 @@ plt<span style="color: #666666">.</span>show()
   <li><a href="._week40-bs054.html">55</a></li>
   <li><a href="._week40-bs055.html">56</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs047.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs047.html b/doc/pub/week40/html/._week40-bs047.html
index 9cf6e0bdf..4b2553667 100644
--- a/doc/pub/week40/html/._week40-bs047.html
+++ b/doc/pub/week40/html/._week40-bs047.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,68 +319,29 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0047"></a>
 <!-- !split -->
-<h2 id="train-and-test-datasets" class="anchor">Train and test datasets </h2>
+<h2 id="neural-network-types" class="anchor">Neural network types </h2>
 
-<p>Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.  </p>
-
-<p>We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.  </p>
-
-<p>It is important that the train and test datasets are drawn randomly from our dataset, to ensure
-no bias in the sampling.  
-Say you are taking measurements of weather data to predict the weather in the coming 5 days.
-You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data
-collected from 12.00 to 24.00.
+<p>An artificial neural network (ANN), is a computational model that
+consists of layers of connected neurons, or nodes or units.  We will
+refer to these interchangeably as units or nodes, and sometimes as
+neurons.
 </p>
 
+<p>It is supposed to mimic a biological nervous system by letting each
+neuron interact with other neurons by sending signals in the form of
+mathematical functions between layers.  A wide variety of different
+ANNs have been developed, but most of them consist of an input layer,
+an output layer and eventual layers in-between, called <em>hidden
+layers</em>. All layers can contain an arbitrary number of nodes, and each
+connection between two nodes is associated with a weight variable.
+</p>
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
-
-<span style="color: #408080; font-style: italic"># one-liner from scikit-learn library</span>
-train_size <span style="color: #666666">=</span> <span style="color: #666666">0.8</span>
-test_size <span style="color: #666666">=</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> train_size
-X_train, X_test, Y_train, Y_test <span style="color: #666666">=</span> train_test_split(inputs, labels, train_size<span style="color: #666666">=</span>train_size,
-                                                    test_size<span style="color: #666666">=</span>test_size)
-
-<span style="color: #408080; font-style: italic"># equivalently in numpy</span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">train_test_split_numpy</span>(inputs, labels, train_size, test_size):
-    n_inputs <span style="color: #666666">=</span> <span style="color: #008000">len</span>(inputs)
-    inputs_shuffled <span style="color: #666666">=</span> inputs<span style="color: #666666">.</span>copy()
-    labels_shuffled <span style="color: #666666">=</span> labels<span style="color: #666666">.</span>copy()
-    
-    np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>shuffle(inputs_shuffled)
-    np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>shuffle(labels_shuffled)
-    
-    train_end <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n_inputs<span style="color: #666666">*</span>train_size)
-    X_train, X_test <span style="color: #666666">=</span> inputs_shuffled[:train_end], inputs_shuffled[train_end:]
-    Y_train, Y_test <span style="color: #666666">=</span> labels_shuffled[:train_end], labels_shuffled[train_end:]
-    
-    <span style="color: #008000; font-weight: bold">return</span> X_train, X_test, Y_train, Y_test
-
-<span style="color: #408080; font-style: italic">#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)</span>
-
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Number of training images: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(<span style="color: #008000">len</span>(X_train)))
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Number of test images: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(<span style="color: #008000">len</span>(X_test)))
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
+<p>Neural networks (also called neural nets) are neural-inspired
+nonlinear models for supervised learning.  As we will see, neural nets
+can be viewed as natural, more powerful extensions of supervised
+learning methods such as linear and logistic regression and soft-max
+methods we discussed earlier.
+</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -397,7 +368,7 @@ X_train, X_test, Y_train, Y_test <span style="color: #666666">=</span> train_tes
   <li><a href="._week40-bs055.html">56</a></li>
   <li><a href="._week40-bs056.html">57</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs048.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs048.html b/doc/pub/week40/html/._week40-bs048.html
index 059016c44..6d7b5d165 100644
--- a/doc/pub/week40/html/._week40-bs048.html
+++ b/doc/pub/week40/html/._week40-bs048.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,47 +319,21 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0048"></a>
 <!-- !split -->
-<h2 id="define-model-and-architecture" class="anchor">Define model and architecture </h2>
+<h2 id="feed-forward-neural-networks" class="anchor">Feed-forward neural networks </h2>
 
-<p>Our simple feed-forward neural network will consist of an <em>input</em> layer, a single <em>hidden</em> layer and an <em>output</em> layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have </p>
-
-<p>$$ z = \sum_{i=1}^n w_i a_i ,$$</p>
-
-<p>$$ y = f(z) ,$$</p>
-
-<p>where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer
-and \( w_i \) is the weight to input \( i \).  
-The activation of the neurons in the input layer is just the features (e.g. a pixel value).  
+<p>The feed-forward neural network (FFNN) was the first and simplest type
+of ANNs that were devised. In this network, the information moves in
+only one direction: forward through the layers.
 </p>
 
-<p>The simplest activation function for a neuron is the <em>Heaviside</em> function:</p>
-
-<p>$$ f(z) = 
-\begin{cases}
-1,  &  z > 0\\
-0,  & \text{otherwise}
-\end{cases}
-$$
+<p>Nodes are represented by circles, while the arrows display the
+connections between the nodes, including the direction of information
+flow. Additionally, each arrow corresponds to a weight variable
+(figure to come).  We observe that each node in a layer is connected
+to <em>all</em> nodes in the subsequent layer, making this a so-called
+<em>fully-connected</em> FFNN.
 </p>
 
-<p>A feed-forward neural network with this activation is known as a <em>perceptron</em>.  
-For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.  
-This activation can be generalized to \( k \) classes (using e.g. the <em>one-against-all</em> strategy), 
-and we call these architectures <em>multiclass perceptrons</em>.  
-</p>
-
-<p>However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and  
-Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.  
-</p>
-
-<p>Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).  
-We will be using the sigmoid function \( \sigma(x) \):  
-</p>
-
-<p>$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$</p>
-
-<p>which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.</p>
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -375,7 +359,7 @@ We will be using the sigmoid function \( \sigma(x) \):
   <li><a href="._week40-bs056.html">57</a></li>
   <li><a href="._week40-bs057.html">58</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs049.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs049.html b/doc/pub/week40/html/._week40-bs049.html
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@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
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         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -308,45 +318,28 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0049"></a>
-<!-- !split   -->
-<h2 id="layers" class="anchor">Layers </h2>
+<!-- !split -->
+<h2 id="convolutional-neural-network" class="anchor">Convolutional Neural Network </h2>
 
-<ul>
-<li> Input</li> 
-</ul>
-<p>Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.  </p>
-
-<ul>
-<li> Hidden layer</li>
-</ul>
-<p>We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.  
-Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.  
+<p>A different variant of FFNNs are <em>convolutional neural networks</em>
+(CNNs), which have a connectivity pattern inspired by the animal
+visual cortex. Individual neurons in the visual cortex only respond to
+stimuli from small sub-regions of the visual field, called a receptive
+field. This makes the neurons well-suited to exploit the strong
+spatially local correlation present in natural images. The response of
+each neuron can be approximated mathematically as a convolution
+operation.  (figure to come)
 </p>
 
-<ul>
-<li> Output</li>
-</ul>
-<p>If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
-which could output 0 or 1 according to the Heaviside function. This would be an example of a <em>hard</em> classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a <em>soft</em> classifier, which outputs the probability of being in class 0 or 1.  
-</p>
-
-<p>For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.  </p>
-
-<p>Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the <em>softmax</em> function:  </p>
-
-<p>$$ P(\text{class \( j \)} \mid \text{input \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
-{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$  
-</p>
-
-<p>i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) the weights of neuron \( j \) to the inputs.  
-The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.  
-The exponent is just the weighted sum of inputs as before:  
-</p>
-
-<p>$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$  </p>
-
-<p>Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
-weights to the output layer.
+<p>Convolutional neural networks emulate the behaviour of neurons in the
+visual cortex by enforcing a <em>local</em> connectivity pattern between
+nodes of adjacent layers: Each node in a convolutional layer is
+connected only to a subset of the nodes in the previous layer, in
+contrast to the fully-connected FFNN.  Often, CNNs consist of several
+convolutional layers that learn local features of the input, with a
+fully-connected layer at the end, which gathers all the local data and
+produces the outputs. They have wide applications in image and video
+recognition.
 </p>
 
 <p>
@@ -374,7 +367,7 @@ weights to the output layer.
   <li><a href="._week40-bs057.html">58</a></li>
   <li><a href="._week40-bs058.html">59</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs050.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs050.html b/doc/pub/week40/html/._week40-bs050.html
index 03e2f6a4a..a6fd6363c 100644
--- a/doc/pub/week40/html/._week40-bs050.html
+++ b/doc/pub/week40/html/._week40-bs050.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -308,58 +318,21 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0050"></a>
-<!-- !split   -->
-<h2 id="weights-and-biases" class="anchor">Weights and biases </h2>
+<!-- !split -->
+<h2 id="recurrent-neural-networks" class="anchor">Recurrent neural networks </h2>
 
-<p>Typically weights are initialized with small values distributed around zero, drawn from a uniform
-or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.  
+<p>So far we have only mentioned ANNs where information flows in one
+direction: forward. <em>Recurrent neural networks</em> on the other hand,
+have connections between nodes that form directed <em>cycles</em>. This
+creates a form of internal memory which are able to capture
+information on what has been calculated before; the output is
+dependent on the previous computations. Recurrent NNs make use of
+sequential information by performing the same task for every element
+in a sequence, where each element depends on previous elements. An
+example of such information is sentences, making recurrent NNs
+especially well-suited for handwriting and speech recognition.
 </p>
 
-<p>Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
-of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \( j \), \( b_j \):  
-</p>
-
-<p>$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$  </p>
-
-<p>The bias weights \( \boldsymbol{b} \) are often initialized to zero, but a small value like \( 0.01 \) ensures all neurons have some output which can be backpropagated in the first training cycle.</p>
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># building our neural network</span>
-
-n_inputs, n_features <span style="color: #666666">=</span> X_train<span style="color: #666666">.</span>shape
-n_hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">50</span>
-n_categories <span style="color: #666666">=</span> <span style="color: #666666">10</span>
-
-<span style="color: #408080; font-style: italic"># we make the weights normally distributed using numpy.random.randn</span>
-
-<span style="color: #408080; font-style: italic"># weights and bias in the hidden layer</span>
-hidden_weights <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_features, n_hidden_neurons)
-hidden_bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n_hidden_neurons) <span style="color: #666666">+</span> <span style="color: #666666">0.01</span>
-
-<span style="color: #408080; font-style: italic"># weights and bias in the output layer</span>
-output_weights <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_hidden_neurons, n_categories)
-output_bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n_categories) <span style="color: #666666">+</span> <span style="color: #666666">0.01</span>
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -385,7 +358,7 @@ output_bias <span style="color: #666666">=</span> np<span style="color: #666666"
   <li><a href="._week40-bs058.html">59</a></li>
   <li><a href="._week40-bs059.html">60</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs051.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs051.html b/doc/pub/week40/html/._week40-bs051.html
index 8ff526df4..8d5d0b948 100644
--- a/doc/pub/week40/html/._week40-bs051.html
+++ b/doc/pub/week40/html/._week40-bs051.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,26 +319,18 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0051"></a>
 <!-- !split -->
-<h2 id="feed-forward-pass" class="anchor">Feed-forward pass </h2>
+<h2 id="other-types-of-networks" class="anchor">Other types of networks </h2>
 
-<p>Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.  
-For each input image we calculate a weighted sum of input features (pixel values) to each neuron \( j \) in the hidden layer \( l \):  
-</p>
-
-<p>$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$</p>
-
-<p>this is then passed through our activation function  </p>
-
-<p>$$ a_{j}^{l} = f(z_{j}^{l}) .$$  </p>
-
-<p>We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:  </p>
-
-<p>$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$  </p>
-
-<p>Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:  </p>
-
-<p>$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$  
+<p>There are many other kinds of ANNs that have been developed. One type
+that is specifically designed for interpolation in multidimensional
+space is the radial basis function (RBF) network. RBFs are typically
+made up of three layers: an input layer, a hidden layer with
+non-linear radial symmetric activation functions and a linear output
+layer (''linear'' here means that each node in the output layer has a
+linear activation function). The layers are normally fully-connected
+and there are no cycles, thus RBFs can be viewed as a type of
+fully-connected FFNN. They are however usually treated as a separate
+type of NN due the unusual activation functions.
 </p>
 
 <p>
@@ -356,7 +358,7 @@ For each input image we calculate a weighted sum of input features (pixel values
   <li><a href="._week40-bs059.html">60</a></li>
   <li><a href="._week40-bs060.html">61</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs052.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs052.html b/doc/pub/week40/html/._week40-bs052.html
index 89a078271..718546bcc 100644
--- a/doc/pub/week40/html/._week40-bs052.html
+++ b/doc/pub/week40/html/._week40-bs052.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -308,95 +318,16 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0052"></a>
-<!-- !split    -->
-<h2 id="matrix-multiplications" class="anchor">Matrix multiplications </h2>
+<!-- !split -->
+<h2 id="multilayer-perceptrons" class="anchor">Multilayer perceptrons  </h2>
 
-<p>Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
-layer have the dimensions  
-\( W_{hidden} = (n_{features}, n_{hidden}) \),
-we can easily feed the network all our training data in one go by taking the matrix product  
+<p>One uses often so-called fully-connected feed-forward neural networks
+with three or more layers (an input layer, one or more hidden layers
+and an output layer) consisting of neurons that have non-linear
+activation functions.
 </p>
 
-<p>$$ X W^{h} = (n_{inputs}, n_{hidden}),$$ </p>
-
-<p>and obtain a matrix that holds the weighted sum of inputs to the hidden layer
-for each input image and each hidden neuron.    
-We also add the bias to obtain a matrix of weighted sums to the hidden layer \( Z^{h} \):  
-</p>
-
-<p>$$ \boldsymbol{z}^{l} = \boldsymbol{X} \boldsymbol{W}^{l} + \boldsymbol{b}^{l} ,$$</p>
-
-<p>meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.  
-This is then passed through the activation:  
-</p>
-
-<p>$$ \boldsymbol{a}^{l} = f(\boldsymbol{z}^l) .$$  </p>
-
-<p>This is fed to the output layer:  </p>
-
-<p>$$ \boldsymbol{z}^{L} = \boldsymbol{a}^{L} \boldsymbol{W}^{L} + \boldsymbol{b}^{L} .$$</p>
-
-<p>Finally we receive our output values for each image and each category by passing it through the softmax function:  </p>
-
-<p>$$ output = softmax (\boldsymbol{z}^{L}) = (n_{inputs}, n_{categories}) .$$</p>
-
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># setup the feed-forward pass, subscript h = hidden layer</span>
-
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sigmoid</span>(x):
-    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1/</span>(<span style="color: #666666">1</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x))
-
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">feed_forward</span>(X):
-    <span style="color: #408080; font-style: italic"># weighted sum of inputs to the hidden layer</span>
-    z_h <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(X, hidden_weights) <span style="color: #666666">+</span> hidden_bias
-    <span style="color: #408080; font-style: italic"># activation in the hidden layer</span>
-    a_h <span style="color: #666666">=</span> sigmoid(z_h)
-    
-    <span style="color: #408080; font-style: italic"># weighted sum of inputs to the output layer</span>
-    z_o <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(a_h, output_weights) <span style="color: #666666">+</span> output_bias
-    <span style="color: #408080; font-style: italic"># softmax output</span>
-    <span style="color: #408080; font-style: italic"># axis 0 holds each input and axis 1 the probabilities of each category</span>
-    exp_term <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(z_o)
-    probabilities <span style="color: #666666">=</span> exp_term <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum(exp_term, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
-    
-    <span style="color: #008000; font-weight: bold">return</span> probabilities
-
-probabilities <span style="color: #666666">=</span> feed_forward(X_train)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;probabilities = (n_inputs, n_categories) = &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(probabilities<span style="color: #666666">.</span>shape))
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;probability that image 0 is in category 0,1,2,...,9 = </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(probabilities[<span style="color: #666666">0</span>]))
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;probabilities sum up to: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(probabilities[<span style="color: #666666">0</span>]<span style="color: #666666">.</span>sum()))
-<span style="color: #008000">print</span>()
-
-<span style="color: #408080; font-style: italic"># we obtain a prediction by taking the class with the highest likelihood</span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">predict</span>(X):
-    probabilities <span style="color: #666666">=</span> feed_forward(X)
-    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>argmax(probabilities, axis<span style="color: #666666">=1</span>)
-
-predictions <span style="color: #666666">=</span> predict(X_train)
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;predictions = (n_inputs) = &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(predictions<span style="color: #666666">.</span>shape))
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;prediction for image 0: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(predictions[<span style="color: #666666">0</span>]))
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;correct label for image 0: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(Y_train[<span style="color: #666666">0</span>]))
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
+<p>Such networks are often called <em>multilayer perceptrons</em> (MLPs).</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -423,7 +354,7 @@ predictions <span style="color: #666666">=</span> predict(X_train)
   <li><a href="._week40-bs060.html">61</a></li>
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs053.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs053.html b/doc/pub/week40/html/._week40-bs053.html
index ae46524be..a562339a2 100644
--- a/doc/pub/week40/html/._week40-bs053.html
+++ b/doc/pub/week40/html/._week40-bs053.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
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         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,33 +319,21 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0053"></a>
 <!-- !split -->
-<h2 id="choose-cost-function-and-optimizer" class="anchor">Choose cost function and optimizer </h2>
+<h2 id="why-multilayer-perceptrons" class="anchor">Why multilayer perceptrons?  </h2>
 
-<p>To measure how well our neural network is doing we need to introduce a cost function.  
-We will call the function that gives the error of a single sample output the <em>loss</em> function, and the function
-that gives the total error of our network across all samples the <em>cost</em> function.
-A typical choice for multiclass classification is the <em>cross-entropy</em> loss, also known as the negative log likelihood.  
+<p>According to the <em>Universal approximation theorem</em>, a feed-forward
+neural network with just a single hidden layer containing a finite
+number of neurons can approximate a continuous multidimensional
+function to arbitrary accuracy, assuming the activation function for
+the hidden layer is a <b>non-constant, bounded and
+monotonically-increasing continuous function</b>.
 </p>
 
-<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:  </p>
-
-<p>$$ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$  </p>
-
-<p>$$ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$  </p>
-
-<p>i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.  </p>
-
-<p>Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector.  
-We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \boldsymbol{x}_i \) in the dataset.
+<p>Note that the requirements on the activation function only applies to
+the hidden layer, the output nodes are always assumed to be linear, so
+as to not restrict the range of output values.
 </p>
 
-<p>In the one-hot representation only one of the terms in the loss function is non-zero, namely the
-probability of the correct category \( c' \)  
-(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong
-you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \boldsymbol{\theta} \) represents the parameters of our network, i.e. all the weights and biases.  
-</p>
-
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -361,7 +359,7 @@ you got the correct label. The probability of category \( c \) is given by the s
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs054.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs054.html b/doc/pub/week40/html/._week40-bs054.html
index ba733f242..1a944b2f6 100644
--- a/doc/pub/week40/html/._week40-bs054.html
+++ b/doc/pub/week40/html/._week40-bs054.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,39 +319,15 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0054"></a>
 <!-- !split -->
-<h2 id="optimizing-the-cost-function" class="anchor">Optimizing the cost function </h2>
+<h2 id="illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" class="anchor">Illustration of a single perceptron model and a multi-perceptron model </h2>
 
-<p>The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is <em>gradient descent</em> and its generalizations. The idea behind gradient descent
-is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a <em>local</em> minimum of the cost function.  
-Each parameter \( \theta \) is iteratively adjusted according to the rule  
-</p>
-
-<p>$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$</p>
-
-<p>where \( \eta \) is known as the <em>learning rate</em>, which controls how big a step we take towards the minimum.  
-This update can be repeated for any number of iterations, or until we are satisfied with the result.  
-</p>
-
-<p>A simple and effective improvement is a variant called <em>Batch Gradient Descent</em>.  
-Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
-on a subset of the data called a <em>minibatch</em>.  
-If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches
-is \( N/M \).  
-We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes:  
-</p>
-
-<p>$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
-</p>
-
-<p>i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.  </p>
-
-<p>This has two important benefits:  </p>
-<ol>
-<li> Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.</li>  
-<li> It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.</li>  
-</ol>
-<p>The various optmization  methods, with codes and algorithms,  are discussed in our lectures on <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_self">Gradient descent approaches</a>.</p>
+<center>  <!-- FIGURE -->
+<hr class="figure">
+<center>
+<p class="caption">Figure 1:  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer. </p>
+</center>
+<p><img src="figures/nns.png" width="600" align="bottom"></p>
+</center>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -367,6 +353,8 @@ We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient th
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs055.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs055.html b/doc/pub/week40/html/._week40-bs055.html
index 45a433859..e520ff00a 100644
--- a/doc/pub/week40/html/._week40-bs055.html
+++ b/doc/pub/week40/html/._week40-bs055.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -308,35 +318,69 @@ MathJax.Hub.Config({
 <div class="container">
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0055"></a>
-<!-- !split   -->
-<h2 id="regularization" class="anchor">Regularization </h2>
+<!-- !split -->
+<h2 id="examples-of-xor-or-and-and-gates" class="anchor">Examples of XOR, OR and AND gates </h2>
 
-<p>It is common to add an extra term to the cost function, proportional
-to the size of the weights.  This is equivalent to constraining the
-size of the weights, so that they do not grow out of control.
-Constraining the size of the weights means that the weights cannot
-grow arbitrarily large to fit the training data, and in this way
-reduces <em>overfitting</em>.
+<p>Let us first try to fit various gates using standard linear
+regression. The gates we are thinking of are the classical XOR, OR and
+AND gates, well-known elements in computer science. The tables here
+show how we can set up the inputs \( x_1 \) and \( x_2 \) in order to yield a
+specific target \( y_i \).
 </p>
 
-<p>We will measure the size of the weights using the so called <em>L2-norm</em>, meaning our cost function becomes:  </p>
 
-<p>$$  \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{N} \sum_{i=1}^N  \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2 
-= \frac{1}{N} \sum_{i=1}^N  \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$  
-</p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
+<span style="color: #BA2121; font-style: italic">Simple code that tests XOR, OR and AND gates with linear regression</span>
+<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
 
-<p>i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.</p>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #408080; font-style: italic"># Design matrix</span>
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The invers of X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-<p>In order to train the model, we need to calculate the derivative of
-the cost function with respect to every bias and weight in the
-network.  In total our network has \( (64 + 1)\times 50=3250 \) weights in
-the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output
-layer (\( +1 \) for the bias), and the gradient must be calculated for
-every parameter.  We use the <em>backpropagation</em> algorithm discussed
-above. This is a clever use of the chain rule that allows us to
-calculate the gradient efficently. 
-</p>
+<span style="color: #408080; font-style: italic"># The XOR gate </span>
+yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
+ThetaXOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+
+<span style="color: #408080; font-style: italic"># The OR gate </span>
+yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
+ThetaOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+
+<span style="color: #408080; font-style: italic"># The OR gate </span>
+yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
+ThetaAND  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>What is happening here?</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -361,6 +405,9 @@ calculate the gradient efficently.
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs056.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs056.html b/doc/pub/week40/html/._week40-bs056.html
index 711be5fb1..0cfa3e896 100644
--- a/doc/pub/week40/html/._week40-bs056.html
+++ b/doc/pub/week40/html/._week40-bs056.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
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-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,40 +319,7 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0056"></a>
 <!-- !split -->
-<h2 id="matrix-multiplication" class="anchor">Matrix  multiplication </h2>
-
-<p>To more efficently train our network these equations are implemented using matrix operations.  
-The error in the output layer is calculated simply as, with \( \boldsymbol{t} \) being our targets,  
-</p>
-
-<p>$$ \delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .$$  </p>
-
-<p>The gradient for the output weights is calculated as  </p>
-
-<p>$$ \nabla W_{L} = \boldsymbol{a}^T \delta_L   = (n_{hidden}, n_{categories}) ,$$</p>
-
-<p>where \( \boldsymbol{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input.  
-Since we are going backwards we have to transpose the activation matrix.  
-</p>
-
-<p>The gradient with respect to the output bias is then  </p>
-
-<p>$$ \nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$  </p>
-
-<p>The error in the hidden layer is  </p>
-
-<p>$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$  </p>
-
-<p>where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean
-that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes
-the <em>Hadamard product</em>, meaning element-wise multiplication.  
-</p>
-
-<p>This again gives us the gradients in the hidden layer:  </p>
-
-<p>$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$  </p>
-
-<p>$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$</p>
+<h2 id="does-logistic-regression-do-a-better-job" class="anchor">Does Logistic Regression do a better Job? </h2>
 
 
 <!-- code=python (!bc pycod) typeset with pygments style "default" -->
@@ -351,74 +328,55 @@ the <em>Hadamard product</em>, meaning element-wise multiplication.
     <div class="inner_cell">
       <div class="input_area">
         <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># to categorical turns our integer vector into a onehot representation</span>
-<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
+  <pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
+<span style="color: #BA2121; font-style: italic">Simple code that tests XOR and OR gates with linear regression</span>
+<span style="color: #BA2121; font-style: italic">and logistic regression</span>
+<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
 
-<span style="color: #408080; font-style: italic"># one-hot in numpy</span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">to_categorical_numpy</span>(integer_vector):
-    n_inputs <span style="color: #666666">=</span> <span style="color: #008000">len</span>(integer_vector)
-    n_categories <span style="color: #666666">=</span> np<span style="color: #666666">.</span>max(integer_vector) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
-    onehot_vector <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n_inputs, n_categories))
-    onehot_vector[<span style="color: #008000">range</span>(n_inputs), integer_vector] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
-    
-    <span style="color: #008000; font-weight: bold">return</span> onehot_vector
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
 
-<span style="color: #408080; font-style: italic">#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)</span>
-Y_train_onehot, Y_test_onehot <span style="color: #666666">=</span> to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
+<span style="color: #408080; font-style: italic"># Design matrix</span>
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The invers of X.TX  matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">feed_forward_train</span>(X):
-    <span style="color: #408080; font-style: italic"># weighted sum of inputs to the hidden layer</span>
-    z_h <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(X, hidden_weights) <span style="color: #666666">+</span> hidden_bias
-    <span style="color: #408080; font-style: italic"># activation in the hidden layer</span>
-    a_h <span style="color: #666666">=</span> sigmoid(z_h)
-    
-    <span style="color: #408080; font-style: italic"># weighted sum of inputs to the output layer</span>
-    z_o <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(a_h, output_weights) <span style="color: #666666">+</span> output_bias
-    <span style="color: #408080; font-style: italic"># softmax output</span>
-    <span style="color: #408080; font-style: italic"># axis 0 holds each input and axis 1 the probabilities of each category</span>
-    exp_term <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(z_o)
-    probabilities <span style="color: #666666">=</span> exp_term <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum(exp_term, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
-    
-    <span style="color: #408080; font-style: italic"># for backpropagation need activations in hidden and output layers</span>
-    <span style="color: #008000; font-weight: bold">return</span> a_h, probabilities
+<span style="color: #408080; font-style: italic"># The XOR gate </span>
+yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
+ThetaXOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">backpropagation</span>(X, Y):
-    a_h, probabilities <span style="color: #666666">=</span> feed_forward_train(X)
-    
-    <span style="color: #408080; font-style: italic"># error in the output layer</span>
-    error_output <span style="color: #666666">=</span> probabilities <span style="color: #666666">-</span> Y
-    <span style="color: #408080; font-style: italic"># error in the hidden layer</span>
-    error_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(error_output, output_weights<span style="color: #666666">.</span>T) <span style="color: #666666">*</span> a_h <span style="color: #666666">*</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> a_h)
-    
-    <span style="color: #408080; font-style: italic"># gradients for the output layer</span>
-    output_weights_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(a_h<span style="color: #666666">.</span>T, error_output)
-    output_bias_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(error_output, axis<span style="color: #666666">=0</span>)
-    
-    <span style="color: #408080; font-style: italic"># gradient for the hidden layer</span>
-    hidden_weights_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(X<span style="color: #666666">.</span>T, error_hidden)
-    hidden_bias_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(error_hidden, axis<span style="color: #666666">=0</span>)
 
-    <span style="color: #008000; font-weight: bold">return</span> output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+<span style="color: #408080; font-style: italic"># The OR gate </span>
+yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
+ThetaOR  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Old accuracy on training data: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(accuracy_score(predict(X_train), Y_train)))
 
-eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
-lmbd <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1000</span>):
-    <span style="color: #408080; font-style: italic"># calculate gradients</span>
-    dWo, dBo, dWh, dBh <span style="color: #666666">=</span> backpropagation(X_train, Y_train_onehot)
-    
-    <span style="color: #408080; font-style: italic"># regularization term gradients</span>
-    dWo <span style="color: #666666">+=</span> lmbd <span style="color: #666666">*</span> output_weights
-    dWh <span style="color: #666666">+=</span> lmbd <span style="color: #666666">*</span> hidden_weights
-    
-    <span style="color: #408080; font-style: italic"># update weights and biases</span>
-    output_weights <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dWo
-    output_bias <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dBo
-    hidden_weights <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dWh
-    hidden_bias <span style="color: #666666">-=</span> eta <span style="color: #666666">*</span> dBh
+<span style="color: #408080; font-style: italic"># The OR gate </span>
+yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
+ThetaAND  <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction  for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
 
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;New accuracy on training data: &quot;</span> <span style="color: #666666">+</span> <span style="color: #008000">str</span>(accuracy_score(predict(X_train), Y_train)))
+<span style="color: #408080; font-style: italic"># Now we change to logistic regression</span>
+
+
+<span style="color: #408080; font-style: italic"># Logistic Regression</span>
+logreg <span style="color: #666666">=</span> LogisticRegression()
+logreg<span style="color: #666666">.</span>fit(X, yOR)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for OR gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yOR)))
+
+logreg<span style="color: #666666">.</span>fit(X, yXOR)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for XOR gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yXOR)))
+
+
+logreg<span style="color: #666666">.</span>fit(X, yAND)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for AND gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yAND)))
 </pre>
 </div>
       </div>
@@ -434,6 +392,7 @@ lmbd <span style="color: #666666">=</span> <span style="color: #666666">0.01</sp
   </div>
 </div>
 
+<p>Not exactly impressive, but somewhat better.</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -457,6 +416,10 @@ lmbd <span style="color: #666666">=</span> <span style="color: #666666">0.01</sp
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs057.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs057.html b/doc/pub/week40/html/._week40-bs057.html
index 9d750cf47..5eacabe04 100644
--- a/doc/pub/week40/html/._week40-bs057.html
+++ b/doc/pub/week40/html/._week40-bs057.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,21 +319,38 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0057"></a>
 <!-- !split -->
-<h2 id="improving-performance" class="anchor">Improving performance </h2>
+<h2 id="adding-neural-networks" class="anchor">Adding Neural Networks </h2>
 
-<p>As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.  
-In order to obtain a network that does something useful, we will have to do a bit more work.  
-</p>
 
-<p>The choice of <em>hyperparameters</em> such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a <em>grid-search</em> is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).  </p>
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># and now neural networks with Scikit-Learn and the XOR</span>
 
-<p>Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an <em>iteration</em>, and a full training period
-going through the entire dataset (\( n/M \) batches) an <em>epoch</em>.
-</p>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPClassifier
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_classification
+X, yXOR <span style="color: #666666">=</span> make_classification(n_samples<span style="color: #666666">=100</span>, random_state<span style="color: #666666">=1</span>)
+FFNN <span style="color: #666666">=</span> MLPClassifier(random_state<span style="color: #666666">=1</span>, max_iter<span style="color: #666666">=300</span>)<span style="color: #666666">.</span>fit(X, yXOR)
+FFNN<span style="color: #666666">.</span>predict_proba(X)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Test set accuracy with Feed Forward Neural Network  for XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>FFNN<span style="color: #666666">.</span>score(X, yXOR)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<p>If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.  
-Andrew Ng goes through some of these considerations in this <a href="https://youtu.be/F1ka6a13S9I" target="_self">video</a>. You can find a summary of the video <a href="https://kevinzakka.github.io/2016/09/26/applying-deep-learning/" target="_self">here</a>.  
-</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -346,6 +373,11 @@ Andrew Ng goes through some of these considerations in this <a href="https://you
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="">...</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs058.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs058.html b/doc/pub/week40/html/._week40-bs058.html
index ba4934190..3c181b868 100644
--- a/doc/pub/week40/html/._week40-bs058.html
+++ b/doc/pub/week40/html/._week40-bs058.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,134 +319,21 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0058"></a>
 <!-- !split -->
-<h2 id="full-object-oriented-implementation" class="anchor">Full object-oriented implementation </h2>
+<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
 
-<p>It is very natural to think of the network as an object, with specific instances of the network
-being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.
+<p>The output \( y \) is produced via the activation function \( f \)</p>
+$$
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
+$$
+
+<p>This function receives \( x_i \) as inputs.
+Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \). 
+In an FFNN of such neurons, the <em>inputs</em> \( x_i \) are the <em>outputs</em> of
+the neurons in the preceding layer. Furthermore, an MLP is
+fully-connected, which means that each neuron receives a weighted sum
+of the outputs of <em>all</em> neurons in the previous layer.
 </p>
 
-
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">class</span> <span style="color: #0000FF; font-weight: bold">NeuralNetwork</span>:
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">__init__</span>(
-            <span style="color: #008000">self</span>,
-            X_data,
-            Y_data,
-            n_hidden_neurons<span style="color: #666666">=50</span>,
-            n_categories<span style="color: #666666">=10</span>,
-            epochs<span style="color: #666666">=10</span>,
-            batch_size<span style="color: #666666">=100</span>,
-            eta<span style="color: #666666">=0.1</span>,
-            lmbd<span style="color: #666666">=0.0</span>):
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>X_data_full <span style="color: #666666">=</span> X_data
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>Y_data_full <span style="color: #666666">=</span> Y_data
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>n_inputs <span style="color: #666666">=</span> X_data<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>]
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>n_features <span style="color: #666666">=</span> X_data<span style="color: #666666">.</span>shape[<span style="color: #666666">1</span>]
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>n_hidden_neurons <span style="color: #666666">=</span> n_hidden_neurons
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>n_categories <span style="color: #666666">=</span> n_categories
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>epochs <span style="color: #666666">=</span> epochs
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>batch_size <span style="color: #666666">=</span> batch_size
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>iterations <span style="color: #666666">=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>n_inputs <span style="color: #666666">//</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>batch_size
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>eta <span style="color: #666666">=</span> eta
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>lmbd <span style="color: #666666">=</span> lmbd
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>create_biases_and_weights()
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">create_biases_and_weights</span>(<span style="color: #008000">self</span>):
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #008000">self</span><span style="color: #666666">.</span>n_features, <span style="color: #008000">self</span><span style="color: #666666">.</span>n_hidden_neurons)
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(<span style="color: #008000">self</span><span style="color: #666666">.</span>n_hidden_neurons) <span style="color: #666666">+</span> <span style="color: #666666">0.01</span>
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #008000">self</span><span style="color: #666666">.</span>n_hidden_neurons, <span style="color: #008000">self</span><span style="color: #666666">.</span>n_categories)
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>output_bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(<span style="color: #008000">self</span><span style="color: #666666">.</span>n_categories) <span style="color: #666666">+</span> <span style="color: #666666">0.01</span>
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">feed_forward</span>(<span style="color: #008000">self</span>):
-        <span style="color: #408080; font-style: italic"># feed-forward for training</span>
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>z_h <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(<span style="color: #008000">self</span><span style="color: #666666">.</span>X_data, <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights) <span style="color: #666666">+</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_bias
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>a_h <span style="color: #666666">=</span> sigmoid(<span style="color: #008000">self</span><span style="color: #666666">.</span>z_h)
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>z_o <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(<span style="color: #008000">self</span><span style="color: #666666">.</span>a_h, <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights) <span style="color: #666666">+</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>output_bias
-
-        exp_term <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #008000">self</span><span style="color: #666666">.</span>z_o)
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>probabilities <span style="color: #666666">=</span> exp_term <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum(exp_term, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">feed_forward_out</span>(<span style="color: #008000">self</span>, X):
-        <span style="color: #408080; font-style: italic"># feed-forward for output</span>
-        z_h <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(X, <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights) <span style="color: #666666">+</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_bias
-        a_h <span style="color: #666666">=</span> sigmoid(z_h)
-
-        z_o <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(a_h, <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights) <span style="color: #666666">+</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>output_bias
-        
-        exp_term <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(z_o)
-        probabilities <span style="color: #666666">=</span> exp_term <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum(exp_term, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
-        <span style="color: #008000; font-weight: bold">return</span> probabilities
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">backpropagation</span>(<span style="color: #008000">self</span>):
-        error_output <span style="color: #666666">=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>probabilities <span style="color: #666666">-</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>Y_data
-        error_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(error_output, <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights<span style="color: #666666">.</span>T) <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>a_h <span style="color: #666666">*</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>a_h)
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(<span style="color: #008000">self</span><span style="color: #666666">.</span>a_h<span style="color: #666666">.</span>T, error_output)
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>output_bias_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(error_output, axis<span style="color: #666666">=0</span>)
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(<span style="color: #008000">self</span><span style="color: #666666">.</span>X_data<span style="color: #666666">.</span>T, error_hidden)
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_bias_gradient <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(error_hidden, axis<span style="color: #666666">=0</span>)
-
-        <span style="color: #008000; font-weight: bold">if</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>lmbd <span style="color: #666666">&gt;</span> <span style="color: #666666">0.0</span>:
-            <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights_gradient <span style="color: #666666">+=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>lmbd <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights
-            <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights_gradient <span style="color: #666666">+=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>lmbd <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights
-
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights <span style="color: #666666">-=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>eta <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>output_weights_gradient
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>output_bias <span style="color: #666666">-=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>eta <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>output_bias_gradient
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights <span style="color: #666666">-=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>eta <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_weights_gradient
-        <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_bias <span style="color: #666666">-=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>eta <span style="color: #666666">*</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>hidden_bias_gradient
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">predict</span>(<span style="color: #008000">self</span>, X):
-        probabilities <span style="color: #666666">=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>feed_forward_out(X)
-        <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>argmax(probabilities, axis<span style="color: #666666">=1</span>)
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">predict_probabilities</span>(<span style="color: #008000">self</span>, X):
-        probabilities <span style="color: #666666">=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>feed_forward_out(X)
-        <span style="color: #008000; font-weight: bold">return</span> probabilities
-
-    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">train</span>(<span style="color: #008000">self</span>):
-        data_indices <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #008000">self</span><span style="color: #666666">.</span>n_inputs)
-
-        <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">self</span><span style="color: #666666">.</span>epochs):
-            <span style="color: #008000; font-weight: bold">for</span> j <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">self</span><span style="color: #666666">.</span>iterations):
-                <span style="color: #408080; font-style: italic"># pick datapoints with replacement</span>
-                chosen_datapoints <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice(
-                    data_indices, size<span style="color: #666666">=</span><span style="color: #008000">self</span><span style="color: #666666">.</span>batch_size, replace<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>
-                )
-
-                <span style="color: #408080; font-style: italic"># minibatch training data</span>
-                <span style="color: #008000">self</span><span style="color: #666666">.</span>X_data <span style="color: #666666">=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>X_data_full[chosen_datapoints]
-                <span style="color: #008000">self</span><span style="color: #666666">.</span>Y_data <span style="color: #666666">=</span> <span style="color: #008000">self</span><span style="color: #666666">.</span>Y_data_full[chosen_datapoints]
-
-                <span style="color: #008000">self</span><span style="color: #666666">.</span>feed_forward()
-                <span style="color: #008000">self</span><span style="color: #666666">.</span>backpropagation()
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
-
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -457,6 +354,10 @@ being realizations of this object with different hyperparameters. An implementat
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs059.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs059.html b/doc/pub/week40/html/._week40-bs059.html
index 5828ef1db..d1b8d3ab2 100644
--- a/doc/pub/week40/html/._week40-bs059.html
+++ b/doc/pub/week40/html/._week40-bs059.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,55 +319,48 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0059"></a>
 <!-- !split -->
-<h2 id="evaluate-model-performance-on-test-data" class="anchor">Evaluate model performance on test data </h2>
+<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
 
-<p>To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.  
-We measure the performance of the network using the <em>accuracy</em> score.  
-The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \).  
+<p>First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),</p>
+
+$$
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
+\tag{7}
+\end{equation}
+$$
+
+<p>Here \( b_i \) is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below.  The value of \( z_i^1 \) is the
+argument to the activation function \( f_i \) of each node \( i \), The
+variable \( M \) stands for all possible inputs to a given node \( i \) in the
+first layer.  We define  the output \( y_i^1 \) of all neurons in layer 1 as
 </p>
 
-<p>$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,$$  </p>
+$$
+\begin{equation}
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\right)
+\tag{8}
+\end{equation}
+$$
 
-<p>where \( I \) is the indicator function, \( 1 \) if \( \tilde{y}_i = y_i \) and \( 0 \) otherwise.</p>
+<p>where we assume that all nodes in the same layer have identical
+activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
+</p>
 
+$$
+\begin{equation}
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+\tag{9}
+\end{equation}
+$$
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;">epochs <span style="color: #666666">=</span> <span style="color: #666666">100</span>
-batch_size <span style="color: #666666">=</span> <span style="color: #666666">100</span>
-
-dnn <span style="color: #666666">=</span> NeuralNetwork(X_train, Y_train_onehot, eta<span style="color: #666666">=</span>eta, lmbd<span style="color: #666666">=</span>lmbd, epochs<span style="color: #666666">=</span>epochs, batch_size<span style="color: #666666">=</span>batch_size,
-                    n_hidden_neurons<span style="color: #666666">=</span>n_hidden_neurons, n_categories<span style="color: #666666">=</span>n_categories)
-dnn<span style="color: #666666">.</span>train()
-test_predict <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>predict(X_test)
-
-<span style="color: #408080; font-style: italic"># accuracy score from scikit library</span>
-<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Accuracy score on test set: &quot;</span>, accuracy_score(Y_test, test_predict))
-
-<span style="color: #408080; font-style: italic"># equivalent in numpy</span>
-<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">accuracy_score_numpy</span>(Y_test, Y_pred):
-    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum(Y_test <span style="color: #666666">==</span> Y_pred) <span style="color: #666666">/</span> <span style="color: #008000">len</span>(Y_test)
-
-<span style="color: #408080; font-style: italic">#print(&quot;Accuracy score on test set: &quot;, accuracy_score_numpy(Y_test, test_predict))</span>
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
+<p>where \( N_l \) is the number of nodes in layer \( l \). When the output of
+all the nodes in the first hidden layer are computed, the values of
+the subsequent layer can be calculated and so forth until the output
+is obtained.
+</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -378,6 +381,10 @@ test_predict <span style="color: #666666">=</span> dnn<span style="color: #66666
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs060.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs060.html b/doc/pub/week40/html/._week40-bs060.html
index 1ba4053c2..e2a7b7560 100644
--- a/doc/pub/week40/html/._week40-bs060.html
+++ b/doc/pub/week40/html/._week40-bs060.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,53 +319,30 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0060"></a>
 <!-- !split -->
-<h2 id="adjust-hyperparameters" class="anchor">Adjust hyperparameters </h2>
+<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
 
-<p>We now perform a grid search to find the optimal hyperparameters for the network.  
-Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate).
-</p>
+<p>The output of neuron \( i \) in layer 2 is thus,</p>
 
+$$
+\begin{align}
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
+\tag{10}\\
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+\tag{11}
+\end{align}
+$$
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;">eta_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
-lmbd_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
-<span style="color: #408080; font-style: italic"># store the models for later use</span>
-DNN_numpy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)), dtype<span style="color: #666666">=</span><span style="color: #008000">object</span>)
+<p>where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads</p>
 
-<span style="color: #408080; font-style: italic"># grid search</span>
-<span style="color: #008000; font-weight: bold">for</span> i, eta <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(eta_vals):
-    <span style="color: #008000; font-weight: bold">for</span> j, lmbd <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmbd_vals):
-        dnn <span style="color: #666666">=</span> NeuralNetwork(X_train, Y_train_onehot, eta<span style="color: #666666">=</span>eta, lmbd<span style="color: #666666">=</span>lmbd, epochs<span style="color: #666666">=</span>epochs, batch_size<span style="color: #666666">=</span>batch_size,
-                            n_hidden_neurons<span style="color: #666666">=</span>n_hidden_neurons, n_categories<span style="color: #666666">=</span>n_categories)
-        dnn<span style="color: #666666">.</span>train()
-        
-        DNN_numpy[i][j] <span style="color: #666666">=</span> dnn
-        
-        test_predict <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>predict(X_test)
-        
-        <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Learning rate  = &quot;</span>, eta)
-        <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Lambda = &quot;</span>, lmbd)
-        <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Accuracy score on test set: &quot;</span>, accuracy_score(Y_test, test_predict))
-        <span style="color: #008000">print</span>()
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
+$$
+\begin{align}
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
+\tag{12}\\
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+  + b_1^3\right]
+\tag{13}
+\end{align}
+$$
 
 
 <p>
@@ -376,6 +363,10 @@ DNN_numpy <span style="color: #666666">=</span> np<span style="color: #666666">.
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs061.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs061.html b/doc/pub/week40/html/._week40-bs061.html
index 28fc32f66..b8d53fc97 100644
--- a/doc/pub/week40/html/._week40-bs061.html
+++ b/doc/pub/week40/html/._week40-bs061.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,63 +319,22 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0061"></a>
 <!-- !split -->
-<h2 id="visualization" class="anchor">Visualization </h2>
+<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
 
+<p>We can generalize this expression to an MLP with \( l \) hidden
+layers. The complete functional form is,
+</p>
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># visual representation of grid search</span>
-<span style="color: #408080; font-style: italic"># uses seaborn heatmap, you can also do this with matplotlib imshow</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
-
-sns<span style="color: #666666">.</span>set()
-
-train_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
-test_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
-
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(eta_vals)):
-    <span style="color: #008000; font-weight: bold">for</span> j <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(lmbd_vals)):
-        dnn <span style="color: #666666">=</span> DNN_numpy[i][j]
-        
-        train_pred <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>predict(X_train) 
-        test_pred <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>predict(X_test)
-
-        train_accuracy[i][j] <span style="color: #666666">=</span> accuracy_score(Y_train, train_pred)
-        test_accuracy[i][j] <span style="color: #666666">=</span> accuracy_score(Y_test, test_pred)
-
-        
-fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
-sns<span style="color: #666666">.</span>heatmap(train_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&quot;viridis&quot;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&quot;Training Accuracy&quot;</span>)
-ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&quot;$\eta$&quot;</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&quot;$\lambda$&quot;</span>)
-plt<span style="color: #666666">.</span>show()
-
-fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
-sns<span style="color: #666666">.</span>heatmap(test_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&quot;viridis&quot;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&quot;Test Accuracy&quot;</span>)
-ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&quot;$\eta$&quot;</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&quot;$\lambda$&quot;</span>)
-plt<span style="color: #666666">.</span>show()
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
+$$
+\begin{align}
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
+\tag{14}
+\end{align}
+$$
 
+<p>which illustrates a basic property of MLPs: The only independent
+variables are the input values \( x_n \).
+</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -384,6 +353,10 @@ plt<span style="color: #666666">.</span>show()
   <li class="active"><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs062.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs062.html b/doc/pub/week40/html/._week40-bs062.html
index 9d9f71970..e5089c37b 100644
--- a/doc/pub/week40/html/._week40-bs062.html
+++ b/doc/pub/week40/html/._week40-bs062.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,60 +319,31 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0062"></a>
 <!-- !split -->
-<h2 id="scikit-learn-implementation" class="anchor">scikit-learn implementation </h2>
+<h2 id="mathematical-model" class="anchor">Mathematical model  </h2>
 
-<p><b>scikit-learn</b> focuses more
-on traditional machine learning methods, such as regression,
-clustering, decision trees, etc. As such, it has only two types of
-neural networks: Multi Layer Perceptron outputting continuous values,
-<em>MPLRegressor</em>, and Multi Layer Perceptron outputting labels,
-<em>MLPClassifier</em>. We will see how simple it is to use these classes.
+<p>This confirms that an MLP, despite its quite convoluted mathematical
+form, is nothing more than an analytic function, specifically a
+mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m \).
 </p>
 
-<p><b>scikit-learn</b> implements a few improvements from our neural network,
-such as early stopping, a varying learning rate, different
-optimization methods, etc. We would therefore expect a better
-performance overall.
+<p>Furthermore, the flexibility and universality of an MLP can be
+illustrated by realizing that the expression is essentially a nested
+sum of scaled activation functions of the form
 </p>
 
+$$
+\begin{equation}
+ f(x) = c_1 f(c_2 x + c_3) + c_4
+\tag{15}
+\end{equation}
+$$
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPClassifier
-<span style="color: #408080; font-style: italic"># store models for later use</span>
-DNN_scikit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)), dtype<span style="color: #666666">=</span><span style="color: #008000">object</span>)
-
-<span style="color: #008000; font-weight: bold">for</span> i, eta <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(eta_vals):
-    <span style="color: #008000; font-weight: bold">for</span> j, lmbd <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmbd_vals):
-        dnn <span style="color: #666666">=</span> MLPClassifier(hidden_layer_sizes<span style="color: #666666">=</span>(n_hidden_neurons), activation<span style="color: #666666">=</span><span style="color: #BA2121">&#39;logistic&#39;</span>,
-                            alpha<span style="color: #666666">=</span>lmbd, learning_rate_init<span style="color: #666666">=</span>eta, max_iter<span style="color: #666666">=</span>epochs)
-        dnn<span style="color: #666666">.</span>fit(X_train, Y_train)
-        
-        DNN_scikit[i][j] <span style="color: #666666">=</span> dnn
-        
-        <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Learning rate  = &quot;</span>, eta)
-        <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Lambda = &quot;</span>, lmbd)
-        <span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Accuracy score on test set: &quot;</span>, dnn<span style="color: #666666">.</span>score(X_test, Y_test))
-        <span style="color: #008000">print</span>()
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
-
+<p>where the parameters \( c_i \) are weights and biases. By adjusting these
+parameters, the activation functions can be shifted up and down or
+left and right, change slope or be rescaled which is the key to the
+flexibility of a neural network.
+</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -380,6 +361,10 @@ DNN_scikit <span style="color: #666666">=</span> np<span style="color: #666666">
   <li><a href="._week40-bs061.html">62</a></li>
   <li class="active"><a href="._week40-bs062.html">63</a></li>
   <li><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs063.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs063.html b/doc/pub/week40/html/._week40-bs063.html
index f3d008613..160f971d8 100644
--- a/doc/pub/week40/html/._week40-bs063.html
+++ b/doc/pub/week40/html/._week40-bs063.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,107 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up a Multi-layer perceptron model for classification',
-               2,
-               None,
-               'setting-up-a-multi-layer-perceptron-model-for-classification'),
-              ('Defining the cost function',
-               2,
-               None,
-               'defining-the-cost-function'),
-              ('Example: binary classification problem',
-               2,
-               None,
-               'example-binary-classification-problem'),
-              ('The Softmax function', 2, None, 'the-softmax-function'),
-              ('Developing a code for doing neural networks with back '
-               'propagation',
-               2,
-               None,
-               'developing-a-code-for-doing-neural-networks-with-back-propagation'),
-              ('Collect and pre-process data',
-               2,
-               None,
-               'collect-and-pre-process-data'),
-              ('Train and test datasets', 2, None, 'train-and-test-datasets'),
-              ('Define model and architecture',
-               2,
-               None,
-               'define-model-and-architecture'),
-              ('Layers', 2, None, 'layers'),
-              ('Weights and biases', 2, None, 'weights-and-biases'),
-              ('Feed-forward pass', 2, None, 'feed-forward-pass'),
-              ('Matrix multiplications', 2, None, 'matrix-multiplications'),
-              ('Choose cost function and optimizer',
-               2,
-               None,
-               'choose-cost-function-and-optimizer'),
-              ('Optimizing the cost function',
-               2,
-               None,
-               'optimizing-the-cost-function'),
-              ('Regularization', 2, None, 'regularization'),
-              ('Matrix  multiplication', 2, None, 'matrix-multiplication'),
-              ('Improving performance', 2, None, 'improving-performance'),
-              ('Full object-oriented implementation',
-               2,
-               None,
-               'full-object-oriented-implementation'),
-              ('Evaluate model performance on test data',
-               2,
-               None,
-               'evaluate-model-performance-on-test-data'),
-              ('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
-              ('Visualization', 2, None, 'visualization'),
-              ('scikit-learn implementation',
-               2,
-               None,
-               'scikit-learn-implementation'),
-              ('Visualization', 2, None, 'visualization')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -228,76 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs049.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix  multiplication</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
-     <!-- navigation toc: --> <li><a href="#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
-     <!-- navigation toc: --> <li><a href="#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -309,62 +319,41 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0063"></a>
 <!-- !split -->
-<h2 id="visualization" class="anchor">Visualization </h2>
+<h3 id="matrix-vector-notation" class="anchor">Matrix-vector notation </h3>
 
-<!-- code=python (!bc pycod) typeset with pygments style "default" -->
-<div class="cell border-box-sizing code_cell rendered">
-  <div class="input">
-    <div class="inner_cell">
-      <div class="input_area">
-        <div class="highlight" style="background: #f8f8f8">
-  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># optional</span>
-<span style="color: #408080; font-style: italic"># visual representation of grid search</span>
-<span style="color: #408080; font-style: italic"># uses seaborn heatmap, could probably do this in matplotlib</span>
-<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
+<p>We can introduce a more convenient notation for the activations in an A NN. </p>
 
-sns<span style="color: #666666">.</span>set()
+<p>Additionally, we can represent the biases and activations
+as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector 
+is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively. 
+</p>
 
-train_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
-test_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
+<p>We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors. 
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
+</p>
+$$
+\begin{equation}
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = 
+ f_2\left(\left[\begin{array}{ccc}
+    w^2_{11} &w^2_{12} &w^2_{13} \\
+    w^2_{21} &w^2_{22} &w^2_{23} \\
+    w^2_{31} &w^2_{32} &w^2_{33} \\
+    \end{array} \right] \cdot
+    \left[\begin{array}{c}
+           y^1_1 \\
+           y^1_2 \\
+           y^1_3 \\
+          \end{array}\right] + 
+    \left[\begin{array}{c}
+           b^2_1 \\
+           b^2_2 \\
+           b^2_3 \\
+          \end{array}\right]\right).
+\tag{16}
+\end{equation}
+$$
 
-<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(eta_vals)):
-    <span style="color: #008000; font-weight: bold">for</span> j <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(lmbd_vals)):
-        dnn <span style="color: #666666">=</span> DNN_scikit[i][j]
-        
-        train_pred <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>predict(X_train) 
-        test_pred <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>predict(X_test)
-
-        train_accuracy[i][j] <span style="color: #666666">=</span> accuracy_score(Y_train, train_pred)
-        test_accuracy[i][j] <span style="color: #666666">=</span> accuracy_score(Y_test, test_pred)
-
-        
-fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
-sns<span style="color: #666666">.</span>heatmap(train_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&quot;viridis&quot;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&quot;Training Accuracy&quot;</span>)
-ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&quot;$\eta$&quot;</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&quot;$\lambda$&quot;</span>)
-plt<span style="color: #666666">.</span>show()
-
-fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
-sns<span style="color: #666666">.</span>heatmap(test_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">&quot;viridis&quot;</span>)
-ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&quot;Test Accuracy&quot;</span>)
-ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&quot;$\eta$&quot;</span>)
-ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&quot;$\lambda$&quot;</span>)
-plt<span style="color: #666666">.</span>show()
-</pre>
-</div>
-      </div>
-    </div>
-  </div>
-  <div class="output_wrapper">
-    <div class="output">
-      <div class="output_area">
-        <div class="output_subarea output_stream output_stdout output_text">          
-        </div>
-      </div>
-    </div>
-  </div>
-</div>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -381,6 +370,11 @@ plt<span style="color: #666666">.</span>show()
   <li><a href="._week40-bs061.html">62</a></li>
   <li><a href="._week40-bs062.html">63</a></li>
   <li class="active"><a href="._week40-bs063.html">64</a></li>
+  <li><a href="._week40-bs064.html">65</a></li>
+  <li><a href="._week40-bs065.html">66</a></li>
+  <li><a href="._week40-bs066.html">67</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
+  <li><a href="._week40-bs064.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
 </div>  <!-- end container -->
diff --git a/doc/pub/week40/html/._week40-bs064.html b/doc/pub/week40/html/._week40-bs064.html
index 5a5c514cb..725401530 100644
--- a/doc/pub/week40/html/._week40-bs064.html
+++ b/doc/pub/week40/html/._week40-bs064.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: From Stochastic Gradient Descent to Neural networks">
-<title>Week 40: From Stochastic Gradient Descent to Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,7 +36,12 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
@@ -59,10 +64,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'slightly-different-approach'),
-              ('Program for stochastic gradient',
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
                2,
                None,
-               'program-for-stochastic-gradient'),
+               'code-with-a-number-of-minibatches-which-varies'),
               ('Replace or not', 2, None, 'replace-or-not'),
               ('Momentum based GD', 2, None, 'momentum-based-gd'),
               ('More on momentum based approaches',
@@ -75,7 +81,14 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                None,
                'second-moment-of-the-gradient'),
               ('RMS prop', 2, None, 'rms-prop'),
-              ('ADAM optimizer', 2, None, 'adam-optimizer'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
@@ -105,16 +118,45 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                'the-syntax-a-dot-b-when-finding-the-dot-product'),
               ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
               ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
               ('Including Stochastic Gradient Descent with Autograd',
                2,
                None,
                'including-stochastic-gradient-descent-with-autograd'),
-              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
-              ('Videos on Neural Networks',
+              ('Same code but now with momentum gradient descent',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -135,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -164,46 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -231,82 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: From Stochastic Gradient Descent to Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#program-for-stochastic-gradient" style="font-size: 80%;"><b>Program for stochastic gradient</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs019.html#adam-optimizer" style="font-size: 80%;"><b>ADAM optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs020.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs021.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs022.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs039.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs061.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs065.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs066.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs067.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs068.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs069.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -318,26 +319,24 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0064"></a>
 <!-- !split -->
-<h2 id="derivative-of-the-cost-function" class="anchor">Derivative of the cost function </h2>
+<h3 id="matrix-vector-notation-and-activation" class="anchor">Matrix-vector notation  and activation </h3>
 
-<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
-
-<p>Let us specialize to the output layer \( l=L \). Our cost function is</p>
-$$
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-$$
-
-<p>The derivative of this function with respect to the weights is</p>
+<p>The activation of node \( i \) in layer 2 is</p>
 
 $$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
-$$
-
-<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
-$$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
+\begin{equation}
+ y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
+ f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
+\tag{17}
+\end{equation}
 $$
 
+<p>This is not just a convenient and compact notation, but also a useful
+and intuitive way to think about MLPs: The output is calculated by a
+series of matrix-vector multiplications and vector additions that are
+used as input to the activation functions. For each operation
+\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
+</p>
 
 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -357,8 +356,6 @@ $$
   <li><a href="._week40-bs065.html">66</a></li>
   <li><a href="._week40-bs066.html">67</a></li>
   <li><a href="._week40-bs067.html">68</a></li>
-  <li><a href="._week40-bs068.html">69</a></li>
-  <li><a href="._week40-bs069.html">70</a></li>
   <li><a href="._week40-bs065.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/._week40-bs065.html b/doc/pub/week40/html/._week40-bs065.html
index a8799639e..9d5c14000 100644
--- a/doc/pub/week40/html/._week40-bs065.html
+++ b/doc/pub/week40/html/._week40-bs065.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: From Stochastic Gradient Descent to Neural networks">
-<title>Week 40: From Stochastic Gradient Descent to Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,7 +36,12 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
@@ -59,10 +64,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'slightly-different-approach'),
-              ('Program for stochastic gradient',
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
                2,
                None,
-               'program-for-stochastic-gradient'),
+               'code-with-a-number-of-minibatches-which-varies'),
               ('Replace or not', 2, None, 'replace-or-not'),
               ('Momentum based GD', 2, None, 'momentum-based-gd'),
               ('More on momentum based approaches',
@@ -75,7 +81,14 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                None,
                'second-moment-of-the-gradient'),
               ('RMS prop', 2, None, 'rms-prop'),
-              ('ADAM optimizer', 2, None, 'adam-optimizer'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
@@ -105,16 +118,45 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                'the-syntax-a-dot-b-when-finding-the-dot-product'),
               ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
               ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
               ('Including Stochastic Gradient Descent with Autograd',
                2,
                None,
                'including-stochastic-gradient-descent-with-autograd'),
-              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
-              ('Videos on Neural Networks',
+              ('Same code but now with momentum gradient descent',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -135,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -164,46 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -231,82 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: From Stochastic Gradient Descent to Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#program-for-stochastic-gradient" style="font-size: 80%;"><b>Program for stochastic gradient</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs019.html#adam-optimizer" style="font-size: 80%;"><b>ADAM optimizer</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs020.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs021.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs022.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs039.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs041.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs042.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs043.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs044.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs045.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs046.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs047.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs048.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs053.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs054.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs055.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs056.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs057.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs058.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs059.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs060.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs061.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs062.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs063.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs064.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs066.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs067.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs068.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs069.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -318,51 +319,20 @@ MathJax.Hub.Config({
 <p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
 <a name="part0065"></a>
 <!-- !split -->
-<h2 id="bringing-it-together-first-back-propagation-equation" class="anchor">Bringing it together, first back propagation equation </h2>
+<h3 id="activation-functions" class="anchor">Activation functions  </h3>
 
-<p>We have thus</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
-$$
-
-<p>Defining</p>
-$$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-$$
-
-<p>and using the Hadamard product of two vectors we can write this as</p>
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-
-<p>This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
+<p>A property that characterizes a neural network, other than its
+connectivity, is the choice of activation function(s).  As described
+in, the following restrictions are imposed on an activation function
+for a FFNN to fulfill the universal approximation theorem
 </p>
 
-<p>Notice that everything in the above equations is easily computed.  In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \).  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-</p>
-
-$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-$$
-
-<p>With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
-$$
-
-
+<ul>
+  <li> Non-constant</li>
+  <li> Bounded</li>
+  <li> Monotonically-increasing</li>
+  <li> Continuous</li>
+</ul>
 <p>
 <!-- navigation buttons at the bottom of the page -->
 <ul class="pagination">
@@ -380,8 +350,6 @@ $$
   <li class="active"><a href="._week40-bs065.html">66</a></li>
   <li><a href="._week40-bs066.html">67</a></li>
   <li><a href="._week40-bs067.html">68</a></li>
-  <li><a href="._week40-bs068.html">69</a></li>
-  <li><a href="._week40-bs069.html">70</a></li>
   <li><a href="._week40-bs066.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
diff --git a/doc/pub/week40/html/week40-bs.html b/doc/pub/week40/html/week40-bs.html
index 07f00c933..5bc847095 100644
--- a/doc/pub/week40/html/week40-bs.html
+++ b/doc/pub/week40/html/week40-bs.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <!-- Bootstrap style: bootstrap -->
 <!-- doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week40-bs --no_mako -->
 <link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
@@ -36,21 +36,127 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -71,11 +177,11 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -100,50 +206,7 @@ doconce format html week40.do.txt --html_style=bootstrap --pygments_html_style=d
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -171,53 +234,80 @@ MathJax.Hub.Config({
       <span class="icon-bar"></span>
       <span class="icon-bar"></span>
     </button>
-    <a class="navbar-brand" href="week40-bs.html">Week 40: Neural networks</a>
+    <a class="navbar-brand" href="week40-bs.html">Week 40: Gradient descent methods (continued) and start Neural networks</a>
   </div>
   <div class="navbar-collapse collapse navbar-responsive-collapse">
     <ul class="nav navbar-nav navbar-right">
       <li class="dropdown">
         <a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
         <ul class="dropdown-menu">
-     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plan-for-week-40" style="font-size: 80%;"><b>Plan for week 40</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs002.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs003.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs005.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs008.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs009.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs010.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs011.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs012.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs014.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs015.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs016.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs017.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs018.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs025.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs026.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs027.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs028.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs029.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer  perceptron (MLP)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs030.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs031.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs033.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs034.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs035.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs036.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs037.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs038.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
-     <!-- navigation toc: --> <li><a href="._week40-bs040.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs001.html#plans-for-week-40" style="font-size: 80%;"><b>Plans for week 40</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs002.html#summary-from-last-week-using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Summary from last week, using gradient descent methods, limitations</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs003.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;"><b>Overview video on Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs004.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs006.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs007.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs008.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs009.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs010.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs011.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs012.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs013.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs014.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;"><b>Code with a Number of Minibatches which varies</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs015.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs016.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs017.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs018.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs019.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs020.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs021.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs022.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs023.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs024.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs025.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs026.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs027.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;"><b>More complicated functions using the elements of their arguments directly</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs028.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs029.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs030.html#and-with-loops" style="font-size: 80%;"><b>And  with loops</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs031.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs032.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs033.html#the-syntax-a-dot-b-when-finding-the-dot-product" style="font-size: 80%;"><b>The syntax a.dot(b) when finding the dot product</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs034.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs035.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs037.html#but-noen-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But noen of these can compete with Newton's method</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs038.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs039.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs040.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs041.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs042.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs043.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs044.html#introducing-jax-https-jax-readthedocs-io-en-latest" style="font-size: 80%;"><b>Introducing "JAX":"https://jax.readthedocs.io/en/latest/"</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs045.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs046.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs047.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs048.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs049.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs050.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs051.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs052.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs053.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs054.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs055.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs056.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs057.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs062.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs063.html#matrix-vector-notation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs064.html#matrix-vector-notation-and-activation" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation  and activation</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs065.html#activation-functions" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs066.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
+     <!-- navigation toc: --> <li><a href="._week40-bs067.html#relevance" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
 
         </ul>
       </li>
@@ -231,7 +321,7 @@ MathJax.Hub.Config({
 <!-- ------------------- main content ---------------------- -->
 <div class="jumbotron">
 <center>
-<h1>Week 40: Neural networks</h1>
+<h1>Week 40: Gradient descent methods (continued) and start Neural networks</h1>
 </center>  <!-- document title -->
 
 <!-- author(s): Morten Hjorth-Jensen -->
@@ -247,7 +337,7 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Oct 8, 2022</h4>
+<h4>October 2-6, 2023</h4>
 </center> <!-- date -->
 <br>
 
@@ -272,7 +362,7 @@ MathJax.Hub.Config({
   <li><a href="._week40-bs008.html">9</a></li>
   <li><a href="._week40-bs009.html">10</a></li>
   <li><a href="">...</a></li>
-  <li><a href="._week40-bs040.html">41</a></li>
+  <li><a href="._week40-bs067.html">68</a></li>
   <li><a href="._week40-bs001.html">&raquo;</a></li>
 </ul>
 <!-- ------------------- end of main content --------------- -->
@@ -286,7 +376,7 @@ MathJax.Hub.Config({
 </footer>
 -->
 <center style="font-size:80%">
-<!-- copyright --> &copy; 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+<!-- copyright --> &copy; 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 </center>
 </body>
 </html>
diff --git a/doc/pub/week40/html/week40-reveal.html b/doc/pub/week40/html/week40-reveal.html
index 2a8d2fa31..f0a25f5a1 100644
--- a/doc/pub/week40/html/week40-reveal.html
+++ b/doc/pub/week40/html/week40-reveal.html
@@ -9,8 +9,8 @@ doconce format html week40-reveal.html week40-reveal reveal --html_slide_theme=b
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 
 <!-- reveal.js: https://lab.hakim.se/reveal-js/ -->
 
@@ -168,7 +168,7 @@ MathJax.Hub.Config({
 <section>
 <!-- ------------------- main content ---------------------- -->
 <center>
-<h1 style="text-align: center;">Week 40: Neural networks</h1>
+<h1 style="text-align: center;">Week 40: Gradient descent methods (continued) and start Neural networks</h1>
 </center>  <!-- document title -->
 
 <!-- author(s): Morten Hjorth-Jensen -->
@@ -184,38 +184,79 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Oct 8, 2022</h4>
+<h4>October 2-6, 2023</h4>
 </center> <!-- date -->
 <br>
 
 
 <center style="font-size:80%">
-<!-- copyright --> &copy; 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+<!-- copyright --> &copy; 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 </center>
 </section>
 
 <section>
-<h2 id="plan-for-week-40">Plan for week 40 </h2>
+<h2 id="plans-for-week-40">Plans for week 40 </h2>
 
-<ul>
-<p><li> Thursday: Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.</li>
-<ul>
-
-<p><li> <a href="https://youtu.be/sCHiGSoG2lE" target="_blank">Video of lecture</a></li>
-</ul>
+<div class="alert alert-block alert-block alert-text-normal">
+<b>Material for the active learning sessions on Tuesday and Wednesday</b>
 <p>
-<p><li> Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.</li>
 <ul>
 
-<p><li> <a href="https://youtu.be/hDTtA7PRRfI" target="_blank">Video of lecture</a></li>
+<p><li> Work on project 1 and discussions on how to structure your report</li>
+
+<p><li> No weekly exercises for week 40, project work only</li>
+
+<p><li> <a href="https://youtu.be/tVW1ZDmZnwM" target="_blank">Video on how to write scientific reports recorded during one of the lab sessions</a></li>
+
+<p><li> A general guideline can be found at <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</tt></a>.</li>
+</ul>
+</div>
+  
+
+<div class="alert alert-block alert-block alert-text-normal">
+<b>Material for the lecture on Thursday October 5, 2023</b>
+<p>
+<ul>
+
+<p><li> Stochastic Gradient descent with examples and automatic differentiation</li>
+
+<p><li> Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.</li>
+
+<p><li> Readings and Videos:</li>
+<ul>
+
+<p><li> These lecture notes</li>
+
+<p><li> For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.</li>
+
+<p><li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 4 on stochastic gradient descent</a></li>
+
+<p><li> For neural networks we recommend Goodfellow et al chapter 6.</li>
+
+<p><li> <a href="https://www.youtube.com/watch?v=sDv4f4s2SB8" target="_blank">Video on gradient descent</a></li>
+
+<p><li> <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI" target="_blank">Video on stochastic gradient descent</a></li>
+
+<p><li> <a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a></li>
+
+<p><li> <a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a></li>
 </ul>
 <p>
 </ul>
-<p>
-<p>Reading suggestions for both days: <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 10</a> and Hastie et al chapter 11.
-For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
-For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
-</p>
+</div>
+</section>
+
+<section>
+<h2 id="summary-from-last-week-using-gradient-descent-methods-limitations">Summary from last week, using gradient descent methods, limitations </h2>
+
+<ul>
+<p><li> <b>Gradient descent (GD) finds local minima of our function</b>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</li>
+<p><li> <b>GD is sensitive to initial conditions</b>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</li>
+<p><li> <b>Gradients are computationally expensive to calculate for large datasets</b>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called &quot;mini batches&quot;. This has the added benefit of introducing stochasticity into our algorithm.</li>
+<p><li> <b>GD is very sensitive to choices of learning rates</b>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</li>
+<p><li> <b>GD treats all directions in parameter space uniformly.</b> Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</li> 
+<p><li> GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</li>
+</ul>
 </section>
 
 <section>
@@ -224,6 +265,636 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.
 <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" target="_blank">What is Stochastic Gradient Descent</a>
 </section>
 
+<section>
+<h2 id="batches-and-mini-batches">Batches and mini-batches </h2>
+
+<p>In gradient descent we compute the cost function and its gradient for all data points we have.</p>
+
+<p>In large-scale applications such as the <a href="https://www.image-net.org/challenges/LSVRC/" target="_blank">ILSVRC challenge</a>, the
+training data can have on order of millions of examples. Hence, it
+seems wasteful to compute the full cost function over the entire
+training set in order to perform only a single parameter update. A
+very common approach to addressing this challenge is to compute the
+gradient over batches of the training data. For example, a typical batch could contain some thousand  examples from
+an  entire training set of several millions. This batch is then used to
+perform a parameter update.
+</p>
+</section>
+
+<section>
+<h2 id="stochastic-gradient-descent-sgd">Stochastic Gradient Descent (SGD) </h2>
+
+<p>In stochastic gradient descent, the extreme case is the case where we
+have only one batch, that is we include the whole data set.
+</p>
+
+<p>This process is called Stochastic Gradient
+Descent (SGD) (or also sometimes on-line gradient descent). This is
+relatively less common to see because in practice due to vectorized
+code optimizations it can be computationally much more efficient to
+evaluate the gradient for 100 examples, than the gradient for one
+example 100 times. Even though SGD technically refers to using a
+single example at a time to evaluate the gradient, you will hear
+people use the term SGD even when referring to mini-batch gradient
+descent (i.e. mentions of MGD for &#8220;Minibatch Gradient Descent&#8221;, or BGD
+for &#8220;Batch gradient descent&#8221; are rare to see), where it is usually
+assumed that mini-batches are used. The size of the mini-batch is a
+hyperparameter but it is not very common to cross-validate or bootstrap it. It is
+usually based on memory constraints (if any), or set to some value,
+e.g. 32, 64 or 128. We use powers of 2 in practice because many
+vectorized operation implementations work faster when their inputs are
+sized in powers of 2.
+</p>
+
+<p>In our notes with  SGD we mean stochastic gradient descent with mini-batches.</p>
+</section>
+
+<section>
+<h2 id="stochastic-gradient-descent">Stochastic Gradient Descent </h2>
+
+<p>Stochastic gradient descent (SGD) and variants thereof address some of
+the shortcomings of the Gradient descent method discussed above.
+</p>
+
+<p>The underlying idea of SGD comes from the observation that the cost
+function, which we want to minimize, can almost always be written as a
+sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \),
+</p>
+<p>&nbsp;<br>
+$$
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}). 
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="computation-of-gradients">Computation of gradients </h2>
+
+<p>This in turn means that the gradient can be
+computed as a sum over \( i \)-gradients 
+</p>
+<p>&nbsp;<br>
+$$
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
+$$
+<p>&nbsp;<br>
+
+<p>Stochasticity/randomness is introduced by only taking the
+gradient on a subset of the data called minibatches.  If there are \( n \)
+data points and the size of each minibatch is \( M \), there will be \( n/M \)
+minibatches. We denote these minibatches by \( B_k \) where
+\( k=1,\cdots,n/M \).
+</p>
+</section>
+
+<section>
+<h2 id="sgd-example">SGD example </h2>
+<p>As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
+and we choose to have \( M=5 \) minibathces,
+then each minibatch contains two data points. In particular we have
+\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
+(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you
+have only a single batch with all data points and on the other extreme,
+you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e
+\( B_k = \mathbf{x}_k \).
+</p>
+
+<p>The idea is now to approximate the gradient by replacing the sum over
+all data points with a sum over the data points in one the minibatches
+picked at random in each gradient descent step 
+</p>
+<p>&nbsp;<br>
+$$
+\nabla_{\beta}
+C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
+c_i(\mathbf{x}_i, \mathbf{\beta}).
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="the-gradient-step">The gradient step </h2>
+
+<p>Thus a gradient descent step now looks like </p>
+<p>&nbsp;<br>
+$$
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
+$$
+<p>&nbsp;<br>
+
+<p>where \( k \) is picked at random with equal
+probability from \( [1,n/M] \). An iteration over the number of
+minibathces (n/M) is commonly referred to as an epoch. Thus it is
+typical to choose a number of epochs and for each epoch iterate over
+the number of minibatches, as exemplified in the code below.
+</p>
+</section>
+
+<section>
+<h2 id="simple-example-code">Simple example code </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span> 
+
+n = <span style="color: #B452CD">100</span> <span style="color: #228B22">#100 datapoints </span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+n_epochs = <span style="color: #B452CD">10</span> <span style="color: #228B22">#number of epochs</span>
+
+j = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">1</span>,n_epochs+<span style="color: #B452CD">1</span>):
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        k = np.random.randint(m) <span style="color: #228B22">#Pick the k-th minibatch at random</span>
+        <span style="color: #228B22">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #228B22">#Compute new suggestion for </span>
+        j += <span style="color: #B452CD">1</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Taking the gradient only on a subset of the data has two important
+benefits. First, it introduces randomness which decreases the chance
+that our opmization scheme gets stuck in a local minima. Second, if
+the size of the minibatches are small relative to the number of
+datapoints (\( M <  n \)), the computation of the gradient is much
+cheaper since we sum over the datapoints in the \( k-th \) minibatch and not
+all \( n \) datapoints.
+</p>
+</section>
+
+<section>
+<h2 id="when-do-we-stop">When do we stop? </h2>
+
+<p>A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the \( \beta \) that
+gave the lowest value.
+</p>
+</section>
+
+<section>
+<h2 id="slightly-different-approach">Slightly different approach </h2>
+
+<p>Another approach is to let the step length \( \gamma_j \) depend on the
+number of epochs in such a way that it becomes very small after a
+reasonable time such that we do not move at all. Such approaches are
+also called scaling. There are many such ways to <a href="https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1" target="_blank">scale the learning
+rate</a>
+and <a href="https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf" target="_blank">discussions here</a>. See
+also
+<a href="https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1" target="_blank"><tt>https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1</tt></a>
+for a discussion of different scaling functions for the learning rate.
+</p>
+</section>
+
+<section>
+<h2 id="time-decay-rate">Time decay rate </h2>
+
+<p>As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function <p>&nbsp;<br>
+$$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$
+<p>&nbsp;<br> goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in <em>time</em> \( t \).</p>
+
+<p>In this way we can fix the number of epochs, compute \( \beta \) and
+evaluate the cost function at the end. Repeating the computation will
+give a different result since the scheme is random by design. Then we
+pick the final \( \beta \) that gives the lowest value of the cost
+function.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span> 
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">step_length</span>(t,t0,t1):
+    <span style="color: #8B008B; font-weight: bold">return</span> t0/(t+t1)
+
+n = <span style="color: #B452CD">100</span> <span style="color: #228B22">#100 datapoints </span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+n_epochs = <span style="color: #B452CD">500</span> <span style="color: #228B22">#number of epochs</span>
+t0 = <span style="color: #B452CD">1.0</span>
+t1 = <span style="color: #B452CD">10</span>
+
+gamma_j = t0/t1
+j = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">1</span>,n_epochs+<span style="color: #B452CD">1</span>):
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        k = np.random.randint(m) <span style="color: #228B22">#Pick the k-th minibatch at random</span>
+        <span style="color: #228B22">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #228B22">#Compute new suggestion for beta</span>
+        t = epoch*m+i
+        gamma_j = step_length(t,t0,t1)
+        j += <span style="color: #B452CD">1</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;gamma_j after %d epochs: %g&quot;</span> % (n_epochs,gamma_j))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="code-with-a-number-of-minibatches-which-varies">Code with a Number of Minibatches which varies </h2>
+
+<p>In the code here we vary the number of mini-batches.</p>
+
+<!-- code=text (!bc pycode) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"># Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
+print(&quot;Own inversion&quot;)
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f&quot;Eigenvalues of Hessian Matrix:{EigValues}&quot;)
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+
+for iter in range(Niterations):
+    gradients = 2.0/n*X.T @ ((X @ theta)-y)
+    theta -= eta*gradients
+print(&quot;theta from own gd&quot;)
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print(&quot;theta from own sdg&quot;)
+print(theta)
+
+plt.plot(xnew, ypredict, &quot;r-&quot;)
+plt.plot(xnew, ypredict2, &quot;b-&quot;)
+plt.plot(x, y ,&#39;ro&#39;)
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r&#39;$x$&#39;)
+plt.ylabel(r&#39;$y$&#39;)
+plt.title(r&#39;Random numbers &#39;)
+plt.show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="replace-or-not">Replace or not </h2>
+
+<p>In the above code, we have use replacement in setting up the
+mini-batches. The discussion
+<a href="https://sebastianraschka.com/faq/docs/sgd-methods.html" target="_blank">here</a> may be
+useful.  
+</p>
+</section>
+
+<section>
+<h2 id="momentum-based-gd">Momentum based GD </h2>
+
+<p>The stochastic gradient descent (SGD) is almost always used with a
+<em>momentum</em> or inertia term that serves as a memory of the direction we
+are moving in parameter space.  This is typically implemented as
+follows
+</p>
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\tag{1}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>where we have introduced a momentum parameter \( \gamma \), with
+\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to
+indicate the gradient is to be taken over a different mini-batch at
+each step. We call this algorithm gradient descent with momentum
+(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a
+running average of recently encountered gradients and
+\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory
+used in the averaging procedure. Consistent with this, when
+\( \gamma=0 \), this just reduces down to ordinary SGD as discussed
+earlier. An equivalent way of writing the updates is
+</p>
+
+<p>&nbsp;<br>
+$$
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+$$
+<p>&nbsp;<br>
+
+<p>where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).</p>
+</section>
+
+<section>
+<h2 id="more-on-momentum-based-approaches">More on momentum based approaches </h2>
+
+<p>Let us try to get more intuition from these equations. It is helpful
+to consider a simple physical analogy with a particle of mass \( m \)
+moving in a viscous medium with drag coefficient \( \mu \) and potential
+\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \),
+then its motion is described by
+</p>
+
+<p>&nbsp;<br>
+$$
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+$$
+<p>&nbsp;<br>
+
+<p>We can discretize this equation in the usual way to get</p>
+
+<p>&nbsp;<br>
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
+<p>&nbsp;<br>
+
+<p>Rearranging this equation, we can rewrite this as</p>
+
+<p>&nbsp;<br>
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="momentum-parameter">Momentum parameter </h2>
+
+<p>Notice that this equation is identical to previous one if we identify
+the position of the particle, \( \mathbf{w} \), with the parameters
+\( \boldsymbol{\theta} \). This allows us to identify the momentum
+parameter and learning rate with the mass of the particle and the
+viscous drag as:
+</p>
+
+<p>&nbsp;<br>
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
+<p>&nbsp;<br>
+
+<p>Thus, as the name suggests, the momentum parameter is proportional to
+the mass of the particle and effectively provides inertia.
+Furthermore, in the large viscosity/small learning rate limit, our
+memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+</p>
+
+<p>Why is momentum useful? SGD momentum helps the gradient descent
+algorithm gain speed in directions with persistent but small gradients
+even in the presence of stochasticity, while suppressing oscillations
+in high-curvature directions. This becomes especially important in
+situations where the landscape is shallow and flat in some directions
+and narrow and steep in others. It has been argued that first-order
+methods (with appropriate initial conditions) can perform comparable
+to more expensive second order methods, especially in the context of
+complex deep learning models.
+</p>
+
+<p>These beneficial properties of momentum can sometimes become even more
+pronounced by using a slight modification of the classical momentum
+algorithm called Nesterov Accelerated Gradient (NAG).
+</p>
+
+<p>In the NAG algorithm, rather than calculating the gradient at the
+current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one
+calculates the gradient at the expected value of the parameters given
+our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma
+\mathbf{v}_{t-1}) \). This yields the NAG update rule
+</p>
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\tag{2}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).</p>
+</section>
+
+<section>
+<h2 id="second-moment-of-the-gradient">Second moment of the gradient </h2>
+
+<p>In stochastic gradient descent, with and without momentum, we still
+have to specify a schedule for tuning the learning rates \( \eta_t \)
+as a function of time.  As discussed in the context of Newton's
+method, this presents a number of dilemmas. The learning rate is
+limited by the steepest direction which can change depending on the
+current position in the landscape. To circumvent this problem, ideally
+our algorithm would keep track of curvature and take large steps in
+shallow, flat directions and small steps in steep, narrow directions.
+Second-order methods accomplish this by calculating or approximating
+the Hessian and normalizing the learning rate by the
+curvature. However, this is very computationally expensive for
+extremely large models. Ideally, we would like to be able to
+adaptively change the step size to match the landscape without paying
+the steep computational price of calculating or approximating
+Hessians.
+</p>
+
+<p>Recently, a number of methods have been introduced that accomplish
+this by tracking not only the gradient, but also the second moment of
+the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
+<a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM</a>.
+</p>
+</section>
+
+<section>
+<h2 id="rms-prop">RMS prop </h2>
+
+<p>In RMS prop, in addition to keeping a running average of the first
+moment of the gradient, we also keep track of the second moment
+denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule
+for RMS prop is given by
+</p>
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\tag{3}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>where \( \beta \) controls the averaging time of the second moment and is
+typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate
+typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8}  \) is a
+small regularization constant to prevent divergences. Multiplication
+and division by vectors is understood as an element-wise operation. It
+is clear from this formula that the learning rate is reduced in
+directions where the norm of the gradient is consistently large. This
+greatly speeds up the convergence by allowing us to use a larger
+learning rate for flat directions.
+</p>
+</section>
+
+<section>
+<h2 id="adam-optimizer-https-arxiv-org-abs-1412-6980"><a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM optimizer</a> </h2>
+
+<p>A related algorithm is the ADAM optimizer. In
+<a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM</a>, we keep a running average of
+both the first and second moment of the gradient and use this
+information to adaptively change the learning rate for different
+parameters.  The method isefficient when working with large
+problems involving lots data and/or parameters.  It is a combination of the
+gradient descent with momentum algorithm and the RMSprop algorithm
+discussed above.
+</p>
+
+<p>In addition to keeping a running average of the first and
+second moments of the gradient
+(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and
+\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM
+performs an additional bias correction to account for the fact that we
+are estimating the first two moments of the gradient using a running
+average (denoted by the hats in the update rule below). The update
+rule for ADAM is given by (where multiplication and division are once
+again understood to be element-wise operations below)
+</p>
+
+<p>&nbsp;<br>
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\tag{4}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\tag{5}
+\end{align}
+$$
+<p>&nbsp;<br>
+
+<p>where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and
+second moment and are typically taken to be \( 0.9 \) and \( 0.99 \)
+respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.
+</p>
+
+<p>Like in RMSprop, the effective step size of a parameter depends on the
+magnitude of its gradient squared.  To understand this better, let us
+rewrite this expression in terms of the variance
+\( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t -
+(\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The
+update rule for this parameter is given by
+</p>
+
+<p>&nbsp;<br>
+$$
+\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 +  m_t^2 }+\epsilon}.
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="algorithms-and-codes-for-adagrad-rmsprop-and-adam">Algorithms and codes for Adagrad, RMSprop and Adam </h2>
+
+<p>The algorithms we have implemented are well described in the text by <a href="https://www.deeplearningbook.org/contents/optimization.html" target="_blank">Goodfellow, Bengio and Courville, chapter 8</a>.</p>
+
+<p>The codes which implement these algorithms are discussed after our presentation of automatic differentiation.</p>
+</section>
+
 <section>
 <h2 id="practical-tips">Practical tips </h2>
 
@@ -267,18 +938,1373 @@ difficulty of converting a computer program into a single expression,
 while numerical differentiation can introduce round-off errors in the
 discretization process and cancellation
 </p>
+
+<p>Python has tools for so-called <b>automatic differentiation</b>.
+Consider the following example
+</p>
+<p>&nbsp;<br>
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
+<p>&nbsp;<br>
+
+<p>which has the following derivative</p>
+<p>&nbsp;<br>
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) 
+$$
+<p>&nbsp;<br>
+
+<p>Using <b>autograd</b> we have</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+
+<span style="color: #228B22"># To do elementwise differentiation:</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> elementwise_grad <span style="color: #8B008B; font-weight: bold">as</span> egrad 
+
+<span style="color: #228B22"># To plot:</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span> 
+
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sin(<span style="color: #B452CD">2</span>*np.pi*x + x**<span style="color: #B452CD">2</span>)
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f_grad_analytic</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.cos(<span style="color: #B452CD">2</span>*np.pi*x + x**<span style="color: #B452CD">2</span>)*(<span style="color: #B452CD">2</span>*np.pi + <span style="color: #B452CD">2</span>*x)
+
+<span style="color: #228B22"># Do the comparison:</span>
+x = np.linspace(<span style="color: #B452CD">0</span>,<span style="color: #B452CD">1</span>,<span style="color: #B452CD">1000</span>)
+
+f_grad = egrad(f)
+
+computed = f_grad(x)
+analytic = f_grad_analytic(x)
+
+plt.title(<span style="color: #CD5555">&#39;Derivative computed from Autograd compared with the analytical derivative&#39;</span>)
+plt.plot(x,computed,label=<span style="color: #CD5555">&#39;autograd&#39;</span>)
+plt.plot(x,analytic,label=<span style="color: #CD5555">&#39;analytic&#39;</span>)
+
+plt.xlabel(<span style="color: #CD5555">&#39;x&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">&#39;y&#39;</span>)
+plt.legend()
+
+plt.show()
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The max absolute difference is: %g&quot;</span>%(np.max(np.abs(computed - analytic))))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 </section>
 
 <section>
-<h2 id="videos-on-neural-networks">Videos on Neural Networks </h2>
+<h2 id="using-autograd">Using autograd </h2>
 
-<a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a>
+<p>Here we
+experiment with what kind of functions Autograd is capable
+of finding the gradient of. The following Python functions are just
+meant to illustrate what Autograd can do, but please feel free to
+experiment with other, possibly more complicated, functions as well.
+</p>
 
-<a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f1</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> x**<span style="color: #B452CD">3</span> + <span style="color: #B452CD">1</span>
+
+f1_grad = grad(f1)
+
+<span style="color: #228B22"># Remember to send in float as argument to the computed gradient from Autograd!</span>
+a = <span style="color: #B452CD">1.0</span>
+
+<span style="color: #228B22"># See the evaluated gradient at a using autograd:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The gradient of f1 evaluated at a = %g using autograd is: %g&quot;</span>%(a,f1_grad(a)))
+
+<span style="color: #228B22"># Compare with the analytical derivative, that is f1&#39;(x) = 3*x**2 </span>
+grad_analytical = <span style="color: #B452CD">3</span>*a**<span style="color: #B452CD">2</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g&quot;</span>%(a,grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 </section>
 
 <section>
-<h2 id="neural-networks">Neural networks </h2>
+<h2 id="autograd-with-more-complicated-functions">Autograd with more complicated functions </h2>
+
+<p>To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f2</span>(x1,x2):
+    <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">3</span>*x1**<span style="color: #B452CD">3</span> + x2*(x1 - <span style="color: #B452CD">5</span>) + <span style="color: #B452CD">1</span>
+
+<span style="color: #228B22"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
+f2_grad_x1 = grad(f2,<span style="color: #B452CD">0</span>)
+
+<span style="color: #228B22"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
+f2_grad_x2 = grad(f2,<span style="color: #B452CD">1</span>)
+
+x1 = <span style="color: #B452CD">1.0</span>
+x2 = <span style="color: #B452CD">3.0</span> 
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Evaluating at x1 = %g, x2 = %g&quot;</span>%(x1,x2))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;-&quot;</span>*<span style="color: #B452CD">30</span>)
+
+<span style="color: #228B22"># Compare with the analytical derivatives:</span>
+
+<span style="color: #228B22"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
+f2_grad_x1_analytical = <span style="color: #B452CD">9</span>*x1**<span style="color: #B452CD">2</span> + x2
+
+<span style="color: #228B22"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
+f2_grad_x2_analytical = x1 - <span style="color: #B452CD">5</span>
+
+<span style="color: #228B22"># See the evaluated derivations:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f2 w.r.t x1: %g&quot;</span>%( f2_grad_x1(x1,x2) ))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f2 w.r.t x1: %g&quot;</span>%( f2_grad_x1(x1,x2) ))
+
+<span style="color: #658b00">print</span>()
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f2 w.r.t x2: %g&quot;</span>%( f2_grad_x2(x1,x2) ))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f2 w.r.t x2: %g&quot;</span>%( f2_grad_x2(x1,x2) ))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.</p>
+</section>
+
+<section>
+<h2 id="more-complicated-functions-using-the-elements-of-their-arguments-directly">More complicated functions using the elements of their arguments directly </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f3</span>(x): <span style="color: #228B22"># Assumes x is an array of length 5 or higher</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">2</span>*x[<span style="color: #B452CD">0</span>] + <span style="color: #B452CD">3</span>*x[<span style="color: #B452CD">1</span>] + <span style="color: #B452CD">5</span>*x[<span style="color: #B452CD">2</span>] + <span style="color: #B452CD">7</span>*x[<span style="color: #B452CD">3</span>] + <span style="color: #B452CD">11</span>*x[<span style="color: #B452CD">4</span>]**<span style="color: #B452CD">2</span>
+
+f3_grad = grad(f3)
+
+x = np.linspace(<span style="color: #B452CD">0</span>,<span style="color: #B452CD">4</span>,<span style="color: #B452CD">5</span>)
+
+<span style="color: #228B22"># Print the computed gradient:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed gradient of f3 is: &quot;</span>, f3_grad(x))
+
+<span style="color: #228B22"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
+f3_grad_analytical = np.array([<span style="color: #B452CD">2</span>, <span style="color: #B452CD">3</span>, <span style="color: #B452CD">5</span>, <span style="color: #B452CD">7</span>, <span style="color: #B452CD">22</span>*x[<span style="color: #B452CD">4</span>]])
+
+<span style="color: #228B22"># Print the analytical gradient:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical gradient of f3 is: &quot;</span>, f3_grad_analytical)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+</p>
+</section>
+
+<section>
+<h2 id="functions-using-mathematical-functions-from-numpy">Functions using mathematical functions from Numpy </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f4</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sqrt(<span style="color: #B452CD">1</span>+x**<span style="color: #B452CD">2</span>) + np.exp(x) + np.sin(<span style="color: #B452CD">2</span>*np.pi*x)
+
+f4_grad = grad(f4)
+
+x = <span style="color: #B452CD">2.7</span>
+
+<span style="color: #228B22"># Print the computed derivative:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f4 at x = %g is: %g&quot;</span>%(x,f4_grad(x)))
+
+<span style="color: #228B22"># The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi</span>
+f4_grad_analytical = x/np.sqrt(<span style="color: #B452CD">1</span> + x**<span style="color: #B452CD">2</span>) + np.exp(x) + np.cos(<span style="color: #B452CD">2</span>*np.pi*x)*<span style="color: #B452CD">2</span>*np.pi
+
+<span style="color: #228B22"># Print the analytical gradient:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical gradient of f4 at x = %g is: %g&quot;</span>%(x,f4_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="more-autograd">More autograd </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f5</span>(x):
+    <span style="color: #8B008B; font-weight: bold">if</span> x &gt;= <span style="color: #B452CD">0</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> x**<span style="color: #B452CD">2</span>
+    <span style="color: #8B008B; font-weight: bold">else</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> -<span style="color: #B452CD">3</span>*x + <span style="color: #B452CD">1</span>
+
+f5_grad = grad(f5)
+
+x = <span style="color: #B452CD">2.7</span>
+
+<span style="color: #228B22"># Print the computed derivative:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f5 at x = %g is: %g&quot;</span>%(x,f5_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="and-with-loops">And  with loops </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f6_for</span>(x):
+    val = <span style="color: #B452CD">0</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">10</span>):
+        val = val + x**i
+    <span style="color: #8B008B; font-weight: bold">return</span> val
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f6_while</span>(x):
+    val = <span style="color: #B452CD">0</span>
+    i = <span style="color: #B452CD">0</span>
+    <span style="color: #8B008B; font-weight: bold">while</span> i &lt; <span style="color: #B452CD">10</span>:
+        val = val + x**i
+        i = i + <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = <span style="color: #B452CD">0.5</span>
+
+<span style="color: #228B22"># Print the computed derivaties of f6_for and f6_while</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f6_for at x = %g is: %g&quot;</span>%(x,f6_for_grad(x)))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f6_while at x = %g is: %g&quot;</span>%(x,f6_while_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #228B22"># Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9</span>
+<span style="color: #228B22"># The analytical derivative is: sum(i*x**(i-1)) </span>
+f6_grad_analytical = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">10</span>):
+    f6_grad_analytical += i*x**(i-<span style="color: #B452CD">1</span>)
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f6 at x = %g is: %g&quot;</span>%(x,f6_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="using-recursion">Using recursion </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f7</span>(n): <span style="color: #228B22"># Assume that n is an integer</span>
+    <span style="color: #8B008B; font-weight: bold">if</span> n == <span style="color: #B452CD">1</span> <span style="color: #8B008B">or</span> n == <span style="color: #B452CD">0</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">else</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> n*f7(n-<span style="color: #B452CD">1</span>)
+
+f7_grad = grad(f7)
+
+n = <span style="color: #B452CD">2.0</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f7 at n = %d is: %g&quot;</span>%(n,f7_grad(n)))
+
+<span style="color: #228B22"># The function f7 is an implementation of the factorial of n.</span>
+<span style="color: #228B22"># By using the product rule, one can find that the derivative is:</span>
+
+f7_grad_analytical = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #658b00">int</span>(n)-<span style="color: #B452CD">1</span>):
+    tmp = <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> k <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #658b00">int</span>(n)-<span style="color: #B452CD">1</span>):
+        <span style="color: #8B008B; font-weight: bold">if</span> k != i:
+            tmp *= (n - k)
+    f7_grad_analytical += tmp
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f7 at n = %d is: %g&quot;</span>%(n,f7_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.</p>
+</section>
+
+<section>
+<h2 id="unsupported-functions">Unsupported functions </h2>
+<p>Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.</p>
+
+<p>Assigning a value to the variable being differentiated with respect to</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f8</span>(x): <span style="color: #228B22"># Assume x is an array</span>
+    x[<span style="color: #B452CD">2</span>] = <span style="color: #B452CD">3</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> x*<span style="color: #B452CD">2</span>
+
+f8_grad = grad(f8)
+
+x = <span style="color: #B452CD">8.4</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f8 is:&quot;</span>,f8_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.</p>
+</section>
+
+<section>
+<h2 id="the-syntax-a-dot-b-when-finding-the-dot-product">The syntax a.dot(b) when finding the dot product </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f9</span>(a): <span style="color: #228B22"># Assume a is an array with 2 elements</span>
+    b = np.array([<span style="color: #B452CD">1.0</span>,<span style="color: #B452CD">2.0</span>])
+    <span style="color: #8B008B; font-weight: bold">return</span> a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([<span style="color: #B452CD">1.0</span>,<span style="color: #B452CD">0.0</span>])
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f9 is:&quot;</span>,f9_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f9_alternative</span>(x): <span style="color: #228B22"># Assume a is an array with 2 elements</span>
+    b = np.array([<span style="color: #B452CD">1.0</span>,<span style="color: #B452CD">2.0</span>])
+    <span style="color: #8B008B; font-weight: bold">return</span> np.dot(x,b) <span style="color: #228B22"># The same as x_1*b_1 + x_2*b_2</span>
+
+f9_alternative_grad = grad(f9_alternative)
+
+x = np.array([<span style="color: #B452CD">3.0</span>,<span style="color: #B452CD">0.0</span>])
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The gradient of f9 is:&quot;</span>,f9_alternative_grad(x))
+
+<span style="color: #228B22"># The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively</span>
+<span style="color: #228B22"># w.r.t x is (b_1, b_2).</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="recommended-to-avoid">Recommended to avoid </h2>
+<p>The documentation recommends to avoid inplace operations such as</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;">a += b
+a -= b
+a*= b
+a /=b
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="using-autograd-with-ols">Using Autograd with OLS </h2>
+
+<p>We conclude the part on optmization by showing how we can make codes
+for linear regression and logistic regression using <b>autograd</b>. The
+first example shows results with ordinary leats squares.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(beta):
+    <span style="color: #8B008B; font-weight: bold">return</span> (<span style="color: #B452CD">1.0</span>/n)*np.sum((y-X @ beta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">1000</span>
+<span style="color: #228B22"># define the gradient</span>
+training_gradient = grad(CostOLS)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+xnew = np.array([[<span style="color: #B452CD">0</span>],[<span style="color: #B452CD">2</span>]])
+Xnew = np.c_[np.ones((<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, <span style="color: #CD5555">&quot;r-&quot;</span>)
+plt.plot(xnew, ypredict2, <span style="color: #CD5555">&quot;b-&quot;</span>)
+plt.plot(x, y ,<span style="color: #CD5555">&#39;ro&#39;</span>)
+plt.axis([<span style="color: #B452CD">0</span>,<span style="color: #B452CD">2.0</span>,<span style="color: #B452CD">0</span>, <span style="color: #B452CD">15.0</span>])
+plt.xlabel(<span style="color: #CD5555">r&#39;$x$&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">r&#39;$y$&#39;</span>)
+plt.title(<span style="color: #CD5555">r&#39;Random numbers &#39;</span>)
+plt.show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="same-code-but-now-with-momentum-gradient-descent">Same code but now with momentum gradient descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(beta):
+    <span style="color: #8B008B; font-weight: bold">return</span> (<span style="color: #B452CD">1.0</span>/n)*np.sum((y-X @ beta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x<span style="color: #228B22">#+np.random.randn(n,1)</span>
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">30</span>
+
+<span style="color: #228B22"># define the gradient</span>
+training_gradient = grad(CostOLS)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+    <span style="color: #658b00">print</span>(<span style="color: #658b00">iter</span>,gradients[<span style="color: #B452CD">0</span>],gradients[<span style="color: #B452CD">1</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+<span style="color: #228B22"># Now improve with momentum gradient descent</span>
+change = <span style="color: #B452CD">0.0</span>
+delta_momentum = <span style="color: #B452CD">0.3</span>
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    <span style="color: #228B22"># calculate gradient</span>
+    gradients = training_gradient(theta)
+    <span style="color: #228B22"># calculate update</span>
+    new_change = eta*gradients+delta_momentum*change
+    <span style="color: #228B22"># take a step</span>
+    theta -= new_change
+    <span style="color: #228B22"># save the change</span>
+    change = new_change
+    <span style="color: #658b00">print</span>(<span style="color: #658b00">iter</span>,gradients[<span style="color: #B452CD">0</span>],gradients[<span style="color: #B452CD">1</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd wth momentum&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="but-noen-of-these-can-compete-with-newton-s-method">But noen of these can compete with Newton's method </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Newton&#39;s method</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(beta):
+    <span style="color: #8B008B; font-weight: bold">return</span> (<span style="color: #B452CD">1.0</span>/n)*np.sum((y-X @ beta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(beta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+<span style="color: #228B22"># Note that here the Hessian does not depend on the parameters beta</span>
+invH = np.linalg.pinv(H)
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+beta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+Niterations = <span style="color: #B452CD">5</span>
+
+<span style="color: #228B22"># define the gradient</span>
+training_gradient = grad(CostOLS)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = training_gradient(beta)
+    beta -= invH @ gradients
+    <span style="color: #658b00">print</span>(<span style="color: #658b00">iter</span>,gradients[<span style="color: #B452CD">0</span>],gradients[<span style="color: #B452CD">1</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;beta from own Newton code&quot;</span>)
+<span style="color: #658b00">print</span>(beta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="including-stochastic-gradient-descent-with-autograd">Including Stochastic Gradient Descent with Autograd </h2>
+<p>In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using <b>autograd</b>.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">1000</span>
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = (<span style="color: #B452CD">1.0</span>/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+xnew = np.array([[<span style="color: #B452CD">0</span>],[<span style="color: #B452CD">2</span>]])
+Xnew = np.c_[np.ones((<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, <span style="color: #CD5555">&quot;r-&quot;</span>)
+plt.plot(xnew, ypredict2, <span style="color: #CD5555">&quot;b-&quot;</span>)
+plt.plot(x, y ,<span style="color: #CD5555">&#39;ro&#39;</span>)
+plt.axis([<span style="color: #B452CD">0</span>,<span style="color: #B452CD">2.0</span>,<span style="color: #B452CD">0</span>, <span style="color: #B452CD">15.0</span>])
+plt.xlabel(<span style="color: #CD5555">r&#39;$x$&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">r&#39;$y$&#39;</span>)
+plt.title(<span style="color: #CD5555">r&#39;Random numbers &#39;</span>)
+plt.show()
+
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+t0, t1 = <span style="color: #B452CD">5</span>, <span style="color: #B452CD">50</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">learning_schedule</span>(t):
+    <span style="color: #8B008B; font-weight: bold">return</span> t0/(t+t1)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+<span style="color: #228B22"># Can you figure out a better way of setting up the contributions to each batch?</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own sdg&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="same-code-but-now-with-momentum-gradient-descent">Same code but now with momentum gradient descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">100</span>
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = (<span style="color: #B452CD">1.0</span>/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+t0, t1 = <span style="color: #B452CD">5</span>, <span style="color: #B452CD">50</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">learning_schedule</span>(t):
+    <span style="color: #8B008B; font-weight: bold">return</span> t0/(t+t1)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+
+change = <span style="color: #B452CD">0.0</span>
+delta_momentum = <span style="color: #B452CD">0.3</span>
+
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        <span style="color: #228B22"># calculate update</span>
+        new_change = eta*gradients+delta_momentum*change
+        <span style="color: #228B22"># take a step</span>
+        theta -= new_change
+        <span style="color: #228B22"># save the change</span>
+        change = new_change
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own sdg with momentum&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="similar-second-order-function-now-problem-but-now-with-adagrad">Similar (second order function now) problem but now with AdaGrad </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">1000</span>
+x = np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">3</span>*x +<span style="color: #B452CD">4</span>*x*x
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+<span style="color: #228B22"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+<span style="color: #228B22"># Guess for unknown parameters theta</span>
+theta = np.random.randn(<span style="color: #B452CD">3</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #228B22"># Value for learning rate</span>
+eta = <span style="color: #B452CD">0.01</span>
+<span style="color: #228B22"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  = <span style="color: #B452CD">1e-8</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    Giter = <span style="color: #B452CD">0.0</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        Giter += gradients*gradients
+        update = gradients*eta/(delta+np.sqrt(Giter))
+        theta -= update
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own AdaGrad&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Running this code we note an almost perfect agreement with the results from matrix inversion.</p>
+</section>
+
+<section>
+<h2 id="rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent">RMSprop for adaptive learning rate with Stochastic Gradient Descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">1000</span>
+x = np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">3</span>*x +<span style="color: #B452CD">4</span>*x*x<span style="color: #228B22"># +np.random.randn(n,1)</span>
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+<span style="color: #228B22"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+<span style="color: #228B22"># Guess for unknown parameters theta</span>
+theta = np.random.randn(<span style="color: #B452CD">3</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #228B22"># Value for learning rate</span>
+eta = <span style="color: #B452CD">0.01</span>
+<span style="color: #228B22"># Value for parameter rho</span>
+rho = <span style="color: #B452CD">0.99</span>
+<span style="color: #228B22"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  = <span style="color: #B452CD">1e-8</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    Giter = <span style="color: #B452CD">0.0</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+	<span style="color: #228B22"># Accumulated gradient</span>
+	<span style="color: #228B22"># Scaling with rho the new and the previous results</span>
+        Giter = (rho*Giter+(<span style="color: #B452CD">1</span>-rho)*gradients*gradients)
+	<span style="color: #228B22"># Taking the diagonal only and inverting</span>
+        update = gradients*eta/(delta+np.sqrt(Giter))
+	<span style="color: #228B22"># Hadamard product</span>
+        theta -= update
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own RMSprop&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf">And finally <a href="https://arxiv.org/pdf/1412.6980.pdf" target="_blank">ADAM</a> </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">1000</span>
+x = np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">3</span>*x +<span style="color: #B452CD">4</span>*x*x<span style="color: #228B22"># +np.random.randn(n,1)</span>
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+<span style="color: #228B22"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+<span style="color: #228B22"># Guess for unknown parameters theta</span>
+theta = np.random.randn(<span style="color: #B452CD">3</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #228B22"># Value for learning rate</span>
+eta = <span style="color: #B452CD">0.01</span>
+<span style="color: #228B22"># Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980</span>
+beta1 = <span style="color: #B452CD">0.9</span>
+beta2 = <span style="color: #B452CD">0.999</span>
+<span style="color: #228B22"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  = <span style="color: #B452CD">1e-7</span>
+<span style="color: #658b00">iter</span> = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    first_moment = <span style="color: #B452CD">0.0</span>
+    second_moment = <span style="color: #B452CD">0.0</span>
+    <span style="color: #658b00">iter</span> += <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        <span style="color: #228B22"># Computing moments first</span>
+        first_moment = beta1*first_moment + (<span style="color: #B452CD">1</span>-beta1)*gradients
+        second_moment = beta2*second_moment+(<span style="color: #B452CD">1</span>-beta2)*gradients*gradients
+        first_term = first_moment/(<span style="color: #B452CD">1.0</span>-beta1**<span style="color: #658b00">iter</span>)
+        second_term = second_moment/(<span style="color: #B452CD">1.0</span>-beta2**<span style="color: #658b00">iter</span>)
+	<span style="color: #228B22"># Scaling with rho the new and the previous results</span>
+        update = eta*first_term/(np.sqrt(second_term)+delta)
+        theta -= update
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own ADAM&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="and-logistic-regression">And Logistic Regression </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">sigmoid</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">0.5</span> * (np.tanh(x / <span style="color: #B452CD">2.</span>) + <span style="color: #B452CD">1</span>)
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">logistic_predictions</span>(weights, inputs):
+    <span style="color: #228B22"># Outputs probability of a label being true according to logistic model.</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> sigmoid(np.dot(inputs, weights))
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">training_loss</span>(weights):
+    <span style="color: #228B22"># Training loss is the negative log-likelihood of the training labels.</span>
+    preds = logistic_predictions(weights, inputs)
+    label_probabilities = preds * targets + (<span style="color: #B452CD">1</span> - preds) * (<span style="color: #B452CD">1</span> - targets)
+    <span style="color: #8B008B; font-weight: bold">return</span> -np.sum(np.log(label_probabilities))
+
+<span style="color: #228B22"># Build a toy dataset.</span>
+inputs = np.array([[<span style="color: #B452CD">0.52</span>, <span style="color: #B452CD">1.12</span>,  <span style="color: #B452CD">0.77</span>],
+                   [<span style="color: #B452CD">0.88</span>, -<span style="color: #B452CD">1.08</span>, <span style="color: #B452CD">0.15</span>],
+                   [<span style="color: #B452CD">0.52</span>, <span style="color: #B452CD">0.06</span>, -<span style="color: #B452CD">1.30</span>],
+                   [<span style="color: #B452CD">0.74</span>, -<span style="color: #B452CD">2.49</span>, <span style="color: #B452CD">1.39</span>]])
+targets = np.array([<span style="color: #8B008B; font-weight: bold">True</span>, <span style="color: #8B008B; font-weight: bold">True</span>, <span style="color: #8B008B; font-weight: bold">False</span>, <span style="color: #8B008B; font-weight: bold">True</span>])
+
+<span style="color: #228B22"># Define a function that returns gradients of training loss using Autograd.</span>
+training_gradient_fun = grad(training_loss)
+
+<span style="color: #228B22"># Optimize weights using gradient descent.</span>
+weights = np.array([<span style="color: #B452CD">0.0</span>, <span style="color: #B452CD">0.0</span>, <span style="color: #B452CD">0.0</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Initial loss:&quot;</span>, training_loss(weights))
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">100</span>):
+    weights -= training_gradient_fun(weights) * <span style="color: #B452CD">0.01</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Trained loss:&quot;</span>, training_loss(weights))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="introducing-jax-https-jax-readthedocs-io-en-latest">Introducing <a href="https://jax.readthedocs.io/en/latest/" target="_blank">JAX</a> </h2>
+
+<p>Presently, instead of using <b>autograd</b>, we recommend using <a href="https://jax.readthedocs.io/en/latest/" target="_blank">JAX</a></p>
+
+<p><b>JAX</b> is Autograd and <a href="https://www.tensorflow.org/xla" target="_blank">XLA (Accelerated Linear Algebra))</a>,
+brought together for high-performance numerical computing and machine learning research.
+It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.
+</p>
+
+<p>Here's a simple example on how you can use <b>JAX</b> to compute the derivate of the logistic function.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="font-size: 80%; line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">jax.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">jnp</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">jax</span> <span style="color: #8B008B; font-weight: bold">import</span> grad, jit, vmap
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">sum_logistic</span>(x):
+  <span style="color: #8B008B; font-weight: bold">return</span> jnp.sum(<span style="color: #B452CD">1.0</span> / (<span style="color: #B452CD">1.0</span> + jnp.exp(-x)))
+
+x_small = jnp.arange(<span style="color: #B452CD">3.</span>)
+derivative_fn = grad(sum_logistic)
+<span style="color: #658b00">print</span>(derivative_fn(x_small))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+</section>
+
+<section>
+<h2 id="introduction-to-neural-networks">Introduction to Neural networks </h2>
 
 <p>Artificial neural networks are computational systems that can learn to
 perform tasks by considering examples, generally without being
@@ -312,7 +2338,7 @@ i.e. has zero output.
 $$
 \begin{equation}
  y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
-\tag{1}
+\tag{6}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -484,7 +2510,7 @@ as to not restrict the range of output values.
 </section>
 
 <section>
-<h2 id="illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model">Illustration of a single perceptropn model and a multi-perceptron model </h2>
+<h2 id="illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model">Illustration of a single perceptron model and a multi-perceptron model </h2>
 
 <center>  <!-- FIGURE -->
 <hr class="figure">
@@ -698,7 +2724,7 @@ of the outputs of <em>all</em> neurons in the previous layer.
 <p>&nbsp;<br>
 $$
 \begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
-\tag{2}
+\tag{7}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -715,7 +2741,7 @@ first layer.  We define  the output \( y_i^1 \) of all neurons in layer 1 as
 $$
 \begin{equation}
  y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\right)
-\tag{3}
+\tag{8}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -729,7 +2755,7 @@ In this case we would identify these functions with a superscript \( l \) for th
 $$
 \begin{equation}
  y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
-\tag{4}
+\tag{9}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -750,9 +2776,9 @@ is obtained.
 $$
 \begin{align}
  y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
-\tag{5}\\
+\tag{10}\\
  &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
-\tag{6}
+\tag{11}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -763,10 +2789,10 @@ $$
 $$
 \begin{align}
  y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
-\tag{7}\\
+\tag{12}\\
  &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
   + b_1^3\right]
-\tag{8}
+\tag{13}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -783,7 +2809,7 @@ layers. The complete functional form is,
 $$
 \begin{align}
 &y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
-\tag{9}
+\tag{14}
 \end{align}
 $$
 <p>&nbsp;<br>
@@ -811,7 +2837,7 @@ sum of scaled activation functions of the form
 $$
 \begin{equation}
  f(x) = c_1 f(c_2 x + c_3) + c_4
-\tag{10}
+\tag{15}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -856,7 +2882,7 @@ $$
            b^2_2 \\
            b^2_3 \\
           \end{array}\right]\right).
-\tag{11}
+\tag{16}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -872,7 +2898,7 @@ $$
 \begin{equation}
  y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
  f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\tag{12}
+\tag{17}
 \end{equation}
 $$
 <p>&nbsp;<br>
@@ -1038,518 +3064,6 @@ plt.show()
 </div>
 </section>
 
-<section>
-<h2 id="the-multilayer-perceptron-mlp">The multilayer  perceptron (MLP) </h2>
-
-<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
-<ol>
-<p><li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
-<p><li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
-<p><li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
-</ol>
-<p>
-<p>As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
-units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-</p>
-
-<p>For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
-</p>
-
-<p>The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
-</p>
-
-<p>The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation \( \hat{z} \), we can think of this as a basis
-expansion of the original inputs \( \hat{x} \). The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in  an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
-</p>
-</section>
-
-<section>
-<h2 id="from-one-to-many-layers-the-universal-approximation-theorem">From one to many layers, the universal approximation theorem </h2>
-
-<p>A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
-</p>
-
-<p>As stated earlier, 
-an important theorem in studies of neural networks, restated without
-proof here, is the <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_blank">universal approximation
-theorem</a>.
-</p>
-
-<p>It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
-</p>
-</section>
-
-<section>
-<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">Deriving the back propagation code for a multilayer perceptron model </h2>
-
-<p>As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_blank">back propagation algorithm</a>.
-</p>
-
-<p>The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
-</p>
-
-<p>To derive these equations let us start with a plain regression problem
-and define our cost function as
-</p>
-
-<p>&nbsp;<br>
-$$
-{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
-$$
-<p>&nbsp;<br>
-
-<p>where the $t_i$s are our \( n \) targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs \( \hat{x} \) are given by \( y_i \).  Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with \( K \)
-classes.
-</p>
-</section>
-
-<section>
-<h2 id="definitions">Definitions </h2>
-
-<p>With our definition of the targets \( \hat{t} \), the outputs of the
-network \( \hat{y} \) and the inputs \( \hat{x} \) we
-define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
-\( l \)-th layer as a function of the bias, the weights which add up from
-the previous layer \( l-1 \) and the forward passes/outputs
-\( \hat{a}^{l-1} \) from the previous layer as
-</p>
-
-<p>&nbsp;<br>
-$$
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
-$$
-<p>&nbsp;<br>
-
-<p>where \( b_k^l \) are the biases from layer \( l \).  Here \( M_{l-1} \)
-represents the total number of nodes/neurons/units of layer \( l-1 \). The
-figure here illustrates this equation.  We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-</p>
-
-<p>&nbsp;<br>
-$$
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-$$
-<p>&nbsp;<br>
-
-<p>With the activation values \( \hat{z}^l \) we can in turn define the
-output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
-and their nodes.  It means we have
-</p>
-
-<p>&nbsp;<br>
-$$
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-$$
-<p>&nbsp;<br>
-</section>
-
-<section>
-<h2 id="derivatives-and-the-chain-rule">Derivatives and the chain rule </h2>
-
-<p>From the definition of the activation \( z_j^l \) we have</p>
-<p>&nbsp;<br>
-$$
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-$$
-<p>&nbsp;<br>
-
-<p>and</p>
-<p>&nbsp;<br>
-$$
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
-$$
-<p>&nbsp;<br>
-
-<p>With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))</p>
-<p>&nbsp;<br>
-$$
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
-$$
-<p>&nbsp;<br>
-</section>
-
-<section>
-<h2 id="derivative-of-the-cost-function">Derivative of the cost function </h2>
-
-<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
-
-<p>Let us specialize to the output layer \( l=L \). Our cost function is</p>
-<p>&nbsp;<br>
-$$
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-$$
-<p>&nbsp;<br>
-
-<p>The derivative of this function with respect to the weights is</p>
-
-<p>&nbsp;<br>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
-$$
-<p>&nbsp;<br>
-
-<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
-<p>&nbsp;<br>
-$$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
-$$
-<p>&nbsp;<br>
-</section>
-
-<section>
-<h2 id="bringing-it-together-first-back-propagation-equation">Bringing it together, first back propagation equation </h2>
-
-<p>We have thus</p>
-<p>&nbsp;<br>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
-$$
-<p>&nbsp;<br>
-
-<p>Defining</p>
-<p>&nbsp;<br>
-$$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-$$
-<p>&nbsp;<br>
-
-<p>and using the Hadamard product of two vectors we can write this as</p>
-<p>&nbsp;<br>
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-<p>&nbsp;<br>
-
-<p>This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
-</p>
-
-<p>Notice that everything in the above equations is easily computed.  In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \).  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-</p>
-
-<p>&nbsp;<br>
-$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-$$
-<p>&nbsp;<br>
-
-<p>With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
-<p>&nbsp;<br>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
-$$
-<p>&nbsp;<br>
-</section>
-
-<section>
-<h2 id="derivatives-in-terms-of-z-j-l">Derivatives in terms of \( z_j^L \) </h2>
-
-<p>It is also easy to see that our previous equation can be written as</p>
-
-<p>&nbsp;<br>
-$$
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
-$$
-<p>&nbsp;<br>
-
-<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely</p>
-<p>&nbsp;<br>
-$$
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
-$$
-<p>&nbsp;<br>
-
-<p>That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. </p>
-</section>
-
-<section>
-<h2 id="bringing-it-together">Bringing it together </h2>
-
-<p>We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b>The starting equations</b>
-<p>
-
-<p>&nbsp;<br>
-$$
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
-\tag{13}
-\end{equation}
-$$
-<p>&nbsp;<br>
-
-<p>and</p>
-<p>&nbsp;<br>
-$$
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\tag{14}
-\end{equation}
-$$
-<p>&nbsp;<br>
-
-<p>and</p>
-
-<p>&nbsp;<br>
-$$
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\tag{15}
-\end{equation}
-$$
-<p>&nbsp;<br>
-</div>
-
-<p>An interesting consequence of the above equations is that when the
-activation \( a_k^{L-1} \) is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
-</p>
-
-<p>Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
-</p>
-
-<p>We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one \( L-1 \) in terms of the errors in the final output layer.
-</p>
-</section>
-
-<section>
-<h2 id="final-back-propagating-equation">Final back propagating equation </h2>
-
-<p>We have that (replacing \( L \) with a general layer \( l \))</p>
-<p>&nbsp;<br>
-$$
-\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
-$$
-<p>&nbsp;<br>
-
-<p>We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have</p>
-
-<p>&nbsp;<br>
-$$
-\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
-$$
-<p>&nbsp;<br>
-
-<p>and recalling that</p>
-<p>&nbsp;<br>
-$$
-z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
-$$
-<p>&nbsp;<br>
-
-<p>with \( M_l \) being the number of nodes in layer \( l \), we obtain</p>
-<p>&nbsp;<br>
-$$
-\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
-$$
-<p>&nbsp;<br>
-
-<p>This is our final equation.</p>
-
-<p>We are now ready to set up the algorithm for back propagation and learning the weights and biases.</p>
-</section>
-
-<section>
-<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
-
-<p>The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>First, we set up the input data \( \hat{x} \) and the activations
-\( \hat{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \hat{a}^1 \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \hat{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \hat{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all</p>
-<p>&nbsp;<br>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-<p>&nbsp;<br>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-<p>&nbsp;<br>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-<p>&nbsp;<br>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-<p>&nbsp;<br>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-<p>&nbsp;<br>
-
-<p>&nbsp;<br>
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-<p>&nbsp;<br>
-</div>
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-</section>
-
-<section>
-<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
-
-<p>The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>First, we set up the input data \( \boldsymbol{x} \) and the activations
-\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \boldsymbol{a}^1 \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \boldsymbol{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all</p>
-<p>&nbsp;<br>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-<p>&nbsp;<br>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-<p>&nbsp;<br>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-<p>&nbsp;<br>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-<p>&nbsp;<br>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-<p>&nbsp;<br>
-
-<p>&nbsp;<br>
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-<p>&nbsp;<br>
-</div>
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-</section>
-
 
 
 </div> <!-- class="slides" -->
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-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <link href="https://cdn.rawgit.com/doconce/doconce/master/bundled/html_styles/style_solarized_box/css/solarized_light_code.css" rel="stylesheet" type="text/css" title="light"/>
 <script src="https://cdn.rawgit.com/doconce/doconce/master/bundled/html_styles/style_solarized_box/js/highlight.pack.js"></script>
 <script>hljs.initHighlightingOnLoad();</script>
@@ -63,21 +63,127 @@ div.toc p,a {
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -98,11 +204,11 @@ div.toc p,a {
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -127,50 +233,7 @@ div.toc p,a {
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -192,7 +255,7 @@ MathJax.Hub.Config({
 
 <!-- ------------------- main content ---------------------- -->
 <center>
-<h1>Week 40: Neural networks</h1>
+<h1>Week 40: Gradient descent methods (continued) and start Neural networks</h1>
 </center>  <!-- document title -->
 
 <!-- author(s): Morten Hjorth-Jensen -->
@@ -208,33 +271,649 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Oct 8, 2022</h4>
+<h4>October 2-6, 2023</h4>
 </center> <!-- date -->
 <br>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="plan-for-week-40">Plan for week 40 </h2>
+<h2 id="plans-for-week-40">Plans for week 40 </h2>
+
+<div class="alert alert-block alert-block alert-text-normal">
+<b>Material for the active learning sessions on Tuesday and Wednesday</b>
+<p>
+<ul>
+  <li> Work on project 1 and discussions on how to structure your report</li>
+  <li> No weekly exercises for week 40, project work only</li>
+  <li> <a href="https://youtu.be/tVW1ZDmZnwM" target="_blank">Video on how to write scientific reports recorded during one of the lab sessions</a></li>
+  <li> A general guideline can be found at <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</tt></a>.</li>
+</ul>
+</div>
+  
+
+<div class="alert alert-block alert-block alert-text-normal">
+<b>Material for the lecture on Thursday October 5, 2023</b>
+<p>
+<ul>
+  <li> Stochastic Gradient descent with examples and automatic differentiation</li>
+  <li> Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.</li>  
+  <li> Readings and Videos:</li>
+<ul>
+    <li> These lecture notes</li>
+    <li> For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.</li>
+    <li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 4 on stochastic gradient descent</a></li>
+    <li> For neural networks we recommend Goodfellow et al chapter 6.</li>
+    <li> <a href="https://www.youtube.com/watch?v=sDv4f4s2SB8" target="_blank">Video on gradient descent</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI" target="_blank">Video on stochastic gradient descent</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a></li>
+</ul>
+</ul>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="summary-from-last-week-using-gradient-descent-methods-limitations">Summary from last week, using gradient descent methods, limitations </h2>
 
 <ul>
-<li> Thursday: Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.</li>
-<ul>
-  <li> <a href="https://youtu.be/sCHiGSoG2lE" target="_blank">Video of lecture</a></li>
+<li> <b>Gradient descent (GD) finds local minima of our function</b>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</li>
+<li> <b>GD is sensitive to initial conditions</b>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</li>
+<li> <b>Gradients are computationally expensive to calculate for large datasets</b>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called &quot;mini batches&quot;. This has the added benefit of introducing stochasticity into our algorithm.</li>
+<li> <b>GD is very sensitive to choices of learning rates</b>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</li>
+<li> <b>GD treats all directions in parameter space uniformly.</b> Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</li> 
+<li> GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</li>
 </ul>
-<li> Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.</li>
-<ul>
-  <li> <a href="https://youtu.be/hDTtA7PRRfI" target="_blank">Video of lecture</a></li>
-</ul>
-</ul>
-<p>Reading suggestions for both days: <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 10</a> and Hastie et al chapter 11.
-For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
-For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
-</p>
-
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="overview-video-on-stochastic-gradient-descent">Overview video on Stochastic Gradient Descent </h2>
 
 <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" target="_blank">What is Stochastic Gradient Descent</a>
 
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="batches-and-mini-batches">Batches and mini-batches </h2>
+
+<p>In gradient descent we compute the cost function and its gradient for all data points we have.</p>
+
+<p>In large-scale applications such as the <a href="https://www.image-net.org/challenges/LSVRC/" target="_blank">ILSVRC challenge</a>, the
+training data can have on order of millions of examples. Hence, it
+seems wasteful to compute the full cost function over the entire
+training set in order to perform only a single parameter update. A
+very common approach to addressing this challenge is to compute the
+gradient over batches of the training data. For example, a typical batch could contain some thousand  examples from
+an  entire training set of several millions. This batch is then used to
+perform a parameter update.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="stochastic-gradient-descent-sgd">Stochastic Gradient Descent (SGD) </h2>
+
+<p>In stochastic gradient descent, the extreme case is the case where we
+have only one batch, that is we include the whole data set.
+</p>
+
+<p>This process is called Stochastic Gradient
+Descent (SGD) (or also sometimes on-line gradient descent). This is
+relatively less common to see because in practice due to vectorized
+code optimizations it can be computationally much more efficient to
+evaluate the gradient for 100 examples, than the gradient for one
+example 100 times. Even though SGD technically refers to using a
+single example at a time to evaluate the gradient, you will hear
+people use the term SGD even when referring to mini-batch gradient
+descent (i.e. mentions of MGD for &#8220;Minibatch Gradient Descent&#8221;, or BGD
+for &#8220;Batch gradient descent&#8221; are rare to see), where it is usually
+assumed that mini-batches are used. The size of the mini-batch is a
+hyperparameter but it is not very common to cross-validate or bootstrap it. It is
+usually based on memory constraints (if any), or set to some value,
+e.g. 32, 64 or 128. We use powers of 2 in practice because many
+vectorized operation implementations work faster when their inputs are
+sized in powers of 2.
+</p>
+
+<p>In our notes with  SGD we mean stochastic gradient descent with mini-batches.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="stochastic-gradient-descent">Stochastic Gradient Descent </h2>
+
+<p>Stochastic gradient descent (SGD) and variants thereof address some of
+the shortcomings of the Gradient descent method discussed above.
+</p>
+
+<p>The underlying idea of SGD comes from the observation that the cost
+function, which we want to minimize, can almost always be written as a
+sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \),
+</p>
+$$
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}). 
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="computation-of-gradients">Computation of gradients </h2>
+
+<p>This in turn means that the gradient can be
+computed as a sum over \( i \)-gradients 
+</p>
+$$
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
+$$
+
+<p>Stochasticity/randomness is introduced by only taking the
+gradient on a subset of the data called minibatches.  If there are \( n \)
+data points and the size of each minibatch is \( M \), there will be \( n/M \)
+minibatches. We denote these minibatches by \( B_k \) where
+\( k=1,\cdots,n/M \).
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="sgd-example">SGD example </h2>
+<p>As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
+and we choose to have \( M=5 \) minibathces,
+then each minibatch contains two data points. In particular we have
+\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
+(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you
+have only a single batch with all data points and on the other extreme,
+you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e
+\( B_k = \mathbf{x}_k \).
+</p>
+
+<p>The idea is now to approximate the gradient by replacing the sum over
+all data points with a sum over the data points in one the minibatches
+picked at random in each gradient descent step 
+</p>
+$$
+\nabla_{\beta}
+C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
+c_i(\mathbf{x}_i, \mathbf{\beta}).
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="the-gradient-step">The gradient step </h2>
+
+<p>Thus a gradient descent step now looks like </p>
+$$
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
+$$
+
+<p>where \( k \) is picked at random with equal
+probability from \( [1,n/M] \). An iteration over the number of
+minibathces (n/M) is commonly referred to as an epoch. Thus it is
+typical to choose a number of epochs and for each epoch iterate over
+the number of minibatches, as exemplified in the code below.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="simple-example-code">Simple example code </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span> 
+
+n = <span style="color: #B452CD">100</span> <span style="color: #228B22">#100 datapoints </span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+n_epochs = <span style="color: #B452CD">10</span> <span style="color: #228B22">#number of epochs</span>
+
+j = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">1</span>,n_epochs+<span style="color: #B452CD">1</span>):
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        k = np.random.randint(m) <span style="color: #228B22">#Pick the k-th minibatch at random</span>
+        <span style="color: #228B22">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #228B22">#Compute new suggestion for </span>
+        j += <span style="color: #B452CD">1</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Taking the gradient only on a subset of the data has two important
+benefits. First, it introduces randomness which decreases the chance
+that our opmization scheme gets stuck in a local minima. Second, if
+the size of the minibatches are small relative to the number of
+datapoints (\( M <  n \)), the computation of the gradient is much
+cheaper since we sum over the datapoints in the \( k-th \) minibatch and not
+all \( n \) datapoints.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="when-do-we-stop">When do we stop? </h2>
+
+<p>A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the \( \beta \) that
+gave the lowest value.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="slightly-different-approach">Slightly different approach </h2>
+
+<p>Another approach is to let the step length \( \gamma_j \) depend on the
+number of epochs in such a way that it becomes very small after a
+reasonable time such that we do not move at all. Such approaches are
+also called scaling. There are many such ways to <a href="https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1" target="_blank">scale the learning
+rate</a>
+and <a href="https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf" target="_blank">discussions here</a>. See
+also
+<a href="https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1" target="_blank"><tt>https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1</tt></a>
+for a discussion of different scaling functions for the learning rate.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="time-decay-rate">Time decay rate </h2>
+
+<p>As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in <em>time</em> \( t \).</p>
+
+<p>In this way we can fix the number of epochs, compute \( \beta \) and
+evaluate the cost function at the end. Repeating the computation will
+give a different result since the scheme is random by design. Then we
+pick the final \( \beta \) that gives the lowest value of the cost
+function.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span> 
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">step_length</span>(t,t0,t1):
+    <span style="color: #8B008B; font-weight: bold">return</span> t0/(t+t1)
+
+n = <span style="color: #B452CD">100</span> <span style="color: #228B22">#100 datapoints </span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+n_epochs = <span style="color: #B452CD">500</span> <span style="color: #228B22">#number of epochs</span>
+t0 = <span style="color: #B452CD">1.0</span>
+t1 = <span style="color: #B452CD">10</span>
+
+gamma_j = t0/t1
+j = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">1</span>,n_epochs+<span style="color: #B452CD">1</span>):
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        k = np.random.randint(m) <span style="color: #228B22">#Pick the k-th minibatch at random</span>
+        <span style="color: #228B22">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #228B22">#Compute new suggestion for beta</span>
+        t = epoch*m+i
+        gamma_j = step_length(t,t0,t1)
+        j += <span style="color: #B452CD">1</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;gamma_j after %d epochs: %g&quot;</span> % (n_epochs,gamma_j))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="code-with-a-number-of-minibatches-which-varies">Code with a Number of Minibatches which varies </h2>
+
+<p>In the code here we vary the number of mini-batches.</p>
+
+<!-- code=text (!bc pycode) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"># Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
+print(&quot;Own inversion&quot;)
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f&quot;Eigenvalues of Hessian Matrix:{EigValues}&quot;)
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+
+for iter in range(Niterations):
+    gradients = 2.0/n*X.T @ ((X @ theta)-y)
+    theta -= eta*gradients
+print(&quot;theta from own gd&quot;)
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print(&quot;theta from own sdg&quot;)
+print(theta)
+
+plt.plot(xnew, ypredict, &quot;r-&quot;)
+plt.plot(xnew, ypredict2, &quot;b-&quot;)
+plt.plot(x, y ,&#39;ro&#39;)
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r&#39;$x$&#39;)
+plt.ylabel(r&#39;$y$&#39;)
+plt.title(r&#39;Random numbers &#39;)
+plt.show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="replace-or-not">Replace or not </h2>
+
+<p>In the above code, we have use replacement in setting up the
+mini-batches. The discussion
+<a href="https://sebastianraschka.com/faq/docs/sgd-methods.html" target="_blank">here</a> may be
+useful.  
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="momentum-based-gd">Momentum based GD </h2>
+
+<p>The stochastic gradient descent (SGD) is almost always used with a
+<em>momentum</em> or inertia term that serves as a memory of the direction we
+are moving in parameter space.  This is typically implemented as
+follows
+</p>
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\label{_auto1}
+\end{align}
+$$
+
+<p>where we have introduced a momentum parameter \( \gamma \), with
+\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to
+indicate the gradient is to be taken over a different mini-batch at
+each step. We call this algorithm gradient descent with momentum
+(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a
+running average of recently encountered gradients and
+\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory
+used in the averaging procedure. Consistent with this, when
+\( \gamma=0 \), this just reduces down to ordinary SGD as discussed
+earlier. An equivalent way of writing the updates is
+</p>
+
+$$
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+$$
+
+<p>where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="more-on-momentum-based-approaches">More on momentum based approaches </h2>
+
+<p>Let us try to get more intuition from these equations. It is helpful
+to consider a simple physical analogy with a particle of mass \( m \)
+moving in a viscous medium with drag coefficient \( \mu \) and potential
+\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \),
+then its motion is described by
+</p>
+
+$$
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+$$
+
+<p>We can discretize this equation in the usual way to get</p>
+
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
+
+<p>Rearranging this equation, we can rewrite this as</p>
+
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="momentum-parameter">Momentum parameter </h2>
+
+<p>Notice that this equation is identical to previous one if we identify
+the position of the particle, \( \mathbf{w} \), with the parameters
+\( \boldsymbol{\theta} \). This allows us to identify the momentum
+parameter and learning rate with the mass of the particle and the
+viscous drag as:
+</p>
+
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
+
+<p>Thus, as the name suggests, the momentum parameter is proportional to
+the mass of the particle and effectively provides inertia.
+Furthermore, in the large viscosity/small learning rate limit, our
+memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+</p>
+
+<p>Why is momentum useful? SGD momentum helps the gradient descent
+algorithm gain speed in directions with persistent but small gradients
+even in the presence of stochasticity, while suppressing oscillations
+in high-curvature directions. This becomes especially important in
+situations where the landscape is shallow and flat in some directions
+and narrow and steep in others. It has been argued that first-order
+methods (with appropriate initial conditions) can perform comparable
+to more expensive second order methods, especially in the context of
+complex deep learning models.
+</p>
+
+<p>These beneficial properties of momentum can sometimes become even more
+pronounced by using a slight modification of the classical momentum
+algorithm called Nesterov Accelerated Gradient (NAG).
+</p>
+
+<p>In the NAG algorithm, rather than calculating the gradient at the
+current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one
+calculates the gradient at the expected value of the parameters given
+our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma
+\mathbf{v}_{t-1}) \). This yields the NAG update rule
+</p>
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\label{_auto2}
+\end{align}
+$$
+
+<p>One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="second-moment-of-the-gradient">Second moment of the gradient </h2>
+
+<p>In stochastic gradient descent, with and without momentum, we still
+have to specify a schedule for tuning the learning rates \( \eta_t \)
+as a function of time.  As discussed in the context of Newton's
+method, this presents a number of dilemmas. The learning rate is
+limited by the steepest direction which can change depending on the
+current position in the landscape. To circumvent this problem, ideally
+our algorithm would keep track of curvature and take large steps in
+shallow, flat directions and small steps in steep, narrow directions.
+Second-order methods accomplish this by calculating or approximating
+the Hessian and normalizing the learning rate by the
+curvature. However, this is very computationally expensive for
+extremely large models. Ideally, we would like to be able to
+adaptively change the step size to match the landscape without paying
+the steep computational price of calculating or approximating
+Hessians.
+</p>
+
+<p>Recently, a number of methods have been introduced that accomplish
+this by tracking not only the gradient, but also the second moment of
+the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
+<a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM</a>.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="rms-prop">RMS prop </h2>
+
+<p>In RMS prop, in addition to keeping a running average of the first
+moment of the gradient, we also keep track of the second moment
+denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule
+for RMS prop is given by
+</p>
+
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\label{_auto3}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+$$
+
+<p>where \( \beta \) controls the averaging time of the second moment and is
+typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate
+typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8}  \) is a
+small regularization constant to prevent divergences. Multiplication
+and division by vectors is understood as an element-wise operation. It
+is clear from this formula that the learning rate is reduced in
+directions where the norm of the gradient is consistently large. This
+greatly speeds up the convergence by allowing us to use a larger
+learning rate for flat directions.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="adam-optimizer-https-arxiv-org-abs-1412-6980"><a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM optimizer</a> </h2>
+
+<p>A related algorithm is the ADAM optimizer. In
+<a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM</a>, we keep a running average of
+both the first and second moment of the gradient and use this
+information to adaptively change the learning rate for different
+parameters.  The method isefficient when working with large
+problems involving lots data and/or parameters.  It is a combination of the
+gradient descent with momentum algorithm and the RMSprop algorithm
+discussed above.
+</p>
+
+<p>In addition to keeping a running average of the first and
+second moments of the gradient
+(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and
+\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM
+performs an additional bias correction to account for the fact that we
+are estimating the first two moments of the gradient using a running
+average (denoted by the hats in the update rule below). The update
+rule for ADAM is given by (where multiplication and division are once
+again understood to be element-wise operations below)
+</p>
+
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\label{_auto4}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\label{_auto5}
+\end{align}
+$$
+
+<p>where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and
+second moment and are typically taken to be \( 0.9 \) and \( 0.99 \)
+respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.
+</p>
+
+<p>Like in RMSprop, the effective step size of a parameter depends on the
+magnitude of its gradient squared.  To understand this better, let us
+rewrite this expression in terms of the variance
+\( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t -
+(\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The
+update rule for this parameter is given by
+</p>
+
+$$
+\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 +  m_t^2 }+\epsilon}.
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="algorithms-and-codes-for-adagrad-rmsprop-and-adam">Algorithms and codes for Adagrad, RMSprop and Adam </h2>
+
+<p>The algorithms we have implemented are well described in the text by <a href="https://www.deeplearningbook.org/contents/optimization.html" target="_blank">Goodfellow, Bengio and Courville, chapter 8</a>.</p>
+
+<p>The codes which implement these algorithms are discussed after our presentation of automatic differentiation.</p>
+
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="practical-tips">Practical tips </h2>
 
@@ -276,15 +955,1363 @@ while numerical differentiation can introduce round-off errors in the
 discretization process and cancellation
 </p>
 
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="videos-on-neural-networks">Videos on Neural Networks </h2>
+<p>Python has tools for so-called <b>automatic differentiation</b>.
+Consider the following example
+</p>
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
 
-<a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a>
+<p>which has the following derivative</p>
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) 
+$$
+
+<p>Using <b>autograd</b> we have</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+
+<span style="color: #228B22"># To do elementwise differentiation:</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> elementwise_grad <span style="color: #8B008B; font-weight: bold">as</span> egrad 
+
+<span style="color: #228B22"># To plot:</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span> 
+
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sin(<span style="color: #B452CD">2</span>*np.pi*x + x**<span style="color: #B452CD">2</span>)
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f_grad_analytic</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.cos(<span style="color: #B452CD">2</span>*np.pi*x + x**<span style="color: #B452CD">2</span>)*(<span style="color: #B452CD">2</span>*np.pi + <span style="color: #B452CD">2</span>*x)
+
+<span style="color: #228B22"># Do the comparison:</span>
+x = np.linspace(<span style="color: #B452CD">0</span>,<span style="color: #B452CD">1</span>,<span style="color: #B452CD">1000</span>)
+
+f_grad = egrad(f)
+
+computed = f_grad(x)
+analytic = f_grad_analytic(x)
+
+plt.title(<span style="color: #CD5555">&#39;Derivative computed from Autograd compared with the analytical derivative&#39;</span>)
+plt.plot(x,computed,label=<span style="color: #CD5555">&#39;autograd&#39;</span>)
+plt.plot(x,analytic,label=<span style="color: #CD5555">&#39;analytic&#39;</span>)
+
+plt.xlabel(<span style="color: #CD5555">&#39;x&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">&#39;y&#39;</span>)
+plt.legend()
+
+plt.show()
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The max absolute difference is: %g&quot;</span>%(np.max(np.abs(computed - analytic))))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split  -->
+<h2 id="using-autograd">Using autograd </h2>
+
+<p>Here we
+experiment with what kind of functions Autograd is capable
+of finding the gradient of. The following Python functions are just
+meant to illustrate what Autograd can do, but please feel free to
+experiment with other, possibly more complicated, functions as well.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f1</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> x**<span style="color: #B452CD">3</span> + <span style="color: #B452CD">1</span>
+
+f1_grad = grad(f1)
+
+<span style="color: #228B22"># Remember to send in float as argument to the computed gradient from Autograd!</span>
+a = <span style="color: #B452CD">1.0</span>
+
+<span style="color: #228B22"># See the evaluated gradient at a using autograd:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The gradient of f1 evaluated at a = %g using autograd is: %g&quot;</span>%(a,f1_grad(a)))
+
+<span style="color: #228B22"># Compare with the analytical derivative, that is f1&#39;(x) = 3*x**2 </span>
+grad_analytical = <span style="color: #B452CD">3</span>*a**<span style="color: #B452CD">2</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g&quot;</span>%(a,grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="neural-networks">Neural networks </h2>
+<h2 id="autograd-with-more-complicated-functions">Autograd with more complicated functions </h2>
+
+<p>To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f2</span>(x1,x2):
+    <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">3</span>*x1**<span style="color: #B452CD">3</span> + x2*(x1 - <span style="color: #B452CD">5</span>) + <span style="color: #B452CD">1</span>
+
+<span style="color: #228B22"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
+f2_grad_x1 = grad(f2,<span style="color: #B452CD">0</span>)
+
+<span style="color: #228B22"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
+f2_grad_x2 = grad(f2,<span style="color: #B452CD">1</span>)
+
+x1 = <span style="color: #B452CD">1.0</span>
+x2 = <span style="color: #B452CD">3.0</span> 
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Evaluating at x1 = %g, x2 = %g&quot;</span>%(x1,x2))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;-&quot;</span>*<span style="color: #B452CD">30</span>)
+
+<span style="color: #228B22"># Compare with the analytical derivatives:</span>
+
+<span style="color: #228B22"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
+f2_grad_x1_analytical = <span style="color: #B452CD">9</span>*x1**<span style="color: #B452CD">2</span> + x2
+
+<span style="color: #228B22"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
+f2_grad_x2_analytical = x1 - <span style="color: #B452CD">5</span>
+
+<span style="color: #228B22"># See the evaluated derivations:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f2 w.r.t x1: %g&quot;</span>%( f2_grad_x1(x1,x2) ))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f2 w.r.t x1: %g&quot;</span>%( f2_grad_x1(x1,x2) ))
+
+<span style="color: #658b00">print</span>()
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f2 w.r.t x2: %g&quot;</span>%( f2_grad_x2(x1,x2) ))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f2 w.r.t x2: %g&quot;</span>%( f2_grad_x2(x1,x2) ))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="more-complicated-functions-using-the-elements-of-their-arguments-directly">More complicated functions using the elements of their arguments directly </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f3</span>(x): <span style="color: #228B22"># Assumes x is an array of length 5 or higher</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">2</span>*x[<span style="color: #B452CD">0</span>] + <span style="color: #B452CD">3</span>*x[<span style="color: #B452CD">1</span>] + <span style="color: #B452CD">5</span>*x[<span style="color: #B452CD">2</span>] + <span style="color: #B452CD">7</span>*x[<span style="color: #B452CD">3</span>] + <span style="color: #B452CD">11</span>*x[<span style="color: #B452CD">4</span>]**<span style="color: #B452CD">2</span>
+
+f3_grad = grad(f3)
+
+x = np.linspace(<span style="color: #B452CD">0</span>,<span style="color: #B452CD">4</span>,<span style="color: #B452CD">5</span>)
+
+<span style="color: #228B22"># Print the computed gradient:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed gradient of f3 is: &quot;</span>, f3_grad(x))
+
+<span style="color: #228B22"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
+f3_grad_analytical = np.array([<span style="color: #B452CD">2</span>, <span style="color: #B452CD">3</span>, <span style="color: #B452CD">5</span>, <span style="color: #B452CD">7</span>, <span style="color: #B452CD">22</span>*x[<span style="color: #B452CD">4</span>]])
+
+<span style="color: #228B22"># Print the analytical gradient:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical gradient of f3 is: &quot;</span>, f3_grad_analytical)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+</p>
+
+<!-- !split  -->
+<h2 id="functions-using-mathematical-functions-from-numpy">Functions using mathematical functions from Numpy </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f4</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sqrt(<span style="color: #B452CD">1</span>+x**<span style="color: #B452CD">2</span>) + np.exp(x) + np.sin(<span style="color: #B452CD">2</span>*np.pi*x)
+
+f4_grad = grad(f4)
+
+x = <span style="color: #B452CD">2.7</span>
+
+<span style="color: #228B22"># Print the computed derivative:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f4 at x = %g is: %g&quot;</span>%(x,f4_grad(x)))
+
+<span style="color: #228B22"># The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi</span>
+f4_grad_analytical = x/np.sqrt(<span style="color: #B452CD">1</span> + x**<span style="color: #B452CD">2</span>) + np.exp(x) + np.cos(<span style="color: #B452CD">2</span>*np.pi*x)*<span style="color: #B452CD">2</span>*np.pi
+
+<span style="color: #228B22"># Print the analytical gradient:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical gradient of f4 at x = %g is: %g&quot;</span>%(x,f4_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="more-autograd">More autograd </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f5</span>(x):
+    <span style="color: #8B008B; font-weight: bold">if</span> x &gt;= <span style="color: #B452CD">0</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> x**<span style="color: #B452CD">2</span>
+    <span style="color: #8B008B; font-weight: bold">else</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> -<span style="color: #B452CD">3</span>*x + <span style="color: #B452CD">1</span>
+
+f5_grad = grad(f5)
+
+x = <span style="color: #B452CD">2.7</span>
+
+<span style="color: #228B22"># Print the computed derivative:</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f5 at x = %g is: %g&quot;</span>%(x,f5_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="and-with-loops">And  with loops </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f6_for</span>(x):
+    val = <span style="color: #B452CD">0</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">10</span>):
+        val = val + x**i
+    <span style="color: #8B008B; font-weight: bold">return</span> val
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f6_while</span>(x):
+    val = <span style="color: #B452CD">0</span>
+    i = <span style="color: #B452CD">0</span>
+    <span style="color: #8B008B; font-weight: bold">while</span> i &lt; <span style="color: #B452CD">10</span>:
+        val = val + x**i
+        i = i + <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = <span style="color: #B452CD">0.5</span>
+
+<span style="color: #228B22"># Print the computed derivaties of f6_for and f6_while</span>
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f6_for at x = %g is: %g&quot;</span>%(x,f6_for_grad(x)))
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f6_while at x = %g is: %g&quot;</span>%(x,f6_while_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #228B22"># Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9</span>
+<span style="color: #228B22"># The analytical derivative is: sum(i*x**(i-1)) </span>
+f6_grad_analytical = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">10</span>):
+    f6_grad_analytical += i*x**(i-<span style="color: #B452CD">1</span>)
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f6 at x = %g is: %g&quot;</span>%(x,f6_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="using-recursion">Using recursion </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f7</span>(n): <span style="color: #228B22"># Assume that n is an integer</span>
+    <span style="color: #8B008B; font-weight: bold">if</span> n == <span style="color: #B452CD">1</span> <span style="color: #8B008B">or</span> n == <span style="color: #B452CD">0</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">else</span>:
+        <span style="color: #8B008B; font-weight: bold">return</span> n*f7(n-<span style="color: #B452CD">1</span>)
+
+f7_grad = grad(f7)
+
+n = <span style="color: #B452CD">2.0</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The computed derivative of f7 at n = %d is: %g&quot;</span>%(n,f7_grad(n)))
+
+<span style="color: #228B22"># The function f7 is an implementation of the factorial of n.</span>
+<span style="color: #228B22"># By using the product rule, one can find that the derivative is:</span>
+
+f7_grad_analytical = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #658b00">int</span>(n)-<span style="color: #B452CD">1</span>):
+    tmp = <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> k <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #658b00">int</span>(n)-<span style="color: #B452CD">1</span>):
+        <span style="color: #8B008B; font-weight: bold">if</span> k != i:
+            tmp *= (n - k)
+    f7_grad_analytical += tmp
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The analytical derivative of f7 at n = %d is: %g&quot;</span>%(n,f7_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="unsupported-functions">Unsupported functions </h2>
+<p>Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.</p>
+
+<p>Assigning a value to the variable being differentiated with respect to</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f8</span>(x): <span style="color: #228B22"># Assume x is an array</span>
+    x[<span style="color: #B452CD">2</span>] = <span style="color: #B452CD">3</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> x*<span style="color: #B452CD">2</span>
+
+f8_grad = grad(f8)
+
+x = <span style="color: #B452CD">8.4</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f8 is:&quot;</span>,f8_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="the-syntax-a-dot-b-when-finding-the-dot-product">The syntax a.dot(b) when finding the dot product </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f9</span>(a): <span style="color: #228B22"># Assume a is an array with 2 elements</span>
+    b = np.array([<span style="color: #B452CD">1.0</span>,<span style="color: #B452CD">2.0</span>])
+    <span style="color: #8B008B; font-weight: bold">return</span> a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([<span style="color: #B452CD">1.0</span>,<span style="color: #B452CD">0.0</span>])
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The derivative of f9 is:&quot;</span>,f9_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">f9_alternative</span>(x): <span style="color: #228B22"># Assume a is an array with 2 elements</span>
+    b = np.array([<span style="color: #B452CD">1.0</span>,<span style="color: #B452CD">2.0</span>])
+    <span style="color: #8B008B; font-weight: bold">return</span> np.dot(x,b) <span style="color: #228B22"># The same as x_1*b_1 + x_2*b_2</span>
+
+f9_alternative_grad = grad(f9_alternative)
+
+x = np.array([<span style="color: #B452CD">3.0</span>,<span style="color: #B452CD">0.0</span>])
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;The gradient of f9 is:&quot;</span>,f9_alternative_grad(x))
+
+<span style="color: #228B22"># The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively</span>
+<span style="color: #228B22"># w.r.t x is (b_1, b_2).</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="recommended-to-avoid">Recommended to avoid </h2>
+<p>The documentation recommends to avoid inplace operations such as</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;">a += b
+a -= b
+a*= b
+a /=b
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="using-autograd-with-ols">Using Autograd with OLS </h2>
+
+<p>We conclude the part on optmization by showing how we can make codes
+for linear regression and logistic regression using <b>autograd</b>. The
+first example shows results with ordinary leats squares.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(beta):
+    <span style="color: #8B008B; font-weight: bold">return</span> (<span style="color: #B452CD">1.0</span>/n)*np.sum((y-X @ beta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">1000</span>
+<span style="color: #228B22"># define the gradient</span>
+training_gradient = grad(CostOLS)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+xnew = np.array([[<span style="color: #B452CD">0</span>],[<span style="color: #B452CD">2</span>]])
+Xnew = np.c_[np.ones((<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, <span style="color: #CD5555">&quot;r-&quot;</span>)
+plt.plot(xnew, ypredict2, <span style="color: #CD5555">&quot;b-&quot;</span>)
+plt.plot(x, y ,<span style="color: #CD5555">&#39;ro&#39;</span>)
+plt.axis([<span style="color: #B452CD">0</span>,<span style="color: #B452CD">2.0</span>,<span style="color: #B452CD">0</span>, <span style="color: #B452CD">15.0</span>])
+plt.xlabel(<span style="color: #CD5555">r&#39;$x$&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">r&#39;$y$&#39;</span>)
+plt.title(<span style="color: #CD5555">r&#39;Random numbers &#39;</span>)
+plt.show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="same-code-but-now-with-momentum-gradient-descent">Same code but now with momentum gradient descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(beta):
+    <span style="color: #8B008B; font-weight: bold">return</span> (<span style="color: #B452CD">1.0</span>/n)*np.sum((y-X @ beta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x<span style="color: #228B22">#+np.random.randn(n,1)</span>
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">30</span>
+
+<span style="color: #228B22"># define the gradient</span>
+training_gradient = grad(CostOLS)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+    <span style="color: #658b00">print</span>(<span style="color: #658b00">iter</span>,gradients[<span style="color: #B452CD">0</span>],gradients[<span style="color: #B452CD">1</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+<span style="color: #228B22"># Now improve with momentum gradient descent</span>
+change = <span style="color: #B452CD">0.0</span>
+delta_momentum = <span style="color: #B452CD">0.3</span>
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    <span style="color: #228B22"># calculate gradient</span>
+    gradients = training_gradient(theta)
+    <span style="color: #228B22"># calculate update</span>
+    new_change = eta*gradients+delta_momentum*change
+    <span style="color: #228B22"># take a step</span>
+    theta -= new_change
+    <span style="color: #228B22"># save the change</span>
+    change = new_change
+    <span style="color: #658b00">print</span>(<span style="color: #658b00">iter</span>,gradients[<span style="color: #B452CD">0</span>],gradients[<span style="color: #B452CD">1</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd wth momentum&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="but-noen-of-these-can-compete-with-newton-s-method">But noen of these can compete with Newton's method </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Newton&#39;s method</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(beta):
+    <span style="color: #8B008B; font-weight: bold">return</span> (<span style="color: #B452CD">1.0</span>/n)*np.sum((y-X @ beta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(beta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+<span style="color: #228B22"># Note that here the Hessian does not depend on the parameters beta</span>
+invH = np.linalg.pinv(H)
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+beta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+Niterations = <span style="color: #B452CD">5</span>
+
+<span style="color: #228B22"># define the gradient</span>
+training_gradient = grad(CostOLS)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = training_gradient(beta)
+    beta -= invH @ gradients
+    <span style="color: #658b00">print</span>(<span style="color: #658b00">iter</span>,gradients[<span style="color: #B452CD">0</span>],gradients[<span style="color: #B452CD">1</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;beta from own Newton code&quot;</span>)
+<span style="color: #658b00">print</span>(beta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="including-stochastic-gradient-descent-with-autograd">Including Stochastic Gradient Descent with Autograd </h2>
+<p>In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using <b>autograd</b>.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">1000</span>
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = (<span style="color: #B452CD">1.0</span>/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+xnew = np.array([[<span style="color: #B452CD">0</span>],[<span style="color: #B452CD">2</span>]])
+Xnew = np.c_[np.ones((<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, <span style="color: #CD5555">&quot;r-&quot;</span>)
+plt.plot(xnew, ypredict2, <span style="color: #CD5555">&quot;b-&quot;</span>)
+plt.plot(x, y ,<span style="color: #CD5555">&#39;ro&#39;</span>)
+plt.axis([<span style="color: #B452CD">0</span>,<span style="color: #B452CD">2.0</span>,<span style="color: #B452CD">0</span>, <span style="color: #B452CD">15.0</span>])
+plt.xlabel(<span style="color: #CD5555">r&#39;$x$&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">r&#39;$y$&#39;</span>)
+plt.title(<span style="color: #CD5555">r&#39;Random numbers &#39;</span>)
+plt.show()
+
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+t0, t1 = <span style="color: #B452CD">5</span>, <span style="color: #B452CD">50</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">learning_schedule</span>(t):
+    <span style="color: #8B008B; font-weight: bold">return</span> t0/(t+t1)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+<span style="color: #228B22"># Can you figure out a better way of setting up the contributions to each batch?</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own sdg&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="same-code-but-now-with-momentum-gradient-descent">Same code but now with momentum gradient descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">100</span>
+x = <span style="color: #B452CD">2</span>*np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">4</span>+<span style="color: #B452CD">3</span>*x+np.random.randn(n,<span style="color: #B452CD">1</span>)
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+<span style="color: #228B22"># Hessian matrix</span>
+H = (<span style="color: #B452CD">2.0</span>/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">f&quot;Eigenvalues of Hessian Matrix:{</span>EigValues<span style="color: #CD5555">}&quot;</span>)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+eta = <span style="color: #B452CD">1.0</span>/np.max(EigValues)
+Niterations = <span style="color: #B452CD">100</span>
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+
+<span style="color: #8B008B; font-weight: bold">for</span> <span style="color: #658b00">iter</span> <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(Niterations):
+    gradients = (<span style="color: #B452CD">1.0</span>/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own gd&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+
+
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+t0, t1 = <span style="color: #B452CD">5</span>, <span style="color: #B452CD">50</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">learning_schedule</span>(t):
+    <span style="color: #8B008B; font-weight: bold">return</span> t0/(t+t1)
+
+theta = np.random.randn(<span style="color: #B452CD">2</span>,<span style="color: #B452CD">1</span>)
+
+change = <span style="color: #B452CD">0.0</span>
+delta_momentum = <span style="color: #B452CD">0.3</span>
+
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        <span style="color: #228B22"># calculate update</span>
+        new_change = eta*gradients+delta_momentum*change
+        <span style="color: #228B22"># take a step</span>
+        theta -= new_change
+        <span style="color: #228B22"># save the change</span>
+        change = new_change
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own sdg with momentum&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="similar-second-order-function-now-problem-but-now-with-adagrad">Similar (second order function now) problem but now with AdaGrad </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">1000</span>
+x = np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">3</span>*x +<span style="color: #B452CD">4</span>*x*x
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+<span style="color: #228B22"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+<span style="color: #228B22"># Guess for unknown parameters theta</span>
+theta = np.random.randn(<span style="color: #B452CD">3</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #228B22"># Value for learning rate</span>
+eta = <span style="color: #B452CD">0.01</span>
+<span style="color: #228B22"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  = <span style="color: #B452CD">1e-8</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    Giter = <span style="color: #B452CD">0.0</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        Giter += gradients*gradients
+        update = gradients*eta/(delta+np.sqrt(Giter))
+        theta -= update
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own AdaGrad&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Running this code we note an almost perfect agreement with the results from matrix inversion.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent">RMSprop for adaptive learning rate with Stochastic Gradient Descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">1000</span>
+x = np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">3</span>*x +<span style="color: #B452CD">4</span>*x*x<span style="color: #228B22"># +np.random.randn(n,1)</span>
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+<span style="color: #228B22"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+<span style="color: #228B22"># Guess for unknown parameters theta</span>
+theta = np.random.randn(<span style="color: #B452CD">3</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #228B22"># Value for learning rate</span>
+eta = <span style="color: #B452CD">0.01</span>
+<span style="color: #228B22"># Value for parameter rho</span>
+rho = <span style="color: #B452CD">0.99</span>
+<span style="color: #228B22"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  = <span style="color: #B452CD">1e-8</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    Giter = <span style="color: #B452CD">0.0</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+	<span style="color: #228B22"># Accumulated gradient</span>
+	<span style="color: #228B22"># Scaling with rho the new and the previous results</span>
+        Giter = (rho*Giter+(<span style="color: #B452CD">1</span>-rho)*gradients*gradients)
+	<span style="color: #228B22"># Taking the diagonal only and inverting</span>
+        update = gradients*eta/(delta+np.sqrt(Giter))
+	<span style="color: #228B22"># Hadamard product</span>
+        theta -= update
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own RMSprop&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf">And finally <a href="https://arxiv.org/pdf/1412.6980.pdf" target="_blank">ADAM</a> </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #228B22"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #228B22"># OLS example</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">random</span> <span style="color: #8B008B; font-weight: bold">import</span> random, seed
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #228B22"># Note change from previous example</span>
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">CostOLS</span>(y,X,theta):
+    <span style="color: #8B008B; font-weight: bold">return</span> np.sum((y-X @ theta)**<span style="color: #B452CD">2</span>)
+
+n = <span style="color: #B452CD">1000</span>
+x = np.random.rand(n,<span style="color: #B452CD">1</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">3</span>*x +<span style="color: #B452CD">4</span>*x*x<span style="color: #228B22"># +np.random.randn(n,1)</span>
+
+X = np.c_[np.ones((n,<span style="color: #B452CD">1</span>)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Own inversion&quot;</span>)
+<span style="color: #658b00">print</span>(theta_linreg)
+
+
+<span style="color: #228B22"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient = grad(CostOLS,<span style="color: #B452CD">2</span>)
+<span style="color: #228B22"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs = <span style="color: #B452CD">50</span>
+M = <span style="color: #B452CD">5</span>   <span style="color: #228B22">#size of each minibatch</span>
+m = <span style="color: #658b00">int</span>(n/M) <span style="color: #228B22">#number of minibatches</span>
+<span style="color: #228B22"># Guess for unknown parameters theta</span>
+theta = np.random.randn(<span style="color: #B452CD">3</span>,<span style="color: #B452CD">1</span>)
+
+<span style="color: #228B22"># Value for learning rate</span>
+eta = <span style="color: #B452CD">0.01</span>
+<span style="color: #228B22"># Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980</span>
+beta1 = <span style="color: #B452CD">0.9</span>
+beta2 = <span style="color: #B452CD">0.999</span>
+<span style="color: #228B22"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  = <span style="color: #B452CD">1e-7</span>
+<span style="color: #658b00">iter</span> = <span style="color: #B452CD">0</span>
+<span style="color: #8B008B; font-weight: bold">for</span> epoch <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(n_epochs):
+    first_moment = <span style="color: #B452CD">0.0</span>
+    second_moment = <span style="color: #B452CD">0.0</span>
+    <span style="color: #658b00">iter</span> += <span style="color: #B452CD">1</span>
+    <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (<span style="color: #B452CD">1.0</span>/M)*training_gradient(yi, xi, theta)
+        <span style="color: #228B22"># Computing moments first</span>
+        first_moment = beta1*first_moment + (<span style="color: #B452CD">1</span>-beta1)*gradients
+        second_moment = beta2*second_moment+(<span style="color: #B452CD">1</span>-beta2)*gradients*gradients
+        first_term = first_moment/(<span style="color: #B452CD">1.0</span>-beta1**<span style="color: #658b00">iter</span>)
+        second_term = second_moment/(<span style="color: #B452CD">1.0</span>-beta2**<span style="color: #658b00">iter</span>)
+	<span style="color: #228B22"># Scaling with rho the new and the previous results</span>
+        update = eta*first_term/(np.sqrt(second_term)+delta)
+        theta -= update
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;theta from own ADAM&quot;</span>)
+<span style="color: #658b00">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="and-logistic-regression">And Logistic Regression </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">autograd.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">autograd</span> <span style="color: #8B008B; font-weight: bold">import</span> grad
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">sigmoid</span>(x):
+    <span style="color: #8B008B; font-weight: bold">return</span> <span style="color: #B452CD">0.5</span> * (np.tanh(x / <span style="color: #B452CD">2.</span>) + <span style="color: #B452CD">1</span>)
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">logistic_predictions</span>(weights, inputs):
+    <span style="color: #228B22"># Outputs probability of a label being true according to logistic model.</span>
+    <span style="color: #8B008B; font-weight: bold">return</span> sigmoid(np.dot(inputs, weights))
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">training_loss</span>(weights):
+    <span style="color: #228B22"># Training loss is the negative log-likelihood of the training labels.</span>
+    preds = logistic_predictions(weights, inputs)
+    label_probabilities = preds * targets + (<span style="color: #B452CD">1</span> - preds) * (<span style="color: #B452CD">1</span> - targets)
+    <span style="color: #8B008B; font-weight: bold">return</span> -np.sum(np.log(label_probabilities))
+
+<span style="color: #228B22"># Build a toy dataset.</span>
+inputs = np.array([[<span style="color: #B452CD">0.52</span>, <span style="color: #B452CD">1.12</span>,  <span style="color: #B452CD">0.77</span>],
+                   [<span style="color: #B452CD">0.88</span>, -<span style="color: #B452CD">1.08</span>, <span style="color: #B452CD">0.15</span>],
+                   [<span style="color: #B452CD">0.52</span>, <span style="color: #B452CD">0.06</span>, -<span style="color: #B452CD">1.30</span>],
+                   [<span style="color: #B452CD">0.74</span>, -<span style="color: #B452CD">2.49</span>, <span style="color: #B452CD">1.39</span>]])
+targets = np.array([<span style="color: #8B008B; font-weight: bold">True</span>, <span style="color: #8B008B; font-weight: bold">True</span>, <span style="color: #8B008B; font-weight: bold">False</span>, <span style="color: #8B008B; font-weight: bold">True</span>])
+
+<span style="color: #228B22"># Define a function that returns gradients of training loss using Autograd.</span>
+training_gradient_fun = grad(training_loss)
+
+<span style="color: #228B22"># Optimize weights using gradient descent.</span>
+weights = np.array([<span style="color: #B452CD">0.0</span>, <span style="color: #B452CD">0.0</span>, <span style="color: #B452CD">0.0</span>])
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Initial loss:&quot;</span>, training_loss(weights))
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">100</span>):
+    weights -= training_gradient_fun(weights) * <span style="color: #B452CD">0.01</span>
+
+<span style="color: #658b00">print</span>(<span style="color: #CD5555">&quot;Trained loss:&quot;</span>, training_loss(weights))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="introducing-jax-https-jax-readthedocs-io-en-latest">Introducing <a href="https://jax.readthedocs.io/en/latest/" target="_blank">JAX</a> </h2>
+
+<p>Presently, instead of using <b>autograd</b>, we recommend using <a href="https://jax.readthedocs.io/en/latest/" target="_blank">JAX</a></p>
+
+<p><b>JAX</b> is Autograd and <a href="https://www.tensorflow.org/xla" target="_blank">XLA (Accelerated Linear Algebra))</a>,
+brought together for high-performance numerical computing and machine learning research.
+It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.
+</p>
+
+<p>Here's a simple example on how you can use <b>JAX</b> to compute the derivate of the logistic function.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #eeeedd">
+  <pre style="line-height: 125%;"><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">jax.numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">jnp</span>
+<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">jax</span> <span style="color: #8B008B; font-weight: bold">import</span> grad, jit, vmap
+
+<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">sum_logistic</span>(x):
+  <span style="color: #8B008B; font-weight: bold">return</span> jnp.sum(<span style="color: #B452CD">1.0</span> / (<span style="color: #B452CD">1.0</span> + jnp.exp(-x)))
+
+x_small = jnp.arange(<span style="color: #B452CD">3.</span>)
+derivative_fn = grad(sum_logistic)
+<span style="color: #658b00">print</span>(derivative_fn(x_small))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="introduction-to-neural-networks">Introduction to Neural networks </h2>
 
 <p>Artificial neural networks are computational systems that can learn to
 perform tasks by considering examples, generally without being
@@ -478,7 +2505,7 @@ as to not restrict the range of output values.
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model">Illustration of a single perceptropn model and a multi-perceptron model </h2>
+<h2 id="illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model">Illustration of a single perceptron model and a multi-perceptron model </h2>
 
 <center>  <!-- FIGURE -->
 <hr class="figure">
@@ -685,7 +2712,7 @@ of the outputs of <em>all</em> neurons in the previous layer.
 
 $$
 \begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
-\label{_auto1}
+\label{_auto6}
 \end{equation}
 $$
 
@@ -730,7 +2757,7 @@ is obtained.
 $$
 \begin{align}
  y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
-\label{_auto2}\\
+\label{_auto7}\\
  &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
 \label{outputLayer2}
 \end{align}
@@ -741,10 +2768,10 @@ $$
 $$
 \begin{align}
  y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
-\label{_auto3}\\
+\label{_auto8}\\
  &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
   + b_1^3\right]
-\label{_auto4}
+\label{_auto9}
 \end{align}
 $$
 
@@ -784,7 +2811,7 @@ sum of scaled activation functions of the form
 $$
 \begin{equation}
  f(x) = c_1 f(c_2 x + c_3) + c_4
-\label{_auto5}
+\label{_auto10}
 \end{equation}
 $$
 
@@ -826,7 +2853,7 @@ $$
            b^2_2 \\
            b^2_3 \\
           \end{array}\right]\right).
-\label{_auto6}
+\label{_auto11}
 \end{equation}
 $$
 
@@ -840,7 +2867,7 @@ $$
 \begin{equation}
  y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
  f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\label{_auto7}
+\label{_auto12}
 \end{equation}
 $$
 
@@ -994,452 +3021,9 @@ plt.show()
 </div>
 
 
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="the-multilayer-perceptron-mlp">The multilayer  perceptron (MLP) </h2>
-
-<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
-<ol>
-<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
-<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
-<li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
-</ol>
-<p>As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
-units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-</p>
-
-<p>For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
-</p>
-
-<p>The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
-</p>
-
-<p>The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation \( \hat{z} \), we can think of this as a basis
-expansion of the original inputs \( \hat{x} \). The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in  an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="from-one-to-many-layers-the-universal-approximation-theorem">From one to many layers, the universal approximation theorem </h2>
-
-<p>A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
-</p>
-
-<p>As stated earlier, 
-an important theorem in studies of neural networks, restated without
-proof here, is the <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_blank">universal approximation
-theorem</a>.
-</p>
-
-<p>It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">Deriving the back propagation code for a multilayer perceptron model </h2>
-
-<p>As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_blank">back propagation algorithm</a>.
-</p>
-
-<p>The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
-</p>
-
-<p>To derive these equations let us start with a plain regression problem
-and define our cost function as
-</p>
-
-$$
-{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
-$$
-
-<p>where the $t_i$s are our \( n \) targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs \( \hat{x} \) are given by \( y_i \).  Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with \( K \)
-classes.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="definitions">Definitions </h2>
-
-<p>With our definition of the targets \( \hat{t} \), the outputs of the
-network \( \hat{y} \) and the inputs \( \hat{x} \) we
-define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
-\( l \)-th layer as a function of the bias, the weights which add up from
-the previous layer \( l-1 \) and the forward passes/outputs
-\( \hat{a}^{l-1} \) from the previous layer as
-</p>
-
-$$
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
-$$
-
-<p>where \( b_k^l \) are the biases from layer \( l \).  Here \( M_{l-1} \)
-represents the total number of nodes/neurons/units of layer \( l-1 \). The
-figure here illustrates this equation.  We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-</p>
-
-$$
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-$$
-
-<p>With the activation values \( \hat{z}^l \) we can in turn define the
-output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
-and their nodes.  It means we have
-</p>
-
-$$
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="derivatives-and-the-chain-rule">Derivatives and the chain rule </h2>
-
-<p>From the definition of the activation \( z_j^l \) we have</p>
-$$
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-$$
-
-<p>and</p>
-$$
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
-$$
-
-<p>With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))</p>
-$$
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="derivative-of-the-cost-function">Derivative of the cost function </h2>
-
-<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
-
-<p>Let us specialize to the output layer \( l=L \). Our cost function is</p>
-$$
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-$$
-
-<p>The derivative of this function with respect to the weights is</p>
-
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
-$$
-
-<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
-$$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="bringing-it-together-first-back-propagation-equation">Bringing it together, first back propagation equation </h2>
-
-<p>We have thus</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
-$$
-
-<p>Defining</p>
-$$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-$$
-
-<p>and using the Hadamard product of two vectors we can write this as</p>
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-
-<p>This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
-</p>
-
-<p>Notice that everything in the above equations is easily computed.  In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \).  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-</p>
-
-$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-$$
-
-<p>With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="derivatives-in-terms-of-z-j-l">Derivatives in terms of \( z_j^L \) </h2>
-
-<p>It is also easy to see that our previous equation can be written as</p>
-
-$$
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
-$$
-
-<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely</p>
-$$
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
-$$
-
-<p>That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. </p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="bringing-it-together">Bringing it together </h2>
-
-<p>We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b>The starting equations</b>
-<p>
-
-$$
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
-\label{_auto8}
-\end{equation}
-$$
-
-<p>and</p>
-$$
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\label{_auto9}
-\end{equation}
-$$
-
-<p>and</p>
-
-$$
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\label{_auto10}
-\end{equation}
-$$
-</div>
-
-
-<p>An interesting consequence of the above equations is that when the
-activation \( a_k^{L-1} \) is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
-</p>
-
-<p>Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
-</p>
-
-<p>We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one \( L-1 \) in terms of the errors in the final output layer.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="final-back-propagating-equation">Final back propagating equation </h2>
-
-<p>We have that (replacing \( L \) with a general layer \( l \))</p>
-$$
-\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
-$$
-
-<p>We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have</p>
-
-$$
-\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
-$$
-
-<p>and recalling that</p>
-$$
-z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
-$$
-
-<p>with \( M_l \) being the number of nodes in layer \( l \), we obtain</p>
-$$
-\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
-$$
-
-<p>This is our final equation.</p>
-
-<p>We are now ready to set up the algorithm for back propagation and learning the weights and biases.</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
-
-<p>The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>First, we set up the input data \( \hat{x} \) and the activations
-\( \hat{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \hat{a}^1 \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \hat{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \hat{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all</p>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-</div>
-
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
-
-<p>The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>First, we set up the input data \( \boldsymbol{x} \) and the activations
-\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \boldsymbol{a}^1 \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \boldsymbol{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all</p>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-</div>
-
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-
 <!-- ------------------- end of main content --------------- -->
 <center style="font-size:80%">
-<!-- copyright --> &copy; 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+<!-- copyright --> &copy; 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 </center>
 </body>
 </html>
diff --git a/doc/pub/week40/html/week40.html b/doc/pub/week40/html/week40.html
index 75d686ec1..cb0187b6a 100644
--- a/doc/pub/week40/html/week40.html
+++ b/doc/pub/week40/html/week40.html
@@ -8,8 +8,8 @@ doconce format html week40.do.txt --pygments_html_style=default --html_style=blo
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
 <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-<meta name="description" content="Week 40: Neural networks">
-<title>Week 40: Neural networks</title>
+<meta name="description" content="Week 40: Gradient descent methods (continued) and start Neural networks">
+<title>Week 40: Gradient descent methods (continued) and start Neural networks</title>
 <style type="text/css">
 /* bloodish style */
 body {
@@ -140,21 +140,127 @@ div.toc p,a {
 
 <!-- tocinfo
 {'highest level': 2,
- 'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
+ 'sections': [('Plans for week 40', 2, None, 'plans-for-week-40'),
+              ('Summary from last week, using gradient descent methods, '
+               'limitations',
+               2,
+               None,
+               'summary-from-last-week-using-gradient-descent-methods-limitations'),
               ('Overview video on Stochastic Gradient Descent',
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
+              ('Stochastic Gradient Descent',
+               2,
+               None,
+               'stochastic-gradient-descent'),
+              ('Computation of gradients', 2, None, 'computation-of-gradients'),
+              ('SGD example', 2, None, 'sgd-example'),
+              ('The gradient step', 2, None, 'the-gradient-step'),
+              ('Simple example code', 2, None, 'simple-example-code'),
+              ('When do we stop?', 2, None, 'when-do-we-stop'),
+              ('Slightly different approach',
+               2,
+               None,
+               'slightly-different-approach'),
+              ('Time decay rate', 2, None, 'time-decay-rate'),
+              ('Code with a Number of Minibatches which varies',
+               2,
+               None,
+               'code-with-a-number-of-minibatches-which-varies'),
+              ('Replace or not', 2, None, 'replace-or-not'),
+              ('Momentum based GD', 2, None, 'momentum-based-gd'),
+              ('More on momentum based approaches',
+               2,
+               None,
+               'more-on-momentum-based-approaches'),
+              ('Momentum parameter', 2, None, 'momentum-parameter'),
+              ('Second moment of the gradient',
+               2,
+               None,
+               'second-moment-of-the-gradient'),
+              ('RMS prop', 2, None, 'rms-prop'),
+              ('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
+               2,
+               None,
+               'adam-optimizer-https-arxiv-org-abs-1412-6980'),
+              ('Algorithms and codes for Adagrad, RMSprop and Adam',
+               2,
+               None,
+               'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
               ('Practical tips', 2, None, 'practical-tips'),
               ('Automatic differentiation',
                2,
                None,
                'automatic-differentiation'),
-              ('Videos on Neural Networks',
+              ('Using autograd', 2, None, 'using-autograd'),
+              ('Autograd with more complicated functions',
                2,
                None,
-               'videos-on-neural-networks'),
-              ('Neural networks', 2, None, 'neural-networks'),
+               'autograd-with-more-complicated-functions'),
+              ('More complicated functions using the elements of their '
+               'arguments directly',
+               2,
+               None,
+               'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
+              ('Functions using mathematical functions from Numpy',
+               2,
+               None,
+               'functions-using-mathematical-functions-from-numpy'),
+              ('More autograd', 2, None, 'more-autograd'),
+              ('And  with loops', 2, None, 'and-with-loops'),
+              ('Using recursion', 2, None, 'using-recursion'),
+              ('Unsupported functions', 2, None, 'unsupported-functions'),
+              ('The syntax a.dot(b) when finding the dot product',
+               2,
+               None,
+               'the-syntax-a-dot-b-when-finding-the-dot-product'),
+              ('Recommended to avoid', 2, None, 'recommended-to-avoid'),
+              ('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ("But noen of these can compete with Newton's method",
+               2,
+               None,
+               'but-noen-of-these-can-compete-with-newton-s-method'),
+              ('Including Stochastic Gradient Descent with Autograd',
+               2,
+               None,
+               'including-stochastic-gradient-descent-with-autograd'),
+              ('Same code but now with momentum gradient descent',
+               2,
+               None,
+               'same-code-but-now-with-momentum-gradient-descent'),
+              ('Similar (second order function now) problem but now with '
+               'AdaGrad',
+               2,
+               None,
+               'similar-second-order-function-now-problem-but-now-with-adagrad'),
+              ('RMSprop for adaptive learning rate with Stochastic Gradient '
+               'Descent',
+               2,
+               None,
+               'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
+              ('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
+               2,
+               None,
+               'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
+              ('And Logistic Regression', 2, None, 'and-logistic-regression'),
+              ('Introducing "JAX":"https://jax.readthedocs.io/en/latest/"',
+               2,
+               None,
+               'introducing-jax-https-jax-readthedocs-io-en-latest'),
+              ('Introduction to Neural networks',
+               2,
+               None,
+               'introduction-to-neural-networks'),
               ('Artificial neurons', 2, None, 'artificial-neurons'),
               ('Neural network types', 2, None, 'neural-network-types'),
               ('Feed-forward neural networks',
@@ -175,11 +281,11 @@ div.toc p,a {
                2,
                None,
                'why-multilayer-perceptrons'),
-              ('Illustration of a single perceptropn model and a '
+              ('Illustration of a single perceptron model and a '
                'multi-perceptron model',
                2,
                None,
-               'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+               'illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model'),
               ('Examples of XOR, OR and AND gates',
                2,
                None,
@@ -204,50 +310,7 @@ div.toc p,a {
                3,
                None,
                'activation-functions-logistic-and-hyperbolic-ones'),
-              ('Relevance', 3, None, 'relevance'),
-              ('The multilayer  perceptron (MLP)',
-               2,
-               None,
-               'the-multilayer-perceptron-mlp'),
-              ('From one to many layers, the universal approximation theorem',
-               2,
-               None,
-               'from-one-to-many-layers-the-universal-approximation-theorem'),
-              ('Deriving the back propagation code for a multilayer perceptron '
-               'model',
-               2,
-               None,
-               'deriving-the-back-propagation-code-for-a-multilayer-perceptron-model'),
-              ('Definitions', 2, None, 'definitions'),
-              ('Derivatives and the chain rule',
-               2,
-               None,
-               'derivatives-and-the-chain-rule'),
-              ('Derivative of the cost function',
-               2,
-               None,
-               'derivative-of-the-cost-function'),
-              ('Bringing it together, first back propagation equation',
-               2,
-               None,
-               'bringing-it-together-first-back-propagation-equation'),
-              ('Derivatives in terms of $z_j^L$',
-               2,
-               None,
-               'derivatives-in-terms-of-z-j-l'),
-              ('Bringing it together', 2, None, 'bringing-it-together'),
-              ('Final back propagating equation',
-               2,
-               None,
-               'final-back-propagating-equation'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm'),
-              ('Setting up the Back propagation algorithm',
-               2,
-               None,
-               'setting-up-the-back-propagation-algorithm')]}
+              ('Relevance', 3, None, 'relevance')]}
 end of tocinfo -->
 
 <body>
@@ -269,7 +332,7 @@ MathJax.Hub.Config({
 
 <!-- ------------------- main content ---------------------- -->
 <center>
-<h1>Week 40: Neural networks</h1>
+<h1>Week 40: Gradient descent methods (continued) and start Neural networks</h1>
 </center>  <!-- document title -->
 
 <!-- author(s): Morten Hjorth-Jensen -->
@@ -285,33 +348,649 @@ MathJax.Hub.Config({
 </center>
 <br>
 <center>
-<h4>Oct 8, 2022</h4>
+<h4>October 2-6, 2023</h4>
 </center> <!-- date -->
 <br>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="plan-for-week-40">Plan for week 40 </h2>
+<h2 id="plans-for-week-40">Plans for week 40 </h2>
+
+<div class="alert alert-block alert-block alert-text-normal">
+<b>Material for the active learning sessions on Tuesday and Wednesday</b>
+<p>
+<ul>
+  <li> Work on project 1 and discussions on how to structure your report</li>
+  <li> No weekly exercises for week 40, project work only</li>
+  <li> <a href="https://youtu.be/tVW1ZDmZnwM" target="_blank">Video on how to write scientific reports recorded during one of the lab sessions</a></li>
+  <li> A general guideline can be found at <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md" target="_blank"><tt>https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</tt></a>.</li>
+</ul>
+</div>
+  
+
+<div class="alert alert-block alert-block alert-text-normal">
+<b>Material for the lecture on Thursday October 5, 2023</b>
+<p>
+<ul>
+  <li> Stochastic Gradient descent with examples and automatic differentiation</li>
+  <li> Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.</li>  
+  <li> Readings and Videos:</li>
+<ul>
+    <li> These lecture notes</li>
+    <li> For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.</li>
+    <li> <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 4 on stochastic gradient descent</a></li>
+    <li> For neural networks we recommend Goodfellow et al chapter 6.</li>
+    <li> <a href="https://www.youtube.com/watch?v=sDv4f4s2SB8" target="_blank">Video on gradient descent</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI" target="_blank">Video on stochastic gradient descent</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a></li>
+    <li> <a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a></li>
+</ul>
+</ul>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="summary-from-last-week-using-gradient-descent-methods-limitations">Summary from last week, using gradient descent methods, limitations </h2>
 
 <ul>
-<li> Thursday: Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.</li>
-<ul>
-  <li> <a href="https://youtu.be/sCHiGSoG2lE" target="_blank">Video of lecture</a></li>
+<li> <b>Gradient descent (GD) finds local minima of our function</b>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</li>
+<li> <b>GD is sensitive to initial conditions</b>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</li>
+<li> <b>Gradients are computationally expensive to calculate for large datasets</b>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called &quot;mini batches&quot;. This has the added benefit of introducing stochasticity into our algorithm.</li>
+<li> <b>GD is very sensitive to choices of learning rates</b>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</li>
+<li> <b>GD treats all directions in parameter space uniformly.</b> Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</li> 
+<li> GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</li>
 </ul>
-<li> Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.</li>
-<ul>
-  <li> <a href="https://youtu.be/hDTtA7PRRfI" target="_blank">Video of lecture</a></li>
-</ul>
-</ul>
-<p>Reading suggestions for both days: <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 10</a> and Hastie et al chapter 11.
-For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
-For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
-</p>
-
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="overview-video-on-stochastic-gradient-descent">Overview video on Stochastic Gradient Descent </h2>
 
 <a href="https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" target="_blank">What is Stochastic Gradient Descent</a>
 
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="batches-and-mini-batches">Batches and mini-batches </h2>
+
+<p>In gradient descent we compute the cost function and its gradient for all data points we have.</p>
+
+<p>In large-scale applications such as the <a href="https://www.image-net.org/challenges/LSVRC/" target="_blank">ILSVRC challenge</a>, the
+training data can have on order of millions of examples. Hence, it
+seems wasteful to compute the full cost function over the entire
+training set in order to perform only a single parameter update. A
+very common approach to addressing this challenge is to compute the
+gradient over batches of the training data. For example, a typical batch could contain some thousand  examples from
+an  entire training set of several millions. This batch is then used to
+perform a parameter update.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="stochastic-gradient-descent-sgd">Stochastic Gradient Descent (SGD) </h2>
+
+<p>In stochastic gradient descent, the extreme case is the case where we
+have only one batch, that is we include the whole data set.
+</p>
+
+<p>This process is called Stochastic Gradient
+Descent (SGD) (or also sometimes on-line gradient descent). This is
+relatively less common to see because in practice due to vectorized
+code optimizations it can be computationally much more efficient to
+evaluate the gradient for 100 examples, than the gradient for one
+example 100 times. Even though SGD technically refers to using a
+single example at a time to evaluate the gradient, you will hear
+people use the term SGD even when referring to mini-batch gradient
+descent (i.e. mentions of MGD for &#8220;Minibatch Gradient Descent&#8221;, or BGD
+for &#8220;Batch gradient descent&#8221; are rare to see), where it is usually
+assumed that mini-batches are used. The size of the mini-batch is a
+hyperparameter but it is not very common to cross-validate or bootstrap it. It is
+usually based on memory constraints (if any), or set to some value,
+e.g. 32, 64 or 128. We use powers of 2 in practice because many
+vectorized operation implementations work faster when their inputs are
+sized in powers of 2.
+</p>
+
+<p>In our notes with  SGD we mean stochastic gradient descent with mini-batches.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="stochastic-gradient-descent">Stochastic Gradient Descent </h2>
+
+<p>Stochastic gradient descent (SGD) and variants thereof address some of
+the shortcomings of the Gradient descent method discussed above.
+</p>
+
+<p>The underlying idea of SGD comes from the observation that the cost
+function, which we want to minimize, can almost always be written as a
+sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \),
+</p>
+$$
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}). 
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="computation-of-gradients">Computation of gradients </h2>
+
+<p>This in turn means that the gradient can be
+computed as a sum over \( i \)-gradients 
+</p>
+$$
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
+$$
+
+<p>Stochasticity/randomness is introduced by only taking the
+gradient on a subset of the data called minibatches.  If there are \( n \)
+data points and the size of each minibatch is \( M \), there will be \( n/M \)
+minibatches. We denote these minibatches by \( B_k \) where
+\( k=1,\cdots,n/M \).
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="sgd-example">SGD example </h2>
+<p>As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
+and we choose to have \( M=5 \) minibathces,
+then each minibatch contains two data points. In particular we have
+\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
+(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you
+have only a single batch with all data points and on the other extreme,
+you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e
+\( B_k = \mathbf{x}_k \).
+</p>
+
+<p>The idea is now to approximate the gradient by replacing the sum over
+all data points with a sum over the data points in one the minibatches
+picked at random in each gradient descent step 
+</p>
+$$
+\nabla_{\beta}
+C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
+c_i(\mathbf{x}_i, \mathbf{\beta}).
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="the-gradient-step">The gradient step </h2>
+
+<p>Thus a gradient descent step now looks like </p>
+$$
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
+$$
+
+<p>where \( k \) is picked at random with equal
+probability from \( [1,n/M] \). An iteration over the number of
+minibathces (n/M) is commonly referred to as an epoch. Thus it is
+typical to choose a number of epochs and for each epoch iterate over
+the number of minibatches, as exemplified in the code below.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="simple-example-code">Simple example code </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span> 
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">10</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
+
+j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
+        <span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #408080; font-style: italic">#Compute new suggestion for </span>
+        j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Taking the gradient only on a subset of the data has two important
+benefits. First, it introduces randomness which decreases the chance
+that our opmization scheme gets stuck in a local minima. Second, if
+the size of the minibatches are small relative to the number of
+datapoints (\( M <  n \)), the computation of the gradient is much
+cheaper since we sum over the datapoints in the \( k-th \) minibatch and not
+all \( n \) datapoints.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="when-do-we-stop">When do we stop? </h2>
+
+<p>A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the \( \beta \) that
+gave the lowest value.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="slightly-different-approach">Slightly different approach </h2>
+
+<p>Another approach is to let the step length \( \gamma_j \) depend on the
+number of epochs in such a way that it becomes very small after a
+reasonable time such that we do not move at all. Such approaches are
+also called scaling. There are many such ways to <a href="https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1" target="_blank">scale the learning
+rate</a>
+and <a href="https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf" target="_blank">discussions here</a>. See
+also
+<a href="https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1" target="_blank"><tt>https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1</tt></a>
+for a discussion of different scaling functions for the learning rate.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="time-decay-rate">Time decay rate </h2>
+
+<p>As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in <em>time</em> \( t \).</p>
+
+<p>In this way we can fix the number of epochs, compute \( \beta \) and
+evaluate the cost function at the end. Repeating the computation will
+give a different result since the scheme is random by design. Then we
+pick the final \( \beta \) that gives the lowest value of the cost
+function.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span> 
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">step_length</span>(t,t0,t1):
+    <span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">500</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
+t0 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+t1 <span style="color: #666666">=</span> <span style="color: #666666">10</span>
+
+gamma_j <span style="color: #666666">=</span> t0<span style="color: #666666">/</span>t1
+j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
+        <span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
+        <span style="color: #408080; font-style: italic">#Compute new suggestion for beta</span>
+        t <span style="color: #666666">=</span> epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i
+        gamma_j <span style="color: #666666">=</span> step_length(t,t0,t1)
+        j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;gamma_j after </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> epochs: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (n_epochs,gamma_j))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="code-with-a-number-of-minibatches-which-varies">Code with a Number of Minibatches which varies </h2>
+
+<p>In the code here we vary the number of mini-batches.</p>
+
+<!-- code=text (!bc pycode) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"># Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
+print(&quot;Own inversion&quot;)
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f&quot;Eigenvalues of Hessian Matrix:{EigValues}&quot;)
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+
+for iter in range(Niterations):
+    gradients = 2.0/n*X.T @ ((X @ theta)-y)
+    theta -= eta*gradients
+print(&quot;theta from own gd&quot;)
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print(&quot;theta from own sdg&quot;)
+print(theta)
+
+plt.plot(xnew, ypredict, &quot;r-&quot;)
+plt.plot(xnew, ypredict2, &quot;b-&quot;)
+plt.plot(x, y ,&#39;ro&#39;)
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r&#39;$x$&#39;)
+plt.ylabel(r&#39;$y$&#39;)
+plt.title(r&#39;Random numbers &#39;)
+plt.show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="replace-or-not">Replace or not </h2>
+
+<p>In the above code, we have use replacement in setting up the
+mini-batches. The discussion
+<a href="https://sebastianraschka.com/faq/docs/sgd-methods.html" target="_blank">here</a> may be
+useful.  
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="momentum-based-gd">Momentum based GD </h2>
+
+<p>The stochastic gradient descent (SGD) is almost always used with a
+<em>momentum</em> or inertia term that serves as a memory of the direction we
+are moving in parameter space.  This is typically implemented as
+follows
+</p>
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\label{_auto1}
+\end{align}
+$$
+
+<p>where we have introduced a momentum parameter \( \gamma \), with
+\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to
+indicate the gradient is to be taken over a different mini-batch at
+each step. We call this algorithm gradient descent with momentum
+(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a
+running average of recently encountered gradients and
+\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory
+used in the averaging procedure. Consistent with this, when
+\( \gamma=0 \), this just reduces down to ordinary SGD as discussed
+earlier. An equivalent way of writing the updates is
+</p>
+
+$$
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+$$
+
+<p>where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="more-on-momentum-based-approaches">More on momentum based approaches </h2>
+
+<p>Let us try to get more intuition from these equations. It is helpful
+to consider a simple physical analogy with a particle of mass \( m \)
+moving in a viscous medium with drag coefficient \( \mu \) and potential
+\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \),
+then its motion is described by
+</p>
+
+$$
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+$$
+
+<p>We can discretize this equation in the usual way to get</p>
+
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
+
+<p>Rearranging this equation, we can rewrite this as</p>
+
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="momentum-parameter">Momentum parameter </h2>
+
+<p>Notice that this equation is identical to previous one if we identify
+the position of the particle, \( \mathbf{w} \), with the parameters
+\( \boldsymbol{\theta} \). This allows us to identify the momentum
+parameter and learning rate with the mass of the particle and the
+viscous drag as:
+</p>
+
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
+
+<p>Thus, as the name suggests, the momentum parameter is proportional to
+the mass of the particle and effectively provides inertia.
+Furthermore, in the large viscosity/small learning rate limit, our
+memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+</p>
+
+<p>Why is momentum useful? SGD momentum helps the gradient descent
+algorithm gain speed in directions with persistent but small gradients
+even in the presence of stochasticity, while suppressing oscillations
+in high-curvature directions. This becomes especially important in
+situations where the landscape is shallow and flat in some directions
+and narrow and steep in others. It has been argued that first-order
+methods (with appropriate initial conditions) can perform comparable
+to more expensive second order methods, especially in the context of
+complex deep learning models.
+</p>
+
+<p>These beneficial properties of momentum can sometimes become even more
+pronounced by using a slight modification of the classical momentum
+algorithm called Nesterov Accelerated Gradient (NAG).
+</p>
+
+<p>In the NAG algorithm, rather than calculating the gradient at the
+current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one
+calculates the gradient at the expected value of the parameters given
+our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma
+\mathbf{v}_{t-1}) \). This yields the NAG update rule
+</p>
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\label{_auto2}
+\end{align}
+$$
+
+<p>One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="second-moment-of-the-gradient">Second moment of the gradient </h2>
+
+<p>In stochastic gradient descent, with and without momentum, we still
+have to specify a schedule for tuning the learning rates \( \eta_t \)
+as a function of time.  As discussed in the context of Newton's
+method, this presents a number of dilemmas. The learning rate is
+limited by the steepest direction which can change depending on the
+current position in the landscape. To circumvent this problem, ideally
+our algorithm would keep track of curvature and take large steps in
+shallow, flat directions and small steps in steep, narrow directions.
+Second-order methods accomplish this by calculating or approximating
+the Hessian and normalizing the learning rate by the
+curvature. However, this is very computationally expensive for
+extremely large models. Ideally, we would like to be able to
+adaptively change the step size to match the landscape without paying
+the steep computational price of calculating or approximating
+Hessians.
+</p>
+
+<p>Recently, a number of methods have been introduced that accomplish
+this by tracking not only the gradient, but also the second moment of
+the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
+<a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM</a>.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="rms-prop">RMS prop </h2>
+
+<p>In RMS prop, in addition to keeping a running average of the first
+moment of the gradient, we also keep track of the second moment
+denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule
+for RMS prop is given by
+</p>
+
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\label{_auto3}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+$$
+
+<p>where \( \beta \) controls the averaging time of the second moment and is
+typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate
+typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8}  \) is a
+small regularization constant to prevent divergences. Multiplication
+and division by vectors is understood as an element-wise operation. It
+is clear from this formula that the learning rate is reduced in
+directions where the norm of the gradient is consistently large. This
+greatly speeds up the convergence by allowing us to use a larger
+learning rate for flat directions.
+</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="adam-optimizer-https-arxiv-org-abs-1412-6980"><a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM optimizer</a> </h2>
+
+<p>A related algorithm is the ADAM optimizer. In
+<a href="https://arxiv.org/abs/1412.6980" target="_blank">ADAM</a>, we keep a running average of
+both the first and second moment of the gradient and use this
+information to adaptively change the learning rate for different
+parameters.  The method isefficient when working with large
+problems involving lots data and/or parameters.  It is a combination of the
+gradient descent with momentum algorithm and the RMSprop algorithm
+discussed above.
+</p>
+
+<p>In addition to keeping a running average of the first and
+second moments of the gradient
+(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and
+\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM
+performs an additional bias correction to account for the fact that we
+are estimating the first two moments of the gradient using a running
+average (denoted by the hats in the update rule below). The update
+rule for ADAM is given by (where multiplication and division are once
+again understood to be element-wise operations below)
+</p>
+
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\label{_auto4}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\label{_auto5}
+\end{align}
+$$
+
+<p>where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and
+second moment and are typically taken to be \( 0.9 \) and \( 0.99 \)
+respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.
+</p>
+
+<p>Like in RMSprop, the effective step size of a parameter depends on the
+magnitude of its gradient squared.  To understand this better, let us
+rewrite this expression in terms of the variance
+\( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t -
+(\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The
+update rule for this parameter is given by
+</p>
+
+$$
+\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 +  m_t^2 }+\epsilon}.
+$$
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="algorithms-and-codes-for-adagrad-rmsprop-and-adam">Algorithms and codes for Adagrad, RMSprop and Adam </h2>
+
+<p>The algorithms we have implemented are well described in the text by <a href="https://www.deeplearningbook.org/contents/optimization.html" target="_blank">Goodfellow, Bengio and Courville, chapter 8</a>.</p>
+
+<p>The codes which implement these algorithms are discussed after our presentation of automatic differentiation.</p>
+
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="practical-tips">Practical tips </h2>
 
@@ -353,15 +1032,1363 @@ while numerical differentiation can introduce round-off errors in the
 discretization process and cancellation
 </p>
 
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="videos-on-neural-networks">Videos on Neural Networks </h2>
+<p>Python has tools for so-called <b>automatic differentiation</b>.
+Consider the following example
+</p>
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
 
-<a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a>
+<p>which has the following derivative</p>
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) 
+$$
+
+<p>Using <b>autograd</b> we have</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+
+<span style="color: #408080; font-style: italic"># To do elementwise differentiation:</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> elementwise_grad <span style="color: #008000; font-weight: bold">as</span> egrad 
+
+<span style="color: #408080; font-style: italic"># To plot:</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span> 
+
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sin(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f_grad_analytic</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi <span style="color: #666666">+</span> <span style="color: #666666">2*</span>x)
+
+<span style="color: #408080; font-style: italic"># Do the comparison:</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">1</span>,<span style="color: #666666">1000</span>)
+
+f_grad <span style="color: #666666">=</span> egrad(f)
+
+computed <span style="color: #666666">=</span> f_grad(x)
+analytic <span style="color: #666666">=</span> f_grad_analytic(x)
+
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&#39;Derivative computed from Autograd compared with the analytical derivative&#39;</span>)
+plt<span style="color: #666666">.</span>plot(x,computed,label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;autograd&#39;</span>)
+plt<span style="color: #666666">.</span>plot(x,analytic,label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;analytic&#39;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;x&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;y&#39;</span>)
+plt<span style="color: #666666">.</span>legend()
+
+plt<span style="color: #666666">.</span>show()
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The max absolute difference is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(computed <span style="color: #666666">-</span> analytic))))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split  -->
+<h2 id="using-autograd">Using autograd </h2>
+
+<p>Here we
+experiment with what kind of functions Autograd is capable
+of finding the gradient of. The following Python functions are just
+meant to illustrate what Autograd can do, but please feel free to
+experiment with other, possibly more complicated, functions as well.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f1</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">**3</span> <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+
+f1_grad <span style="color: #666666">=</span> grad(f1)
+
+<span style="color: #408080; font-style: italic"># Remember to send in float as argument to the computed gradient from Autograd!</span>
+a <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+
+<span style="color: #408080; font-style: italic"># See the evaluated gradient at a using autograd:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The gradient of f1 evaluated at a = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> using autograd is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(a,f1_grad(a)))
+
+<span style="color: #408080; font-style: italic"># Compare with the analytical derivative, that is f1&#39;(x) = 3*x**2 </span>
+grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">3*</span>a<span style="color: #666666">**2</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The gradient of f1 evaluated at a = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> by finding the analytic expression is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(a,grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
 
-<a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="neural-networks">Neural networks </h2>
+<h2 id="autograd-with-more-complicated-functions">Autograd with more complicated functions </h2>
+
+<p>To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f2</span>(x1,x2):
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">3*</span>x1<span style="color: #666666">**3</span> <span style="color: #666666">+</span> x2<span style="color: #666666">*</span>(x1 <span style="color: #666666">-</span> <span style="color: #666666">5</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+
+<span style="color: #408080; font-style: italic"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
+f2_grad_x1 <span style="color: #666666">=</span> grad(f2,<span style="color: #666666">0</span>)
+
+<span style="color: #408080; font-style: italic"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
+f2_grad_x2 <span style="color: #666666">=</span> grad(f2,<span style="color: #666666">1</span>)
+
+x1 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+x2 <span style="color: #666666">=</span> <span style="color: #666666">3.0</span> 
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Evaluating at x1 = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">, x2 = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x1,x2))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;-&quot;</span><span style="color: #666666">*30</span>)
+
+<span style="color: #408080; font-style: italic"># Compare with the analytical derivatives:</span>
+
+<span style="color: #408080; font-style: italic"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
+f2_grad_x1_analytical <span style="color: #666666">=</span> <span style="color: #666666">9*</span>x1<span style="color: #666666">**2</span> <span style="color: #666666">+</span> x2
+
+<span style="color: #408080; font-style: italic"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
+f2_grad_x2_analytical <span style="color: #666666">=</span> x1 <span style="color: #666666">-</span> <span style="color: #666666">5</span>
+
+<span style="color: #408080; font-style: italic"># See the evaluated derivations:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f2 w.r.t x1: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x1(x1,x2) ))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f2 w.r.t x1: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x1(x1,x2) ))
+
+<span style="color: #008000">print</span>()
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f2 w.r.t x2: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x2(x1,x2) ))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f2 w.r.t x2: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>( f2_grad_x2(x1,x2) ))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="more-complicated-functions-using-the-elements-of-their-arguments-directly">More complicated functions using the elements of their arguments directly </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f3</span>(x): <span style="color: #408080; font-style: italic"># Assumes x is an array of length 5 or higher</span>
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">2*</span>x[<span style="color: #666666">0</span>] <span style="color: #666666">+</span> <span style="color: #666666">3*</span>x[<span style="color: #666666">1</span>] <span style="color: #666666">+</span> <span style="color: #666666">5*</span>x[<span style="color: #666666">2</span>] <span style="color: #666666">+</span> <span style="color: #666666">7*</span>x[<span style="color: #666666">3</span>] <span style="color: #666666">+</span> <span style="color: #666666">11*</span>x[<span style="color: #666666">4</span>]<span style="color: #666666">**2</span>
+
+f3_grad <span style="color: #666666">=</span> grad(f3)
+
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">4</span>,<span style="color: #666666">5</span>)
+
+<span style="color: #408080; font-style: italic"># Print the computed gradient:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed gradient of f3 is: &quot;</span>, f3_grad(x))
+
+<span style="color: #408080; font-style: italic"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
+f3_grad_analytical <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">2</span>, <span style="color: #666666">3</span>, <span style="color: #666666">5</span>, <span style="color: #666666">7</span>, <span style="color: #666666">22*</span>x[<span style="color: #666666">4</span>]])
+
+<span style="color: #408080; font-style: italic"># Print the analytical gradient:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical gradient of f3 is: &quot;</span>, f3_grad_analytical)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+</p>
+
+<!-- !split  -->
+<h2 id="functions-using-mathematical-functions-from-numpy">Functions using mathematical functions from Numpy </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f4</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sqrt(<span style="color: #666666">1+</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>exp(x) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>sin(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)
+
+f4_grad <span style="color: #666666">=</span> grad(f4)
+
+x <span style="color: #666666">=</span> <span style="color: #666666">2.7</span>
+
+<span style="color: #408080; font-style: italic"># Print the computed derivative:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f4 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f4_grad(x)))
+
+<span style="color: #408080; font-style: italic"># The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi</span>
+f4_grad_analytical <span style="color: #666666">=</span> x<span style="color: #666666">/</span>np<span style="color: #666666">.</span>sqrt(<span style="color: #666666">1</span> <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>exp(x) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)<span style="color: #666666">*2*</span>np<span style="color: #666666">.</span>pi
+
+<span style="color: #408080; font-style: italic"># Print the analytical gradient:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical gradient of f4 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f4_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="more-autograd">More autograd </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f5</span>(x):
+    <span style="color: #008000; font-weight: bold">if</span> x <span style="color: #666666">&gt;=</span> <span style="color: #666666">0</span>:
+        <span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">**2</span>
+    <span style="color: #008000; font-weight: bold">else</span>:
+        <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">-3*</span>x <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+
+f5_grad <span style="color: #666666">=</span> grad(f5)
+
+x <span style="color: #666666">=</span> <span style="color: #666666">2.7</span>
+
+<span style="color: #408080; font-style: italic"># Print the computed derivative:</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f5 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f5_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="and-with-loops">And  with loops </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f6_for</span>(x):
+    val <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">10</span>):
+        val <span style="color: #666666">=</span> val <span style="color: #666666">+</span> x<span style="color: #666666">**</span>i
+    <span style="color: #008000; font-weight: bold">return</span> val
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f6_while</span>(x):
+    val <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+    <span style="color: #008000; font-weight: bold">while</span> i <span style="color: #666666">&lt;</span> <span style="color: #666666">10</span>:
+        val <span style="color: #666666">=</span> val <span style="color: #666666">+</span> x<span style="color: #666666">**</span>i
+        i <span style="color: #666666">=</span> i <span style="color: #666666">+</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">return</span> val
+
+f6_for_grad <span style="color: #666666">=</span> grad(f6_for)
+f6_while_grad <span style="color: #666666">=</span> grad(f6_while)
+
+x <span style="color: #666666">=</span> <span style="color: #666666">0.5</span>
+
+<span style="color: #408080; font-style: italic"># Print the computed derivaties of f6_for and f6_while</span>
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f6_for at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f6_for_grad(x)))
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f6_while at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f6_while_grad(x)))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #408080; font-style: italic"># Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9</span>
+<span style="color: #408080; font-style: italic"># The analytical derivative is: sum(i*x**(i-1)) </span>
+f6_grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">10</span>):
+    f6_grad_analytical <span style="color: #666666">+=</span> i<span style="color: #666666">*</span>x<span style="color: #666666">**</span>(i<span style="color: #666666">-1</span>)
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f6 at x = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(x,f6_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="using-recursion">Using recursion </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f7</span>(n): <span style="color: #408080; font-style: italic"># Assume that n is an integer</span>
+    <span style="color: #008000; font-weight: bold">if</span> n <span style="color: #666666">==</span> <span style="color: #666666">1</span> <span style="color: #AA22FF; font-weight: bold">or</span> n <span style="color: #666666">==</span> <span style="color: #666666">0</span>:
+        <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">else</span>:
+        <span style="color: #008000; font-weight: bold">return</span> n<span style="color: #666666">*</span>f7(n<span style="color: #666666">-1</span>)
+
+f7_grad <span style="color: #666666">=</span> grad(f7)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">2.0</span>
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The computed derivative of f7 at n = </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(n,f7_grad(n)))
+
+<span style="color: #408080; font-style: italic"># The function f7 is an implementation of the factorial of n.</span>
+<span style="color: #408080; font-style: italic"># By using the product rule, one can find that the derivative is:</span>
+
+f7_grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">int</span>(n)<span style="color: #666666">-1</span>):
+    tmp <span style="color: #666666">=</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">for</span> k <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">int</span>(n)<span style="color: #666666">-1</span>):
+        <span style="color: #008000; font-weight: bold">if</span> k <span style="color: #666666">!=</span> i:
+            tmp <span style="color: #666666">*=</span> (n <span style="color: #666666">-</span> k)
+    f7_grad_analytical <span style="color: #666666">+=</span> tmp
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The analytical derivative of f7 at n = </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>(n,f7_grad_analytical))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="unsupported-functions">Unsupported functions </h2>
+<p>Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.</p>
+
+<p>Assigning a value to the variable being differentiated with respect to</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f8</span>(x): <span style="color: #408080; font-style: italic"># Assume x is an array</span>
+    x[<span style="color: #666666">2</span>] <span style="color: #666666">=</span> <span style="color: #666666">3</span>
+    <span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">*2</span>
+
+f8_grad <span style="color: #666666">=</span> grad(f8)
+
+x <span style="color: #666666">=</span> <span style="color: #666666">8.4</span>
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f8 is:&quot;</span>,f8_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="the-syntax-a-dot-b-when-finding-the-dot-product">The syntax a.dot(b) when finding the dot product </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f9</span>(a): <span style="color: #408080; font-style: italic"># Assume a is an array with 2 elements</span>
+    b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1.0</span>,<span style="color: #666666">2.0</span>])
+    <span style="color: #008000; font-weight: bold">return</span> a<span style="color: #666666">.</span>dot(b)
+
+f9_grad <span style="color: #666666">=</span> grad(f9)
+
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1.0</span>,<span style="color: #666666">0.0</span>])
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The derivative of f9 is:&quot;</span>,f9_grad(x))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f9_alternative</span>(x): <span style="color: #408080; font-style: italic"># Assume a is an array with 2 elements</span>
+    b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1.0</span>,<span style="color: #666666">2.0</span>])
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>dot(x,b) <span style="color: #408080; font-style: italic"># The same as x_1*b_1 + x_2*b_2</span>
+
+f9_alternative_grad <span style="color: #666666">=</span> grad(f9_alternative)
+
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">3.0</span>,<span style="color: #666666">0.0</span>])
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;The gradient of f9 is:&quot;</span>,f9_alternative_grad(x))
+
+<span style="color: #408080; font-style: italic"># The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively</span>
+<span style="color: #408080; font-style: italic"># w.r.t x is (b_1, b_2).</span>
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="recommended-to-avoid">Recommended to avoid </h2>
+<p>The documentation recommends to avoid inplace operations such as</p>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;">a <span style="color: #666666">+=</span> b
+a <span style="color: #666666">-=</span> b
+a<span style="color: #666666">*=</span> b
+a <span style="color: #666666">/=</span>b
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="using-autograd-with-ols">Using Autograd with OLS </h2>
+
+<p>We conclude the part on optmization by showing how we can make codes
+for linear regression and logistic regression using <b>autograd</b>. The
+first example shows results with ordinary leats squares.
+</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(beta):
+    <span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> beta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+<span style="color: #408080; font-style: italic"># define the gradient</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> training_gradient(theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
+Xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
+ypredict <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta)
+ypredict2 <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta_linreg)
+
+plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">&quot;b-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Random numbers &#39;</span>)
+plt<span style="color: #666666">.</span>show()
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="same-code-but-now-with-momentum-gradient-descent">Same code but now with momentum gradient descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients for OLS</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(beta):
+    <span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> beta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #408080; font-style: italic">#+np.random.randn(n,1)</span>
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">30</span>
+
+<span style="color: #408080; font-style: italic"># define the gradient</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> training_gradient(theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+    <span style="color: #008000">print</span>(<span style="color: #008000">iter</span>,gradients[<span style="color: #666666">0</span>],gradients[<span style="color: #666666">1</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+<span style="color: #408080; font-style: italic"># Now improve with momentum gradient descent</span>
+change <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+delta_momentum <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    <span style="color: #408080; font-style: italic"># calculate gradient</span>
+    gradients <span style="color: #666666">=</span> training_gradient(theta)
+    <span style="color: #408080; font-style: italic"># calculate update</span>
+    new_change <span style="color: #666666">=</span> eta<span style="color: #666666">*</span>gradients<span style="color: #666666">+</span>delta_momentum<span style="color: #666666">*</span>change
+    <span style="color: #408080; font-style: italic"># take a step</span>
+    theta <span style="color: #666666">-=</span> new_change
+    <span style="color: #408080; font-style: italic"># save the change</span>
+    change <span style="color: #666666">=</span> new_change
+    <span style="color: #008000">print</span>(<span style="color: #008000">iter</span>,gradients[<span style="color: #666666">0</span>],gradients[<span style="color: #666666">1</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd wth momentum&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="but-noen-of-these-can-compete-with-newton-s-method">But noen of these can compete with Newton's method </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Newton&#39;s method</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(beta):
+    <span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> beta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+beta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(beta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+<span style="color: #408080; font-style: italic"># Note that here the Hessian does not depend on the parameters beta</span>
+invH <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(H)
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">5</span>
+
+<span style="color: #408080; font-style: italic"># define the gradient</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> training_gradient(beta)
+    beta <span style="color: #666666">-=</span> invH <span style="color: #666666">@</span> gradients
+    <span style="color: #008000">print</span>(<span style="color: #008000">iter</span>,gradients[<span style="color: #666666">0</span>],gradients[<span style="color: #666666">1</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;beta from own Newton code&quot;</span>)
+<span style="color: #008000">print</span>(beta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="including-stochastic-gradient-descent-with-autograd">Including Stochastic Gradient Descent with Autograd </h2>
+<p>In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using <b>autograd</b>.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>training_gradient(y, X, theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
+Xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
+ypredict <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta)
+ypredict2 <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta_linreg)
+
+plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">&quot;b-&quot;</span>)
+plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Random numbers &#39;</span>)
+plt<span style="color: #666666">.</span>show()
+
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+t0, t1 <span style="color: #666666">=</span> <span style="color: #666666">5</span>, <span style="color: #666666">50</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">learning_schedule</span>(t):
+    <span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+<span style="color: #408080; font-style: italic"># Can you figure out a better way of setting up the contributions to each batch?</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        eta <span style="color: #666666">=</span> learning_schedule(epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i)
+        theta <span style="color: #666666">=</span> theta <span style="color: #666666">-</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own sdg&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="same-code-but-now-with-momentum-gradient-descent">Same code but now with momentum gradient descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using SGD</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+<span style="color: #408080; font-style: italic"># Hessian matrix</span>
+H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> XT_X
+EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
+Niterations <span style="color: #666666">=</span> <span style="color: #666666">100</span>
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+
+<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
+    gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>n)<span style="color: #666666">*</span>training_gradient(y, X, theta)
+    theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own gd&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+
+
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+t0, t1 <span style="color: #666666">=</span> <span style="color: #666666">5</span>, <span style="color: #666666">50</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">learning_schedule</span>(t):
+    <span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
+
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
+
+change <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+delta_momentum <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
+
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        eta <span style="color: #666666">=</span> learning_schedule(epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i)
+        <span style="color: #408080; font-style: italic"># calculate update</span>
+        new_change <span style="color: #666666">=</span> eta<span style="color: #666666">*</span>gradients<span style="color: #666666">+</span>delta_momentum<span style="color: #666666">*</span>change
+        <span style="color: #408080; font-style: italic"># take a step</span>
+        theta <span style="color: #666666">-=</span> new_change
+        <span style="color: #408080; font-style: italic"># save the change</span>
+        change <span style="color: #666666">=</span> new_change
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own sdg with momentum&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="similar-second-order-function-now-problem-but-now-with-adagrad">Similar (second order function now) problem but now with AdaGrad </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+3*</span>x <span style="color: #666666">+4*</span>x<span style="color: #666666">*</span>x
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">*</span>x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+<span style="color: #408080; font-style: italic"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+<span style="color: #408080; font-style: italic"># Guess for unknown parameters theta</span>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">3</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #408080; font-style: italic"># Value for learning rate</span>
+eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
+<span style="color: #408080; font-style: italic"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  <span style="color: #666666">=</span> <span style="color: #666666">1e-8</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    Giter <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        Giter <span style="color: #666666">+=</span> gradients<span style="color: #666666">*</span>gradients
+        update <span style="color: #666666">=</span> gradients<span style="color: #666666">*</span>eta<span style="color: #666666">/</span>(delta<span style="color: #666666">+</span>np<span style="color: #666666">.</span>sqrt(Giter))
+        theta <span style="color: #666666">-=</span> update
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own AdaGrad&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+<p>Running this code we note an almost perfect agreement with the results from matrix inversion.</p>
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent">RMSprop for adaptive learning rate with Stochastic Gradient Descent </h2>
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+3*</span>x <span style="color: #666666">+4*</span>x<span style="color: #666666">*</span>x<span style="color: #408080; font-style: italic"># +np.random.randn(n,1)</span>
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">*</span>x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+<span style="color: #408080; font-style: italic"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+<span style="color: #408080; font-style: italic"># Guess for unknown parameters theta</span>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">3</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #408080; font-style: italic"># Value for learning rate</span>
+eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
+<span style="color: #408080; font-style: italic"># Value for parameter rho</span>
+rho <span style="color: #666666">=</span> <span style="color: #666666">0.99</span>
+<span style="color: #408080; font-style: italic"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  <span style="color: #666666">=</span> <span style="color: #666666">1e-8</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    Giter <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+	<span style="color: #408080; font-style: italic"># Accumulated gradient</span>
+	<span style="color: #408080; font-style: italic"># Scaling with rho the new and the previous results</span>
+        Giter <span style="color: #666666">=</span> (rho<span style="color: #666666">*</span>Giter<span style="color: #666666">+</span>(<span style="color: #666666">1-</span>rho)<span style="color: #666666">*</span>gradients<span style="color: #666666">*</span>gradients)
+	<span style="color: #408080; font-style: italic"># Taking the diagonal only and inverting</span>
+        update <span style="color: #666666">=</span> gradients<span style="color: #666666">*</span>eta<span style="color: #666666">/</span>(delta<span style="color: #666666">+</span>np<span style="color: #666666">.</span>sqrt(Giter))
+	<span style="color: #408080; font-style: italic"># Hadamard product</span>
+        theta <span style="color: #666666">-=</span> update
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own RMSprop&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf">And finally <a href="https://arxiv.org/pdf/1412.6980.pdf" target="_blank">ADAM</a> </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent</span>
+<span style="color: #408080; font-style: italic"># OLS example</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #408080; font-style: italic"># Note change from previous example</span>
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">CostOLS</span>(y,X,theta):
+    <span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y<span style="color: #666666">-</span>X <span style="color: #666666">@</span> theta)<span style="color: #666666">**2</span>)
+
+n <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+3*</span>x <span style="color: #666666">+4*</span>x<span style="color: #666666">*</span>x<span style="color: #408080; font-style: italic"># +np.random.randn(n,1)</span>
+
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x, x<span style="color: #666666">*</span>x]
+XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
+theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(XT_X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Own inversion&quot;</span>)
+<span style="color: #008000">print</span>(theta_linreg)
+
+
+<span style="color: #408080; font-style: italic"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
+training_gradient <span style="color: #666666">=</span> grad(CostOLS,<span style="color: #666666">2</span>)
+<span style="color: #408080; font-style: italic"># Define parameters for Stochastic Gradient Descent</span>
+n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
+M <span style="color: #666666">=</span> <span style="color: #666666">5</span>   <span style="color: #408080; font-style: italic">#size of each minibatch</span>
+m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
+<span style="color: #408080; font-style: italic"># Guess for unknown parameters theta</span>
+theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">3</span>,<span style="color: #666666">1</span>)
+
+<span style="color: #408080; font-style: italic"># Value for learning rate</span>
+eta <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
+<span style="color: #408080; font-style: italic"># Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980</span>
+beta1 <span style="color: #666666">=</span> <span style="color: #666666">0.9</span>
+beta2 <span style="color: #666666">=</span> <span style="color: #666666">0.999</span>
+<span style="color: #408080; font-style: italic"># Including AdaGrad parameter to avoid possible division by zero</span>
+delta  <span style="color: #666666">=</span> <span style="color: #666666">1e-7</span>
+<span style="color: #008000">iter</span> <span style="color: #666666">=</span> <span style="color: #666666">0</span>
+<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
+    first_moment <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    second_moment <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+    <span style="color: #008000">iter</span> <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
+    <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
+        random_index <span style="color: #666666">=</span> M<span style="color: #666666">*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
+        xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+</span>M]
+        yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+</span>M]
+        gradients <span style="color: #666666">=</span> (<span style="color: #666666">1.0/</span>M)<span style="color: #666666">*</span>training_gradient(yi, xi, theta)
+        <span style="color: #408080; font-style: italic"># Computing moments first</span>
+        first_moment <span style="color: #666666">=</span> beta1<span style="color: #666666">*</span>first_moment <span style="color: #666666">+</span> (<span style="color: #666666">1-</span>beta1)<span style="color: #666666">*</span>gradients
+        second_moment <span style="color: #666666">=</span> beta2<span style="color: #666666">*</span>second_moment<span style="color: #666666">+</span>(<span style="color: #666666">1-</span>beta2)<span style="color: #666666">*</span>gradients<span style="color: #666666">*</span>gradients
+        first_term <span style="color: #666666">=</span> first_moment<span style="color: #666666">/</span>(<span style="color: #666666">1.0-</span>beta1<span style="color: #666666">**</span><span style="color: #008000">iter</span>)
+        second_term <span style="color: #666666">=</span> second_moment<span style="color: #666666">/</span>(<span style="color: #666666">1.0-</span>beta2<span style="color: #666666">**</span><span style="color: #008000">iter</span>)
+	<span style="color: #408080; font-style: italic"># Scaling with rho the new and the previous results</span>
+        update <span style="color: #666666">=</span> eta<span style="color: #666666">*</span>first_term<span style="color: #666666">/</span>(np<span style="color: #666666">.</span>sqrt(second_term)<span style="color: #666666">+</span>delta)
+        theta <span style="color: #666666">-=</span> update
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;theta from own ADAM&quot;</span>)
+<span style="color: #008000">print</span>(theta)
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="and-logistic-regression">And Logistic Regression </h2>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sigmoid</span>(x):
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">0.5</span> <span style="color: #666666">*</span> (np<span style="color: #666666">.</span>tanh(x <span style="color: #666666">/</span> <span style="color: #666666">2.</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">logistic_predictions</span>(weights, inputs):
+    <span style="color: #408080; font-style: italic"># Outputs probability of a label being true according to logistic model.</span>
+    <span style="color: #008000; font-weight: bold">return</span> sigmoid(np<span style="color: #666666">.</span>dot(inputs, weights))
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">training_loss</span>(weights):
+    <span style="color: #408080; font-style: italic"># Training loss is the negative log-likelihood of the training labels.</span>
+    preds <span style="color: #666666">=</span> logistic_predictions(weights, inputs)
+    label_probabilities <span style="color: #666666">=</span> preds <span style="color: #666666">*</span> targets <span style="color: #666666">+</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> preds) <span style="color: #666666">*</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> targets)
+    <span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">-</span>np<span style="color: #666666">.</span>sum(np<span style="color: #666666">.</span>log(label_probabilities))
+
+<span style="color: #408080; font-style: italic"># Build a toy dataset.</span>
+inputs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0.52</span>, <span style="color: #666666">1.12</span>,  <span style="color: #666666">0.77</span>],
+                   [<span style="color: #666666">0.88</span>, <span style="color: #666666">-1.08</span>, <span style="color: #666666">0.15</span>],
+                   [<span style="color: #666666">0.52</span>, <span style="color: #666666">0.06</span>, <span style="color: #666666">-1.30</span>],
+                   [<span style="color: #666666">0.74</span>, <span style="color: #666666">-2.49</span>, <span style="color: #666666">1.39</span>]])
+targets <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #008000; font-weight: bold">True</span>, <span style="color: #008000; font-weight: bold">True</span>, <span style="color: #008000; font-weight: bold">False</span>, <span style="color: #008000; font-weight: bold">True</span>])
+
+<span style="color: #408080; font-style: italic"># Define a function that returns gradients of training loss using Autograd.</span>
+training_gradient_fun <span style="color: #666666">=</span> grad(training_loss)
+
+<span style="color: #408080; font-style: italic"># Optimize weights using gradient descent.</span>
+weights <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0.0</span>, <span style="color: #666666">0.0</span>, <span style="color: #666666">0.0</span>])
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Initial loss:&quot;</span>, training_loss(weights))
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">100</span>):
+    weights <span style="color: #666666">-=</span> training_gradient_fun(weights) <span style="color: #666666">*</span> <span style="color: #666666">0.01</span>
+
+<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Trained loss:&quot;</span>, training_loss(weights))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="introducing-jax-https-jax-readthedocs-io-en-latest">Introducing <a href="https://jax.readthedocs.io/en/latest/" target="_blank">JAX</a> </h2>
+
+<p>Presently, instead of using <b>autograd</b>, we recommend using <a href="https://jax.readthedocs.io/en/latest/" target="_blank">JAX</a></p>
+
+<p><b>JAX</b> is Autograd and <a href="https://www.tensorflow.org/xla" target="_blank">XLA (Accelerated Linear Algebra))</a>,
+brought together for high-performance numerical computing and machine learning research.
+It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.
+</p>
+
+<p>Here's a simple example on how you can use <b>JAX</b> to compute the derivate of the logistic function.</p>
+
+
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="cell border-box-sizing code_cell rendered">
+  <div class="input">
+    <div class="inner_cell">
+      <div class="input_area">
+        <div class="highlight" style="background: #f8f8f8">
+  <pre style="line-height: 125%;"><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">jax.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">jnp</span>
+<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">jax</span> <span style="color: #008000; font-weight: bold">import</span> grad, jit, vmap
+
+<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sum_logistic</span>(x):
+  <span style="color: #008000; font-weight: bold">return</span> jnp<span style="color: #666666">.</span>sum(<span style="color: #666666">1.0</span> <span style="color: #666666">/</span> (<span style="color: #666666">1.0</span> <span style="color: #666666">+</span> jnp<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x)))
+
+x_small <span style="color: #666666">=</span> jnp<span style="color: #666666">.</span>arange(<span style="color: #666666">3.</span>)
+derivative_fn <span style="color: #666666">=</span> grad(sum_logistic)
+<span style="color: #008000">print</span>(derivative_fn(x_small))
+</pre>
+</div>
+      </div>
+    </div>
+  </div>
+  <div class="output_wrapper">
+    <div class="output">
+      <div class="output_area">
+        <div class="output_subarea output_stream output_stdout output_text">          
+        </div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+<!-- !split --><br><br><br><br><br><br><br><br><br><br>
+<h2 id="introduction-to-neural-networks">Introduction to Neural networks </h2>
 
 <p>Artificial neural networks are computational systems that can learn to
 perform tasks by considering examples, generally without being
@@ -555,7 +2582,7 @@ as to not restrict the range of output values.
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model">Illustration of a single perceptropn model and a multi-perceptron model </h2>
+<h2 id="illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model">Illustration of a single perceptron model and a multi-perceptron model </h2>
 
 <center>  <!-- FIGURE -->
 <hr class="figure">
@@ -762,7 +2789,7 @@ of the outputs of <em>all</em> neurons in the previous layer.
 
 $$
 \begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
-\label{_auto1}
+\label{_auto6}
 \end{equation}
 $$
 
@@ -807,7 +2834,7 @@ is obtained.
 $$
 \begin{align}
  y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
-\label{_auto2}\\
+\label{_auto7}\\
  &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
 \label{outputLayer2}
 \end{align}
@@ -818,10 +2845,10 @@ $$
 $$
 \begin{align}
  y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
-\label{_auto3}\\
+\label{_auto8}\\
  &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
   + b_1^3\right]
-\label{_auto4}
+\label{_auto9}
 \end{align}
 $$
 
@@ -861,7 +2888,7 @@ sum of scaled activation functions of the form
 $$
 \begin{equation}
  f(x) = c_1 f(c_2 x + c_3) + c_4
-\label{_auto5}
+\label{_auto10}
 \end{equation}
 $$
 
@@ -903,7 +2930,7 @@ $$
            b^2_2 \\
            b^2_3 \\
           \end{array}\right]\right).
-\label{_auto6}
+\label{_auto11}
 \end{equation}
 $$
 
@@ -917,7 +2944,7 @@ $$
 \begin{equation}
  y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
  f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\label{_auto7}
+\label{_auto12}
 \end{equation}
 $$
 
@@ -1071,452 +3098,9 @@ plt<span style="color: #666666">.</span>show()
 </div>
 
 
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="the-multilayer-perceptron-mlp">The multilayer  perceptron (MLP) </h2>
-
-<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
-<ol>
-<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
-<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
-<li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
-</ol>
-<p>As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
-units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-</p>
-
-<p>For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
-</p>
-
-<p>The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
-</p>
-
-<p>The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation \( \hat{z} \), we can think of this as a basis
-expansion of the original inputs \( \hat{x} \). The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in  an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="from-one-to-many-layers-the-universal-approximation-theorem">From one to many layers, the universal approximation theorem </h2>
-
-<p>A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
-</p>
-
-<p>As stated earlier, 
-an important theorem in studies of neural networks, restated without
-proof here, is the <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_blank">universal approximation
-theorem</a>.
-</p>
-
-<p>It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">Deriving the back propagation code for a multilayer perceptron model </h2>
-
-<p>As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_blank">back propagation algorithm</a>.
-</p>
-
-<p>The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
-</p>
-
-<p>To derive these equations let us start with a plain regression problem
-and define our cost function as
-</p>
-
-$$
-{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
-$$
-
-<p>where the $t_i$s are our \( n \) targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs \( \hat{x} \) are given by \( y_i \).  Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with \( K \)
-classes.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="definitions">Definitions </h2>
-
-<p>With our definition of the targets \( \hat{t} \), the outputs of the
-network \( \hat{y} \) and the inputs \( \hat{x} \) we
-define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
-\( l \)-th layer as a function of the bias, the weights which add up from
-the previous layer \( l-1 \) and the forward passes/outputs
-\( \hat{a}^{l-1} \) from the previous layer as
-</p>
-
-$$
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
-$$
-
-<p>where \( b_k^l \) are the biases from layer \( l \).  Here \( M_{l-1} \)
-represents the total number of nodes/neurons/units of layer \( l-1 \). The
-figure here illustrates this equation.  We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-</p>
-
-$$
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-$$
-
-<p>With the activation values \( \hat{z}^l \) we can in turn define the
-output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
-and their nodes.  It means we have
-</p>
-
-$$
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="derivatives-and-the-chain-rule">Derivatives and the chain rule </h2>
-
-<p>From the definition of the activation \( z_j^l \) we have</p>
-$$
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-$$
-
-<p>and</p>
-$$
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
-$$
-
-<p>With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))</p>
-$$
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="derivative-of-the-cost-function">Derivative of the cost function </h2>
-
-<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
-
-<p>Let us specialize to the output layer \( l=L \). Our cost function is</p>
-$$
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-$$
-
-<p>The derivative of this function with respect to the weights is</p>
-
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
-$$
-
-<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
-$$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="bringing-it-together-first-back-propagation-equation">Bringing it together, first back propagation equation </h2>
-
-<p>We have thus</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
-$$
-
-<p>Defining</p>
-$$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-$$
-
-<p>and using the Hadamard product of two vectors we can write this as</p>
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-
-<p>This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
-</p>
-
-<p>Notice that everything in the above equations is easily computed.  In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \).  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-</p>
-
-$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-$$
-
-<p>With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
-$$
-
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="derivatives-in-terms-of-z-j-l">Derivatives in terms of \( z_j^L \) </h2>
-
-<p>It is also easy to see that our previous equation can be written as</p>
-
-$$
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
-$$
-
-<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely</p>
-$$
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
-$$
-
-<p>That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. </p>
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="bringing-it-together">Bringing it together </h2>
-
-<p>We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b>The starting equations</b>
-<p>
-
-$$
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
-\label{_auto8}
-\end{equation}
-$$
-
-<p>and</p>
-$$
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\label{_auto9}
-\end{equation}
-$$
-
-<p>and</p>
-
-$$
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\label{_auto10}
-\end{equation}
-$$
-</div>
-
-
-<p>An interesting consequence of the above equations is that when the
-activation \( a_k^{L-1} \) is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
-</p>
-
-<p>Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
-</p>
-
-<p>We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one \( L-1 \) in terms of the errors in the final output layer.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="final-back-propagating-equation">Final back propagating equation </h2>
-
-<p>We have that (replacing \( L \) with a general layer \( l \))</p>
-$$
-\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
-$$
-
-<p>We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have</p>
-
-$$
-\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
-$$
-
-<p>and recalling that</p>
-$$
-z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
-$$
-
-<p>with \( M_l \) being the number of nodes in layer \( l \), we obtain</p>
-$$
-\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
-$$
-
-<p>This is our final equation.</p>
-
-<p>We are now ready to set up the algorithm for back propagation and learning the weights and biases.</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
-
-<p>The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>First, we set up the input data \( \hat{x} \) and the activations
-\( \hat{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \hat{a}^1 \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \hat{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \hat{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all</p>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-</div>
-
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-
-<!-- !split --><br><br><br><br><br><br><br><br><br><br>
-<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
-
-<p>The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>First, we set up the input data \( \boldsymbol{x} \) and the activations
-\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \boldsymbol{a}^1 \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \boldsymbol{a}^l \) for
-\( l=2,3,\dots,L \).
-</p>
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all</p>
-$$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-</div>
-
-
-<div class="alert alert-block alert-block alert-text-normal">
-<b></b>
-<p>
-<p>Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules</p>
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-</div>
-
-
-<p>The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-</p>
-
 <!-- ------------------- end of main content --------------- -->
 <center style="font-size:80%">
-<!-- copyright --> &copy; 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
+<!-- copyright --> &copy; 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
 </center>
 </body>
 </html>
diff --git a/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz b/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz
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index adaa6e460..70ab38246 100644
--- a/doc/pub/week40/ipynb/week40.ipynb
+++ b/doc/pub/week40/ipynb/week40.ipynb
@@ -2,56 +2,100 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "8ca827ff",
+   "id": "c410abdb",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
     "doconce format html week40.do.txt --no_mako -->\n",
-    "<!-- dom:TITLE: Week 40: Neural networks -->"
+    "<!-- dom:TITLE: Week 40: Gradient descent methods (continued) and start Neural networks -->"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "ae73291a",
+   "id": "ffdc5797",
    "metadata": {
     "editable": true
    },
    "source": [
-    "# Week 40: Neural networks\n",
+    "# Week 40: Gradient descent methods (continued) and start Neural networks\n",
     "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA\n",
     "\n",
-    "Date: **Oct 8, 2022**\n",
-    "\n",
-    "Copyright 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license"
+    "Date: **October 2-6, 2023**"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "ad8aa02d",
+   "id": "4bab315b",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Plan for week 40\n",
+    "## Plans for week 40\n",
     "\n",
-    "* Thursday: Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.\n",
+    "**Material for the active learning sessions on Tuesday and Wednesday.**\n",
     "\n",
-    "  * [Video of lecture](https://youtu.be/sCHiGSoG2lE)\n",
+    "  * Work on project 1 and discussions on how to structure your report\n",
     "\n",
-    "* Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.\n",
+    "  * No weekly exercises for week 40, project work only\n",
     "\n",
-    "  * [Video of lecture](https://youtu.be/hDTtA7PRRfI)\n",
+    "  * [Video on how to write scientific reports recorded during one of the lab sessions](https://youtu.be/tVW1ZDmZnwM)\n",
     "\n",
-    "Reading suggestions for both days: [Aurelien Geron's chapter 10](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf) and Hastie et al chapter 11.\n",
-    "For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.\n",
-    "For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4"
+    "  * A general guideline can be found at <https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md>.\n",
+    "\n",
+    "  \n",
+    "\n",
+    "**Material for the lecture on Thursday October 5, 2023.**\n",
+    "\n",
+    "  * Stochastic Gradient descent with examples and automatic differentiation\n",
+    "\n",
+    "  * Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.  \n",
+    "\n",
+    "  * Readings and Videos:\n",
+    "\n",
+    "    * These lecture notes\n",
+    "\n",
+    "    * For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.\n",
+    "\n",
+    "    * [Aurelien Geron's chapter 4 on stochastic gradient descent](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf)\n",
+    "\n",
+    "    * For neural networks we recommend Goodfellow et al chapter 6.\n",
+    "\n",
+    "    * [Video on gradient descent](https://www.youtube.com/watch?v=sDv4f4s2SB8)\n",
+    "\n",
+    "    * [Video on stochastic gradient descent](https://www.youtube.com/watch?v=vMh0zPT0tLI)\n",
+    "\n",
+    "    * [Neural Networks demystified](https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs)\n",
+    "\n",
+    "    * [Building Neural Networks from scratch](https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "c608bf7a",
+   "id": "1ba76140",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Summary from last week, using gradient descent methods, limitations\n",
+    "\n",
+    "* **Gradient descent (GD) finds local minima of our function**. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.\n",
+    "\n",
+    "* **GD is sensitive to initial conditions**. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.\n",
+    "\n",
+    "* **Gradients are computationally expensive to calculate for large datasets**. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \\propto \\sum_{i=1}^n (y_i - \\mathbf{w}^T\\cdot\\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called \"mini batches\". This has the added benefit of introducing stochasticity into our algorithm.\n",
+    "\n",
+    "* **GD is very sensitive to choices of learning rates**. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape.\n",
+    "\n",
+    "* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. \n",
+    "\n",
+    "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a8b56c00",
    "metadata": {
     "editable": true
    },
@@ -63,7 +107,994 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d15f6f05",
+   "id": "eba32497",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Batches and mini-batches\n",
+    "\n",
+    "In gradient descent we compute the cost function and its gradient for all data points we have.\n",
+    "\n",
+    "In large-scale applications such as the [ILSVRC challenge](https://www.image-net.org/challenges/LSVRC/), the\n",
+    "training data can have on order of millions of examples. Hence, it\n",
+    "seems wasteful to compute the full cost function over the entire\n",
+    "training set in order to perform only a single parameter update. A\n",
+    "very common approach to addressing this challenge is to compute the\n",
+    "gradient over batches of the training data. For example, a typical batch could contain some thousand  examples from\n",
+    "an  entire training set of several millions. This batch is then used to\n",
+    "perform a parameter update."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55578599",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Stochastic Gradient Descent (SGD)\n",
+    "\n",
+    "In stochastic gradient descent, the extreme case is the case where we\n",
+    "have only one batch, that is we include the whole data set.\n",
+    "\n",
+    "This process is called Stochastic Gradient\n",
+    "Descent (SGD) (or also sometimes on-line gradient descent). This is\n",
+    "relatively less common to see because in practice due to vectorized\n",
+    "code optimizations it can be computationally much more efficient to\n",
+    "evaluate the gradient for 100 examples, than the gradient for one\n",
+    "example 100 times. Even though SGD technically refers to using a\n",
+    "single example at a time to evaluate the gradient, you will hear\n",
+    "people use the term SGD even when referring to mini-batch gradient\n",
+    "descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD\n",
+    "for “Batch gradient descent” are rare to see), where it is usually\n",
+    "assumed that mini-batches are used. The size of the mini-batch is a\n",
+    "hyperparameter but it is not very common to cross-validate or bootstrap it. It is\n",
+    "usually based on memory constraints (if any), or set to some value,\n",
+    "e.g. 32, 64 or 128. We use powers of 2 in practice because many\n",
+    "vectorized operation implementations work faster when their inputs are\n",
+    "sized in powers of 2.\n",
+    "\n",
+    "In our notes with  SGD we mean stochastic gradient descent with mini-batches."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "140607b7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Stochastic Gradient Descent\n",
+    "\n",
+    "Stochastic gradient descent (SGD) and variants thereof address some of\n",
+    "the shortcomings of the Gradient descent method discussed above.\n",
+    "\n",
+    "The underlying idea of SGD comes from the observation that the cost\n",
+    "function, which we want to minimize, can almost always be written as a\n",
+    "sum over $n$ data points $\\{\\mathbf{x}_i\\}_{i=1}^n$,"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c1a00332",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n",
+    "\\mathbf{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2c04bdee",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Computation of gradients\n",
+    "\n",
+    "This in turn means that the gradient can be\n",
+    "computed as a sum over $i$-gradients"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "087684a4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n",
+    "\\mathbf{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e0362df4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Stochasticity/randomness is introduced by only taking the\n",
+    "gradient on a subset of the data called minibatches.  If there are $n$\n",
+    "data points and the size of each minibatch is $M$, there will be $n/M$\n",
+    "minibatches. We denote these minibatches by $B_k$ where\n",
+    "$k=1,\\cdots,n/M$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "24051a4e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## SGD example\n",
+    "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n",
+    "and we choose to have $M=5$ minibathces,\n",
+    "then each minibatch contains two data points. In particular we have\n",
+    "$B_1 = (\\mathbf{x}_1,\\mathbf{x}_2), \\cdots, B_5 =\n",
+    "(\\mathbf{x}_9,\\mathbf{x}_{10})$. Note that if you choose $M=1$ you\n",
+    "have only a single batch with all data points and on the other extreme,\n",
+    "you may choose $M=n$ resulting in a minibatch for each datapoint, i.e\n",
+    "$B_k = \\mathbf{x}_k$.\n",
+    "\n",
+    "The idea is now to approximate the gradient by replacing the sum over\n",
+    "all data points with a sum over the data points in one the minibatches\n",
+    "picked at random in each gradient descent step"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7723f927",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\nabla_{\\beta}\n",
+    "C(\\mathbf{\\beta}) = \\sum_{i=1}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n",
+    "\\mathbf{\\beta}) \\rightarrow \\sum_{i \\in B_k}^n \\nabla_\\beta\n",
+    "c_i(\\mathbf{x}_i, \\mathbf{\\beta}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "59221981",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The gradient step\n",
+    "\n",
+    "Thus a gradient descent step now looks like"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a7d27b48",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n",
+    "\\mathbf{\\beta})\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c7595344",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $k$ is picked at random with equal\n",
+    "probability from $[1,n/M]$. An iteration over the number of\n",
+    "minibathces (n/M) is commonly referred to as an epoch. Thus it is\n",
+    "typical to choose a number of epochs and for each epoch iterate over\n",
+    "the number of minibatches, as exemplified in the code below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0d7024b5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Simple example code"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "0f9dc38b",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np \n",
+    "\n",
+    "n = 100 #100 datapoints \n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "n_epochs = 10 #number of epochs\n",
+    "\n",
+    "j = 0\n",
+    "for epoch in range(1,n_epochs+1):\n",
+    "    for i in range(m):\n",
+    "        k = np.random.randint(m) #Pick the k-th minibatch at random\n",
+    "        #Compute the gradient using the data in minibatch Bk\n",
+    "        #Compute new suggestion for \n",
+    "        j += 1"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "df447303",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Taking the gradient only on a subset of the data has two important\n",
+    "benefits. First, it introduces randomness which decreases the chance\n",
+    "that our opmization scheme gets stuck in a local minima. Second, if\n",
+    "the size of the minibatches are small relative to the number of\n",
+    "datapoints ($M <  n$), the computation of the gradient is much\n",
+    "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n",
+    "all $n$ datapoints."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "976aef35",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## When do we stop?\n",
+    "\n",
+    "A natural question is when do we stop the search for a new minimum?\n",
+    "One possibility is to compute the full gradient after a given number\n",
+    "of epochs and check if the norm of the gradient is smaller than some\n",
+    "threshold and stop if true. However, the condition that the gradient\n",
+    "is zero is valid also for local minima, so this would only tell us\n",
+    "that we are close to a local/global minimum. However, we could also\n",
+    "evaluate the cost function at this point, store the result and\n",
+    "continue the search. If the test kicks in at a later stage we can\n",
+    "compare the values of the cost function and keep the $\\beta$ that\n",
+    "gave the lowest value."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0fab1ae1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Slightly different approach\n",
+    "\n",
+    "Another approach is to let the step length $\\gamma_j$ depend on the\n",
+    "number of epochs in such a way that it becomes very small after a\n",
+    "reasonable time such that we do not move at all. Such approaches are\n",
+    "also called scaling. There are many such ways to [scale the learning\n",
+    "rate](https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1)\n",
+    "and [discussions here](https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf). See\n",
+    "also\n",
+    "<https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1>\n",
+    "for a discussion of different scaling functions for the learning rate."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2db0116b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Time decay rate\n",
+    "\n",
+    "As an example, let $e = 0,1,2,3,\\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \\cdot m + i$ where $m$ is the number of minibatches and $i=0,\\cdots,m-1$. Then the function $$\\gamma_j(t; t_0, t_1) = \\frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$.\n",
+    "\n",
+    "In this way we can fix the number of epochs, compute $\\beta$ and\n",
+    "evaluate the cost function at the end. Repeating the computation will\n",
+    "give a different result since the scheme is random by design. Then we\n",
+    "pick the final $\\beta$ that gives the lowest value of the cost\n",
+    "function."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "a9ca6f9a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np \n",
+    "\n",
+    "def step_length(t,t0,t1):\n",
+    "    return t0/(t+t1)\n",
+    "\n",
+    "n = 100 #100 datapoints \n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "n_epochs = 500 #number of epochs\n",
+    "t0 = 1.0\n",
+    "t1 = 10\n",
+    "\n",
+    "gamma_j = t0/t1\n",
+    "j = 0\n",
+    "for epoch in range(1,n_epochs+1):\n",
+    "    for i in range(m):\n",
+    "        k = np.random.randint(m) #Pick the k-th minibatch at random\n",
+    "        #Compute the gradient using the data in minibatch Bk\n",
+    "        #Compute new suggestion for beta\n",
+    "        t = epoch*m+i\n",
+    "        gamma_j = step_length(t,t0,t1)\n",
+    "        j += 1\n",
+    "\n",
+    "print(\"gamma_j after %d epochs: %g\" % (n_epochs,gamma_j))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fcf9b69b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Code with a Number of Minibatches which varies\n",
+    "\n",
+    "In the code here we vary the number of mini-batches."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "861b050f",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "%matplotlib inline\n",
+    "\n",
+    "# Importing various packages\n",
+    "from math import exp, sqrt\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "n = 100\n",
+    "x = 2*np.random.rand(n,1)\n",
+    "y = 4+3*x+np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "# Hessian matrix\n",
+    "H = (2.0/n)* XT_X\n",
+    "EigValues, EigVectors = np.linalg.eig(H)\n",
+    "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "eta = 1.0/np.max(EigValues)\n",
+    "Niterations = 1000\n",
+    "\n",
+    "\n",
+    "for iter in range(Niterations):\n",
+    "    gradients = 2.0/n*X.T @ ((X @ theta)-y)\n",
+    "    theta -= eta*gradients\n",
+    "print(\"theta from own gd\")\n",
+    "print(theta)\n",
+    "\n",
+    "xnew = np.array([[0],[2]])\n",
+    "Xnew = np.c_[np.ones((2,1)), xnew]\n",
+    "ypredict = Xnew.dot(theta)\n",
+    "ypredict2 = Xnew.dot(theta_linreg)\n",
+    "\n",
+    "n_epochs = 50\n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "t0, t1 = 5, 50\n",
+    "\n",
+    "def learning_schedule(t):\n",
+    "    return t0/(t+t1)\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "\n",
+    "for epoch in range(n_epochs):\n",
+    "# Can you figure out a better way of setting up the contributions to each batch?\n",
+    "    for i in range(m):\n",
+    "        random_index = M*np.random.randint(m)\n",
+    "        xi = X[random_index:random_index+M]\n",
+    "        yi = y[random_index:random_index+M]\n",
+    "        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)\n",
+    "        eta = learning_schedule(epoch*m+i)\n",
+    "        theta = theta - eta*gradients\n",
+    "print(\"theta from own sdg\")\n",
+    "print(theta)\n",
+    "\n",
+    "plt.plot(xnew, ypredict, \"r-\")\n",
+    "plt.plot(xnew, ypredict2, \"b-\")\n",
+    "plt.plot(x, y ,'ro')\n",
+    "plt.axis([0,2.0,0, 15.0])\n",
+    "plt.xlabel(r'$x$')\n",
+    "plt.ylabel(r'$y$')\n",
+    "plt.title(r'Random numbers ')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "97311aec",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Replace or not\n",
+    "\n",
+    "In the above code, we have use replacement in setting up the\n",
+    "mini-batches. The discussion\n",
+    "[here](https://sebastianraschka.com/faq/docs/sgd-methods.html) may be\n",
+    "useful."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "423ddc16",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Momentum based GD\n",
+    "\n",
+    "The stochastic gradient descent (SGD) is almost always used with a\n",
+    "*momentum* or inertia term that serves as a memory of the direction we\n",
+    "are moving in parameter space.  This is typically implemented as\n",
+    "follows"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a4f85670",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t) \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f15ea450",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto1\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation} \n",
+    "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t},\n",
+    "\\label{_auto1} \\tag{1}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "233d7b7e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have introduced a momentum parameter $\\gamma$, with\n",
+    "$0\\le\\gamma\\le 1$, and for brevity we dropped the explicit notation to\n",
+    "indicate the gradient is to be taken over a different mini-batch at\n",
+    "each step. We call this algorithm gradient descent with momentum\n",
+    "(GDM). From these equations, it is clear that $\\mathbf{v}_t$ is a\n",
+    "running average of recently encountered gradients and\n",
+    "$(1-\\gamma)^{-1}$ sets the characteristic time scale for the memory\n",
+    "used in the averaging procedure. Consistent with this, when\n",
+    "$\\gamma=0$, this just reduces down to ordinary SGD as discussed\n",
+    "earlier. An equivalent way of writing the updates is"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b923e7d5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\Delta \\boldsymbol{\\theta}_{t+1} = \\gamma \\Delta \\boldsymbol{\\theta}_t -\\ \\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "60980ded",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where we have defined $\\Delta \\boldsymbol{\\theta}_{t}= \\boldsymbol{\\theta}_t-\\boldsymbol{\\theta}_{t-1}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cc771e70",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More on momentum based approaches\n",
+    "\n",
+    "Let us try to get more intuition from these equations. It is helpful\n",
+    "to consider a simple physical analogy with a particle of mass $m$\n",
+    "moving in a viscous medium with drag coefficient $\\mu$ and potential\n",
+    "$E(\\mathbf{w})$. If we denote the particle's position by $\\mathbf{w}$,\n",
+    "then its motion is described by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "859f6ffc",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "m {d^2 \\mathbf{w} \\over dt^2} + \\mu {d \\mathbf{w} \\over dt }= -\\nabla_w E(\\mathbf{w}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "064cc085",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "We can discretize this equation in the usual way to get"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "47d13c3c",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "m { \\mathbf{w}_{t+\\Delta t}-2 \\mathbf{w}_{t} +\\mathbf{w}_{t-\\Delta t} \\over (\\Delta t)^2}+\\mu {\\mathbf{w}_{t+\\Delta t}- \\mathbf{w}_{t} \\over \\Delta t} = -\\nabla_w E(\\mathbf{w}).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0ca67954",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Rearranging this equation, we can rewrite this as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ea9f63a8",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\Delta \\mathbf{w}_{t +\\Delta t}= - { (\\Delta t)^2 \\over m +\\mu \\Delta t} \\nabla_w E(\\mathbf{w})+ {m \\over m +\\mu \\Delta t} \\Delta \\mathbf{w}_t.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "35146ea5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Momentum parameter\n",
+    "\n",
+    "Notice that this equation is identical to previous one if we identify\n",
+    "the position of the particle, $\\mathbf{w}$, with the parameters\n",
+    "$\\boldsymbol{\\theta}$. This allows us to identify the momentum\n",
+    "parameter and learning rate with the mass of the particle and the\n",
+    "viscous drag as:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "82c87bb1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\gamma= {m \\over m +\\mu \\Delta t }, \\qquad \\eta = {(\\Delta t)^2 \\over m +\\mu \\Delta t}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "75415fca",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Thus, as the name suggests, the momentum parameter is proportional to\n",
+    "the mass of the particle and effectively provides inertia.\n",
+    "Furthermore, in the large viscosity/small learning rate limit, our\n",
+    "memory time scales as $(1-\\gamma)^{-1} \\approx m/(\\mu \\Delta t)$.\n",
+    "\n",
+    "Why is momentum useful? SGD momentum helps the gradient descent\n",
+    "algorithm gain speed in directions with persistent but small gradients\n",
+    "even in the presence of stochasticity, while suppressing oscillations\n",
+    "in high-curvature directions. This becomes especially important in\n",
+    "situations where the landscape is shallow and flat in some directions\n",
+    "and narrow and steep in others. It has been argued that first-order\n",
+    "methods (with appropriate initial conditions) can perform comparable\n",
+    "to more expensive second order methods, especially in the context of\n",
+    "complex deep learning models.\n",
+    "\n",
+    "These beneficial properties of momentum can sometimes become even more\n",
+    "pronounced by using a slight modification of the classical momentum\n",
+    "algorithm called Nesterov Accelerated Gradient (NAG).\n",
+    "\n",
+    "In the NAG algorithm, rather than calculating the gradient at the\n",
+    "current parameters, $\\nabla_\\theta E(\\boldsymbol{\\theta}_t)$, one\n",
+    "calculates the gradient at the expected value of the parameters given\n",
+    "our current momentum, $\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma\n",
+    "\\mathbf{v}_{t-1})$. This yields the NAG update rule"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "59892cd6",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma \\mathbf{v}_{t-1}) \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a01225ea",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto2\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation} \n",
+    "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t}.\n",
+    "\\label{_auto2} \\tag{2}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e2c9f57b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\\gamma$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1672a79e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Second moment of the gradient\n",
+    "\n",
+    "In stochastic gradient descent, with and without momentum, we still\n",
+    "have to specify a schedule for tuning the learning rates $\\eta_t$\n",
+    "as a function of time.  As discussed in the context of Newton's\n",
+    "method, this presents a number of dilemmas. The learning rate is\n",
+    "limited by the steepest direction which can change depending on the\n",
+    "current position in the landscape. To circumvent this problem, ideally\n",
+    "our algorithm would keep track of curvature and take large steps in\n",
+    "shallow, flat directions and small steps in steep, narrow directions.\n",
+    "Second-order methods accomplish this by calculating or approximating\n",
+    "the Hessian and normalizing the learning rate by the\n",
+    "curvature. However, this is very computationally expensive for\n",
+    "extremely large models. Ideally, we would like to be able to\n",
+    "adaptively change the step size to match the landscape without paying\n",
+    "the steep computational price of calculating or approximating\n",
+    "Hessians.\n",
+    "\n",
+    "Recently, a number of methods have been introduced that accomplish\n",
+    "this by tracking not only the gradient, but also the second moment of\n",
+    "the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and\n",
+    "[ADAM](https://arxiv.org/abs/1412.6980)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6d4032f9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## RMS prop\n",
+    "\n",
+    "In RMS prop, in addition to keeping a running average of the first\n",
+    "moment of the gradient, we also keep track of the second moment\n",
+    "denoted by $\\mathbf{s}_t=\\mathbb{E}[\\mathbf{g}_t^2]$. The update rule\n",
+    "for RMS prop is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "63cde9f3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto3\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n",
+    "\\label{_auto3} \\tag{3}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6f8a52c2",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbf{s}_t =\\beta \\mathbf{s}_{t-1} +(1-\\beta)\\mathbf{g}_t^2 \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9edf087d",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\mathbf{g}_t \\over \\sqrt{\\mathbf{s}_t +\\epsilon}}, \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7eff676b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\beta$ controls the averaging time of the second moment and is\n",
+    "typically taken to be about $\\beta=0.9$, $\\eta_t$ is a learning rate\n",
+    "typically chosen to be $10^{-3}$, and $\\epsilon\\sim 10^{-8} $ is a\n",
+    "small regularization constant to prevent divergences. Multiplication\n",
+    "and division by vectors is understood as an element-wise operation. It\n",
+    "is clear from this formula that the learning rate is reduced in\n",
+    "directions where the norm of the gradient is consistently large. This\n",
+    "greatly speeds up the convergence by allowing us to use a larger\n",
+    "learning rate for flat directions."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3fcb1068",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## [ADAM optimizer](https://arxiv.org/abs/1412.6980)\n",
+    "\n",
+    "A related algorithm is the ADAM optimizer. In\n",
+    "[ADAM](https://arxiv.org/abs/1412.6980), we keep a running average of\n",
+    "both the first and second moment of the gradient and use this\n",
+    "information to adaptively change the learning rate for different\n",
+    "parameters.  The method isefficient when working with large\n",
+    "problems involving lots data and/or parameters.  It is a combination of the\n",
+    "gradient descent with momentum algorithm and the RMSprop algorithm\n",
+    "discussed above.\n",
+    "\n",
+    "In addition to keeping a running average of the first and\n",
+    "second moments of the gradient\n",
+    "(i.e. $\\mathbf{m}_t=\\mathbb{E}[\\mathbf{g}_t]$ and\n",
+    "$\\mathbf{s}_t=\\mathbb{E}[\\mathbf{g}^2_t]$, respectively), ADAM\n",
+    "performs an additional bias correction to account for the fact that we\n",
+    "are estimating the first two moments of the gradient using a running\n",
+    "average (denoted by the hats in the update rule below). The update\n",
+    "rule for ADAM is given by (where multiplication and division are once\n",
+    "again understood to be element-wise operations below)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "31b034e1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto4\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n",
+    "\\label{_auto4} \\tag{4}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "571e9a91",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbf{m}_t = \\beta_1 \\mathbf{m}_{t-1} + (1-\\beta_1) \\mathbf{g}_t \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fb5883fa",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\mathbf{s}_t =\\beta_2 \\mathbf{s}_{t-1} +(1-\\beta_2)\\mathbf{g}_t^2 \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ebffe7a1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\mathbf{m}}_t={\\mathbf{m}_t \\over 1-\\beta_1^t} \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a513bd7",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\mathbf{s}}_t ={\\mathbf{s}_t \\over1-\\beta_2^t} \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d49bc312",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\boldsymbol{\\mathbf{m}}_t \\over \\sqrt{\\boldsymbol{\\mathbf{s}}_t} +\\epsilon}, \\nonumber\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6f4e5040",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "<!-- Equation labels as ordinary links -->\n",
+    "<div id=\"_auto5\"></div>\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation} \n",
+    "\\label{_auto5} \\tag{5}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4771881e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "where $\\beta_1$ and $\\beta_2$ set the memory lifetime of the first and\n",
+    "second moment and are typically taken to be $0.9$ and $0.99$\n",
+    "respectively, and $\\eta$ and $\\epsilon$ are identical to RMSprop.\n",
+    "\n",
+    "Like in RMSprop, the effective step size of a parameter depends on the\n",
+    "magnitude of its gradient squared.  To understand this better, let us\n",
+    "rewrite this expression in terms of the variance\n",
+    "$\\boldsymbol{\\sigma}_t^2 = \\boldsymbol{\\mathbf{s}}_t -\n",
+    "(\\boldsymbol{\\mathbf{m}}_t)^2$. Consider a single parameter $\\theta_t$. The\n",
+    "update rule for this parameter is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a0d438e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "\\Delta \\theta_{t+1}= -\\eta_t { \\boldsymbol{m}_t \\over \\sqrt{\\sigma_t^2 +  m_t^2 }+\\epsilon}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6cbb721b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Algorithms and codes for Adagrad, RMSprop and Adam\n",
+    "\n",
+    "The algorithms we have implemented are well described in the text by [Goodfellow, Bengio and Courville, chapter 8](https://www.deeplearningbook.org/contents/optimization.html).\n",
+    "\n",
+    "The codes which implement these algorithms are discussed after our presentation of automatic differentiation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e7d8b851",
    "metadata": {
     "editable": true
    },
@@ -83,7 +1114,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e70b1a1",
+   "id": "75afab2b",
    "metadata": {
     "editable": true
    },
@@ -113,31 +1144,1349 @@
     "Symbolic differentiation can lead to inefficient code and faces the\n",
     "difficulty of converting a computer program into a single expression,\n",
     "while numerical differentiation can introduce round-off errors in the\n",
-    "discretization process and cancellation"
+    "discretization process and cancellation\n",
+    "\n",
+    "Python has tools for so-called **automatic differentiation**.\n",
+    "Consider the following example"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "39e8a2e6",
+   "id": "c551bfeb",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Videos on Neural Networks\n",
-    "\n",
-    "[Neural Networks demystified](https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs)\n",
-    "\n",
-    "[Building Neural Networks from scratch](https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex)"
+    "$$\n",
+    "f(x) = \\sin\\left(2\\pi x + x^2\\right)\n",
+    "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4634e20f",
+   "id": "f2cbdd82",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Neural networks\n",
+    "which has the following derivative"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "22e5d8ce",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "$$\n",
+    "f'(x) = \\cos\\left(2\\pi x + x^2\\right)\\left(2\\pi + 2x\\right)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d2d352b4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Using **autograd** we have"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "19f1b95c",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "\n",
+    "# To do elementwise differentiation:\n",
+    "from autograd import elementwise_grad as egrad \n",
+    "\n",
+    "# To plot:\n",
+    "import matplotlib.pyplot as plt \n",
+    "\n",
+    "\n",
+    "def f(x):\n",
+    "    return np.sin(2*np.pi*x + x**2)\n",
+    "\n",
+    "def f_grad_analytic(x):\n",
+    "    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)\n",
+    "\n",
+    "# Do the comparison:\n",
+    "x = np.linspace(0,1,1000)\n",
+    "\n",
+    "f_grad = egrad(f)\n",
+    "\n",
+    "computed = f_grad(x)\n",
+    "analytic = f_grad_analytic(x)\n",
+    "\n",
+    "plt.title('Derivative computed from Autograd compared with the analytical derivative')\n",
+    "plt.plot(x,computed,label='autograd')\n",
+    "plt.plot(x,analytic,label='analytic')\n",
+    "\n",
+    "plt.xlabel('x')\n",
+    "plt.ylabel('y')\n",
+    "plt.legend()\n",
+    "\n",
+    "plt.show()\n",
+    "\n",
+    "print(\"The max absolute difference is: %g\"%(np.max(np.abs(computed - analytic))))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f3a495be",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Using autograd\n",
+    "\n",
+    "Here we\n",
+    "experiment with what kind of functions Autograd is capable\n",
+    "of finding the gradient of. The following Python functions are just\n",
+    "meant to illustrate what Autograd can do, but please feel free to\n",
+    "experiment with other, possibly more complicated, functions as well."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "5c856602",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "\n",
+    "def f1(x):\n",
+    "    return x**3 + 1\n",
+    "\n",
+    "f1_grad = grad(f1)\n",
+    "\n",
+    "# Remember to send in float as argument to the computed gradient from Autograd!\n",
+    "a = 1.0\n",
+    "\n",
+    "# See the evaluated gradient at a using autograd:\n",
+    "print(\"The gradient of f1 evaluated at a = %g using autograd is: %g\"%(a,f1_grad(a)))\n",
+    "\n",
+    "# Compare with the analytical derivative, that is f1'(x) = 3*x**2 \n",
+    "grad_analytical = 3*a**2\n",
+    "print(\"The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g\"%(a,grad_analytical))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b361074b",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Autograd with more complicated functions\n",
+    "\n",
+    "To differentiate with respect to two (or more) arguments of a Python\n",
+    "function, Autograd need to know at which variable the function if\n",
+    "being differentiated with respect to."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "a458a151",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f2(x1,x2):\n",
+    "    return 3*x1**3 + x2*(x1 - 5) + 1\n",
+    "\n",
+    "# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1\n",
+    "f2_grad_x1 = grad(f2,0)\n",
+    "\n",
+    "# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad\n",
+    "f2_grad_x2 = grad(f2,1)\n",
+    "\n",
+    "x1 = 1.0\n",
+    "x2 = 3.0 \n",
+    "\n",
+    "print(\"Evaluating at x1 = %g, x2 = %g\"%(x1,x2))\n",
+    "print(\"-\"*30)\n",
+    "\n",
+    "# Compare with the analytical derivatives:\n",
+    "\n",
+    "# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:\n",
+    "f2_grad_x1_analytical = 9*x1**2 + x2\n",
+    "\n",
+    "# Derivative of f2 w.r.t x2 is: x1 - 5:\n",
+    "f2_grad_x2_analytical = x1 - 5\n",
+    "\n",
+    "# See the evaluated derivations:\n",
+    "print(\"The derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n",
+    "print(\"The analytical derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n",
+    "\n",
+    "print()\n",
+    "\n",
+    "print(\"The derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))\n",
+    "print(\"The analytical derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "946c37f1",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a00d38e5",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More complicated functions using the elements of their arguments directly"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "d8c2a448",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f3(x): # Assumes x is an array of length 5 or higher\n",
+    "    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2\n",
+    "\n",
+    "f3_grad = grad(f3)\n",
+    "\n",
+    "x = np.linspace(0,4,5)\n",
+    "\n",
+    "# Print the computed gradient:\n",
+    "print(\"The computed gradient of f3 is: \", f3_grad(x))\n",
+    "\n",
+    "# The analytical gradient is: (2, 3, 5, 7, 22*x[4])\n",
+    "f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])\n",
+    "\n",
+    "# Print the analytical gradient:\n",
+    "print(\"The analytical gradient of f3 is: \", f3_grad_analytical)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "026d8733",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Note that in this case, when sending an array as input argument, the\n",
+    "output from Autograd is another array. This is the true gradient of\n",
+    "the function, as opposed to the function in the previous example. By\n",
+    "using arrays to represent the variables, the output from Autograd\n",
+    "might be easier to work with, as the output is closer to what one\n",
+    "could expect form a gradient-evaluting function."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1a4dca11",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Functions using mathematical functions from Numpy"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "c10b664f",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f4(x):\n",
+    "    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)\n",
+    "\n",
+    "f4_grad = grad(f4)\n",
+    "\n",
+    "x = 2.7\n",
+    "\n",
+    "# Print the computed derivative:\n",
+    "print(\"The computed derivative of f4 at x = %g is: %g\"%(x,f4_grad(x)))\n",
+    "\n",
+    "# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi\n",
+    "f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi\n",
+    "\n",
+    "# Print the analytical gradient:\n",
+    "print(\"The analytical gradient of f4 at x = %g is: %g\"%(x,f4_grad_analytical))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07436a71",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## More autograd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "1840a5d2",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f5(x):\n",
+    "    if x >= 0:\n",
+    "        return x**2\n",
+    "    else:\n",
+    "        return -3*x + 1\n",
+    "\n",
+    "f5_grad = grad(f5)\n",
+    "\n",
+    "x = 2.7\n",
+    "\n",
+    "# Print the computed derivative:\n",
+    "print(\"The computed derivative of f5 at x = %g is: %g\"%(x,f5_grad(x)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "87ee8137",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## And  with loops"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "f1b25f09",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f6_for(x):\n",
+    "    val = 0\n",
+    "    for i in range(10):\n",
+    "        val = val + x**i\n",
+    "    return val\n",
+    "\n",
+    "def f6_while(x):\n",
+    "    val = 0\n",
+    "    i = 0\n",
+    "    while i < 10:\n",
+    "        val = val + x**i\n",
+    "        i = i + 1\n",
+    "    return val\n",
+    "\n",
+    "f6_for_grad = grad(f6_for)\n",
+    "f6_while_grad = grad(f6_while)\n",
+    "\n",
+    "x = 0.5\n",
+    "\n",
+    "# Print the computed derivaties of f6_for and f6_while\n",
+    "print(\"The computed derivative of f6_for at x = %g is: %g\"%(x,f6_for_grad(x)))\n",
+    "print(\"The computed derivative of f6_while at x = %g is: %g\"%(x,f6_while_grad(x)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "5fa2802b",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9\n",
+    "# The analytical derivative is: sum(i*x**(i-1)) \n",
+    "f6_grad_analytical = 0\n",
+    "for i in range(10):\n",
+    "    f6_grad_analytical += i*x**(i-1)\n",
+    "\n",
+    "print(\"The analytical derivative of f6 at x = %g is: %g\"%(x,f6_grad_analytical))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "eb66fab4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Using recursion"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "965c8bbb",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "\n",
+    "def f7(n): # Assume that n is an integer\n",
+    "    if n == 1 or n == 0:\n",
+    "        return 1\n",
+    "    else:\n",
+    "        return n*f7(n-1)\n",
+    "\n",
+    "f7_grad = grad(f7)\n",
+    "\n",
+    "n = 2.0\n",
+    "\n",
+    "print(\"The computed derivative of f7 at n = %d is: %g\"%(n,f7_grad(n)))\n",
+    "\n",
+    "# The function f7 is an implementation of the factorial of n.\n",
+    "# By using the product rule, one can find that the derivative is:\n",
+    "\n",
+    "f7_grad_analytical = 0\n",
+    "for i in range(int(n)-1):\n",
+    "    tmp = 1\n",
+    "    for k in range(int(n)-1):\n",
+    "        if k != i:\n",
+    "            tmp *= (n - k)\n",
+    "    f7_grad_analytical += tmp\n",
+    "\n",
+    "print(\"The analytical derivative of f7 at n = %d is: %g\"%(n,f7_grad_analytical))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2e6e0c8a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "42adbcc3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Unsupported functions\n",
+    "Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.\n",
+    "\n",
+    "Assigning a value to the variable being differentiated with respect to"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "6ca4a5dc",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f8(x): # Assume x is an array\n",
+    "    x[2] = 3\n",
+    "    return x*2\n",
+    "\n",
+    "f8_grad = grad(f8)\n",
+    "\n",
+    "x = 8.4\n",
+    "\n",
+    "print(\"The derivative of f8 is:\",f8_grad(x))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5d816052",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "73f2e7e4",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## The syntax a.dot(b) when finding the dot product"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "2ead27d5",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f9(a): # Assume a is an array with 2 elements\n",
+    "    b = np.array([1.0,2.0])\n",
+    "    return a.dot(b)\n",
+    "\n",
+    "f9_grad = grad(f9)\n",
+    "\n",
+    "x = np.array([1.0,0.0])\n",
+    "\n",
+    "print(\"The derivative of f9 is:\",f9_grad(x))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1edcb932",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Here we are told that the 'dot' function does not belong to Autograd's\n",
+    "version of a Numpy array.  To overcome this, an alternative syntax\n",
+    "which also computed the dot product can be used:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "05897777",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "def f9_alternative(x): # Assume a is an array with 2 elements\n",
+    "    b = np.array([1.0,2.0])\n",
+    "    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2\n",
+    "\n",
+    "f9_alternative_grad = grad(f9_alternative)\n",
+    "\n",
+    "x = np.array([3.0,0.0])\n",
+    "\n",
+    "print(\"The gradient of f9 is:\",f9_alternative_grad(x))\n",
+    "\n",
+    "# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively\n",
+    "# w.r.t x is (b_1, b_2)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2c899815",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Recommended to avoid\n",
+    "The documentation recommends to avoid inplace operations such as"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "fd05063c",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "a += b\n",
+    "a -= b\n",
+    "a*= b\n",
+    "a /=b"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "94d6f0d9",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Using Autograd with OLS\n",
+    "\n",
+    "We conclude the part on optmization by showing how we can make codes\n",
+    "for linear regression and logistic regression using **autograd**. The\n",
+    "first example shows results with ordinary leats squares."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "d002c672",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients for OLS\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "def CostOLS(beta):\n",
+    "    return (1.0/n)*np.sum((y-X @ beta)**2)\n",
+    "\n",
+    "n = 100\n",
+    "x = 2*np.random.rand(n,1)\n",
+    "y = 4+3*x+np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "# Hessian matrix\n",
+    "H = (2.0/n)* XT_X\n",
+    "EigValues, EigVectors = np.linalg.eig(H)\n",
+    "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "eta = 1.0/np.max(EigValues)\n",
+    "Niterations = 1000\n",
+    "# define the gradient\n",
+    "training_gradient = grad(CostOLS)\n",
+    "\n",
+    "for iter in range(Niterations):\n",
+    "    gradients = training_gradient(theta)\n",
+    "    theta -= eta*gradients\n",
+    "print(\"theta from own gd\")\n",
+    "print(theta)\n",
+    "\n",
+    "xnew = np.array([[0],[2]])\n",
+    "Xnew = np.c_[np.ones((2,1)), xnew]\n",
+    "ypredict = Xnew.dot(theta)\n",
+    "ypredict2 = Xnew.dot(theta_linreg)\n",
+    "\n",
+    "plt.plot(xnew, ypredict, \"r-\")\n",
+    "plt.plot(xnew, ypredict2, \"b-\")\n",
+    "plt.plot(x, y ,'ro')\n",
+    "plt.axis([0,2.0,0, 15.0])\n",
+    "plt.xlabel(r'$x$')\n",
+    "plt.ylabel(r'$y$')\n",
+    "plt.title(r'Random numbers ')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e72279fe",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Same code but now with momentum gradient descent"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "62aa2606",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients for OLS\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "def CostOLS(beta):\n",
+    "    return (1.0/n)*np.sum((y-X @ beta)**2)\n",
+    "\n",
+    "n = 100\n",
+    "x = 2*np.random.rand(n,1)\n",
+    "y = 4+3*x#+np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "# Hessian matrix\n",
+    "H = (2.0/n)* XT_X\n",
+    "EigValues, EigVectors = np.linalg.eig(H)\n",
+    "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "eta = 1.0/np.max(EigValues)\n",
+    "Niterations = 30\n",
+    "\n",
+    "# define the gradient\n",
+    "training_gradient = grad(CostOLS)\n",
+    "\n",
+    "for iter in range(Niterations):\n",
+    "    gradients = training_gradient(theta)\n",
+    "    theta -= eta*gradients\n",
+    "    print(iter,gradients[0],gradients[1])\n",
+    "print(\"theta from own gd\")\n",
+    "print(theta)\n",
+    "\n",
+    "# Now improve with momentum gradient descent\n",
+    "change = 0.0\n",
+    "delta_momentum = 0.3\n",
+    "for iter in range(Niterations):\n",
+    "    # calculate gradient\n",
+    "    gradients = training_gradient(theta)\n",
+    "    # calculate update\n",
+    "    new_change = eta*gradients+delta_momentum*change\n",
+    "    # take a step\n",
+    "    theta -= new_change\n",
+    "    # save the change\n",
+    "    change = new_change\n",
+    "    print(iter,gradients[0],gradients[1])\n",
+    "print(\"theta from own gd wth momentum\")\n",
+    "print(theta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bad8e42f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## But noen of these can compete with Newton's method"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "13a572a8",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Newton's method\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "def CostOLS(beta):\n",
+    "    return (1.0/n)*np.sum((y-X @ beta)**2)\n",
+    "\n",
+    "n = 100\n",
+    "x = 2*np.random.rand(n,1)\n",
+    "y = 4+3*x+np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x]\n",
+    "XT_X = X.T @ X\n",
+    "beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(beta_linreg)\n",
+    "# Hessian matrix\n",
+    "H = (2.0/n)* XT_X\n",
+    "# Note that here the Hessian does not depend on the parameters beta\n",
+    "invH = np.linalg.pinv(H)\n",
+    "EigValues, EigVectors = np.linalg.eig(H)\n",
+    "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n",
+    "\n",
+    "beta = np.random.randn(2,1)\n",
+    "Niterations = 5\n",
+    "\n",
+    "# define the gradient\n",
+    "training_gradient = grad(CostOLS)\n",
+    "\n",
+    "for iter in range(Niterations):\n",
+    "    gradients = training_gradient(beta)\n",
+    "    beta -= invH @ gradients\n",
+    "    print(iter,gradients[0],gradients[1])\n",
+    "print(\"beta from own Newton code\")\n",
+    "print(beta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5b2c9e3a",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Including Stochastic Gradient Descent with Autograd\n",
+    "In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using **autograd**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "830370bf",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients using SGD\n",
+    "# OLS example\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "# Note change from previous example\n",
+    "def CostOLS(y,X,theta):\n",
+    "    return np.sum((y-X @ theta)**2)\n",
+    "\n",
+    "n = 100\n",
+    "x = 2*np.random.rand(n,1)\n",
+    "y = 4+3*x+np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "# Hessian matrix\n",
+    "H = (2.0/n)* XT_X\n",
+    "EigValues, EigVectors = np.linalg.eig(H)\n",
+    "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "eta = 1.0/np.max(EigValues)\n",
+    "Niterations = 1000\n",
+    "\n",
+    "# Note that we request the derivative wrt third argument (theta, 2 here)\n",
+    "training_gradient = grad(CostOLS,2)\n",
+    "\n",
+    "for iter in range(Niterations):\n",
+    "    gradients = (1.0/n)*training_gradient(y, X, theta)\n",
+    "    theta -= eta*gradients\n",
+    "print(\"theta from own gd\")\n",
+    "print(theta)\n",
+    "\n",
+    "xnew = np.array([[0],[2]])\n",
+    "Xnew = np.c_[np.ones((2,1)), xnew]\n",
+    "ypredict = Xnew.dot(theta)\n",
+    "ypredict2 = Xnew.dot(theta_linreg)\n",
+    "\n",
+    "plt.plot(xnew, ypredict, \"r-\")\n",
+    "plt.plot(xnew, ypredict2, \"b-\")\n",
+    "plt.plot(x, y ,'ro')\n",
+    "plt.axis([0,2.0,0, 15.0])\n",
+    "plt.xlabel(r'$x$')\n",
+    "plt.ylabel(r'$y$')\n",
+    "plt.title(r'Random numbers ')\n",
+    "plt.show()\n",
+    "\n",
+    "n_epochs = 50\n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "t0, t1 = 5, 50\n",
+    "def learning_schedule(t):\n",
+    "    return t0/(t+t1)\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "\n",
+    "for epoch in range(n_epochs):\n",
+    "# Can you figure out a better way of setting up the contributions to each batch?\n",
+    "    for i in range(m):\n",
+    "        random_index = M*np.random.randint(m)\n",
+    "        xi = X[random_index:random_index+M]\n",
+    "        yi = y[random_index:random_index+M]\n",
+    "        gradients = (1.0/M)*training_gradient(yi, xi, theta)\n",
+    "        eta = learning_schedule(epoch*m+i)\n",
+    "        theta = theta - eta*gradients\n",
+    "print(\"theta from own sdg\")\n",
+    "print(theta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e580483f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Same code but now with momentum gradient descent"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "68895d3f",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients using SGD\n",
+    "# OLS example\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "# Note change from previous example\n",
+    "def CostOLS(y,X,theta):\n",
+    "    return np.sum((y-X @ theta)**2)\n",
+    "\n",
+    "n = 100\n",
+    "x = 2*np.random.rand(n,1)\n",
+    "y = 4+3*x+np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "# Hessian matrix\n",
+    "H = (2.0/n)* XT_X\n",
+    "EigValues, EigVectors = np.linalg.eig(H)\n",
+    "print(f\"Eigenvalues of Hessian Matrix:{EigValues}\")\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "eta = 1.0/np.max(EigValues)\n",
+    "Niterations = 100\n",
+    "\n",
+    "# Note that we request the derivative wrt third argument (theta, 2 here)\n",
+    "training_gradient = grad(CostOLS,2)\n",
+    "\n",
+    "for iter in range(Niterations):\n",
+    "    gradients = (1.0/n)*training_gradient(y, X, theta)\n",
+    "    theta -= eta*gradients\n",
+    "print(\"theta from own gd\")\n",
+    "print(theta)\n",
+    "\n",
+    "\n",
+    "n_epochs = 50\n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "t0, t1 = 5, 50\n",
+    "def learning_schedule(t):\n",
+    "    return t0/(t+t1)\n",
+    "\n",
+    "theta = np.random.randn(2,1)\n",
+    "\n",
+    "change = 0.0\n",
+    "delta_momentum = 0.3\n",
+    "\n",
+    "for epoch in range(n_epochs):\n",
+    "    for i in range(m):\n",
+    "        random_index = M*np.random.randint(m)\n",
+    "        xi = X[random_index:random_index+M]\n",
+    "        yi = y[random_index:random_index+M]\n",
+    "        gradients = (1.0/M)*training_gradient(yi, xi, theta)\n",
+    "        eta = learning_schedule(epoch*m+i)\n",
+    "        # calculate update\n",
+    "        new_change = eta*gradients+delta_momentum*change\n",
+    "        # take a step\n",
+    "        theta -= new_change\n",
+    "        # save the change\n",
+    "        change = new_change\n",
+    "print(\"theta from own sdg with momentum\")\n",
+    "print(theta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "01c29c9e",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Similar (second order function now) problem but now with AdaGrad"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "36a00e5a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent\n",
+    "# OLS example\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "# Note change from previous example\n",
+    "def CostOLS(y,X,theta):\n",
+    "    return np.sum((y-X @ theta)**2)\n",
+    "\n",
+    "n = 1000\n",
+    "x = np.random.rand(n,1)\n",
+    "y = 2.0+3*x +4*x*x\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x, x*x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "\n",
+    "\n",
+    "# Note that we request the derivative wrt third argument (theta, 2 here)\n",
+    "training_gradient = grad(CostOLS,2)\n",
+    "# Define parameters for Stochastic Gradient Descent\n",
+    "n_epochs = 50\n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "# Guess for unknown parameters theta\n",
+    "theta = np.random.randn(3,1)\n",
+    "\n",
+    "# Value for learning rate\n",
+    "eta = 0.01\n",
+    "# Including AdaGrad parameter to avoid possible division by zero\n",
+    "delta  = 1e-8\n",
+    "for epoch in range(n_epochs):\n",
+    "    Giter = 0.0\n",
+    "    for i in range(m):\n",
+    "        random_index = M*np.random.randint(m)\n",
+    "        xi = X[random_index:random_index+M]\n",
+    "        yi = y[random_index:random_index+M]\n",
+    "        gradients = (1.0/M)*training_gradient(yi, xi, theta)\n",
+    "        Giter += gradients*gradients\n",
+    "        update = gradients*eta/(delta+np.sqrt(Giter))\n",
+    "        theta -= update\n",
+    "print(\"theta from own AdaGrad\")\n",
+    "print(theta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "18ccca46",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "Running this code we note an almost perfect agreement with the results from matrix inversion."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e076c773",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## RMSprop for adaptive learning rate with Stochastic Gradient Descent"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "cb4ad1d3",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent\n",
+    "# OLS example\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "# Note change from previous example\n",
+    "def CostOLS(y,X,theta):\n",
+    "    return np.sum((y-X @ theta)**2)\n",
+    "\n",
+    "n = 1000\n",
+    "x = np.random.rand(n,1)\n",
+    "y = 2.0+3*x +4*x*x# +np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x, x*x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "\n",
+    "\n",
+    "# Note that we request the derivative wrt third argument (theta, 2 here)\n",
+    "training_gradient = grad(CostOLS,2)\n",
+    "# Define parameters for Stochastic Gradient Descent\n",
+    "n_epochs = 50\n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "# Guess for unknown parameters theta\n",
+    "theta = np.random.randn(3,1)\n",
+    "\n",
+    "# Value for learning rate\n",
+    "eta = 0.01\n",
+    "# Value for parameter rho\n",
+    "rho = 0.99\n",
+    "# Including AdaGrad parameter to avoid possible division by zero\n",
+    "delta  = 1e-8\n",
+    "for epoch in range(n_epochs):\n",
+    "    Giter = 0.0\n",
+    "    for i in range(m):\n",
+    "        random_index = M*np.random.randint(m)\n",
+    "        xi = X[random_index:random_index+M]\n",
+    "        yi = y[random_index:random_index+M]\n",
+    "        gradients = (1.0/M)*training_gradient(yi, xi, theta)\n",
+    "\t# Accumulated gradient\n",
+    "\t# Scaling with rho the new and the previous results\n",
+    "        Giter = (rho*Giter+(1-rho)*gradients*gradients)\n",
+    "\t# Taking the diagonal only and inverting\n",
+    "        update = gradients*eta/(delta+np.sqrt(Giter))\n",
+    "\t# Hadamard product\n",
+    "        theta -= update\n",
+    "print(\"theta from own RMSprop\")\n",
+    "print(theta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2253fa34",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## And finally [ADAM](https://arxiv.org/pdf/1412.6980.pdf)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "92b3454a",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "# Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent\n",
+    "# OLS example\n",
+    "from random import random, seed\n",
+    "import numpy as np\n",
+    "import autograd.numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from autograd import grad\n",
+    "\n",
+    "# Note change from previous example\n",
+    "def CostOLS(y,X,theta):\n",
+    "    return np.sum((y-X @ theta)**2)\n",
+    "\n",
+    "n = 1000\n",
+    "x = np.random.rand(n,1)\n",
+    "y = 2.0+3*x +4*x*x# +np.random.randn(n,1)\n",
+    "\n",
+    "X = np.c_[np.ones((n,1)), x, x*x]\n",
+    "XT_X = X.T @ X\n",
+    "theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)\n",
+    "print(\"Own inversion\")\n",
+    "print(theta_linreg)\n",
+    "\n",
+    "\n",
+    "# Note that we request the derivative wrt third argument (theta, 2 here)\n",
+    "training_gradient = grad(CostOLS,2)\n",
+    "# Define parameters for Stochastic Gradient Descent\n",
+    "n_epochs = 50\n",
+    "M = 5   #size of each minibatch\n",
+    "m = int(n/M) #number of minibatches\n",
+    "# Guess for unknown parameters theta\n",
+    "theta = np.random.randn(3,1)\n",
+    "\n",
+    "# Value for learning rate\n",
+    "eta = 0.01\n",
+    "# Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980\n",
+    "beta1 = 0.9\n",
+    "beta2 = 0.999\n",
+    "# Including AdaGrad parameter to avoid possible division by zero\n",
+    "delta  = 1e-7\n",
+    "iter = 0\n",
+    "for epoch in range(n_epochs):\n",
+    "    first_moment = 0.0\n",
+    "    second_moment = 0.0\n",
+    "    iter += 1\n",
+    "    for i in range(m):\n",
+    "        random_index = M*np.random.randint(m)\n",
+    "        xi = X[random_index:random_index+M]\n",
+    "        yi = y[random_index:random_index+M]\n",
+    "        gradients = (1.0/M)*training_gradient(yi, xi, theta)\n",
+    "        # Computing moments first\n",
+    "        first_moment = beta1*first_moment + (1-beta1)*gradients\n",
+    "        second_moment = beta2*second_moment+(1-beta2)*gradients*gradients\n",
+    "        first_term = first_moment/(1.0-beta1**iter)\n",
+    "        second_term = second_moment/(1.0-beta2**iter)\n",
+    "\t# Scaling with rho the new and the previous results\n",
+    "        update = eta*first_term/(np.sqrt(second_term)+delta)\n",
+    "        theta -= update\n",
+    "print(\"theta from own ADAM\")\n",
+    "print(theta)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c2025d97",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## And Logistic Regression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "id": "3f6d8746",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import autograd.numpy as np\n",
+    "from autograd import grad\n",
+    "\n",
+    "def sigmoid(x):\n",
+    "    return 0.5 * (np.tanh(x / 2.) + 1)\n",
+    "\n",
+    "def logistic_predictions(weights, inputs):\n",
+    "    # Outputs probability of a label being true according to logistic model.\n",
+    "    return sigmoid(np.dot(inputs, weights))\n",
+    "\n",
+    "def training_loss(weights):\n",
+    "    # Training loss is the negative log-likelihood of the training labels.\n",
+    "    preds = logistic_predictions(weights, inputs)\n",
+    "    label_probabilities = preds * targets + (1 - preds) * (1 - targets)\n",
+    "    return -np.sum(np.log(label_probabilities))\n",
+    "\n",
+    "# Build a toy dataset.\n",
+    "inputs = np.array([[0.52, 1.12,  0.77],\n",
+    "                   [0.88, -1.08, 0.15],\n",
+    "                   [0.52, 0.06, -1.30],\n",
+    "                   [0.74, -2.49, 1.39]])\n",
+    "targets = np.array([True, True, False, True])\n",
+    "\n",
+    "# Define a function that returns gradients of training loss using Autograd.\n",
+    "training_gradient_fun = grad(training_loss)\n",
+    "\n",
+    "# Optimize weights using gradient descent.\n",
+    "weights = np.array([0.0, 0.0, 0.0])\n",
+    "print(\"Initial loss:\", training_loss(weights))\n",
+    "for i in range(100):\n",
+    "    weights -= training_gradient_fun(weights) * 0.01\n",
+    "\n",
+    "print(\"Trained loss:\", training_loss(weights))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "716627e3",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Introducing [JAX](https://jax.readthedocs.io/en/latest/)\n",
+    "\n",
+    "Presently, instead of using **autograd**, we recommend using [JAX](https://jax.readthedocs.io/en/latest/)\n",
+    "\n",
+    "**JAX** is Autograd and [XLA (Accelerated Linear Algebra))](https://www.tensorflow.org/xla),\n",
+    "brought together for high-performance numerical computing and machine learning research.\n",
+    "It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.\n",
+    "\n",
+    "Here's a simple example on how you can use **JAX** to compute the derivate of the logistic function."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "id": "5c938af4",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
+    "import jax.numpy as jnp\n",
+    "from jax import grad, jit, vmap\n",
+    "\n",
+    "def sum_logistic(x):\n",
+    "  return jnp.sum(1.0 / (1.0 + jnp.exp(-x)))\n",
+    "\n",
+    "x_small = jnp.arange(3.)\n",
+    "derivative_fn = grad(sum_logistic)\n",
+    "print(derivative_fn(x_small))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b087cc5f",
+   "metadata": {
+    "editable": true
+   },
+   "source": [
+    "## Introduction to Neural networks\n",
     "\n",
     "Artificial neural networks are computational systems that can learn to\n",
     "perform tasks by considering examples, generally without being\n",
@@ -150,7 +2499,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6d3d26d8",
+   "id": "c040b49e",
    "metadata": {
     "editable": true
    },
@@ -174,7 +2523,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "65f64bcc",
+   "id": "663a7548",
    "metadata": {
     "editable": true
    },
@@ -185,14 +2534,14 @@
     "$$\n",
     "\\begin{equation}\n",
     " y = f\\left(\\sum_{i=1}^n w_ix_i\\right) = f(u)\n",
-    "\\label{artificialNeuron} \\tag{1}\n",
+    "\\label{artificialNeuron} \\tag{6}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "7b35b206",
+   "id": "7d41caae",
    "metadata": {
     "editable": true
    },
@@ -232,7 +2581,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df226094",
+   "id": "8cd5fa1c",
    "metadata": {
     "editable": true
    },
@@ -261,7 +2610,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9b5e03a4",
+   "id": "0b2c6e40",
    "metadata": {
     "editable": true
    },
@@ -282,7 +2631,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d2e639d",
+   "id": "90e946b7",
    "metadata": {
     "editable": true
    },
@@ -311,7 +2660,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ee9119f3",
+   "id": "1964f9e3",
    "metadata": {
     "editable": true
    },
@@ -332,7 +2681,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50ed8fbb",
+   "id": "faf981a1",
    "metadata": {
     "editable": true
    },
@@ -353,7 +2702,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3f81f6ef",
+   "id": "3667182a",
    "metadata": {
     "editable": true
    },
@@ -370,7 +2719,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9db6da61",
+   "id": "5dd1a89f",
    "metadata": {
     "editable": true
    },
@@ -391,12 +2740,12 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c56753f6",
+   "id": "da5b8927",
    "metadata": {
     "editable": true
    },
    "source": [
-    "## Illustration of a single perceptropn model and a multi-perceptron model\n",
+    "## Illustration of a single perceptron model and a multi-perceptron model\n",
     "\n",
     "<!-- dom:FIGURE: [figures/nns.png, width=600 frac=0.8]  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer. -->\n",
     "<!-- begin figure -->\n",
@@ -407,7 +2756,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60f30853",
+   "id": "3bba849e",
    "metadata": {
     "editable": true
    },
@@ -423,8 +2772,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
-   "id": "f08c97b7",
+   "execution_count": 27,
+   "id": "de11d95e",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -465,7 +2814,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "80c31197",
+   "id": "b0477050",
    "metadata": {
     "editable": true
    },
@@ -475,7 +2824,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e112a16c",
+   "id": "1d72d90e",
    "metadata": {
     "editable": true
    },
@@ -485,16 +2834,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
-   "id": "1a9d5253",
+   "execution_count": 28,
+   "id": "501aa7b5",
    "metadata": {
     "collapsed": false,
     "editable": true
    },
    "outputs": [],
    "source": [
-    "%matplotlib inline\n",
-    "\n",
     "\"\"\"\n",
     "Simple code that tests XOR and OR gates with linear regression\n",
     "and logistic regression\n",
@@ -548,7 +2895,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fab2a2d9",
+   "id": "03908b42",
    "metadata": {
     "editable": true
    },
@@ -558,7 +2905,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "99255609",
+   "id": "91971469",
    "metadata": {
     "editable": true
    },
@@ -568,8 +2915,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
-   "id": "56adbeea",
+   "execution_count": 29,
+   "id": "f1717531",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -589,7 +2936,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "759f3f91",
+   "id": "05726714",
    "metadata": {
     "editable": true
    },
@@ -601,7 +2948,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b79179de",
+   "id": "1cf57e1c",
    "metadata": {
     "editable": true
    },
@@ -613,7 +2960,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a91fb9a7",
+   "id": "57743f5e",
    "metadata": {
     "editable": true
    },
@@ -628,7 +2975,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d8ea83b1",
+   "id": "2d3f8338",
    "metadata": {
     "editable": true
    },
@@ -640,24 +2987,24 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6b1e44d9",
+   "id": "20be0ccb",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto1\"></div>\n",
+    "<div id=\"_auto6\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} z_i^1 = \\sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1\n",
-    "\\label{_auto1} \\tag{2}\n",
+    "\\label{_auto6} \\tag{7}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "501afd5c",
+   "id": "d289b4c8",
    "metadata": {
     "editable": true
    },
@@ -672,7 +3019,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8b1ee2e4",
+   "id": "498c2494",
    "metadata": {
     "editable": true
    },
@@ -683,14 +3030,14 @@
     "$$\n",
     "\\begin{equation}\n",
     " y_i^1 = f(z_i^1) = f\\left(\\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\\right)\n",
-    "\\label{outputLayer1} \\tag{3}\n",
+    "\\label{outputLayer1} \\tag{8}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b8d08814",
+   "id": "77995e5d",
    "metadata": {
     "editable": true
    },
@@ -702,7 +3049,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56b376b4",
+   "id": "ef353d76",
    "metadata": {
     "editable": true
    },
@@ -713,14 +3060,14 @@
     "$$\n",
     "\\begin{equation}\n",
     " y_i^l = f^l(u_i^l) = f^l\\left(\\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\\right)\n",
-    "\\label{generalLayer} \\tag{4}\n",
+    "\\label{generalLayer} \\tag{9}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "82892d19",
+   "id": "0a25d2f4",
    "metadata": {
     "editable": true
    },
@@ -733,7 +3080,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fa0f17cb",
+   "id": "d7d29703",
    "metadata": {
     "editable": true
    },
@@ -745,25 +3092,25 @@
   },
   {
    "cell_type": "markdown",
-   "id": "465dbb92",
+   "id": "94eddeb9",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto2\"></div>\n",
+    "<div id=\"_auto7\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     " y_i^2 = f^2\\left(\\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\\right) \n",
-    "\\label{_auto2} \\tag{5}\n",
+    "\\label{_auto7} \\tag{10}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "fd7c9e3b",
+   "id": "f047f4c6",
    "metadata": {
     "editable": true
    },
@@ -774,14 +3121,14 @@
     "$$\n",
     "\\begin{equation} \n",
     " = f^2\\left[\\sum_{j=1}^N w_{ij}^2f^1\\left(\\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\\right) + b_i^2\\right]\n",
-    "\\label{outputLayer2} \\tag{6}\n",
+    "\\label{outputLayer2} \\tag{11}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "c4b1ef36",
+   "id": "91d4806e",
    "metadata": {
     "editable": true
    },
@@ -791,44 +3138,44 @@
   },
   {
    "cell_type": "markdown",
-   "id": "72602a84",
+   "id": "7342e125",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto3\"></div>\n",
+    "<div id=\"_auto8\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     " y_i^3 = f^3\\left(\\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\\right) \n",
-    "\\label{_auto3} \\tag{7}\n",
+    "\\label{_auto8} \\tag{12}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b1c97247",
+   "id": "5068e976",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto4\"></div>\n",
+    "<div id=\"_auto9\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation} \n",
     " = f_3\\left[\\sum_{j} w_{ij}^3 f^2\\left(\\sum_{k} w_{jk}^2 f^1\\left(\\sum_{m} w_{km}^1 x_m + b_k^1\\right) + b_j^2\\right)\n",
     "  + b_1^3\\right]\n",
-    "\\label{_auto4} \\tag{8}\n",
+    "\\label{_auto9} \\tag{13}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "259354e9",
+   "id": "b51a241d",
    "metadata": {
     "editable": true
    },
@@ -841,7 +3188,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df2526c6",
+   "id": "5a7b4915",
    "metadata": {
     "editable": true
    },
@@ -852,14 +3199,14 @@
     "$$\n",
     "\\begin{equation}\n",
     "y^{l+1}_i = f^{l+1}\\left[\\!\\sum_{j=1}^{N_l} w_{ij}^3 f^l\\left(\\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\\left(\\dots f^1\\left(\\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\\right)\\dots\\right)+b_k^2\\right)+b_1^3\\right] \n",
-    "\\label{completeNN} \\tag{9}\n",
+    "\\label{completeNN} \\tag{14}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "27327f03",
+   "id": "c3215b7f",
    "metadata": {
     "editable": true
    },
@@ -870,7 +3217,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "39aefd8a",
+   "id": "f92eedb0",
    "metadata": {
     "editable": true
    },
@@ -889,25 +3236,25 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b9dc9947",
+   "id": "b658fa6d",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto5\"></div>\n",
+    "<div id=\"_auto10\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     " f(x) = c_1 f(c_2 x + c_3) + c_4\n",
-    "\\label{_auto5} \\tag{10}\n",
+    "\\label{_auto10} \\tag{15}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "40392627",
+   "id": "8506281a",
    "metadata": {
     "editable": true
    },
@@ -920,7 +3267,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "013de6fc",
+   "id": "0ad3f400",
    "metadata": {
     "editable": true
    },
@@ -940,13 +3287,13 @@
   },
   {
    "cell_type": "markdown",
-   "id": "966a0291",
+   "id": "7b431efc",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto6\"></div>\n",
+    "<div id=\"_auto11\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
@@ -966,14 +3313,14 @@
     "           b^2_2 \\\\\n",
     "           b^2_3 \\\\\n",
     "          \\end{array}\\right]\\right).\n",
-    "\\label{_auto6} \\tag{11}\n",
+    "\\label{_auto11} \\tag{16}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "2861079a",
+   "id": "d129b057",
    "metadata": {
     "editable": true
    },
@@ -985,26 +3332,26 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d29ad367",
+   "id": "7af14562",
    "metadata": {
     "editable": true
    },
    "source": [
     "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto7\"></div>\n",
+    "<div id=\"_auto12\"></div>\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
     " y^2_i = f_2\\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\\Bigr) = \n",
     " f_2\\left(\\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\\right).\n",
-    "\\label{_auto7} \\tag{12}\n",
+    "\\label{_auto12} \\tag{17}\n",
     "\\end{equation}\n",
     "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "53138f1b",
+   "id": "0b7e127c",
    "metadata": {
     "editable": true
    },
@@ -1018,7 +3365,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8f1c228e",
+   "id": "91266ae3",
    "metadata": {
     "editable": true
    },
@@ -1041,7 +3388,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb855c0d",
+   "id": "54728fbd",
    "metadata": {
     "editable": true
    },
@@ -1060,7 +3407,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "949c265a",
+   "id": "17b851fb",
    "metadata": {
     "editable": true
    },
@@ -1072,7 +3419,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "29628583",
+   "id": "6e17f015",
    "metadata": {
     "editable": true
    },
@@ -1082,7 +3429,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24182193",
+   "id": "574fbcd0",
    "metadata": {
     "editable": true
    },
@@ -1094,7 +3441,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d482a6f8",
+   "id": "daa971d1",
    "metadata": {
     "editable": true
    },
@@ -1110,8 +3457,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
-   "id": "35195eee",
+   "execution_count": 30,
+   "id": "c12bc7fe",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1190,943 +3537,6 @@
     "\n",
     "plt.show()"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "9b92dc7a",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## The multilayer  perceptron (MLP)\n",
-    "\n",
-    "The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of\n",
-    "1. A neural network with one or more layers of nodes between the input and the output nodes.\n",
-    "\n",
-    "2. The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.\n",
-    "\n",
-    "3. The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.\n",
-    "\n",
-    "As a convention it is normal to call  a  network with one layer of input units, one layer of hidden\n",
-    "units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.\n",
-    "\n",
-    "For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.\n",
-    "Hereafter we will call the various entities of a layer for nodes.\n",
-    "There are also no connections within a single layer.\n",
-    "\n",
-    "The number of input nodes does not need to equal the number of output\n",
-    "nodes. This applies also to the hidden layers. Each layer may have its\n",
-    "own number of nodes and activation functions.\n",
-    "\n",
-    "The hidden layers have their name from the fact that they are not\n",
-    "linked to observables and as we will see below when we define the\n",
-    "so-called activation $\\hat{z}$, we can think of this as a basis\n",
-    "expansion of the original inputs $\\hat{x}$. The difference however\n",
-    "between neural networks and say linear regression is that now these\n",
-    "basis functions (which will correspond to the weights in the network)\n",
-    "are learned from data. This results in  an important difference between\n",
-    "neural networks and deep learning approaches on one side and methods\n",
-    "like logistic regression or linear regression and their modifications on the other side."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "26655442",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## From one to many layers, the universal approximation theorem\n",
-    "\n",
-    "A neural network with only one layer, what we called the simple\n",
-    "perceptron, is best suited if we have a standard binary model with\n",
-    "clear (linear) boundaries between the outcomes. As such it could\n",
-    "equally well be replaced by standard linear regression or logistic\n",
-    "regression. Networks with one or more hidden layers approximate\n",
-    "systems with more complex boundaries.\n",
-    "\n",
-    "As stated earlier, \n",
-    "an important theorem in studies of neural networks, restated without\n",
-    "proof here, is the [universal approximation\n",
-    "theorem](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf).\n",
-    "\n",
-    "It states that a feed-forward network with a single hidden layer\n",
-    "containing a finite number of neurons can approximate continuous\n",
-    "functions on compact subsets of real functions. The theorem thus\n",
-    "states that simple neural networks can represent a wide variety of\n",
-    "interesting functions when given appropriate parameters. It is the\n",
-    "multilayer feedforward architecture itself which gives neural networks\n",
-    "the potential of being universal approximators."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "fc64cba3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Deriving the back propagation code for a multilayer perceptron model\n",
-    "\n",
-    "As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.\n",
-    "The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible.\n",
-    "This leads us to the famous [back propagation algorithm](https://www.nature.com/articles/323533a0).\n",
-    "\n",
-    "The questions we want to ask are how do changes in the biases and the\n",
-    "weights in our network change the cost function and how can we use the\n",
-    "final output to modify the weights?\n",
-    "\n",
-    "To derive these equations let us start with a plain regression problem\n",
-    "and define our cost function as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "431da3ce",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "{\\cal C}(\\hat{W})  =  \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "613007e9",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where the $t_i$s are our $n$ targets (the values we want to\n",
-    "reproduce), while the outputs of the network after having propagated\n",
-    "all inputs $\\hat{x}$ are given by $y_i$.  Below we will demonstrate\n",
-    "how the basic equations arising from the back propagation algorithm\n",
-    "can be modified in order to study classification problems with $K$\n",
-    "classes."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "cd5293a2",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Definitions\n",
-    "\n",
-    "With our definition of the targets $\\hat{t}$, the outputs of the\n",
-    "network $\\hat{y}$ and the inputs $\\hat{x}$ we\n",
-    "define now the activation $z_j^l$ of node/neuron/unit $j$ of the\n",
-    "$l$-th layer as a function of the bias, the weights which add up from\n",
-    "the previous layer $l-1$ and the forward passes/outputs\n",
-    "$\\hat{a}^{l-1}$ from the previous layer as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d8b4f0d4",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "z_j^l = \\sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "7aba185e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "where $b_k^l$ are the biases from layer $l$.  Here $M_{l-1}$\n",
-    "represents the total number of nodes/neurons/units of layer $l-1$. The\n",
-    "figure here illustrates this equation.  We can rewrite this in a more\n",
-    "compact form as the matrix-vector products we discussed earlier,"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "c361d0ad",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{z}^l = \\left(\\hat{W}^l\\right)^T\\hat{a}^{l-1}+\\hat{b}^l.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "1e191428",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "With the activation values $\\hat{z}^l$ we can in turn define the\n",
-    "output of layer $l$ as $\\hat{a}^l = f(\\hat{z}^l)$ where $f$ is our\n",
-    "activation function. In the examples here we will use the sigmoid\n",
-    "function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers\n",
-    "and their nodes.  It means we have"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "6ddcc355",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "a_j^l = f(z_j^l) = \\frac{1}{1+\\exp{-(z_j^l)}}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "6e88ab07",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Derivatives and the chain rule\n",
-    "\n",
-    "From the definition of the activation $z_j^l$ we have"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "cc3f2444",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial z_j^l}{\\partial w_{ij}^l} = a_i^{l-1},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ca738ee5",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "and"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8af44287",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial z_j^l}{\\partial a_i^{l-1}} = w_{ji}^l.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "891c9a61",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "With our definition of the activation function we have that (note that this function depends only on $z_j^l$)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ab795d6c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial a_j^l}{\\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "2187f447",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Derivative of the cost function\n",
-    "\n",
-    "With these definitions we can now compute the derivative of the cost function in terms of the weights.\n",
-    "\n",
-    "Let us specialize to the output layer $l=L$. Our cost function is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "6f41842f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "{\\cal C}(\\hat{W^L})  =  \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - t_i\\right)^2,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "fa9eeb5e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "The derivative of this function with respect to the weights is"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "05028ad2",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L}  =  \\left(a_j^L - t_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ca023599",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "The last partial derivative can easily be computed and reads (by applying the chain rule)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "14fec26a",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "32f33277",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Bringing it together, first back propagation equation\n",
-    "\n",
-    "We have thus"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a2987cad",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L}  =  \\left(a_j^L - t_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "4a52b0c3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Defining"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "3bc93ec2",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - t_j\\right) = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "3bb5312c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "and using the Hadamard product of two vectors we can write this as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f09c731a",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\hat{\\delta}^L = f'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\hat{a}^L)}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f238f8c2",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "This is an important expression. The second term on the right handside\n",
-    "measures how fast the cost function is changing as a function of the $j$th\n",
-    "output activation.  If, for example, the cost function doesn't depend\n",
-    "much on a particular output node $j$, then $\\delta_j^L$ will be small,\n",
-    "which is what we would expect. The first term on the right, measures\n",
-    "how fast the activation function $f$ is changing at a given activation\n",
-    "value $z_j^L$.\n",
-    "\n",
-    "Notice that everything in the above equations is easily computed.  In\n",
-    "particular, we compute $z_j^L$ while computing the behaviour of the\n",
-    "network, and it is only a small additional overhead to compute\n",
-    "$f'(z^L_j)$.  The exact form of the derivative with respect to the\n",
-    "output depends on the form of the cost function.\n",
-    "However, provided the cost function is known there should be little\n",
-    "trouble in calculating"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ac511dcb",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "3847f7e6",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "With the definition of $\\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ea65e479",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L}  =  \\delta_j^La_k^{L-1}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "e51ddf74",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Derivatives in terms of $z_j^L$\n",
-    "\n",
-    "It is also easy to see that our previous equation can be written as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ae461202",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^L =\\frac{\\partial {\\cal C}}{\\partial z_j^L}= \\frac{\\partial {\\cal C}}{\\partial a_j^L}\\frac{\\partial a_j^L}{\\partial z_j^L},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "bcac59ae",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "9335ba62",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L}\\frac{\\partial b_j^L}{\\partial z_j^L}=\\frac{\\partial {\\cal C}}{\\partial b_j^L},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "5960bea5",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "That is, the error $\\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "dfbaf66f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Bringing it together\n",
-    "\n",
-    "We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are\n",
-    "\n",
-    "**The starting equations.**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a0d9cd4b",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto8\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L}  =  \\delta_j^La_k^{L-1},\n",
-    "\\label{_auto8} \\tag{13}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "5e0680db",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "and"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "64c3b16e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto9\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n",
-    "\\label{_auto9} \\tag{14}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "833196e3",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "and"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "14b4b39f",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "<!-- Equation labels as ordinary links -->\n",
-    "<div id=\"_auto10\"></div>\n",
-    "\n",
-    "$$\n",
-    "\\begin{equation}\n",
-    "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L},\n",
-    "\\label{_auto10} \\tag{15}\n",
-    "\\end{equation}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "0b7bb9aa",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "An interesting consequence of the above equations is that when the\n",
-    "activation $a_k^{L-1}$ is small, the gradient term, that is the\n",
-    "derivative of the cost function with respect to the weights, will also\n",
-    "tend to be small. We say then that the weight learns slowly, meaning\n",
-    "that it changes slowly when we minimize the weights via say gradient\n",
-    "descent. In this case we say the system learns slowly.\n",
-    "\n",
-    "Another interesting feature is that is when the activation function,\n",
-    "represented by the sigmoid function here, is rather flat when we move towards\n",
-    "its end values $0$ and $1$ (see the above Python codes). In these\n",
-    "cases, the derivatives of the activation function will also be close\n",
-    "to zero, meaning again that the gradients will be small and the\n",
-    "network learns slowly again.\n",
-    "\n",
-    "We need a fourth equation and we are set. We are going to propagate\n",
-    "backwards in order to the determine the weights and biases. In order\n",
-    "to do so we need to represent the error in the layer before the final\n",
-    "one $L-1$ in terms of the errors in the final output layer."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "544eaaca",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Final back propagating equation\n",
-    "\n",
-    "We have that (replacing $L$ with a general layer $l$)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d2fe95ae",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^l =\\frac{\\partial {\\cal C}}{\\partial z_j^l}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "78d9aa7e",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "e24f3888",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^l =\\sum_k \\frac{\\partial {\\cal C}}{\\partial z_k^{l+1}}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}}=\\sum_k \\delta_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "88fbf723",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "and recalling that"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "04702c25",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "z_j^{l+1} = \\sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b1a6c794",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "with $M_l$ being the number of nodes in layer $l$, we obtain"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "c3e3f6fa",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^l =\\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d0e7310a",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "This is our final equation.\n",
-    "\n",
-    "We are now ready to set up the algorithm for back propagation and learning the weights and biases."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "4d66f8ea",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Setting up the Back propagation algorithm\n",
-    "\n",
-    "The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n",
-    "\n",
-    "First, we set up the input data $\\hat{x}$ and the activations\n",
-    "$\\hat{z}_1$ of the input layer and compute the activation function and\n",
-    "the pertinent outputs $\\hat{a}^1$.\n",
-    "\n",
-    "Secondly, we perform then the feed forward till we reach the output\n",
-    "layer and compute all $\\hat{z}_l$ of the input layer and compute the\n",
-    "activation function and the pertinent outputs $\\hat{a}^l$ for\n",
-    "$l=2,3,\\dots,L$.\n",
-    "\n",
-    "Thereafter we compute the ouput error $\\hat{\\delta}^L$ by computing all"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "e57af7d8",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f78aa920",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,2$ as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "94645be6",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b57fccda",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8cfe6f8a",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "w_{jk}^l\\leftarrow  = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8f4719ad",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "e6ee70a8",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n",
-    "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "340e3e53",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "## Setting up the Back propagation algorithm\n",
-    "\n",
-    "The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n",
-    "\n",
-    "First, we set up the input data $\\boldsymbol{x}$ and the activations\n",
-    "$\\boldsymbol{z}_1$ of the input layer and compute the activation function and\n",
-    "the pertinent outputs $\\boldsymbol{a}^1$.\n",
-    "\n",
-    "Secondly, we perform then the feed forward till we reach the output\n",
-    "layer and compute all $\\boldsymbol{z}_l$ of the input layer and compute the\n",
-    "activation function and the pertinent outputs $\\boldsymbol{a}^l$ for\n",
-    "$l=2,3,\\dots,L$.\n",
-    "\n",
-    "Thereafter we compute the ouput error $\\boldsymbol{\\delta}^L$ by computing all"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b5c97dd4",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "e54f9f8b",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,2$ as"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "022dd3d1",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ef912762",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f223a69a",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "w_{jk}^l\\leftarrow  = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "94c4e93c",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "$$\n",
-    "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "adf5fb95",
-   "metadata": {
-    "editable": true
-   },
-   "source": [
-    "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n",
-    "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training."
-   ]
   }
  ],
  "metadata": {},
diff --git a/doc/src/week40/backupweek40.do.txt b/doc/src/week40/backupweek40.do.txt
new file mode 100644
index 000000000..22899c0e5
--- /dev/null
+++ b/doc/src/week40/backupweek40.do.txt
@@ -0,0 +1,1128 @@
+TITLE: Week 40: Neural networks
+AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo, Norway & Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
+DATE: Week 40, 2023
+
+
+!split
+===== Plan for week 40 =====
+
+* Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.
+* Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.
+
+Reading suggestions for both days: "Aurelien Geron's chapter 10":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" and Hastie et al chapter 11.
+For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
+For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
+
+
+
+!split
+===== Overview video on Stochastic Gradient Descent =====
+
+"What is Stochastic Gradient Descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer"
+
+
+
+
+!split
+===== Practical tips =====
+
+* _Randomize the data when making mini-batches_. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
+
+* _Transform your inputs_. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.
+
+* _Monitor the out-of-sample performance._ Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings.
+	
+* _Adaptive optimization methods don't always have good generalization._ Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.
+
+Geron's text, see chapter 11, has several interesting discussions.
+
+
+
+!split
+===== Automatic differentiation =====
+
+"Automatic differentiation (AD)":"https://en.wikipedia.org/wiki/Automatic_differentiation", 
+also called algorithmic
+differentiation or computational differentiation,is a set of
+techniques to numerically evaluate the derivative of a function
+specified by a computer program. AD exploits the fact that every
+computer program, no matter how complicated, executes a sequence of
+elementary arithmetic operations (addition, subtraction,
+multiplication, division, etc.) and elementary functions (exp, log,
+sin, cos, etc.). By applying the chain rule repeatedly to these
+operations, derivatives of arbitrary order can be computed
+automatically, accurately to working precision, and using at most a
+small constant factor more arithmetic operations than the original
+program.
+
+Automatic differentiation is neither:
+
+* Symbolic differentiation, nor
+* Numerical differentiation (the method of finite differences).
+
+Symbolic differentiation can lead to inefficient code and faces the
+difficulty of converting a computer program into a single expression,
+while numerical differentiation can introduce round-off errors in the
+discretization process and cancellation
+
+
+
+
+!split
+===== Videos on Neural Networks =====
+
+
+"Neural Networks demystified":"https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs"
+
+"Building Neural Networks from scratch":"https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex"
+
+
+
+
+!split
+===== Neural networks =====
+
+Artificial neural networks are computational systems that can learn to
+perform tasks by considering examples, generally without being
+programmed with any task-specific rules. It is supposed to mimic a
+biological system, wherein neurons interact by sending signals in the
+form of mathematical functions between layers. All layers can contain
+an arbitrary number of neurons, and each connection is represented by
+a weight variable.
+
+
+!split
+===== Artificial neurons  =====
+
+The field of artificial neural networks has a long history of
+development, and is closely connected with the advancement of computer
+science and computers in general. A model of artificial neurons was
+first developed by McCulloch and Pitts in 1943 to study signal
+processing in the brain and has later been refined by others. The
+general idea is to mimic neural networks in the human brain, which is
+composed of billions of neurons that communicate with each other by
+sending electrical signals.  Each neuron accumulates its incoming
+signals, which must exceed an activation threshold to yield an
+output. If the threshold is not overcome, the neuron remains inactive,
+i.e. has zero output.
+
+This behaviour has inspired a simple mathematical model for an artificial neuron.
+
+!bt
+\begin{equation}
+ y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
+ label{artificialNeuron}
+\end{equation}
+!et
+Here, the output $y$ of the neuron is the value of its activation function, which have as input
+a weighted sum of signals $x_i, \dots ,x_n$ received by $n$ other neurons.
+
+Conceptually, it is helpful to divide neural networks into four
+categories:
+o general purpose neural networks for supervised learning,
+o neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),
+o neural networks for sequential data such as Recurrent Neural Networks (RNNs), and
+o neural networks for unsupervised learning such as Deep Boltzmann Machines.
+
+
+In natural science, DNNs and CNNs have already found numerous
+applications. In statistical physics, they have been applied to detect
+phase transitions in 2D Ising and Potts models, lattice gauge
+theories, and different phases of polymers, or solving the
+Navier-Stokes equation in weather forecasting.  Deep learning has also
+found interesting applications in quantum physics. Various quantum
+phase transitions can be detected and studied using DNNs and CNNs,
+topological phases, and even non-equilibrium many-body
+localization. Representing quantum states as DNNs quantum state
+tomography are among some of the impressive achievements to reveal the
+potential of DNNs to facilitate the study of quantum systems.
+
+In quantum information theory, it has been shown that one can perform
+gate decompositions with the help of neural. 
+
+The applications are not limited to the natural sciences. There is a
+plethora of applications in essentially all disciplines, from the
+humanities to life science and medicine.
+
+!split
+===== Neural network types =====
+
+An artificial neural network (ANN), is a computational model that
+consists of layers of connected neurons, or nodes or units.  We will
+refer to these interchangeably as units or nodes, and sometimes as
+neurons.
+
+It is supposed to mimic a biological nervous system by letting each
+neuron interact with other neurons by sending signals in the form of
+mathematical functions between layers.  A wide variety of different
+ANNs have been developed, but most of them consist of an input layer,
+an output layer and eventual layers in-between, called *hidden
+layers*. All layers can contain an arbitrary number of nodes, and each
+connection between two nodes is associated with a weight variable.
+
+Neural networks (also called neural nets) are neural-inspired
+nonlinear models for supervised learning.  As we will see, neural nets
+can be viewed as natural, more powerful extensions of supervised
+learning methods such as linear and logistic regression and soft-max
+methods we discussed earlier.
+
+
+!split
+===== Feed-forward neural networks =====
+
+The feed-forward neural network (FFNN) was the first and simplest type
+of ANNs that were devised. In this network, the information moves in
+only one direction: forward through the layers.
+
+Nodes are represented by circles, while the arrows display the
+connections between the nodes, including the direction of information
+flow. Additionally, each arrow corresponds to a weight variable
+(figure to come).  We observe that each node in a layer is connected
+to *all* nodes in the subsequent layer, making this a so-called
+*fully-connected* FFNN.
+
+
+
+!split
+===== Convolutional Neural Network =====
+
+A different variant of FFNNs are *convolutional neural networks*
+(CNNs), which have a connectivity pattern inspired by the animal
+visual cortex. Individual neurons in the visual cortex only respond to
+stimuli from small sub-regions of the visual field, called a receptive
+field. This makes the neurons well-suited to exploit the strong
+spatially local correlation present in natural images. The response of
+each neuron can be approximated mathematically as a convolution
+operation.  (figure to come)
+
+Convolutional neural networks emulate the behaviour of neurons in the
+visual cortex by enforcing a *local* connectivity pattern between
+nodes of adjacent layers: Each node in a convolutional layer is
+connected only to a subset of the nodes in the previous layer, in
+contrast to the fully-connected FFNN.  Often, CNNs consist of several
+convolutional layers that learn local features of the input, with a
+fully-connected layer at the end, which gathers all the local data and
+produces the outputs. They have wide applications in image and video
+recognition.
+
+!split
+===== Recurrent neural networks =====
+
+So far we have only mentioned ANNs where information flows in one
+direction: forward. *Recurrent neural networks* on the other hand,
+have connections between nodes that form directed *cycles*. This
+creates a form of internal memory which are able to capture
+information on what has been calculated before; the output is
+dependent on the previous computations. Recurrent NNs make use of
+sequential information by performing the same task for every element
+in a sequence, where each element depends on previous elements. An
+example of such information is sentences, making recurrent NNs
+especially well-suited for handwriting and speech recognition.
+
+!split
+===== Other types of networks =====
+
+There are many other kinds of ANNs that have been developed. One type
+that is specifically designed for interpolation in multidimensional
+space is the radial basis function (RBF) network. RBFs are typically
+made up of three layers: an input layer, a hidden layer with
+non-linear radial symmetric activation functions and a linear output
+layer (''linear'' here means that each node in the output layer has a
+linear activation function). The layers are normally fully-connected
+and there are no cycles, thus RBFs can be viewed as a type of
+fully-connected FFNN. They are however usually treated as a separate
+type of NN due the unusual activation functions.
+
+!split
+===== Multilayer perceptrons  =====
+
+One uses often so-called fully-connected feed-forward neural networks
+with three or more layers (an input layer, one or more hidden layers
+and an output layer) consisting of neurons that have non-linear
+activation functions.
+
+Such networks are often called *multilayer perceptrons* (MLPs).
+
+!split
+===== Why multilayer perceptrons?  =====
+
+According to the *Universal approximation theorem*, a feed-forward
+neural network with just a single hidden layer containing a finite
+number of neurons can approximate a continuous multidimensional
+function to arbitrary accuracy, assuming the activation function for
+the hidden layer is a _non-constant, bounded and
+monotonically-increasing continuous function_.
+
+Note that the requirements on the activation function only applies to
+the hidden layer, the output nodes are always assumed to be linear, so
+as to not restrict the range of output values.
+
+
+!split
+=====  Illustration of a single perceptropn model and a multi-perceptron model =====
+
+FIGURE: [figures/nns.png, width=600 frac=0.8]  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer.
+
+
+!split
+===== Examples of XOR, OR and AND gates =====
+
+
+
+Let us first try to fit various gates using standard linear
+regression. The gates we are thinking of are the classical XOR, OR and
+AND gates, well-known elements in computer science. The tables here
+show how we can set up the inputs $x_1$ and $x_2$ in order to yield a
+specific target $y_i$.
+
+
+
+
+!bc pycod
+"""
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
+
+import numpy as np
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX  matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX  matrix:{Xinv}")
+
+# The XOR gate 
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR  = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction  for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate 
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR  = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction  for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate 
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND  = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")
+!ec
+
+What is happening here?
+
+!split
+===== Does Logistic Regression do a better Job? =====
+
+!bc pycod
+"""
+Simple code that tests XOR and OR gates with linear regression
+and logistic regression
+"""
+
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LogisticRegression
+import numpy as np
+
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX  matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX  matrix:{Xinv}")
+
+# The XOR gate 
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR  = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction  for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate 
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR  = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction  for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate 
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND  = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")
+
+# Now we change to logistic regression
+
+
+# Logistic Regression
+logreg = LogisticRegression()
+logreg.fit(X, yOR)
+print("Test set accuracy with Logistic Regression for OR gate: {:.2f}".format(logreg.score(X,yOR)))
+
+logreg.fit(X, yXOR)
+print("Test set accuracy with Logistic Regression for XOR gate: {:.2f}".format(logreg.score(X,yXOR)))
+
+
+logreg.fit(X, yAND)
+print("Test set accuracy with Logistic Regression for AND gate: {:.2f}".format(logreg.score(X,yAND)))
+!ec
+
+Not exactly impressive, but somewhat better.
+
+!split
+===== Adding Neural Networks =====
+
+!bc pycod
+
+# and now neural networks with Scikit-Learn and the XOR
+
+from sklearn.neural_network import MLPClassifier
+from sklearn.datasets import make_classification
+X, yXOR = make_classification(n_samples=100, random_state=1)
+FFNN = MLPClassifier(random_state=1, max_iter=300).fit(X, yXOR)
+FFNN.predict_proba(X)
+print(f"Test set accuracy with Feed Forward Neural Network  for XOR gate:{FFNN.score(X, yXOR)}")
+
+!ec
+
+
+
+!split
+===== Mathematical model  =====
+
+The output $y$ is produced via the activation function $f$
+!bt
+\[
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
+\]
+!et
+This function receives $x_i$ as inputs.
+Here the activation $z=(\sum_{i=1}^n w_ix_i+b_i)$. 
+In an FFNN of such neurons, the *inputs* $x_i$ are the *outputs* of
+the neurons in the preceding layer. Furthermore, an MLP is
+fully-connected, which means that each neuron receives a weighted sum
+of the outputs of *all* neurons in the previous layer.
+
+!split
+===== Mathematical model  =====
+
+First, for each node $i$ in the first hidden layer, we calculate a weighted sum $z_i^1$ of the input coordinates $x_j$,
+
+!bt
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
+\end{equation}
+!et
+
+Here $b_i$ is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below.  The value of $z_i^1$ is the
+argument to the activation function $f_i$ of each node $i$, The
+variable $M$ stands for all possible inputs to a given node $i$ in the
+first layer.  We define  the output $y_i^1$ of all neurons in layer 1 as
+
+!bt
+\begin{equation}
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\right)
+ label{outputLayer1}
+\end{equation}
+!et
+
+where we assume that all nodes in the same layer have identical
+activation functions, hence the notation $f$. In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript $l$ for the $l$-th layer,
+
+!bt
+\begin{equation}
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+ label{generalLayer}
+\end{equation}
+!et
+
+where $N_l$ is the number of nodes in layer $l$. When the output of
+all the nodes in the first hidden layer are computed, the values of
+the subsequent layer can be calculated and so forth until the output
+is obtained.
+
+
+
+!split
+===== Mathematical model  =====
+
+The output of neuron $i$ in layer 2 is thus,
+
+!bt
+\begin{align}
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) \\
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+ label{outputLayer2}
+\end{align}
+!et
+where we have substituted $y_k^1$ with the inputs $x_k$. Finally, the ANN output reads
+
+!bt
+\begin{align}
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) \\
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+  + b_1^3\right]
+\end{align}
+!et
+
+!split
+===== Mathematical model  =====
+
+We can generalize this expression to an MLP with $l$ hidden
+layers. The complete functional form is,
+
+!bt
+\begin{align}
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
+ label{completeNN}
+\end{align}
+!et
+
+which illustrates a basic property of MLPs: The only independent
+variables are the input values $x_n$.
+
+!split
+===== Mathematical model  =====
+
+This confirms that an MLP, despite its quite convoluted mathematical
+form, is nothing more than an analytic function, specifically a
+mapping of real-valued vectors $\hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m$.
+
+Furthermore, the flexibility and universality of an MLP can be
+illustrated by realizing that the expression is essentially a nested
+sum of scaled activation functions of the form
+
+!bt
+\begin{equation}
+ f(x) = c_1 f(c_2 x + c_3) + c_4
+\end{equation}
+!et
+
+where the parameters $c_i$ are weights and biases. By adjusting these
+parameters, the activation functions can be shifted up and down or
+left and right, change slope or be rescaled which is the key to the
+flexibility of a neural network.
+
+!split
+=== Matrix-vector notation ===
+
+We can introduce a more convenient notation for the activations in an A NN. 
+
+Additionally, we can represent the biases and activations
+as layer-wise column vectors $\hat{b}_l$ and $\hat{y}_l$, so that the $i$-th element of each vector 
+is the bias $b_i^l$ and activation $y_i^l$ of node $i$ in layer $l$ respectively. 
+
+We have that $\mathrm{W}_l$ is an $N_{l-1} \times N_l$ matrix, while $\hat{b}_l$ and $\hat{y}_l$ are $N_l \times 1$ column vectors. 
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
+!bt
+\begin{equation}
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = 
+ f_2\left(\left[\begin{array}{ccc}
+    w^2_{11} &w^2_{12} &w^2_{13} \\
+    w^2_{21} &w^2_{22} &w^2_{23} \\
+    w^2_{31} &w^2_{32} &w^2_{33} \\
+    \end{array} \right] \cdot
+    \left[\begin{array}{c}
+           y^1_1 \\
+           y^1_2 \\
+           y^1_3 \\
+          \end{array}\right] + 
+    \left[\begin{array}{c}
+           b^2_1 \\
+           b^2_2 \\
+           b^2_3 \\
+          \end{array}\right]\right).
+\end{equation}
+!et
+
+!split
+=== Matrix-vector notation  and activation ===
+
+The activation of node $i$ in layer 2 is
+
+!bt
+\begin{equation}
+ y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
+ f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
+\end{equation}
+!et 
+
+This is not just a convenient and compact notation, but also a useful
+and intuitive way to think about MLPs: The output is calculated by a
+series of matrix-vector multiplications and vector additions that are
+used as input to the activation functions. For each operation
+$\mathrm{W}_l \hat{y}_{l-1}$ we move forward one layer.
+
+
+!split
+=== Activation functions  ===
+
+
+A property that characterizes a neural network, other than its
+connectivity, is the choice of activation function(s).  As described
+in, the following restrictions are imposed on an activation function
+for a FFNN to fulfill the universal approximation theorem
+
+  * Non-constant
+
+  * Bounded
+
+  * Monotonically-increasing
+
+  * Continuous
+
+!split
+=== Activation functions, Logistic and Hyperbolic ones  ===
+
+The second requirement excludes all linear functions. Furthermore, in
+a MLP with only linear activation functions, each layer simply
+performs a linear transformation of its inputs.
+
+Regardless of the number of layers, the output of the NN will be
+nothing but a linear function of the inputs. Thus we need to introduce
+some kind of non-linearity to the NN to be able to fit non-linear
+functions Typical examples are the logistic *Sigmoid*
+
+!bt
+\[
+ f(x) = \frac{1}{1 + e^{-x}},
+\]
+!et
+and the *hyperbolic tangent* function
+!bt
+\[
+ f(x) = \tanh(x)
+\]
+!et
+
+!split
+=== Relevance === 
+
+The *sigmoid* function are more biologically plausible because the
+output of inactive neurons are zero. Such activation function are
+called *one-sided*. However, it has been shown that the hyperbolic
+tangent performs better than the sigmoid for training MLPs.  has
+become the most popular for *deep neural networks*
+
+!bc pycod
+"""The sigmoid function (or the logistic curve) is a 
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
+
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""Sine Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.sin(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sine function')
+
+plt.show()
+
+"""Plots a graph of the squashing function used by a rectified linear
+unit"""
+z = numpy.arange(-2, 2, .1)
+zero = numpy.zeros(len(z))
+y = numpy.max([zero, z], axis=0)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, y)
+ax.set_ylim([-2.0, 2.0])
+ax.set_xlim([-2.0, 2.0])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('Rectified linear unit')
+
+plt.show()
+!ec
+
+
+!split
+===== The multilayer  perceptron (MLP) =====
+
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
+o A neural network with one or more layers of nodes between the input and the output nodes.
+o The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
+o The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
+
+As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
+units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
+
+For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
+Hereafter we will call the various entities of a layer for nodes.
+There are also no connections within a single layer.
+
+The number of input nodes does not need to equal the number of output
+nodes. This applies also to the hidden layers. Each layer may have its
+own number of nodes and activation functions.
+
+The hidden layers have their name from the fact that they are not
+linked to observables and as we will see below when we define the
+so-called activation $\hat{z}$, we can think of this as a basis
+expansion of the original inputs $\hat{x}$. The difference however
+between neural networks and say linear regression is that now these
+basis functions (which will correspond to the weights in the network)
+are learned from data. This results in  an important difference between
+neural networks and deep learning approaches on one side and methods
+like logistic regression or linear regression and their modifications on the other side.
+
+
+!split
+=====  From one to many layers, the universal approximation theorem =====
+
+
+A neural network with only one layer, what we called the simple
+perceptron, is best suited if we have a standard binary model with
+clear (linear) boundaries between the outcomes. As such it could
+equally well be replaced by standard linear regression or logistic
+regression. Networks with one or more hidden layers approximate
+systems with more complex boundaries.
+
+As stated earlier, 
+an important theorem in studies of neural networks, restated without
+proof here, is the "universal approximation
+theorem":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf".
+
+It states that a feed-forward network with a single hidden layer
+containing a finite number of neurons can approximate continuous
+functions on compact subsets of real functions. The theorem thus
+states that simple neural networks can represent a wide variety of
+interesting functions when given appropriate parameters. It is the
+multilayer feedforward architecture itself which gives neural networks
+the potential of being universal approximators.
+
+
+!split
+=====  Deriving the back propagation code for a multilayer perceptron model =====
+
+
+
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
+The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible.
+This leads us to the famous "back propagation algorithm":"https://www.nature.com/articles/323533a0".
+
+The questions we want to ask are how do changes in the biases and the
+weights in our network change the cost function and how can we use the
+final output to modify the weights?
+
+To derive these equations let us start with a plain regression problem
+and define our cost function as
+
+!bt
+\[
+{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
+\]
+!et
+
+where the $t_i$s are our $n$ targets (the values we want to
+reproduce), while the outputs of the network after having propagated
+all inputs $\hat{x}$ are given by $y_i$.  Below we will demonstrate
+how the basic equations arising from the back propagation algorithm
+can be modified in order to study classification problems with $K$
+classes.
+
+!split
+===== Definitions =====
+
+With our definition of the targets $\hat{t}$, the outputs of the
+network $\hat{y}$ and the inputs $\hat{x}$ we
+define now the activation $z_j^l$ of node/neuron/unit $j$ of the
+$l$-th layer as a function of the bias, the weights which add up from
+the previous layer $l-1$ and the forward passes/outputs
+$\hat{a}^{l-1}$ from the previous layer as
+
+
+!bt
+\[
+z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
+\]
+!et
+
+where $b_k^l$ are the biases from layer $l$.  Here $M_{l-1}$
+represents the total number of nodes/neurons/units of layer $l-1$. The
+figure here illustrates this equation.  We can rewrite this in a more
+compact form as the matrix-vector products we discussed earlier,
+
+!bt
+\[
+\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
+\]
+!et
+
+With the activation values $\hat{z}^l$ we can in turn define the
+output of layer $l$ as $\hat{a}^l = f(\hat{z}^l)$ where $f$ is our
+activation function. In the examples here we will use the sigmoid
+function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers
+and their nodes.  It means we have
+
+!bt
+\[
+a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
+\]
+!et
+
+
+!split
+===== Derivatives and the chain rule =====
+
+From the definition of the activation $z_j^l$ we have
+!bt
+\[
+\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
+\]
+!et
+and
+!bt
+\[
+\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
+\]
+!et
+
+With our definition of the activation function we have that (note that this function depends only on $z_j^l$)
+!bt
+\[
+\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
+\]
+!et
+
+
+!split
+===== Derivative of the cost function =====
+
+With these definitions we can now compute the derivative of the cost function in terms of the weights.
+
+Let us specialize to the output layer $l=L$. Our cost function is
+!bt
+\[
+{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
+\]
+!et
+The derivative of this function with respect to the weights is
+
+!bt
+\[
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
+\]
+!et
+The last partial derivative can easily be computed and reads (by applying the chain rule)
+!bt
+\[
+\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
+\]
+!et
+
+
+
+!split
+===== Bringing it together, first back propagation equation =====
+
+We have thus
+!bt
+\[
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
+\]
+!et
+
+Defining
+!bt
+\[
+\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+\]
+!et
+and using the Hadamard product of two vectors we can write this as
+!bt
+\[
+\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
+\]
+!et
+
+This is an important expression. The second term on the right handside
+measures how fast the cost function is changing as a function of the $j$th
+output activation.  If, for example, the cost function doesn't depend
+much on a particular output node $j$, then $\delta_j^L$ will be small,
+which is what we would expect. The first term on the right, measures
+how fast the activation function $f$ is changing at a given activation
+value $z_j^L$.
+
+Notice that everything in the above equations is easily computed.  In
+particular, we compute $z_j^L$ while computing the behaviour of the
+network, and it is only a small additional overhead to compute
+$f'(z^L_j)$.  The exact form of the derivative with respect to the
+output depends on the form of the cost function.
+However, provided the cost function is known there should be little
+trouble in calculating
+
+!bt
+\[
+\frac{\partial {\cal C}}{\partial (a_j^L)}
+\]
+!et
+
+With the definition of $\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+!bt
+\[
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
+\]
+!et
+
+!split
+===== Derivatives in terms of $z_j^L$ =====
+
+It is also easy to see that our previous equation can be written as
+
+!bt
+\[
+\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
+\]
+!et
+which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely
+!bt
+\[
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
+\]
+!et
+That is, the error $\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias. 
+!split
+===== Bringing it together =====
+
+We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
+
+!bblock The starting equations
+
+!bt
+\begin{equation}
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
+\end{equation}
+!et
+and
+!bt
+\begin{equation}
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+\end{equation}
+!et
+and
+
+!bt
+\begin{equation}
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
+\end{equation}
+!et
+!eblock
+
+
+An interesting consequence of the above equations is that when the
+activation $a_k^{L-1}$ is small, the gradient term, that is the
+derivative of the cost function with respect to the weights, will also
+tend to be small. We say then that the weight learns slowly, meaning
+that it changes slowly when we minimize the weights via say gradient
+descent. In this case we say the system learns slowly.
+
+Another interesting feature is that is when the activation function,
+represented by the sigmoid function here, is rather flat when we move towards
+its end values $0$ and $1$ (see the above Python codes). In these
+cases, the derivatives of the activation function will also be close
+to zero, meaning again that the gradients will be small and the
+network learns slowly again.
+
+
+
+We need a fourth equation and we are set. We are going to propagate
+backwards in order to the determine the weights and biases. In order
+to do so we need to represent the error in the layer before the final
+one $L-1$ in terms of the errors in the final output layer.
+
+!split
+===== Final back propagating equation =====
+
+We have that (replacing $L$ with a general layer $l$)
+!bt
+\[
+\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
+\]
+!et
+We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have
+
+!bt
+\[
+\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
+\]
+!et
+and recalling that
+!bt
+\[
+z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
+\]
+!et
+with $M_l$ being the number of nodes in layer $l$, we obtain
+!bt
+\[
+\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
+\]
+!et
+This is our final equation.
+
+We are now ready to set up the algorithm for back propagation and learning the weights and biases.
+
+!split
+===== Setting up the Back propagation algorithm =====
+
+
+
+The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
+
+!bblock
+First, we set up the input data $\hat{x}$ and the activations
+$\hat{z}_1$ of the input layer and compute the activation function and
+the pertinent outputs $\hat{a}^1$.
+!eblock
+
+!bblock
+Secondly, we perform then the feed forward till we reach the output
+layer and compute all $\hat{z}_l$ of the input layer and compute the
+activation function and the pertinent outputs $\hat{a}^l$ for
+$l=2,3,\dots,L$.
+!eblock
+
+!bblock
+Thereafter we compute the ouput error $\hat{\delta}^L$ by computing all
+!bt
+\[
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
+\]
+!et
+!eblock
+
+!bblock
+Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as
+!bt
+\[
+\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
+\]
+!et
+!eblock
+
+!bblock
+Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules
+!bt
+\[
+w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
+\]
+!et
+
+!bt
+\[
+b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
+\]
+!et
+!eblock
+
+The parameter $\eta$ is the learning parameter discussed in connection with the gradient descent methods.
+Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
+
+
+!split
+===== Setting up the Back propagation algorithm =====
+
+
+
+The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
+
+!bblock
+First, we set up the input data $\bm{x}$ and the activations
+$\bm{z}_1$ of the input layer and compute the activation function and
+the pertinent outputs $\bm{a}^1$.
+!eblock
+
+!bblock
+Secondly, we perform then the feed forward till we reach the output
+layer and compute all $\bm{z}_l$ of the input layer and compute the
+activation function and the pertinent outputs $\bm{a}^l$ for
+$l=2,3,\dots,L$.
+!eblock
+
+!bblock
+Thereafter we compute the ouput error $\bm{\delta}^L$ by computing all
+!bt
+\[
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
+\]
+!et
+!eblock
+
+!bblock
+Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as
+!bt
+\[
+\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
+\]
+!et
+!eblock
+
+!bblock
+Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules
+!bt
+\[
+w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
+\]
+!et
+
+!bt
+\[
+b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
+\]
+!et
+!eblock
+
+The parameter $\eta$ is the learning parameter discussed in connection with the gradient descent methods.
+Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
+
+
+
+
diff --git a/doc/src/week40/week40.do.txt b/doc/src/week40/week40.do.txt
index 22899c0e5..dcf213c5b 100644
--- a/doc/src/week40/week40.do.txt
+++ b/doc/src/week40/week40.do.txt
@@ -1,19 +1,52 @@
-TITLE: Week 40: Neural networks
+TITLE: Week 40: Gradient descent methods (continued) and start Neural networks
 AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo, Norway & Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
-DATE: Week 40, 2023
+DATE: October 2-6, 2023
+
 
 
 !split
-===== Plan for week 40 =====
+===== Plans for week 40 =====
 
-* Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.
-* Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.
 
-Reading suggestions for both days: "Aurelien Geron's chapter 10":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" and Hastie et al chapter 11.
-For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
-For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
+!bblock  Material for the active learning sessions on Tuesday and Wednesday
+  * Work on project 1 and discussions on how to structure your report
+  * No weekly exercises for week 40, project work only
+  * "Video on how to write scientific reports recorded during one of the lab sessions":"https://youtu.be/tVW1ZDmZnwM"
+  * A general guideline can be found at URL:"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md".
+!eblock  
 
 
+!bblock Material for the lecture on Thursday October 5, 2023
+  * Stochastic Gradient descent with examples and automatic differentiation
+  * Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.  
+  * Readings and Videos:
+    * These lecture notes
+    * For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.
+    * "Aurelien Geron's chapter 4 on stochastic gradient descent":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf"
+    * For neural networks we recommend Goodfellow et al chapter 6.
+    * "Video on gradient descent":"https://www.youtube.com/watch?v=sDv4f4s2SB8"
+    * "Video on stochastic gradient descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI"
+    * "Neural Networks demystified":"https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs"
+    * "Building Neural Networks from scratch":"https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex"
+!eblock
+
+
+
+!split
+===== Summary from last week, using gradient descent methods, limitations =====
+
+* _Gradient descent (GD) finds local minima of our function_. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
+
+* _GD is sensitive to initial conditions_. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
+
+* _Gradients are computationally expensive to calculate for large datasets_. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called ``mini batches''. This has the added benefit of introducing stochasticity into our algorithm.
+
+* _GD is very sensitive to choices of learning rates_. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape.
+
+* _GD treats all directions in parameter space uniformly._ Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. 
+	
+* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.
+
 
 !split
 ===== Overview video on Stochastic Gradient Descent =====
@@ -21,6 +54,527 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.
 "What is Stochastic Gradient Descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer"
 
 
+!split
+=====  Batches and mini-batches =====
+
+In gradient descent we compute the cost function and its gradient for all data points we have.
+
+In large-scale applications such as the "ILSVRC challenge":"https://www.image-net.org/challenges/LSVRC/", the
+training data can have on order of millions of examples. Hence, it
+seems wasteful to compute the full cost function over the entire
+training set in order to perform only a single parameter update. A
+very common approach to addressing this challenge is to compute the
+gradient over batches of the training data. For example, a typical batch could contain some thousand  examples from
+an  entire training set of several millions. This batch is then used to
+perform a parameter update.
+
+!split
+===== Stochastic Gradient Descent (SGD) =====
+
+In stochastic gradient descent, the extreme case is the case where we
+have only one batch, that is we include the whole data set.
+
+This process is called Stochastic Gradient
+Descent (SGD) (or also sometimes on-line gradient descent). This is
+relatively less common to see because in practice due to vectorized
+code optimizations it can be computationally much more efficient to
+evaluate the gradient for 100 examples, than the gradient for one
+example 100 times. Even though SGD technically refers to using a
+single example at a time to evaluate the gradient, you will hear
+people use the term SGD even when referring to mini-batch gradient
+descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD
+for “Batch gradient descent” are rare to see), where it is usually
+assumed that mini-batches are used. The size of the mini-batch is a
+hyperparameter but it is not very common to cross-validate or bootstrap it. It is
+usually based on memory constraints (if any), or set to some value,
+e.g. 32, 64 or 128. We use powers of 2 in practice because many
+vectorized operation implementations work faster when their inputs are
+sized in powers of 2.
+
+In our notes with  SGD we mean stochastic gradient descent with mini-batches.
+
+
+!split
+===== Stochastic Gradient Descent =====
+
+Stochastic gradient descent (SGD) and variants thereof address some of
+the shortcomings of the Gradient descent method discussed above.
+
+The underlying idea of SGD comes from the observation that the cost
+function, which we want to minimize, can almost always be written as a
+sum over $n$ data points $\{\mathbf{x}_i\}_{i=1}^n$,
+!bt
+\[
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}). 
+\]
+!et
+
+!split
+===== Computation of gradients =====
+
+This in turn means that the gradient can be
+computed as a sum over $i$-gradients 
+!bt
+\[
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
+\]
+!et
+
+Stochasticity/randomness is introduced by only taking the
+gradient on a subset of the data called minibatches.  If there are $n$
+data points and the size of each minibatch is $M$, there will be $n/M$
+minibatches. We denote these minibatches by $B_k$ where
+$k=1,\cdots,n/M$.
+
+
+
+!split
+===== SGD example =====
+As an example, suppose we have $10$ data points $(\mathbf{x}_1,\cdots, \mathbf{x}_{10})$ 
+and we choose to have $M=5$ minibathces,
+then each minibatch contains two data points. In particular we have
+$B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
+(\mathbf{x}_9,\mathbf{x}_{10})$. Note that if you choose $M=1$ you
+have only a single batch with all data points and on the other extreme,
+you may choose $M=n$ resulting in a minibatch for each datapoint, i.e
+$B_k = \mathbf{x}_k$.
+
+The idea is now to approximate the gradient by replacing the sum over
+all data points with a sum over the data points in one the minibatches
+picked at random in each gradient descent step 
+!bt
+\[
+\nabla_{\beta}
+C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
+c_i(\mathbf{x}_i, \mathbf{\beta}).
+\]
+!et
+
+!split
+===== The gradient step =====
+
+Thus a gradient descent step now looks like 
+!bt
+\[
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
+\]
+!et
+
+where $k$ is picked at random with equal
+probability from $[1,n/M]$. An iteration over the number of
+minibathces (n/M) is commonly referred to as an epoch. Thus it is
+typical to choose a number of epochs and for each epoch iterate over
+the number of minibatches, as exemplified in the code below.
+
+!split
+===== Simple example code =====
+
+!bc pycod
+import numpy as np 
+
+n = 100 #100 datapoints 
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+n_epochs = 10 #number of epochs
+
+j = 0
+for epoch in range(1,n_epochs+1):
+    for i in range(m):
+        k = np.random.randint(m) #Pick the k-th minibatch at random
+        #Compute the gradient using the data in minibatch Bk
+        #Compute new suggestion for 
+        j += 1
+!ec
+
+Taking the gradient only on a subset of the data has two important
+benefits. First, it introduces randomness which decreases the chance
+that our opmization scheme gets stuck in a local minima. Second, if
+the size of the minibatches are small relative to the number of
+datapoints ($M <  n$), the computation of the gradient is much
+cheaper since we sum over the datapoints in the $k-th$ minibatch and not
+all $n$ datapoints.
+
+!split
+===== When do we stop? =====
+
+A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the $\beta$ that
+gave the lowest value.
+
+!split
+===== Slightly different approach =====
+
+Another approach is to let the step length $\gamma_j$ depend on the
+number of epochs in such a way that it becomes very small after a
+reasonable time such that we do not move at all. Such approaches are
+also called scaling. There are many such ways to "scale the learning
+rate":"https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1"
+and "discussions here":"https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf". See
+also
+URL:"https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1"
+for a discussion of different scaling functions for the learning rate.
+
+!split
+===== Time decay rate =====
+
+As an example, let $e = 0,1,2,3,\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \cdot m + i$ where $m$ is the number of minibatches and $i=0,\cdots,m-1$. Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$.
+
+In this way we can fix the number of epochs, compute $\beta$ and
+evaluate the cost function at the end. Repeating the computation will
+give a different result since the scheme is random by design. Then we
+pick the final $\beta$ that gives the lowest value of the cost
+function.
+
+!bc pycod
+import numpy as np 
+
+def step_length(t,t0,t1):
+    return t0/(t+t1)
+
+n = 100 #100 datapoints 
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+n_epochs = 500 #number of epochs
+t0 = 1.0
+t1 = 10
+
+gamma_j = t0/t1
+j = 0
+for epoch in range(1,n_epochs+1):
+    for i in range(m):
+        k = np.random.randint(m) #Pick the k-th minibatch at random
+        #Compute the gradient using the data in minibatch Bk
+        #Compute new suggestion for beta
+        t = epoch*m+i
+        gamma_j = step_length(t,t0,t1)
+        j += 1
+
+print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
+!ec
+
+
+
+
+
+
+!split
+===== Code with a Number of Minibatches which varies =====
+
+In the code here we vary the number of mini-batches.
+!bc pycode
+# Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+
+for iter in range(Niterations):
+    gradients = 2.0/n*X.T @ ((X @ theta)-y)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print("theta from own sdg")
+print(theta)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
+
+!ec
+
+
+
+!split
+===== Replace or not =====
+
+In the above code, we have use replacement in setting up the
+mini-batches. The discussion
+"here":"https://sebastianraschka.com/faq/docs/sgd-methods.html" may be
+useful.  
+
+
+!split
+===== Momentum based GD =====
+
+The stochastic gradient descent (SGD) is almost always used with a
+*momentum* or inertia term that serves as a memory of the direction we
+are moving in parameter space.  This is typically implemented as
+follows
+
+!bt
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\end{align}
+!et
+
+where we have introduced a momentum parameter $\gamma$, with
+$0\le\gamma\le 1$, and for brevity we dropped the explicit notation to
+indicate the gradient is to be taken over a different mini-batch at
+each step. We call this algorithm gradient descent with momentum
+(GDM). From these equations, it is clear that $\mathbf{v}_t$ is a
+running average of recently encountered gradients and
+$(1-\gamma)^{-1}$ sets the characteristic time scale for the memory
+used in the averaging procedure. Consistent with this, when
+$\gamma=0$, this just reduces down to ordinary SGD as discussed
+earlier. An equivalent way of writing the updates is
+
+!bt
+\[
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+\]
+!et
+where we have defined $\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}$.
+
+!split
+===== More on momentum based approaches =====
+
+Let us try to get more intuition from these equations. It is helpful
+to consider a simple physical analogy with a particle of mass $m$
+moving in a viscous medium with drag coefficient $\mu$ and potential
+$E(\mathbf{w})$. If we denote the particle's position by $\mathbf{w}$,
+then its motion is described by
+
+!bt
+\[
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+\]
+!et
+
+We can discretize this equation in the usual way to get
+
+!bt
+\[
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+\]
+!et
+
+Rearranging this equation, we can rewrite this as
+
+!bt
+\[
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+\]
+!et
+
+!split
+===== Momentum parameter =====
+
+Notice that this equation is identical to previous one if we identify
+the position of the particle, $\mathbf{w}$, with the parameters
+$\boldsymbol{\theta}$. This allows us to identify the momentum
+parameter and learning rate with the mass of the particle and the
+viscous drag as:
+
+!bt
+\[
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+\]
+!et
+
+Thus, as the name suggests, the momentum parameter is proportional to
+the mass of the particle and effectively provides inertia.
+Furthermore, in the large viscosity/small learning rate limit, our
+memory time scales as $(1-\gamma)^{-1} \approx m/(\mu \Delta t)$.
+
+Why is momentum useful? SGD momentum helps the gradient descent
+algorithm gain speed in directions with persistent but small gradients
+even in the presence of stochasticity, while suppressing oscillations
+in high-curvature directions. This becomes especially important in
+situations where the landscape is shallow and flat in some directions
+and narrow and steep in others. It has been argued that first-order
+methods (with appropriate initial conditions) can perform comparable
+to more expensive second order methods, especially in the context of
+complex deep learning models.
+
+These beneficial properties of momentum can sometimes become even more
+pronounced by using a slight modification of the classical momentum
+algorithm called Nesterov Accelerated Gradient (NAG).
+
+In the NAG algorithm, rather than calculating the gradient at the
+current parameters, $\nabla_\theta E(\boldsymbol{\theta}_t)$, one
+calculates the gradient at the expected value of the parameters given
+our current momentum, $\nabla_\theta E(\boldsymbol{\theta}_t +\gamma
+\mathbf{v}_{t-1})$. This yields the NAG update rule
+
+!bt
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\end{align}
+!et
+
+One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\gamma$.
+
+
+!split
+===== Second moment of the gradient =====
+
+
+In stochastic gradient descent, with and without momentum, we still
+have to specify a schedule for tuning the learning rates $\eta_t$
+as a function of time.  As discussed in the context of Newton's
+method, this presents a number of dilemmas. The learning rate is
+limited by the steepest direction which can change depending on the
+current position in the landscape. To circumvent this problem, ideally
+our algorithm would keep track of curvature and take large steps in
+shallow, flat directions and small steps in steep, narrow directions.
+Second-order methods accomplish this by calculating or approximating
+the Hessian and normalizing the learning rate by the
+curvature. However, this is very computationally expensive for
+extremely large models. Ideally, we would like to be able to
+adaptively change the step size to match the landscape without paying
+the steep computational price of calculating or approximating
+Hessians.
+
+Recently, a number of methods have been introduced that accomplish
+this by tracking not only the gradient, but also the second moment of
+the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
+"ADAM":"https://arxiv.org/abs/1412.6980".
+
+!split
+===== RMS prop =====
+
+In RMS prop, in addition to keeping a running average of the first
+moment of the gradient, we also keep track of the second moment
+denoted by $\mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2]$. The update rule
+for RMS prop is given by
+
+!bt
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+!et
+
+where $\beta$ controls the averaging time of the second moment and is
+typically taken to be about $\beta=0.9$, $\eta_t$ is a learning rate
+typically chosen to be $10^{-3}$, and $\epsilon\sim 10^{-8} $ is a
+small regularization constant to prevent divergences. Multiplication
+and division by vectors is understood as an element-wise operation. It
+is clear from this formula that the learning rate is reduced in
+directions where the norm of the gradient is consistently large. This
+greatly speeds up the convergence by allowing us to use a larger
+learning rate for flat directions.
+
+
+!split
+===== "ADAM optimizer":"https://arxiv.org/abs/1412.6980" =====
+
+A related algorithm is the ADAM optimizer. In
+"ADAM":"https://arxiv.org/abs/1412.6980", we keep a running average of
+both the first and second moment of the gradient and use this
+information to adaptively change the learning rate for different
+parameters.  The method isefficient when working with large
+problems involving lots data and/or parameters.  It is a combination of the
+gradient descent with momentum algorithm and the RMSprop algorithm
+discussed above.
+
+In addition to keeping a running average of the first and
+second moments of the gradient
+(i.e. $\mathbf{m}_t=\mathbb{E}[\mathbf{g}_t]$ and
+$\mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t]$, respectively), ADAM
+performs an additional bias correction to account for the fact that we
+are estimating the first two moments of the gradient using a running
+average (denoted by the hats in the update rule below). The update
+rule for ADAM is given by (where multiplication and division are once
+again understood to be element-wise operations below)
+
+!bt
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\bm{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\bm{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \bm{\mathbf{m}}_t \over \sqrt{\bm{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\end{align}
+!et
+
+where $\beta_1$ and $\beta_2$ set the memory lifetime of the first and
+second moment and are typically taken to be $0.9$ and $0.99$
+respectively, and $\eta$ and $\epsilon$ are identical to RMSprop.
+
+Like in RMSprop, the effective step size of a parameter depends on the
+magnitude of its gradient squared.  To understand this better, let us
+rewrite this expression in terms of the variance
+$\boldsymbol{\sigma}_t^2 = \bm{\mathbf{s}}_t -
+(\bm{\mathbf{m}}_t)^2$. Consider a single parameter $\theta_t$. The
+update rule for this parameter is given by
+
+!bt
+\[
+\Delta \theta_{t+1}= -\eta_t { \bm{m}_t \over \sqrt{\sigma_t^2 +  m_t^2 }+\epsilon}.
+\]
+!et
+
+!split
+===== Algorithms and codes for Adagrad, RMSprop and Adam =====
+
+The algorithms we have implemented are well described in the text by "Goodfellow, Bengio and Courville, chapter 8":"https://www.deeplearningbook.org/contents/optimization.html".
+
+The codes which implement these algorithms are discussed after our presentation of automatic differentiation.
 
 
 !split
@@ -67,20 +621,929 @@ discretization process and cancellation
 
 
 
+Python has tools for so-called _automatic differentiation_.
+Consider the following example
+!bt
+\[
+f(x) = \sin\left(2\pi x + x^2\right)
+\]
+!et
+which has the following derivative
+!bt
+\[
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) 
+\]
+!et
+Using _autograd_ we have
+
+!bc pycod
+import autograd.numpy as np
+
+# To do elementwise differentiation:
+from autograd import elementwise_grad as egrad 
+
+# To plot:
+import matplotlib.pyplot as plt 
+
+
+def f(x):
+    return np.sin(2*np.pi*x + x**2)
+
+def f_grad_analytic(x):
+    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)
+
+# Do the comparison:
+x = np.linspace(0,1,1000)
+
+f_grad = egrad(f)
+
+computed = f_grad(x)
+analytic = f_grad_analytic(x)
+
+plt.title('Derivative computed from Autograd compared with the analytical derivative')
+plt.plot(x,computed,label='autograd')
+plt.plot(x,analytic,label='analytic')
+
+plt.xlabel('x')
+plt.ylabel('y')
+plt.legend()
+
+plt.show()
+
+print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))
+!ec
+
+!split 
+===== Using autograd =====
+
+Here we
+experiment with what kind of functions Autograd is capable
+of finding the gradient of. The following Python functions are just
+meant to illustrate what Autograd can do, but please feel free to
+experiment with other, possibly more complicated, functions as well.
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+
+def f1(x):
+    return x**3 + 1
+
+f1_grad = grad(f1)
+
+# Remember to send in float as argument to the computed gradient from Autograd!
+a = 1.0
+
+# See the evaluated gradient at a using autograd:
+print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
+
+# Compare with the analytical derivative, that is f1'(x) = 3*x**2 
+grad_analytical = 3*a**2
+print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
+!ec
+
 
 !split
-===== Videos on Neural Networks =====
+===== Autograd with more complicated functions =====
+
+To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f2(x1,x2):
+    return 3*x1**3 + x2*(x1 - 5) + 1
+
+# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
+f2_grad_x1 = grad(f2,0)
+
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
+
+x1 = 1.0
+x2 = 3.0 
+
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
+
+# Compare with the analytical derivatives:
+
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical = 9*x1**2 + x2
+
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 - 5
+
+# See the evaluated derivations:
+print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+
+print()
+
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+!ec
+
+Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
+
+
+!split
+=====  More complicated functions using the elements of their arguments directly =====
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f3(x): # Assumes x is an array of length 5 or higher
+    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
+
+f3_grad = grad(f3)
+
+x = np.linspace(0,4,5)
+
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
+
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
+!ec
+
+Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+
+!split 
+===== Functions using mathematical functions from Numpy =====
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f4(x):
+    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+
+f4_grad = grad(f4)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+
+# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
+f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
+
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
+!ec
+
+
+!split
+===== More autograd =====
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f5(x):
+    if x >= 0:
+        return x**2
+    else:
+        return -3*x + 1
+
+f5_grad = grad(f5)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+!ec
+
+
+!split
+===== And  with loops =====
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f6_for(x):
+    val = 0
+    for i in range(10):
+        val = val + x**i
+    return val
+
+def f6_while(x):
+    val = 0
+    i = 0
+    while i < 10:
+        val = val + x**i
+        i = i + 1
+    return val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = 0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+!ec
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9
+# The analytical derivative is: sum(i*x**(i-1)) 
+f6_grad_analytical = 0
+for i in range(10):
+    f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+!ec
+
+!split
+===== Using recursion =====
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+
+def f7(n): # Assume that n is an integer
+    if n == 1 or n == 0:
+        return 1
+    else:
+        return n*f7(n-1)
+
+f7_grad = grad(f7)
+
+n = 2.0
+
+print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
+
+# The function f7 is an implementation of the factorial of n.
+# By using the product rule, one can find that the derivative is:
+
+f7_grad_analytical = 0
+for i in range(int(n)-1):
+    tmp = 1
+    for k in range(int(n)-1):
+        if k != i:
+            tmp *= (n - k)
+    f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+
+!ec
+Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
+
+!split
+===== Unsupported functions =====
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
+Assigning a value to the variable being differentiated with respect to
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f8(x): # Assume x is an array
+    x[2] = 3
+    return x*2
+
+f8_grad = grad(f8)
+
+x = 8.4
+
+print("The derivative of f8 is:",f8_grad(x))
+!ec
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
+
+!split
+===== The syntax a.dot(b) when finding the dot product =====
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f9(a): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([1.0,0.0])
+
+print("The derivative of f9 is:",f9_grad(x))
+!ec
+
+Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+def f9_alternative(x): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2
+
+f9_alternative_grad = grad(f9_alternative)
+
+x = np.array([3.0,0.0])
+
+print("The gradient of f9 is:",f9_alternative_grad(x))
+
+# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively
+# w.r.t x is (b_1, b_2).
+!ec
+
+!split
+===== Recommended to avoid =====
+The documentation recommends to avoid inplace operations such as
+!bc pycod
+a += b
+a -= b
+a*= b
+a /=b
+!ec
+
+
+!split
+=====  Using Autograd with OLS =====
+
+We conclude the part on optmization by showing how we can make codes
+for linear regression and logistic regression using _autograd_. The
+first example shows results with ordinary leats squares.
+
+!bc pycod
+# Using Autograd to calculate gradients for OLS
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+def CostOLS(beta):
+    return (1.0/n)*np.sum((y-X @ beta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+# define the gradient
+training_gradient = grad(CostOLS)
+
+for iter in range(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
+
+!ec
+
+
+!split
+===== Same code but now with momentum gradient descent =====
+!bc pycod
+# Using Autograd to calculate gradients for OLS
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+def CostOLS(beta):
+    return (1.0/n)*np.sum((y-X @ beta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x#+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 30
+
+# define the gradient
+training_gradient = grad(CostOLS)
+
+for iter in range(Niterations):
+    gradients = training_gradient(theta)
+    theta -= eta*gradients
+    print(iter,gradients[0],gradients[1])
+print("theta from own gd")
+print(theta)
+
+# Now improve with momentum gradient descent
+change = 0.0
+delta_momentum = 0.3
+for iter in range(Niterations):
+    # calculate gradient
+    gradients = training_gradient(theta)
+    # calculate update
+    new_change = eta*gradients+delta_momentum*change
+    # take a step
+    theta -= new_change
+    # save the change
+    change = new_change
+    print(iter,gradients[0],gradients[1])
+print("theta from own gd wth momentum")
+print(theta)
+
+!ec
+
+!split
+===== But noen of these can compete with Newton's method =====
+
+!bc pycod
+# Using Newton's method
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+def CostOLS(beta):
+    return (1.0/n)*np.sum((y-X @ beta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(beta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+# Note that here the Hessian does not depend on the parameters beta
+invH = np.linalg.pinv(H)
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+beta = np.random.randn(2,1)
+Niterations = 5
+
+# define the gradient
+training_gradient = grad(CostOLS)
+
+for iter in range(Niterations):
+    gradients = training_gradient(beta)
+    beta -= invH @ gradients
+    print(iter,gradients[0],gradients[1])
+print("beta from own Newton code")
+print(beta)
+!ec
+
+
+!split
+===== Including Stochastic Gradient Descent with Autograd =====
+In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using _autograd_.
+
+!bc pycod
+# Using Autograd to calculate gradients using SGD
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+    return np.sum((y-X @ theta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 1000
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+
+for iter in range(Niterations):
+    gradients = (1.0/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+# Can you figure out a better way of setting up the contributions to each batch?
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (1.0/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print("theta from own sdg")
+print(theta)
+
+
+!ec
+
+
+!split
+===== Same code but now with momentum gradient descent =====
+!bc pycod
+# Using Autograd to calculate gradients using SGD
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+    return np.sum((y-X @ theta)**2)
+
+n = 100
+x = 2*np.random.rand(n,1)
+y = 4+3*x+np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+# Hessian matrix
+H = (2.0/n)* XT_X
+EigValues, EigVectors = np.linalg.eig(H)
+print(f"Eigenvalues of Hessian Matrix:{EigValues}")
+
+theta = np.random.randn(2,1)
+eta = 1.0/np.max(EigValues)
+Niterations = 100
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+
+for iter in range(Niterations):
+    gradients = (1.0/n)*training_gradient(y, X, theta)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+change = 0.0
+delta_momentum = 0.3
+
+for epoch in range(n_epochs):
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (1.0/M)*training_gradient(yi, xi, theta)
+        eta = learning_schedule(epoch*m+i)
+        # calculate update
+        new_change = eta*gradients+delta_momentum*change
+        # take a step
+        theta -= new_change
+        # save the change
+        change = new_change
+print("theta from own sdg with momentum")
+print(theta)
+!ec
+
+
+!split
+===== Similar (second order function now) problem but now with AdaGrad =====
+!bc pycod
+# Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+    return np.sum((y-X @ theta)**2)
+
+n = 1000
+x = np.random.rand(n,1)
+y = 2.0+3*x +4*x*x
+
+X = np.c_[np.ones((n,1)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+# Define parameters for Stochastic Gradient Descent
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+# Guess for unknown parameters theta
+theta = np.random.randn(3,1)
+
+# Value for learning rate
+eta = 0.01
+# Including AdaGrad parameter to avoid possible division by zero
+delta  = 1e-8
+for epoch in range(n_epochs):
+    Giter = 0.0
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (1.0/M)*training_gradient(yi, xi, theta)
+        Giter += gradients*gradients
+        update = gradients*eta/(delta+np.sqrt(Giter))
+        theta -= update
+print("theta from own AdaGrad")
+print(theta)
+
+
+!ec
+
+Running this code we note an almost perfect agreement with the results from matrix inversion.
+
+!split
+=====  RMSprop for adaptive learning rate with Stochastic Gradient Descent =====
+!bc pycod
+# Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+    return np.sum((y-X @ theta)**2)
+
+n = 1000
+x = np.random.rand(n,1)
+y = 2.0+3*x +4*x*x# +np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+# Define parameters for Stochastic Gradient Descent
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+# Guess for unknown parameters theta
+theta = np.random.randn(3,1)
+
+# Value for learning rate
+eta = 0.01
+# Value for parameter rho
+rho = 0.99
+# Including AdaGrad parameter to avoid possible division by zero
+delta  = 1e-8
+for epoch in range(n_epochs):
+    Giter = 0.0
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (1.0/M)*training_gradient(yi, xi, theta)
+	# Accumulated gradient
+	# Scaling with rho the new and the previous results
+        Giter = (rho*Giter+(1-rho)*gradients*gradients)
+	# Taking the diagonal only and inverting
+        update = gradients*eta/(delta+np.sqrt(Giter))
+	# Hadamard product
+        theta -= update
+print("theta from own RMSprop")
+print(theta)
+!ec
+
+!split
+===== And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf" =====
+
+!bc pycod
+# Using Autograd to calculate gradients using RMSprop  and Stochastic Gradient descent
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+    return np.sum((y-X @ theta)**2)
+
+n = 1000
+x = np.random.rand(n,1)
+y = 2.0+3*x +4*x*x# +np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+# Define parameters for Stochastic Gradient Descent
+n_epochs = 50
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+# Guess for unknown parameters theta
+theta = np.random.randn(3,1)
+
+# Value for learning rate
+eta = 0.01
+# Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980
+beta1 = 0.9
+beta2 = 0.999
+# Including AdaGrad parameter to avoid possible division by zero
+delta  = 1e-7
+iter = 0
+for epoch in range(n_epochs):
+    first_moment = 0.0
+    second_moment = 0.0
+    iter += 1
+    for i in range(m):
+        random_index = M*np.random.randint(m)
+        xi = X[random_index:random_index+M]
+        yi = y[random_index:random_index+M]
+        gradients = (1.0/M)*training_gradient(yi, xi, theta)
+        # Computing moments first
+        first_moment = beta1*first_moment + (1-beta1)*gradients
+        second_moment = beta2*second_moment+(1-beta2)*gradients*gradients
+        first_term = first_moment/(1.0-beta1**iter)
+        second_term = second_moment/(1.0-beta2**iter)
+	# Scaling with rho the new and the previous results
+        update = eta*first_term/(np.sqrt(second_term)+delta)
+        theta -= update
+print("theta from own ADAM")
+print(theta)
+!ec
+
+!split
+===== And Logistic Regression =====
+
+!bc pycod
+import autograd.numpy as np
+from autograd import grad
+
+def sigmoid(x):
+    return 0.5 * (np.tanh(x / 2.) + 1)
+
+def logistic_predictions(weights, inputs):
+    # Outputs probability of a label being true according to logistic model.
+    return sigmoid(np.dot(inputs, weights))
+
+def training_loss(weights):
+    # Training loss is the negative log-likelihood of the training labels.
+    preds = logistic_predictions(weights, inputs)
+    label_probabilities = preds * targets + (1 - preds) * (1 - targets)
+    return -np.sum(np.log(label_probabilities))
+
+# Build a toy dataset.
+inputs = np.array([[0.52, 1.12,  0.77],
+                   [0.88, -1.08, 0.15],
+                   [0.52, 0.06, -1.30],
+                   [0.74, -2.49, 1.39]])
+targets = np.array([True, True, False, True])
+
+# Define a function that returns gradients of training loss using Autograd.
+training_gradient_fun = grad(training_loss)
+
+# Optimize weights using gradient descent.
+weights = np.array([0.0, 0.0, 0.0])
+print("Initial loss:", training_loss(weights))
+for i in range(100):
+    weights -= training_gradient_fun(weights) * 0.01
+
+print("Trained loss:", training_loss(weights))
+!ec
+
+
+!split
+===== Introducing "JAX":"https://jax.readthedocs.io/en/latest/" =====
+
+Presently, instead of using _autograd_, we recommend using "JAX":"https://jax.readthedocs.io/en/latest/"
+
+_JAX_ is Autograd and "XLA (Accelerated Linear Algebra))":"https://www.tensorflow.org/xla",
+brought together for high-performance numerical computing and machine learning research.
+It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.
+
+Here's a simple example on how you can use _JAX_ to compute the derivate of the logistic function.
+
+!bc pycod
+import jax.numpy as jnp
+from jax import grad, jit, vmap
+
+def sum_logistic(x):
+  return jnp.sum(1.0 / (1.0 + jnp.exp(-x)))
+
+x_small = jnp.arange(3.)
+derivative_fn = grad(sum_logistic)
+print(derivative_fn(x_small))
+
+!ec
+
+
+
 
 
-"Neural Networks demystified":"https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs"
 
-"Building Neural Networks from scratch":"https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex"
 
 
 
 
 !split
-===== Neural networks =====
+===== Introduction to Neural networks =====
 
 Artificial neural networks are computational systems that can learn to
 perform tasks by considering examples, generally without being
@@ -259,7 +1722,7 @@ as to not restrict the range of output values.
 
 
 !split
-=====  Illustration of a single perceptropn model and a multi-perceptron model =====
+=====  Illustration of a single perceptron model and a multi-perceptron model =====
 
 FIGURE: [figures/nns.png, width=600 frac=0.8]  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer.
 
@@ -684,445 +2147,3 @@ plt.show()
 !ec
 
 
-!split
-===== The multilayer  perceptron (MLP) =====
-
-The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
-o A neural network with one or more layers of nodes between the input and the output nodes.
-o The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
-o The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
-
-As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
-units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-
-For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
-
-The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
-
-The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation $\hat{z}$, we can think of this as a basis
-expansion of the original inputs $\hat{x}$. The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in  an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
-
-
-!split
-=====  From one to many layers, the universal approximation theorem =====
-
-
-A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
-
-As stated earlier, 
-an important theorem in studies of neural networks, restated without
-proof here, is the "universal approximation
-theorem":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf".
-
-It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
-
-
-!split
-=====  Deriving the back propagation code for a multilayer perceptron model =====
-
-
-
-As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous "back propagation algorithm":"https://www.nature.com/articles/323533a0".
-
-The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
-
-To derive these equations let us start with a plain regression problem
-and define our cost function as
-
-!bt
-\[
-{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
-\]
-!et
-
-where the $t_i$s are our $n$ targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs $\hat{x}$ are given by $y_i$.  Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with $K$
-classes.
-
-!split
-===== Definitions =====
-
-With our definition of the targets $\hat{t}$, the outputs of the
-network $\hat{y}$ and the inputs $\hat{x}$ we
-define now the activation $z_j^l$ of node/neuron/unit $j$ of the
-$l$-th layer as a function of the bias, the weights which add up from
-the previous layer $l-1$ and the forward passes/outputs
-$\hat{a}^{l-1}$ from the previous layer as
-
-
-!bt
-\[
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
-\]
-!et
-
-where $b_k^l$ are the biases from layer $l$.  Here $M_{l-1}$
-represents the total number of nodes/neurons/units of layer $l-1$. The
-figure here illustrates this equation.  We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-
-!bt
-\[
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-\]
-!et
-
-With the activation values $\hat{z}^l$ we can in turn define the
-output of layer $l$ as $\hat{a}^l = f(\hat{z}^l)$ where $f$ is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers
-and their nodes.  It means we have
-
-!bt
-\[
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-\]
-!et
-
-
-!split
-===== Derivatives and the chain rule =====
-
-From the definition of the activation $z_j^l$ we have
-!bt
-\[
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-\]
-!et
-and
-!bt
-\[
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
-\]
-!et
-
-With our definition of the activation function we have that (note that this function depends only on $z_j^l$)
-!bt
-\[
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
-\]
-!et
-
-
-!split
-===== Derivative of the cost function =====
-
-With these definitions we can now compute the derivative of the cost function in terms of the weights.
-
-Let us specialize to the output layer $l=L$. Our cost function is
-!bt
-\[
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-\]
-!et
-The derivative of this function with respect to the weights is
-
-!bt
-\[
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
-\]
-!et
-The last partial derivative can easily be computed and reads (by applying the chain rule)
-!bt
-\[
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
-\]
-!et
-
-
-
-!split
-===== Bringing it together, first back propagation equation =====
-
-We have thus
-!bt
-\[
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
-\]
-!et
-
-Defining
-!bt
-\[
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\]
-!et
-and using the Hadamard product of two vectors we can write this as
-!bt
-\[
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-\]
-!et
-
-This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node $j$, then $\delta_j^L$ will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function $f$ is changing at a given activation
-value $z_j^L$.
-
-Notice that everything in the above equations is easily computed.  In
-particular, we compute $z_j^L$ while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-$f'(z^L_j)$.  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-
-!bt
-\[
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-\]
-!et
-
-With the definition of $\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely
-!bt
-\[
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
-\]
-!et
-
-!split
-===== Derivatives in terms of $z_j^L$ =====
-
-It is also easy to see that our previous equation can be written as
-
-!bt
-\[
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
-\]
-!et
-which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely
-!bt
-\[
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
-\]
-!et
-That is, the error $\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias. 
-!split
-===== Bringing it together =====
-
-We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
-
-!bblock The starting equations
-
-!bt
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
-\end{equation}
-!et
-and
-!bt
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\end{equation}
-!et
-and
-
-!bt
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\end{equation}
-!et
-!eblock
-
-
-An interesting consequence of the above equations is that when the
-activation $a_k^{L-1}$ is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
-
-Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values $0$ and $1$ (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
-
-
-
-We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one $L-1$ in terms of the errors in the final output layer.
-
-!split
-===== Final back propagating equation =====
-
-We have that (replacing $L$ with a general layer $l$)
-!bt
-\[
-\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
-\]
-!et
-We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have
-
-!bt
-\[
-\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
-\]
-!et
-and recalling that
-!bt
-\[
-z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
-\]
-!et
-with $M_l$ being the number of nodes in layer $l$, we obtain
-!bt
-\[
-\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
-\]
-!et
-This is our final equation.
-
-We are now ready to set up the algorithm for back propagation and learning the weights and biases.
-
-!split
-===== Setting up the Back propagation algorithm =====
-
-
-
-The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
-
-!bblock
-First, we set up the input data $\hat{x}$ and the activations
-$\hat{z}_1$ of the input layer and compute the activation function and
-the pertinent outputs $\hat{a}^1$.
-!eblock
-
-!bblock
-Secondly, we perform then the feed forward till we reach the output
-layer and compute all $\hat{z}_l$ of the input layer and compute the
-activation function and the pertinent outputs $\hat{a}^l$ for
-$l=2,3,\dots,L$.
-!eblock
-
-!bblock
-Thereafter we compute the ouput error $\hat{\delta}^L$ by computing all
-!bt
-\[
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-\]
-!et
-!eblock
-
-!bblock
-Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as
-!bt
-\[
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-\]
-!et
-!eblock
-
-!bblock
-Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules
-!bt
-\[
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-\]
-!et
-
-!bt
-\[
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-\]
-!et
-!eblock
-
-The parameter $\eta$ is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-
-
-!split
-===== Setting up the Back propagation algorithm =====
-
-
-
-The four equations  above  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
-
-!bblock
-First, we set up the input data $\bm{x}$ and the activations
-$\bm{z}_1$ of the input layer and compute the activation function and
-the pertinent outputs $\bm{a}^1$.
-!eblock
-
-!bblock
-Secondly, we perform then the feed forward till we reach the output
-layer and compute all $\bm{z}_l$ of the input layer and compute the
-activation function and the pertinent outputs $\bm{a}^l$ for
-$l=2,3,\dots,L$.
-!eblock
-
-!bblock
-Thereafter we compute the ouput error $\bm{\delta}^L$ by computing all
-!bt
-\[
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
-\]
-!et
-!eblock
-
-!bblock
-Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as
-!bt
-\[
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-\]
-!et
-!eblock
-
-!bblock
-Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules
-!bt
-\[
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-\]
-!et
-
-!bt
-\[
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-\]
-!et
-!eblock
-
-The parameter $\eta$ is the learning parameter discussed in connection with the gradient descent methods.
-Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.
-
-
-
-