Morten Hjorth-Jensen
406 lines
24 KiB
HTML
406 lines
24 KiB
HTML
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<title>Week 40: From Stochastic Gradient Descent to Neural networks</title>
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'sections': [('Plan for week 40', 2, None, 'plan-for-week-40'),
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('Overview video on Stochastic Gradient Descent',
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2,
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None,
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'overview-video-on-stochastic-gradient-descent'),
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('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
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('Stochastic Gradient Descent (SGD)',
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2,
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None,
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'stochastic-gradient-descent-sgd'),
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('Stochastic Gradient Descent',
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2,
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None,
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('Computation of gradients', 2, None, 'computation-of-gradients'),
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('The gradient step', 2, None, 'the-gradient-step'),
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('Slightly different approach',
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('Program for stochastic gradient',
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2,
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None,
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('Momentum based GD', 2, None, 'momentum-based-gd'),
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('More on momentum based approaches',
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None,
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'more-on-momentum-based-approaches'),
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('Momentum parameter', 2, None, 'momentum-parameter'),
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('Second moment of the gradient',
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None,
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'second-moment-of-the-gradient'),
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('RMS prop', 2, None, 'rms-prop'),
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('Practical tips', 2, None, 'practical-tips'),
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('Automatic differentiation',
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('And with loops', 2, None, 'and-with-loops'),
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('Using recursion', 2, None, 'using-recursion'),
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('The syntax a.dot(b) when finding the dot product',
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('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
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('Does Logistic Regression do a better Job?',
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('Derivative of the cost function',
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<a class="navbar-brand" href="week40-bs.html">Week 40: From Stochastic Gradient Descent to Neural networks</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week40-bs003.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs004.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs005.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs006.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs007.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs008.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs009.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs010.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs011.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs012.html#program-for-stochastic-gradient" style="font-size: 80%;"><b>Program for stochastic gradient</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs013.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs014.html#more-on-momentum-based-approaches" style="font-size: 80%;"><b>More on momentum based approaches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs015.html#momentum-parameter" style="font-size: 80%;"><b>Momentum parameter</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs016.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs017.html#rms-prop" style="font-size: 80%;"><b>RMS prop</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs018.html#adam-optimizer" style="font-size: 80%;"><b>ADAM optimizer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs019.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs020.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs021.html#using-autograd" style="font-size: 80%;"><b>Using autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs022.html#autograd-with-more-complicated-functions" style="font-size: 80%;"><b>Autograd with more complicated functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.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>
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<!-- navigation toc: --> <li><a href="._week40-bs024.html#functions-using-mathematical-functions-from-numpy" style="font-size: 80%;"><b>Functions using mathematical functions from Numpy</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs025.html#more-autograd" style="font-size: 80%;"><b>More autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs026.html#and-with-loops" style="font-size: 80%;"><b>And with loops</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs027.html#using-recursion" style="font-size: 80%;"><b>Using recursion</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#unsupported-functions" style="font-size: 80%;"><b>Unsupported functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs029.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>
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<!-- navigation toc: --> <li><a href="._week40-bs030.html#recommended-to-avoid" style="font-size: 80%;"><b>Recommended to avoid</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs031.html#using-autograd-with-ols" style="font-size: 80%;"><b>Using Autograd with OLS</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs032.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs033.html#and-logistic-regression" style="font-size: 80%;"><b>And Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs034.html#videos-on-neural-networks" style="font-size: 80%;"><b>Videos on Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs035.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs036.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs037.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs038.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs040.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs041.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs043.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs044.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>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs053.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs054.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs055.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs057.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs058.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs059.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>
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<!-- navigation toc: --> <li><a href="._week40-bs060.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>
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<!-- navigation toc: --> <li><a href="._week40-bs061.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs062.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs064.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs065.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs066.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs067.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs068.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
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</ul>
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</li>
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</ul>
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</div>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
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<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>
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$$
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\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
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\tag{7}
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\end{equation}
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$$
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<p>Here \( b_i \) is the so-called bias which is normally needed in
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case of zero activation weights or inputs. How to fix the biases and
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the weights will be discussed below. The value of \( z_i^1 \) is the
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argument to the activation function \( f_i \) of each node \( i \), The
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variable \( M \) stands for all possible inputs to a given node \( i \) in the
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first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
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</p>
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$$
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\begin{equation}
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y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
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\tag{8}
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\end{equation}
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$$
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<p>where we assume that all nodes in the same layer have identical
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activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
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In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
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</p>
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$$
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\begin{equation}
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y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
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\tag{9}
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\end{equation}
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$$
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<p>where \( N_l \) is the number of nodes in layer \( l \). When the output of
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all the nodes in the first hidden layer are computed, the values of
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the subsequent layer can be calculated and so forth until the output
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is obtained.
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</p>
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<p>
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