Morten Hjorth-Jensen
401 lines
24 KiB
HTML
401 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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('Momentum based GD', 2, None, 'momentum-based-gd'),
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('More on momentum based approaches',
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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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('RMS prop', 2, None, 'rms-prop'),
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('Practical tips', 2, None, 'practical-tips'),
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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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<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="#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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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0064"></a>
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<!-- !split -->
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<h2 id="bringing-it-together-first-back-propagation-equation" class="anchor">Bringing it together, first back propagation equation </h2>
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<p>We have thus</p>
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$$
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\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
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$$
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<p>Defining</p>
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$$
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\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
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$$
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<p>and using the Hadamard product of two vectors we can write this as</p>
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$$
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\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
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$$
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<p>This is an important expression. The second term on the right handside
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measures how fast the cost function is changing as a function of the $j$th
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output activation. If, for example, the cost function doesn't depend
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much on a particular output node \( j \), then \( \delta_j^L \) will be small,
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which is what we would expect. The first term on the right, measures
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how fast the activation function \( f \) is changing at a given activation
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value \( z_j^L \).
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</p>
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<p>Notice that everything in the above equations is easily computed. In
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particular, we compute \( z_j^L \) while computing the behaviour of the
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network, and it is only a small additional overhead to compute
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\( f'(z^L_j) \). The exact form of the derivative with respect to the
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output depends on the form of the cost function.
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However, provided the cost function is known there should be little
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trouble in calculating
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</p>
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$$
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\frac{\partial {\cal C}}{\partial (a_j^L)}
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$$
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<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>
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$$
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\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
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$$
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