302 lines
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HTML
302 lines
17 KiB
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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#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs004.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs005.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-bs006.html#neural-networks" style="font-size: 80%;"><b>Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs007.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs008.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-bs009.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-bs010.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-bs011.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-bs012.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-bs013.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs014.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-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>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week40-bs018.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-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs024.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs025.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-bs026.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs027.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-bs028.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs029.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-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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week40-bs032.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week40-bs037.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-bs038.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-bs040.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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<!-- 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>
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</ul>
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</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0031"></a>
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<!-- !split -->
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<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>
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<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.
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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.
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This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_self">back propagation algorithm</a>.
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</p>
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<p>The questions we want to ask are how do changes in the biases and the
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weights in our network change the cost function and how can we use the
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final output to modify the weights?
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</p>
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<p>To derive these equations let us start with a plain regression problem
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and define our cost function as
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</p>
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$$
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{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
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$$
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<p>where the $t_i$s are our \( n \) targets (the values we want to
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reproduce), while the outputs of the network after having propagated
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all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
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how the basic equations arising from the back propagation algorithm
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can be modified in order to study classification problems with \( K \)
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classes.
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</p>
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<p>
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