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mhjensen a20960aa4a updated can notes
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs018.html#___sec17" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions, Logistic and Hyperbolic ones</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs019.html#___sec18" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Relevance</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs020.html#___sec19" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs022.html#___sec21" 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="._NeuralNet-bs025.html#___sec24" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs027.html#___sec26" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs028.html#___sec27" style="font-size: 80%;"><b>Bringing it together</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs029.html#___sec28" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs031.html#___sec30" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model for classification</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs035.html#___sec34" style="font-size: 80%;"><b>Developing a code for doing neural networks with back propagation</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs054.html#___sec53" style="font-size: 80%;"><b>Building neural networks in Tensorflow and Keras</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs058.html#___sec57" style="font-size: 80%;"><b>Optimizing and using gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs060.html#___sec59" style="font-size: 80%;"><b>Which activation function should I use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs061.html#___sec60" style="font-size: 80%;"><b>Is the Logistic activation function (Sigmoid) our choice?</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs065.html#___sec64" style="font-size: 80%;"><b>A top-down perspective on Neural networks</b></a></li>
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<h2 id="___sec25" class="anchor">Bringing it together, first back propagation equation </h2>
<p>
We have thus
$$
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
$$
<p>
Defining
$$
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
$$
and using the Hadamard product of two vectors we can write this as
$$
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
$$
<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>
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
$$
\frac{\partial {\cal C}}{\partial (a_j^L)}
$$
<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
$$
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
$$
<p>
<p>
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