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mhjensen c2be6efa93 update on cnn
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs001.html#___sec0" style="font-size: 80%;"><b>Neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs002.html#___sec1" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs008.html#___sec7" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs009.html#___sec8" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs010.html#___sec9" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs011.html#___sec10" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs012.html#___sec11" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs013.html#___sec12" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs014.html#___sec13" style="font-size: 80%;"><b>Mathematical model</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs015.html#___sec14" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs016.html#___sec15" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation and activation</a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs017.html#___sec16" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Activation functions</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs021.html#___sec20" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs023.html#___sec22" style="font-size: 80%;"><b>Definitions</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs024.html#___sec23" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
<!-- 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="._NeuralNet-bs026.html#___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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs030.html#___sec29" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs032.html#___sec31" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs033.html#___sec32" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs034.html#___sec33" style="font-size: 80%;"><b>The Softmax function</b></a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs036.html#___sec35" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs037.html#___sec36" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs038.html#___sec37" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs039.html#___sec38" style="font-size: 80%;"><b>Layers</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs040.html#___sec39" style="font-size: 80%;"><b>Weights and biases</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs041.html#___sec40" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs042.html#___sec41" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs043.html#___sec42" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs044.html#___sec43" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec44" style="font-size: 80%;"><b>Regularization</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs047.html#___sec46" style="font-size: 80%;"><b>Improving performance</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs048.html#___sec47" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs049.html#___sec48" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs050.html#___sec49" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs051.html#___sec50" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs052.html#___sec51" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs053.html#___sec52" style="font-size: 80%;"><b>Visualization</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs054.html#___sec53" style="font-size: 80%;"><b>Building neural networks in Tensorflow and Keras</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs055.html#___sec54" style="font-size: 80%;"><b>Tensorflow</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs056.html#___sec55" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs057.html#___sec56" style="font-size: 80%;"><b>Using TensorFlow backend</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs058.html#___sec57" style="font-size: 80%;"><b>Optimizing and using gradient descent</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs059.html#___sec58" style="font-size: 80%;"><b>Using Keras</b></a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs062.html#___sec61" style="font-size: 80%;"><b>The derivative of the Logistic funtion</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs063.html#___sec62" style="font-size: 80%;"><b>The RELU function family</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs064.html#___sec63" style="font-size: 80%;"><b>Which activation function should we use?</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs065.html#___sec64" style="font-size: 80%;"><b>A top-down perspective on Neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs066.html#___sec65" style="font-size: 80%;"><b>Limitations of supervised learning with deep networks</b></a></li>
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<h2 id="___sec44" class="anchor">Regularization </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>
We will measure the size of the weights using the so called <em>L2-norm</em>, meaning our cost function becomes:
$$ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2
= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
<p>
i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
<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>
<p>
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