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mhjensen a20960aa4a updated can notes
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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>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs010.html#___sec9" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- 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="#___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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs043.html#___sec42" style="font-size: 80%;"><b>Choose cost function and optimizer</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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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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>
<!-- navigation toc: --> <li><a href="._NeuralNet-bs062.html#___sec61" style="font-size: 80%;"><b>The derivative of the Logistic funtion</b></a></li>
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<!-- 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="___sec19" class="anchor">The multilayer perceptron (MLP) </h2>
<p>
The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
<ol>
<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
<li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
</ol>
As a convention it is normal to call a network with one layer of input units, one layer of hidden
units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
<p>
For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
Hereafter we will call the various entities of a layer for nodes.
There are also no connections within a single layer.
<p>
The number of input nodes does not need to equal the number of output
nodes. This applies also to the hidden layers. Each layer may have its
own number of nodes and activation functions.
<p>
The hidden layers have their name from the fact that they are not
linked to observables and as we will see below when we define the
so-called activation \( \hat{z} \), we can think of this as a basis
expansion of the original inputs \( \hat{x} \). The difference however
between neural networks and say linear regression is that now these
basis functions (which will correspond to the weights in the network)
are learned from data. This results in an important difference between
neural networks and deep learning approaches on one side and methods
like logistic regression or linear regression and their modifications on the other side.
<p>
<p>
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