Morten Hjorth-Jensen
396 lines
24 KiB
HTML
396 lines
24 KiB
HTML
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<a class="navbar-brand" href="week40-bs.html">Week 40: 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#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="._week40-bs031.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-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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<!-- navigation toc: --> <li><a href="._week40-bs041.html#setting-up-a-multi-layer-perceptron-model-for-classification" 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="._week40-bs042.html#defining-the-cost-function" style="font-size: 80%;"><b>Defining the cost function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs043.html#example-binary-classification-problem" style="font-size: 80%;"><b>Example: binary classification problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs044.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs045.html#developing-a-code-for-doing-neural-networks-with-back-propagation" 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="._week40-bs046.html#collect-and-pre-process-data" style="font-size: 80%;"><b>Collect and pre-process data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#train-and-test-datasets" style="font-size: 80%;"><b>Train and test datasets</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs048.html#define-model-and-architecture" style="font-size: 80%;"><b>Define model and architecture</b></a></li>
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<!-- navigation toc: --> <li><a href="#layers" style="font-size: 80%;"><b>Layers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs050.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs051.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs053.html#choose-cost-function-and-optimizer" style="font-size: 80%;"><b>Choose cost function and optimizer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs054.html#optimizing-the-cost-function" style="font-size: 80%;"><b>Optimizing the cost function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs055.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs057.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs058.html#full-object-oriented-implementation" style="font-size: 80%;"><b>Full object-oriented implementation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs059.html#evaluate-model-performance-on-test-data" style="font-size: 80%;"><b>Evaluate model performance on test data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs060.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs062.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="layers" class="anchor">Layers </h2>
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<ul>
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<li> Input</li>
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<p>Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons. </p>
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<ul>
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<li> Hidden layer</li>
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</ul>
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<p>We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
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Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.
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</p>
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<ul>
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<li> Output</li>
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</ul>
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<p>If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
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which could output 0 or 1 according to the Heaviside function. This would be an example of a <em>hard</em> classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a <em>soft</em> classifier, which outputs the probability of being in class 0 or 1.
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</p>
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<p>For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class. </p>
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<p>Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the <em>softmax</em> function: </p>
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<p>$$ P(\text{class \( j \)} \mid \text{input \( \boldsymbol{a} \)}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
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{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$
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</p>
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<p>i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \boldsymbol{a} \), with \( \boldsymbol{w}_j \) the weights of neuron \( j \) to the inputs.
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The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.
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The exponent is just the weighted sum of inputs as before:
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</p>
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<p>$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$ </p>
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<p>Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
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weights to the output layer.
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</p>
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<p>
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