408 lines
25 KiB
HTML
408 lines
25 KiB
HTML
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('Lecture Thursday October 12',
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<a class="navbar-brand" href="week41-bs.html">Week 41 Neural networks and constructing a neural network code</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="._week41-bs001.html#plan-for-week-41" style="font-size: 80%;"><b>Plan for week 41</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs002.html#lecture-thursday-october-12" style="font-size: 80%;"><b>Lecture Thursday October 12</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs003.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs004.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs005.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs006.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="._week41-bs007.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs008.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs009.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="._week41-bs010.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs011.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs012.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs013.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="._week41-bs014.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="._week41-bs015.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs020.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs020.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs020.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs020.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs020.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs021.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs022.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="._week41-bs023.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs024.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="._week41-bs025.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs026.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="._week41-bs027.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="._week41-bs028.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="._week41-bs029.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs030.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="._week41-bs031.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="._week41-bs032.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="._week41-bs033.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="._week41-bs034.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs035.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="._week41-bs038.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="._week41-bs038.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="._week41-bs038.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="._week41-bs039.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="._week41-bs040.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="._week41-bs041.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="._week41-bs042.html#the-softmax-function" style="font-size: 80%;"><b>The Softmax function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs043.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="._week41-bs044.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="._week41-bs045.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="._week41-bs046.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="._week41-bs047.html#layers" style="font-size: 80%;"><b>Layers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs048.html#weights-and-biases" style="font-size: 80%;"><b>Weights and biases</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs049.html#feed-forward-pass" style="font-size: 80%;"><b>Feed-forward pass</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs050.html#matrix-multiplications" style="font-size: 80%;"><b>Matrix multiplications</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs051.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="#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="._week41-bs053.html#regularization" style="font-size: 80%;"><b>Regularization</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs054.html#matrix-multiplication" style="font-size: 80%;"><b>Matrix multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs055.html#improving-performance" style="font-size: 80%;"><b>Improving performance</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs056.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="._week41-bs057.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="._week41-bs058.html#adjust-hyperparameters" style="font-size: 80%;"><b>Adjust hyperparameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs061.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs060.html#scikit-learn-implementation" style="font-size: 80%;"><b>scikit-learn implementation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs061.html#visualization" style="font-size: 80%;"><b>Visualization</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs062.html#testing-our-code-for-the-xor-or-and-and-gates" style="font-size: 80%;"><b>Testing our code for the XOR, OR and AND gates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs063.html#the-and-and-xor-gates" style="font-size: 80%;"><b>The AND and XOR Gates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs064.html#representing-the-data-sets" style="font-size: 80%;"><b>Representing the Data Sets</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs065.html#setting-up-the-neural-network" style="font-size: 80%;"><b>Setting up the Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs066.html#the-code-using-scikit-learn" style="font-size: 80%;"><b>The Code using Scikit-Learn</b></a></li>
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</ul>
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</li>
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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="part0052"></a>
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<!-- !split -->
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<h2 id="optimizing-the-cost-function" class="anchor">Optimizing the cost function </h2>
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<p>The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is <em>gradient descent</em> and its generalizations. The idea behind gradient descent
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is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a <em>local</em> minimum of the cost function.
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Each parameter \( \theta \) is iteratively adjusted according to the rule
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</p>
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<p>$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$</p>
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<p>where \( \eta \) is known as the <em>learning rate</em>, which controls how big a step we take towards the minimum.
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This update can be repeated for any number of iterations, or until we are satisfied with the result.
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</p>
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<p>A simple and effective improvement is a variant called <em>Batch Gradient Descent</em>.
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Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
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on a subset of the data called a <em>minibatch</em>.
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If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches
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is \( N/M \).
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We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes:
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</p>
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<p>$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
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\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
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</p>
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<p>i.e. instead of averaging the loss over the entire dataset, we average over a minibatch. </p>
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<p>This has two important benefits: </p>
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<ol>
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<li> Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.</li>
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<li> It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.</li>
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</ol>
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<p>The various optmization methods, with codes and algorithms, are discussed in our lectures on <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_self">Gradient descent approaches</a>.</p>
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