Morten Hjorth-Jensen
4062 lines
219 KiB
HTML
4062 lines
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
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<li class="toctree-l1"><a class="reference internal" href="statistics.html">1. Elements of Probability Theory and Statistical Data Analysis</a></li>
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<li class="toctree-l1"><a class="reference internal" href="linalg.html">2. Linear Algebra, Handling of Arrays and more Python Features</a></li>
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter2.html">4. Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter3.html">5. Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter4.html">6. Logistic Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapteroptimization.html">7. Optimization, the central part of any Machine Learning algortithm</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter5.html">8. Support Vector Machines, overarching aims</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter6.html">9. Decision trees, overarching aims</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter7.html">10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter8.html">11. Basic ideas of the Principal Component Analysis (PCA)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="clustering.html">12. Clustering and Unsupervised Learning</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Deep Learning Methods</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
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<ul class="current nav bd-sidenav">
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.html">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek35.html">Exercises week 35</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 37</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Gradient descent methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 38</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Statistical analysis, bias-variance tradeoff and resampling methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek39.html">Exercises week 39</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week39.html">Week 39: Resampling methods and logistic regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week40.html">Week 40: Gradient descent methods (continued) and start Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week41.html">Week 41 Neural networks and constructing a neural network code</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek41.html">Exercises week 41</a></li>
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Week 42 Constructing a Neural Network code with examples</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek42.html">Exercises week 42</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week43.html">Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Projects</span></p>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 6 (midnight), 2025</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project2.html">Project 2 on Machine Learning, deadline November 10 (Midnight)</a></li>
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<h1>Week 42 Constructing a Neural Network code with examples</h1>
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<div id="print-main-content">
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<h2> Contents </h2>
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<nav aria-label="Page">
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<ul class="visible nav section-nav flex-column">
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-october-13-2025">Lecture October 13, 2025</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#readings-and-videos">Readings and videos</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-the-lab-sessions-on-tuesday-and-wednesday">Material for the lab sessions on Tuesday and Wednesday</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-material-writing-a-code-which-implements-a-feed-forward-neural-network">Lecture material: Writing a code which implements a feed-forward neural network</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematics-of-deep-learning">Mathematics of deep learning</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reminder-on-books-with-hands-on-material-and-codes">Reminder on books with hands-on material and codes</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reading-recommendations">Reading recommendations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reminder-from-last-week-first-network-example-simple-percepetron-with-one-input">Reminder from last week: First network example, simple percepetron with one input</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-simple-neural-network-with-no-hidden-layer">Layout of a simple neural network with no hidden layer</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#optimizing-the-parameters">Optimizing the parameters</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adding-a-hidden-layer">Adding a hidden layer</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-simple-neural-network-with-one-hidden-layer">Layout of a simple neural network with one hidden layer</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-derivatives">The derivatives</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#important-observations">Important observations</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-training">The training</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-example">Code example</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-neural-network-and-the-back-propagation-equations">Simple neural network and the back propagation equations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-with-two-hidden-noeds-and-one-output-node">Layout of a simple neural network with two input nodes, one hidden layer with two hidden noeds and one output node</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-ouput-layer">The ouput layer</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compact-expressions">Compact expressions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#output-layer">Output layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#explicit-derivatives">Explicit derivatives</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-of-the-hidden-layer">Derivatives of the hidden layer</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-expression">Final expression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#completing-the-list">Completing the list</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-expressions-for-the-biases-of-the-hidden-layer">Final expressions for the biases of the hidden layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-expressions">Gradient expressions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-equations-for-a-neural-network">Setting up the equations for a neural network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-neural-network-with-three-hidden-layers-last-layer-l-l-4-first-layer-l-0">Layout of a neural network with three hidden layers (last layer = <span class="math notranslate nohighlight">\(l=L=4\)</span>, first layer <span class="math notranslate nohighlight">\(l=0\)</span>)</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#definitions">Definitions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inputs-to-the-activation-function">Inputs to the activation function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-input-to-first-hidden-layer-l-1-from-input-layer-l-0">Layout of input to first hidden layer <span class="math notranslate nohighlight">\(l=1\)</span> from input layer <span class="math notranslate nohighlight">\(l=0\)</span></a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-and-the-chain-rule">Derivatives and the chain rule</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-of-the-cost-function">Derivative of the cost function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-back-propagation-equations-for-a-neural-network">The back propagation equations for a neural network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analyzing-the-last-results">Analyzing the last results</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-considerations">More considerations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-in-terms-of-z-j-l">Derivatives in terms of <span class="math notranslate nohighlight">\(z_j^L\)</span></a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bringing-it-together">Bringing it together</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-back-propagating-equation">Final back propagating equation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-the-chain-rule-and-summing-over-all-k-entries">Using the chain rule and summing over all <span class="math notranslate nohighlight">\(k\)</span> entries</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-1">Setting up the back propagation algorithm, part 1</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-2">Setting up the back propagation algorithm, part 2</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-3">Setting up the Back propagation algorithm, part 3</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#updating-the-gradients">Updating the gradients</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions">Activation functions</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions-logistic-and-hyperbolic-ones">Activation functions, Logistic and Hyperbolic ones</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#relevance">Relevance</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#vanishing-gradients">Vanishing gradients</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exploding-gradients">Exploding gradients</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#is-the-logistic-activation-function-sigmoid-our-choice">Is the Logistic activation function (Sigmoid) our choice?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logistic-function-as-the-root-of-problems">Logistic function as the root of problems</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-derivative-of-the-logistic-funtion">The derivative of the Logistic funtion</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#insights-from-the-paper-by-glorot-and-bengio">Insights from the paper by Glorot and Bengio</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-relu-function-family">The RELU function family</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#elu-function">ELU function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-activation-function-should-we-use">Which activation function should we use?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-on-activation-functions-output-layers">More on activation functions, output layers</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fine-tuning-neural-network-hyperparameters">Fine-tuning neural network hyperparameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hidden-layers">Hidden layers</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#batch-normalization">Batch Normalization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dropout">Dropout</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-clipping">Gradient Clipping</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-top-down-perspective-on-neural-networks">A top-down perspective on Neural networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-top-down-perspectives">More top-down perspectives</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#limitations-of-supervised-learning-with-deep-networks">Limitations of supervised learning with deep networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#limitations-of-nns">Limitations of NNs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#homogeneous-data">Homogeneous data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-limitations">More limitations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-a-multi-layer-perceptron-model-for-classification">Setting up a Multi-layer perceptron model for classification</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-cost-function">Defining the cost function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-binary-classification-problem">Example: binary classification problem</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-softmax-function">The Softmax function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#developing-a-code-for-doing-neural-networks-with-back-propagation">Developing a code for doing neural networks with back propagation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#collect-and-pre-process-data">Collect and pre-process data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#train-and-test-datasets">Train and test datasets</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-model-and-architecture">Define model and architecture</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layers">Layers</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weights-and-biases">Weights and biases</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#feed-forward-pass">Feed-forward pass</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-multiplications">Matrix multiplications</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choose-cost-function-and-optimizer">Choose cost function and optimizer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#optimizing-the-cost-function">Optimizing the cost function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization">Regularization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-multiplication">Matrix multiplication</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#improving-performance">Improving performance</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#full-object-oriented-implementation">Full object-oriented implementation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluate-model-performance-on-test-data">Evaluate model performance on test data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adjust-hyperparameters">Adjust hyperparameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#visualization">Visualization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#scikit-learn-implementation">scikit-learn implementation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Visualization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-neural-networks-in-tensorflow-and-keras">Building neural networks in Tensorflow and Keras</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tensorflow">Tensorflow</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-keras">Using Keras</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Collect and pre-process data</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-neural-network-code">Building a neural network code</a><ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-rate-methods">Learning rate methods</a></li>
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-the-above-learning-rate-schedulers">Usage of the above learning rate schedulers</a></li>
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-functions">Cost functions</a></li>
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Activation functions</a></li>
|
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-neural-network">The Neural Network</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#multiclass-classification">Multiclass classification</a></li>
|
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</ul>
|
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</li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#testing-the-xor-gate-and-other-gates">Testing the XOR gate and other gates</a></li>
|
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</ul>
|
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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doconce format html week42.do.txt --no_mako -->
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<!-- dom:TITLE: Week 42 Constructing a Neural Network code with examples --><section class="tex2jax_ignore mathjax_ignore" id="week-42-constructing-a-neural-network-code-with-examples">
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<h1>Week 42 Constructing a Neural Network code with examples<a class="headerlink" href="#week-42-constructing-a-neural-network-code-with-examples" title="Link to this heading">#</a></h1>
|
||
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo, Norway</p>
|
||
<p>Date: <strong>October 13-17, 2025</strong></p>
|
||
<section id="lecture-october-13-2025">
|
||
<h2>Lecture October 13, 2025<a class="headerlink" href="#lecture-october-13-2025" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Building our own Feed-forward Neural Network and discussion of project 2</p></li>
|
||
<li><p>Project 2 is available at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/2025/Project2/ipynb/Project2.ipynb">CompPhysics/MachineLearning</a></p></li>
|
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</ol>
|
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</section>
|
||
<section id="readings-and-videos">
|
||
<h2>Readings and videos<a class="headerlink" href="#readings-and-videos" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>These lecture notes</p></li>
|
||
<li><p>Video of lecture at <a class="reference external" href="https://youtu.be/eqyNrEYRXnY">https://youtu.be/eqyNrEYRXnY</a></p></li>
|
||
<li><p>Whiteboard notes at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2025/FYSSTKweek42.pdf">CompPhysics/MachineLearning</a></p></li>
|
||
<li><p>For a more in depth discussion on neural networks we recommend Goodfellow et al chapters 6 and 7. For the optimization part, see chapter 8.</p></li>
|
||
<li><p>Neural Networks demystified at <a class="reference external" href="https://www.youtube.com/watch?v=bxe2T-V8XRs&amp;list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&amp;ab_channel=WelchLabs">https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs</a></p></li>
|
||
<li><p>Building Neural Networks from scratch at <a class="reference external" href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&amp;ab_channel=sentdex">https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex</a></p></li>
|
||
<li><p>Video on Neural Networks at <a class="reference external" href="https://www.youtube.com/watch?v=CqOfi41LfDw">https://www.youtube.com/watch?v=CqOfi41LfDw</a></p></li>
|
||
<li><p>Video on the back propagation algorithm at <a class="reference external" href="https://www.youtube.com/watch?v=Ilg3gGewQ5U">https://www.youtube.com/watch?v=Ilg3gGewQ5U</a></p></li>
|
||
</ol>
|
||
<p>I also recommend Michael Nielsen’s intuitive approach to the neural networks and the universal approximation theorem, see the slides at <a class="reference external" href="http://neuralnetworksanddeeplearning.com/chap4.html">http://neuralnetworksanddeeplearning.com/chap4.html</a>.</p>
|
||
</section>
|
||
<section id="material-for-the-lab-sessions-on-tuesday-and-wednesday">
|
||
<h2>Material for the lab sessions on Tuesday and Wednesday<a class="headerlink" href="#material-for-the-lab-sessions-on-tuesday-and-wednesday" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Exercises on writing a code for neural networks, back propagation part, see exercises for week 42 at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek42.html">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/exercisesweek42.html</a></p></li>
|
||
<li><p>Discussion of project 2</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="lecture-material-writing-a-code-which-implements-a-feed-forward-neural-network">
|
||
<h2>Lecture material: Writing a code which implements a feed-forward neural network<a class="headerlink" href="#lecture-material-writing-a-code-which-implements-a-feed-forward-neural-network" title="Link to this heading">#</a></h2>
|
||
<p>Last week we discussed the basics of neural networks and deep learning
|
||
and the basics of automatic differentiation. We looked also at
|
||
examples on how compute the parameters of a simple network with scalar
|
||
inputs and ouputs and no or just one hidden layers.</p>
|
||
<p>We ended our discussions with the derivation of the equations for a
|
||
neural network with one hidden layers and two input variables and two
|
||
hidden nodes but only one output node. We did almost finish the derivation of the back propagation algorithm.</p>
|
||
</section>
|
||
<section id="mathematics-of-deep-learning">
|
||
<h2>Mathematics of deep learning<a class="headerlink" href="#mathematics-of-deep-learning" title="Link to this heading">#</a></h2>
|
||
<p><strong>Two recent books online.</strong></p>
|
||
<ol class="arabic simple">
|
||
<li><p><a class="reference external" href="https://arxiv.org/abs/2105.04026">The Modern Mathematics of Deep Learning, by Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen</a>, published as <a class="reference external" href="https://doi.org/10.1017/9781009025096.002">Mathematical Aspects of Deep Learning, pp. 1-111. Cambridge University Press, 2022</a></p></li>
|
||
<li><p><a class="reference external" href="https://doi.org/10.48550/arXiv.2310.20360">Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory, Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger</a></p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="reminder-on-books-with-hands-on-material-and-codes">
|
||
<h2>Reminder on books with hands-on material and codes<a class="headerlink" href="#reminder-on-books-with-hands-on-material-and-codes" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p><a class="reference external" href="https://sebastianraschka.com/blog/2022/ml-pytorch-book.html">Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch</a></p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="reading-recommendations">
|
||
<h2>Reading recommendations<a class="headerlink" href="#reading-recommendations" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Rashkca et al., chapter 11, jupyter-notebook sent separately, from <a class="reference external" href="https://github.com/rasbt/machine-learning-book">GitHub</a></p></li>
|
||
<li><p>Goodfellow et al, chapter 6 and 7 contain most of the neural network background.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="reminder-from-last-week-first-network-example-simple-percepetron-with-one-input">
|
||
<h2>Reminder from last week: First network example, simple percepetron with one input<a class="headerlink" href="#reminder-from-last-week-first-network-example-simple-percepetron-with-one-input" title="Link to this heading">#</a></h2>
|
||
<p>As yet another example we define now a simple perceptron model with
|
||
all quantities given by scalars. We consider only one input variable
|
||
<span class="math notranslate nohighlight">\(x\)</span> and one target value <span class="math notranslate nohighlight">\(y\)</span>. We define an activation function
|
||
<span class="math notranslate nohighlight">\(\sigma_1\)</span> which takes as input</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_1 = w_1x+b_1,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(w_1\)</span> is the weight and <span class="math notranslate nohighlight">\(b_1\)</span> is the bias. These are the
|
||
parameters we want to optimize. The output is <span class="math notranslate nohighlight">\(a_1=\sigma(z_1)\)</span> (see
|
||
graph from whiteboard notes). This output is then fed into the
|
||
<strong>cost/loss</strong> function, which we here for the sake of simplicity just
|
||
define as the squared error</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(x;w_1,b_1)=\frac{1}{2}(a_1-y)^2.
|
||
\]</div>
|
||
</section>
|
||
<section id="layout-of-a-simple-neural-network-with-no-hidden-layer">
|
||
<h2>Layout of a simple neural network with no hidden layer<a class="headerlink" href="#layout-of-a-simple-neural-network-with-no-hidden-layer" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/simplenn1.png, width=900 frac=1.0] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/simplenn1.png" width="900"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="optimizing-the-parameters">
|
||
<h2>Optimizing the parameters<a class="headerlink" href="#optimizing-the-parameters" title="Link to this heading">#</a></h2>
|
||
<p>In setting up the feed forward and back propagation parts of the
|
||
algorithm, we need now the derivative of the various variables we want
|
||
to train.</p>
|
||
<p>We need</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_1} \hspace{0.1cm}\mathrm{and}\hspace{0.1cm}\frac{\partial C}{\partial b_1}.
|
||
\]</div>
|
||
<p>Using the chain rule we find</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_1}=\frac{\partial C}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial w_1}=(a_1-y)\sigma_1'x,
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial b_1}=\frac{\partial C}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial b_1}=(a_1-y)\sigma_1',
|
||
\]</div>
|
||
<p>which we later will just define as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial a_1}\frac{\partial a_1}{\partial z_1}=\delta_1.
|
||
\]</div>
|
||
</section>
|
||
<section id="adding-a-hidden-layer">
|
||
<h2>Adding a hidden layer<a class="headerlink" href="#adding-a-hidden-layer" title="Link to this heading">#</a></h2>
|
||
<p>We change our simple model to (see graph)
|
||
a network with just one hidden layer but with scalar variables only.</p>
|
||
<p>Our output variable changes to <span class="math notranslate nohighlight">\(a_2\)</span> and <span class="math notranslate nohighlight">\(a_1\)</span> is now the output from the hidden node and <span class="math notranslate nohighlight">\(a_0=x\)</span>.
|
||
We have then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_1 = w_1a_0+b_1 \hspace{0.1cm} \wedge a_1 = \sigma_1(z_1),
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_2 = w_2a_1+b_2 \hspace{0.1cm} \wedge a_2 = \sigma_2(z_2),
|
||
\]</div>
|
||
<p>and the cost function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(x;\boldsymbol{\Theta})=\frac{1}{2}(a_2-y)^2,
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(\boldsymbol{\Theta}=[w_1,w_2,b_1,b_2]\)</span>.</p>
|
||
</section>
|
||
<section id="layout-of-a-simple-neural-network-with-one-hidden-layer">
|
||
<h2>Layout of a simple neural network with one hidden layer<a class="headerlink" href="#layout-of-a-simple-neural-network-with-one-hidden-layer" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/simplenn2.png, width=900 frac=1.0] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/simplenn2.png" width="900"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="the-derivatives">
|
||
<h2>The derivatives<a class="headerlink" href="#the-derivatives" title="Link to this heading">#</a></h2>
|
||
<p>The derivatives are now, using the chain rule again</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_2}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial w_2}=(a_2-y)\sigma_2'a_1=\delta_2a_1,
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial b_2}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial b_2}=(a_2-y)\sigma_2'=\delta_2,
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_1}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial w_1}=(a_2-y)\sigma_2'a_1\sigma_1'a_0,
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial b_1}=\frac{\partial C}{\partial a_2}\frac{\partial a_2}{\partial z_2}\frac{\partial z_2}{\partial a_1}\frac{\partial a_1}{\partial z_1}\frac{\partial z_1}{\partial b_1}=(a_2-y)\sigma_2'\sigma_1'=\delta_1.
|
||
\]</div>
|
||
<p>Can you generalize this to more than one hidden layer?</p>
|
||
</section>
|
||
<section id="important-observations">
|
||
<h2>Important observations<a class="headerlink" href="#important-observations" title="Link to this heading">#</a></h2>
|
||
<p>From the above equations we see that the derivatives of the activation
|
||
functions play a central role. If they vanish, the training may
|
||
stop. This is called the vanishing gradient problem, see discussions below. If they become
|
||
large, the parameters <span class="math notranslate nohighlight">\(w_i\)</span> and <span class="math notranslate nohighlight">\(b_i\)</span> may simply go to infinity. This
|
||
is referenced as the exploding gradient problem.</p>
|
||
</section>
|
||
<section id="the-training">
|
||
<h2>The training<a class="headerlink" href="#the-training" title="Link to this heading">#</a></h2>
|
||
<p>The training of the parameters is done through various gradient descent approximations with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{i}\leftarrow w_{i}- \eta \delta_i a_{i-1},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b_i \leftarrow b_i-\eta \delta_i,
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(\eta\)</span> is the learning rate.</p>
|
||
<p>One iteration consists of one feed forward step and one back-propagation step. Each back-propagation step does one update of the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\Theta}\)</span>.</p>
|
||
<p>For the first hidden layer <span class="math notranslate nohighlight">\(a_{i-1}=a_0=x\)</span> for this simple model.</p>
|
||
</section>
|
||
<section id="code-example">
|
||
<h2>Code example<a class="headerlink" href="#code-example" title="Link to this heading">#</a></h2>
|
||
<p>The code here implements the above model with one hidden layer and
|
||
scalar variables for the same function we studied in the previous
|
||
example. The code is however set up so that we can add multiple
|
||
inputs <span class="math notranslate nohighlight">\(x\)</span> and target values <span class="math notranslate nohighlight">\(y\)</span>. Note also that we have the
|
||
possibility of defining a feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> with more than just
|
||
one column for the input values. This will turn useful in our next example. We have also defined matrices and vectors for all of our operations although it is not necessary here.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
|
||
# We use the Sigmoid function as activation function
|
||
def sigmoid(z):
|
||
return 1.0/(1.0+np.exp(-z))
|
||
|
||
def forwardpropagation(x):
|
||
# weighted sum of inputs to the hidden layer
|
||
z_1 = np.matmul(x, w_1) + b_1
|
||
# activation in the hidden layer
|
||
a_1 = sigmoid(z_1)
|
||
# weighted sum of inputs to the output layer
|
||
z_2 = np.matmul(a_1, w_2) + b_2
|
||
a_2 = z_2
|
||
return a_1, a_2
|
||
|
||
def backpropagation(x, y):
|
||
a_1, a_2 = forwardpropagation(x)
|
||
# parameter delta for the output layer, note that a_2=z_2 and its derivative wrt z_2 is just 1
|
||
delta_2 = a_2 - y
|
||
print(0.5*((a_2-y)**2))
|
||
# delta for the hidden layer
|
||
delta_1 = np.matmul(delta_2, w_2.T) * a_1 * (1 - a_1)
|
||
# gradients for the output layer
|
||
output_weights_gradient = np.matmul(a_1.T, delta_2)
|
||
output_bias_gradient = np.sum(delta_2, axis=0)
|
||
# gradient for the hidden layer
|
||
hidden_weights_gradient = np.matmul(x.T, delta_1)
|
||
hidden_bias_gradient = np.sum(delta_1, axis=0)
|
||
return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
|
||
|
||
|
||
# ensure the same random numbers appear every time
|
||
np.random.seed(0)
|
||
# Input variable
|
||
x = np.array([4.0],dtype=np.float64)
|
||
# Target values
|
||
y = 2*x+1.0
|
||
|
||
# Defining the neural network, only scalars here
|
||
n_inputs = x.shape
|
||
n_features = 1
|
||
n_hidden_neurons = 1
|
||
n_outputs = 1
|
||
|
||
# Initialize the network
|
||
# weights and bias in the hidden layer
|
||
w_1 = np.random.randn(n_features, n_hidden_neurons)
|
||
b_1 = np.zeros(n_hidden_neurons) + 0.01
|
||
|
||
# weights and bias in the output layer
|
||
w_2 = np.random.randn(n_hidden_neurons, n_outputs)
|
||
b_2 = np.zeros(n_outputs) + 0.01
|
||
|
||
eta = 0.1
|
||
for i in range(50):
|
||
# calculate gradients
|
||
derivW2, derivB2, derivW1, derivB1 = backpropagation(x, y)
|
||
# update weights and biases
|
||
w_2 -= eta * derivW2
|
||
b_2 -= eta * derivB2
|
||
w_1 -= eta * derivW1
|
||
b_1 -= eta * derivB1
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We see that after some few iterations (the results do depend on the learning rate however), we get an error which is rather small.</p>
|
||
</section>
|
||
<section id="simple-neural-network-and-the-back-propagation-equations">
|
||
<h2>Simple neural network and the back propagation equations<a class="headerlink" href="#simple-neural-network-and-the-back-propagation-equations" title="Link to this heading">#</a></h2>
|
||
<p>Let us now try to increase our level of ambition and attempt at setting
|
||
up the equations for a neural network with two input nodes, one hidden
|
||
layer with two hidden nodes and one output layer with one output node/neuron only (see graph)..</p>
|
||
<p>We need to define the following parameters and variables with the input layer (layer <span class="math notranslate nohighlight">\((0)\)</span>)
|
||
where we label the nodes <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span></p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
x_1 = a_1^{(0)} \wedge x_2 = a_2^{(0)}.
|
||
\]</div>
|
||
<p>The hidden layer (layer <span class="math notranslate nohighlight">\((1)\)</span>) has nodes which yield the outputs <span class="math notranslate nohighlight">\(a_1^{(1)}\)</span> and <span class="math notranslate nohighlight">\(a_2^{(1)}\)</span>) with weight <span class="math notranslate nohighlight">\(\boldsymbol{w}\)</span> and bias <span class="math notranslate nohighlight">\(\boldsymbol{b}\)</span> parameters</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{ij}^{(1)}=\left\{w_{11}^{(1)},w_{12}^{(1)},w_{21}^{(1)},w_{22}^{(1)}\right\} \wedge b^{(1)}=\left\{b_1^{(1)},b_2^{(1)}\right\}.
|
||
\]</div>
|
||
</section>
|
||
<section id="layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-with-two-hidden-noeds-and-one-output-node">
|
||
<h2>Layout of a simple neural network with two input nodes, one hidden layer with two hidden noeds and one output node<a class="headerlink" href="#layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-with-two-hidden-noeds-and-one-output-node" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/simplenn3.png, width=900 frac=1.0] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/simplenn3.png" width="900"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="the-ouput-layer">
|
||
<h2>The ouput layer<a class="headerlink" href="#the-ouput-layer" title="Link to this heading">#</a></h2>
|
||
<p>We have the ouput layer given by layer label <span class="math notranslate nohighlight">\((2)\)</span> with output <span class="math notranslate nohighlight">\(a^{(2)}\)</span> and weights and biases to be determined given by the variables</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{i}^{(2)}=\left\{w_{1}^{(2)},w_{2}^{(2)}\right\} \wedge b^{(2)}.
|
||
\]</div>
|
||
<p>Our output is <span class="math notranslate nohighlight">\(\tilde{y}=a^{(2)}\)</span> and we define a generic cost function <span class="math notranslate nohighlight">\(C(a^{(2)},y;\boldsymbol{\Theta})\)</span> where <span class="math notranslate nohighlight">\(y\)</span> is the target value (a scalar here).
|
||
The parameters we need to optimize are given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\Theta}=\left\{w_{11}^{(1)},w_{12}^{(1)},w_{21}^{(1)},w_{22}^{(1)},w_{1}^{(2)},w_{2}^{(2)},b_1^{(1)},b_2^{(1)},b^{(2)}\right\}.
|
||
\]</div>
|
||
</section>
|
||
<section id="compact-expressions">
|
||
<h2>Compact expressions<a class="headerlink" href="#compact-expressions" title="Link to this heading">#</a></h2>
|
||
<p>We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions.
|
||
The inputs to the first hidden layer are</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{bmatrix}z_1^{(1)} \\ z_2^{(1)} \end{bmatrix}=\left(\begin{bmatrix}w_{11}^{(1)} & w_{12}^{(1)}\\ w_{21}^{(1)} &w_{22}^{(1)} \end{bmatrix}\right)^{T}\begin{bmatrix}a_1^{(0)} \\ a_2^{(0)} \end{bmatrix}+\begin{bmatrix}b_1^{(1)} \\ b_2^{(1)} \end{bmatrix},
|
||
\end{split}\]</div>
|
||
<p>with outputs</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{bmatrix}a_1^{(1)} \\ a_2^{(1)} \end{bmatrix}=\begin{bmatrix}\sigma^{(1)}(z_1^{(1)}) \\ \sigma^{(1)}(z_2^{(1)}) \end{bmatrix}.
|
||
\end{split}\]</div>
|
||
</section>
|
||
<section id="output-layer">
|
||
<h2>Output layer<a class="headerlink" href="#output-layer" title="Link to this heading">#</a></h2>
|
||
<p>For the final output layer we have the inputs to the final activation function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z^{(2)} = w_{1}^{(2)}a_1^{(1)} +w_{2}^{(2)}a_2^{(1)}+b^{(2)},
|
||
\]</div>
|
||
<p>resulting in the output</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
a^{(2)}=\sigma^{(2)}(z^{(2)}).
|
||
\]</div>
|
||
</section>
|
||
<section id="explicit-derivatives">
|
||
<h2>Explicit derivatives<a class="headerlink" href="#explicit-derivatives" title="Link to this heading">#</a></h2>
|
||
<p>In total we have nine parameters which we need to train. Using the
|
||
chain rule (or just the back-propagation algorithm) we can find all
|
||
derivatives. Since we will use automatic differentiation in reverse
|
||
mode, we start with the derivatives of the cost function with respect
|
||
to the parameters of the output layer, namely</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_{i}^{(2)}}=\frac{\partial C}{\partial a^{(2)}}\frac{\partial a^{(2)}}{\partial z^{(2)}}\frac{\partial z^{(2)}}{\partial w_{i}^{(2)}}=\delta^{(2)}a_i^{(1)},
|
||
\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta^{(2)}=\frac{\partial C}{\partial a^{(2)}}\frac{\partial a^{(2)}}{\partial z^{(2)}}
|
||
\]</div>
|
||
<p>and finally</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial b^{(2)}}=\frac{\partial C}{\partial a^{(2)}}\frac{\partial a^{(2)}}{\partial z^{(2)}}\frac{\partial z^{(2)}}{\partial b^{(2)}}=\delta^{(2)}.
|
||
\]</div>
|
||
</section>
|
||
<section id="derivatives-of-the-hidden-layer">
|
||
<h2>Derivatives of the hidden layer<a class="headerlink" href="#derivatives-of-the-hidden-layer" title="Link to this heading">#</a></h2>
|
||
<p>Using the chain rule we have the following expressions for say one of the weight parameters (it is easy to generalize to the other weight parameters)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_{11}^{(1)}}=\frac{\partial C}{\partial a^{(2)}}\frac{\partial a^{(2)}}{\partial z^{(2)}}
|
||
\frac{\partial z^{(2)}}{\partial z_1^{(1)}}\frac{\partial z_1^{(1)}}{\partial w_{11}^{(1)}}= \delta^{(2)}\frac{\partial z^{(2)}}{\partial z_1^{(1)}}\frac{\partial z_1^{(1)}}{\partial w_{11}^{(1)}},
|
||
\]</div>
|
||
<p>which, noting that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z^{(2)} =w_1^{(2)}a_1^{(1)}+w_2^{(2)}a_2^{(1)}+b^{(2)},
|
||
\]</div>
|
||
<p>allows us to rewrite</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial z^{(2)}}{\partial z_1^{(1)}}\frac{\partial z_1^{(1)}}{\partial w_{11}^{(1)}}=w_1^{(2)}\frac{\partial a_1^{(1)}}{\partial z_1^{(1)}}a_1^{(1)}.
|
||
\]</div>
|
||
</section>
|
||
<section id="final-expression">
|
||
<h2>Final expression<a class="headerlink" href="#final-expression" title="Link to this heading">#</a></h2>
|
||
<p>Defining</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_1^{(1)}=w_1^{(2)}\frac{\partial a_1^{(1)}}{\partial z_1^{(1)}}\delta^{(2)},
|
||
\]</div>
|
||
<p>we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_{11}^{(1)}}=\delta_1^{(1)}a_1^{(1)}.
|
||
\]</div>
|
||
<p>Similarly, we obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_{12}^{(1)}}=\delta_1^{(1)}a_2^{(1)}.
|
||
\]</div>
|
||
</section>
|
||
<section id="completing-the-list">
|
||
<h2>Completing the list<a class="headerlink" href="#completing-the-list" title="Link to this heading">#</a></h2>
|
||
<p>Similarly, we find</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_{21}^{(1)}}=\delta_2^{(1)}a_1^{(1)},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial w_{22}^{(1)}}=\delta_2^{(1)}a_2^{(1)},
|
||
\]</div>
|
||
<p>where we have defined</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_2^{(1)}=w_2^{(2)}\frac{\partial a_2^{(1)}}{\partial z_2^{(1)}}\delta^{(2)}.
|
||
\]</div>
|
||
</section>
|
||
<section id="final-expressions-for-the-biases-of-the-hidden-layer">
|
||
<h2>Final expressions for the biases of the hidden layer<a class="headerlink" href="#final-expressions-for-the-biases-of-the-hidden-layer" title="Link to this heading">#</a></h2>
|
||
<p>For the sake of completeness, we list the derivatives of the biases, which are</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial b_{1}^{(1)}}=\delta_1^{(1)},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial C}{\partial b_{2}^{(1)}}=\delta_2^{(1)}.
|
||
\]</div>
|
||
<p>As we will see below, these expressions can be generalized in a more compact form.</p>
|
||
</section>
|
||
<section id="gradient-expressions">
|
||
<h2>Gradient expressions<a class="headerlink" href="#gradient-expressions" title="Link to this heading">#</a></h2>
|
||
<p>For this specific model, with just one output node and two hidden
|
||
nodes, the gradient descent equations take the following form for output layer</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{i}^{(2)}\leftarrow w_{i}^{(2)}- \eta \delta^{(2)} a_{i}^{(1)},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b^{(2)} \leftarrow b^{(2)}-\eta \delta^{(2)},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{ij}^{(1)}\leftarrow w_{ij}^{(1)}- \eta \delta_{i}^{(1)} a_{j}^{(0)},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b_{i}^{(1)} \leftarrow b_{i}^{(1)}-\eta \delta_{i}^{(1)},
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\eta\)</span> is the learning rate.</p>
|
||
</section>
|
||
<section id="setting-up-the-equations-for-a-neural-network">
|
||
<h2>Setting up the equations for a neural network<a class="headerlink" href="#setting-up-the-equations-for-a-neural-network" title="Link to this heading">#</a></h2>
|
||
<p>The questions we want to ask are how do changes in the biases and the
|
||
weights in our network change the cost function and how can we use the
|
||
final output to modify the weights and biases?</p>
|
||
<p>To derive these equations let us start with a plain regression problem
|
||
and define our cost function as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
{\cal C}(\boldsymbol{\Theta}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2,
|
||
\]</div>
|
||
<p>where the <span class="math notranslate nohighlight">\(y_i\)</span>s are our <span class="math notranslate nohighlight">\(n\)</span> targets (the values we want to
|
||
reproduce), while the outputs of the network after having propagated
|
||
all inputs <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> are given by <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}_i\)</span>.</p>
|
||
</section>
|
||
<section id="layout-of-a-neural-network-with-three-hidden-layers-last-layer-l-l-4-first-layer-l-0">
|
||
<h2>Layout of a neural network with three hidden layers (last layer = <span class="math notranslate nohighlight">\(l=L=4\)</span>, first layer <span class="math notranslate nohighlight">\(l=0\)</span>)<a class="headerlink" href="#layout-of-a-neural-network-with-three-hidden-layers-last-layer-l-l-4-first-layer-l-0" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/nn2.png, width=900 frac=1.0] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/nn2.png" width="900"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="definitions">
|
||
<h2>Definitions<a class="headerlink" href="#definitions" title="Link to this heading">#</a></h2>
|
||
<p>With our definition of the targets <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>, the outputs of the
|
||
network <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> and the inputs <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> we
|
||
define now the activation <span class="math notranslate nohighlight">\(z_j^l\)</span> of node/neuron/unit <span class="math notranslate nohighlight">\(j\)</span> of the
|
||
<span class="math notranslate nohighlight">\(l\)</span>-th layer as a function of the bias, the weights which add up from
|
||
the previous layer <span class="math notranslate nohighlight">\(l-1\)</span> and the forward passes/outputs
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{a}^{l-1}\)</span> from the previous layer as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(b_k^l\)</span> are the biases from layer <span class="math notranslate nohighlight">\(l\)</span>. Here <span class="math notranslate nohighlight">\(M_{l-1}\)</span>
|
||
represents the total number of nodes/neurons/units of layer <span class="math notranslate nohighlight">\(l-1\)</span>. The
|
||
figure in the whiteboard notes illustrates this equation. We can rewrite this in a more
|
||
compact form as the matrix-vector products we discussed earlier,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{z}^l = \left(\boldsymbol{W}^l\right)^T\boldsymbol{a}^{l-1}+\boldsymbol{b}^l.
|
||
\]</div>
|
||
</section>
|
||
<section id="inputs-to-the-activation-function">
|
||
<h2>Inputs to the activation function<a class="headerlink" href="#inputs-to-the-activation-function" title="Link to this heading">#</a></h2>
|
||
<p>With the activation values <span class="math notranslate nohighlight">\(\boldsymbol{z}^l\)</span> we can in turn define the
|
||
output of layer <span class="math notranslate nohighlight">\(l\)</span> as <span class="math notranslate nohighlight">\(\boldsymbol{a}^l = \sigma(\boldsymbol{z}^l)\)</span> where <span class="math notranslate nohighlight">\(\sigma\)</span> is our
|
||
activation function. In the examples here we will use the sigmoid
|
||
function discussed in our logistic regression lectures. We will also use the same activation function <span class="math notranslate nohighlight">\(\sigma\)</span> for all layers
|
||
and their nodes. It means we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
a_j^l = \sigma(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
|
||
\]</div>
|
||
</section>
|
||
<section id="layout-of-input-to-first-hidden-layer-l-1-from-input-layer-l-0">
|
||
<h2>Layout of input to first hidden layer <span class="math notranslate nohighlight">\(l=1\)</span> from input layer <span class="math notranslate nohighlight">\(l=0\)</span><a class="headerlink" href="#layout-of-input-to-first-hidden-layer-l-1-from-input-layer-l-0" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/structure.png, width=900 frac=1.0] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/structure.png" width="900"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="derivatives-and-the-chain-rule">
|
||
<h2>Derivatives and the chain rule<a class="headerlink" href="#derivatives-and-the-chain-rule" title="Link to this heading">#</a></h2>
|
||
<p>From the definition of the input variable to the activation function, that is <span class="math notranslate nohighlight">\(z_j^l\)</span> we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
|
||
\]</div>
|
||
<p>With our definition of the activation function we have that (note that this function depends only on <span class="math notranslate nohighlight">\(z_j^l\)</span>)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=\sigma(z_j^l)(1-\sigma(z_j^l)).
|
||
\]</div>
|
||
</section>
|
||
<section id="derivative-of-the-cost-function">
|
||
<h2>Derivative of the cost function<a class="headerlink" href="#derivative-of-the-cost-function" title="Link to this heading">#</a></h2>
|
||
<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
|
||
<p>Let us specialize to the output layer <span class="math notranslate nohighlight">\(l=L\)</span>. Our cost function is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
{\cal C}(\boldsymbol{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2,
|
||
\]</div>
|
||
<p>The derivative of this function with respect to the weights is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial{\cal C}(\boldsymbol{\Theta}^L)}{\partial w_{ij}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{ij}^{L}},
|
||
\]</div>
|
||
<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial a_j^L}{\partial w_{ij}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{ij}^{L}}=a_j^L(1-a_j^L)a_i^{L-1}.
|
||
\]</div>
|
||
</section>
|
||
<section id="the-back-propagation-equations-for-a-neural-network">
|
||
<h2>The back propagation equations for a neural network<a class="headerlink" href="#the-back-propagation-equations-for-a-neural-network" title="Link to this heading">#</a></h2>
|
||
<p>We have thus</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial{\cal C}((\boldsymbol{\Theta}^L)}{\partial w_{ij}^L} = \left(a_j^L - y_j\right)a_j^L(1-a_j^L)a_i^{L-1},
|
||
\]</div>
|
||
<p>Defining</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - y_j\right) = \sigma'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
|
||
\]</div>
|
||
<p>and using the Hadamard product of two vectors we can write this as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\delta}^L = \sigma'(\boldsymbol{z}^L)\circ\frac{\partial {\cal C}}{\partial (\boldsymbol{a}^L)}.
|
||
\]</div>
|
||
</section>
|
||
<section id="analyzing-the-last-results">
|
||
<h2>Analyzing the last results<a class="headerlink" href="#analyzing-the-last-results" title="Link to this heading">#</a></h2>
|
||
<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 <span class="math notranslate nohighlight">\(j\)</span>th
|
||
output activation. If, for example, the cost function doesn’t depend
|
||
much on a particular output node <span class="math notranslate nohighlight">\(j\)</span>, then <span class="math notranslate nohighlight">\(\delta_j^L\)</span> will be small,
|
||
which is what we would expect. The first term on the right, measures
|
||
how fast the activation function <span class="math notranslate nohighlight">\(f\)</span> is changing at a given activation
|
||
value <span class="math notranslate nohighlight">\(z_j^L\)</span>.</p>
|
||
</section>
|
||
<section id="more-considerations">
|
||
<h2>More considerations<a class="headerlink" href="#more-considerations" title="Link to this heading">#</a></h2>
|
||
<p>Notice that everything in the above equations is easily computed. In
|
||
particular, we compute <span class="math notranslate nohighlight">\(z_j^L\)</span> while computing the behaviour of the
|
||
network, and it is only a small additional overhead to compute
|
||
<span class="math notranslate nohighlight">\(\sigma'(z^L_j)\)</span>. 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</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial {\cal C}}{\partial (a_j^L)}
|
||
\]</div>
|
||
<p>With the definition of <span class="math notranslate nohighlight">\(\delta_j^L\)</span> we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial{\cal C}}{\partial w_{ij}^L} = \delta_j^La_i^{L-1}.
|
||
\]</div>
|
||
</section>
|
||
<section id="derivatives-in-terms-of-z-j-l">
|
||
<h2>Derivatives in terms of <span class="math notranslate nohighlight">\(z_j^L\)</span><a class="headerlink" href="#derivatives-in-terms-of-z-j-l" title="Link to this heading">#</a></h2>
|
||
<p>It is also easy to see that our previous equation can be written as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
|
||
\]</div>
|
||
<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases <span class="math notranslate nohighlight">\(b_j^L\)</span>, namely</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
|
||
\]</div>
|
||
<p>That is, the error <span class="math notranslate nohighlight">\(\delta_j^L\)</span> is exactly equal to the rate of change of the cost function as a function of the bias.</p>
|
||
</section>
|
||
<section id="bringing-it-together">
|
||
<h2>Bringing it together<a class="headerlink" href="#bringing-it-together" title="Link to this heading">#</a></h2>
|
||
<p>We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto1"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\frac{\partial{\cal C}(\boldsymbol{W^L})}{\partial w_{ij}^L} = \delta_j^La_i^{L-1},
|
||
\label{_auto1} \tag{1}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>and</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto2"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\delta_j^L = \sigma'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
|
||
\label{_auto2} \tag{2}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>and</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto3"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
|
||
\label{_auto3} \tag{3}
|
||
\end{equation}
|
||
\]</div>
|
||
</section>
|
||
<section id="final-back-propagating-equation">
|
||
<h2>Final back propagating equation<a class="headerlink" href="#final-back-propagating-equation" title="Link to this heading">#</a></h2>
|
||
<p>We have that (replacing <span class="math notranslate nohighlight">\(L\)</span> with a general layer <span class="math notranslate nohighlight">\(l\)</span>)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
|
||
\]</div>
|
||
<p>We want to express this in terms of the equations for layer <span class="math notranslate nohighlight">\(l+1\)</span>.</p>
|
||
</section>
|
||
<section id="using-the-chain-rule-and-summing-over-all-k-entries">
|
||
<h2>Using the chain rule and summing over all <span class="math notranslate nohighlight">\(k\)</span> entries<a class="headerlink" href="#using-the-chain-rule-and-summing-over-all-k-entries" title="Link to this heading">#</a></h2>
|
||
<p>We obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
|
||
\]</div>
|
||
<p>and recalling that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(M_l\)</span> being the number of nodes in layer <span class="math notranslate nohighlight">\(l\)</span>, we obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l),
|
||
\]</div>
|
||
<p>This is our final equation.</p>
|
||
<p>We are now ready to set up the algorithm for back propagation and learning the weights and biases.</p>
|
||
</section>
|
||
<section id="setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">
|
||
<h2>Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations" title="Link to this heading">#</a></h2>
|
||
<p><strong>The architecture (our model).</strong></p>
|
||
<ol class="arabic simple">
|
||
<li><p>Set up your inputs and outputs (scalars, vectors, matrices or higher-order arrays)</p></li>
|
||
<li><p>Define the number of hidden layers and hidden nodes</p></li>
|
||
<li><p>Define activation functions for hidden layers and output layers</p></li>
|
||
<li><p>Define optimizer (plan learning rate, momentum, ADAgrad, RMSprop, ADAM etc) and array of initial learning rates</p></li>
|
||
<li><p>Define cost function and possible regularization terms with hyperparameters</p></li>
|
||
<li><p>Initialize weights and biases</p></li>
|
||
<li><p>Fix number of iterations for the feed forward part and back propagation part</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="setting-up-the-back-propagation-algorithm-part-1">
|
||
<h2>Setting up the back propagation algorithm, part 1<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-1" title="Link to this heading">#</a></h2>
|
||
<p>The four equations provide us with a way of computing the gradients of the cost function. Let us write this out in the form of an algorithm.</p>
|
||
<p><strong>First</strong>, we set up the input data <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> and the activations
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{z}_1\)</span> of the input layer and compute the activation function and
|
||
the pertinent outputs <span class="math notranslate nohighlight">\(\boldsymbol{a}^1\)</span>.</p>
|
||
<p><strong>Secondly</strong>, we perform then the feed forward till we reach the output
|
||
layer and compute all <span class="math notranslate nohighlight">\(\boldsymbol{z}_l\)</span> of the input layer and compute the
|
||
activation function and the pertinent outputs <span class="math notranslate nohighlight">\(\boldsymbol{a}^l\)</span> for
|
||
<span class="math notranslate nohighlight">\(l=1,2,3,\dots,L\)</span>.</p>
|
||
<p><strong>Notation</strong>: The first hidden layer has <span class="math notranslate nohighlight">\(l=1\)</span> as label and the final output layer has <span class="math notranslate nohighlight">\(l=L\)</span>.</p>
|
||
</section>
|
||
<section id="setting-up-the-back-propagation-algorithm-part-2">
|
||
<h2>Setting up the back propagation algorithm, part 2<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-2" title="Link to this heading">#</a></h2>
|
||
<p>Thereafter we compute the ouput error <span class="math notranslate nohighlight">\(\boldsymbol{\delta}^L\)</span> by computing all</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^L = \sigma'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
|
||
\]</div>
|
||
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l).
|
||
\]</div>
|
||
</section>
|
||
<section id="setting-up-the-back-propagation-algorithm-part-3">
|
||
<h2>Setting up the Back propagation algorithm, part 3<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-3" title="Link to this heading">#</a></h2>
|
||
<p>Finally, we update the weights and the biases using gradient descent
|
||
for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> (the first hidden layer) and update the weights and biases
|
||
according to the rules</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{ij}^l\leftarrow = w_{ij}^l- \eta \delta_j^la_i^{l-1},
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(\eta\)</span> being the learning rate.</p>
|
||
</section>
|
||
<section id="updating-the-gradients">
|
||
<h2>Updating the gradients<a class="headerlink" href="#updating-the-gradients" title="Link to this heading">#</a></h2>
|
||
<p>With the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l),
|
||
\]</div>
|
||
<p>we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> and update the weights and biases according to the rules</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{ij}^l\leftarrow = w_{ij}^l- \eta \delta_j^la_i^{l-1},
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
|
||
\]</div>
|
||
</section>
|
||
<section id="activation-functions">
|
||
<h2>Activation functions<a class="headerlink" href="#activation-functions" title="Link to this heading">#</a></h2>
|
||
<p>A property that characterizes a neural network, other than its
|
||
connectivity, is the choice of activation function(s). The following
|
||
restrictions are imposed on an activation function for an FFNN to
|
||
fulfill the universal approximation theorem</p>
|
||
<ul class="simple">
|
||
<li><p>Non-constant</p></li>
|
||
<li><p>Bounded</p></li>
|
||
<li><p>Monotonically-increasing</p></li>
|
||
<li><p>Continuous</p></li>
|
||
</ul>
|
||
<section id="activation-functions-logistic-and-hyperbolic-ones">
|
||
<h3>Activation functions, Logistic and Hyperbolic ones<a class="headerlink" href="#activation-functions-logistic-and-hyperbolic-ones" title="Link to this heading">#</a></h3>
|
||
<p>The second requirement excludes all linear functions. Furthermore, in
|
||
a MLP with only linear activation functions, each layer simply
|
||
performs a linear transformation of its inputs.</p>
|
||
<p>Regardless of the number of layers, the output of the NN will be
|
||
nothing but a linear function of the inputs. Thus we need to introduce
|
||
some kind of non-linearity to the NN to be able to fit non-linear
|
||
functions Typical examples are the logistic <em>Sigmoid</em></p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sigma(x) = \frac{1}{1 + e^{-x}},
|
||
\]</div>
|
||
<p>and the <em>hyperbolic tangent</em> function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sigma(x) = \tanh(x)
|
||
\]</div>
|
||
</section>
|
||
</section>
|
||
<section id="relevance">
|
||
<h2>Relevance<a class="headerlink" href="#relevance" title="Link to this heading">#</a></h2>
|
||
<p>The <em>sigmoid</em> function are more biologically plausible because the
|
||
output of inactive neurons are zero. Such activation function are
|
||
called <em>one-sided</em>. However, it has been shown that the hyperbolic
|
||
tangent performs better than the sigmoid for training MLPs. has
|
||
become the most popular for <em>deep neural networks</em></p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>%matplotlib inline
|
||
|
||
"""The sigmoid function (or the logistic curve) is a
|
||
function that takes any real number, z, and outputs a number (0,1).
|
||
It is useful in neural networks for assigning weights on a relative scale.
|
||
The value z is the weighted sum of parameters involved in the learning algorithm."""
|
||
|
||
import numpy
|
||
import matplotlib.pyplot as plt
|
||
import math as mt
|
||
|
||
z = numpy.arange(-5, 5, .1)
|
||
sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
|
||
sigma = sigma_fn(z)
|
||
|
||
fig = plt.figure()
|
||
ax = fig.add_subplot(111)
|
||
ax.plot(z, sigma)
|
||
ax.set_ylim([-0.1, 1.1])
|
||
ax.set_xlim([-5,5])
|
||
ax.grid(True)
|
||
ax.set_xlabel('z')
|
||
ax.set_title('sigmoid function')
|
||
|
||
plt.show()
|
||
|
||
"""Step Function"""
|
||
z = numpy.arange(-5, 5, .02)
|
||
step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
|
||
step = step_fn(z)
|
||
|
||
fig = plt.figure()
|
||
ax = fig.add_subplot(111)
|
||
ax.plot(z, step)
|
||
ax.set_ylim([-0.5, 1.5])
|
||
ax.set_xlim([-5,5])
|
||
ax.grid(True)
|
||
ax.set_xlabel('z')
|
||
ax.set_title('step function')
|
||
|
||
plt.show()
|
||
|
||
"""Sine Function"""
|
||
z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
|
||
t = numpy.sin(z)
|
||
|
||
fig = plt.figure()
|
||
ax = fig.add_subplot(111)
|
||
ax.plot(z, t)
|
||
ax.set_ylim([-1.0, 1.0])
|
||
ax.set_xlim([-2*mt.pi,2*mt.pi])
|
||
ax.grid(True)
|
||
ax.set_xlabel('z')
|
||
ax.set_title('sine function')
|
||
|
||
plt.show()
|
||
|
||
"""Plots a graph of the squashing function used by a rectified linear
|
||
unit"""
|
||
z = numpy.arange(-2, 2, .1)
|
||
zero = numpy.zeros(len(z))
|
||
y = numpy.max([zero, z], axis=0)
|
||
|
||
fig = plt.figure()
|
||
ax = fig.add_subplot(111)
|
||
ax.plot(z, y)
|
||
ax.set_ylim([-2.0, 2.0])
|
||
ax.set_xlim([-2.0, 2.0])
|
||
ax.grid(True)
|
||
ax.set_xlabel('z')
|
||
ax.set_title('Rectified linear unit')
|
||
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="vanishing-gradients">
|
||
<h2>Vanishing gradients<a class="headerlink" href="#vanishing-gradients" title="Link to this heading">#</a></h2>
|
||
<p>The Back propagation algorithm we derived above works by going from
|
||
the output layer to the input layer, propagating the error gradient on
|
||
the way. Once the algorithm has computed the gradient of the cost
|
||
function with regards to each parameter in the network, it uses these
|
||
gradients to update each parameter with a Gradient Descent (GD) step.</p>
|
||
<p>Unfortunately for us, the gradients often get smaller and smaller as
|
||
the algorithm progresses down to the first hidden layers. As a result,
|
||
the GD update leaves the lower layer connection weights virtually
|
||
unchanged, and training never converges to a good solution. This is
|
||
known in the literature as <strong>the vanishing gradients problem</strong>.</p>
|
||
</section>
|
||
<section id="exploding-gradients">
|
||
<h2>Exploding gradients<a class="headerlink" href="#exploding-gradients" title="Link to this heading">#</a></h2>
|
||
<p>In other cases, the opposite can happen, namely the the gradients can
|
||
grow bigger and bigger. The result is that many of the layers get
|
||
large updates of the weights the algorithm diverges. This is the
|
||
<strong>exploding gradients problem</strong>, which is mostly encountered in
|
||
recurrent neural networks. More generally, deep neural networks suffer
|
||
from unstable gradients, different layers may learn at widely
|
||
different speeds</p>
|
||
</section>
|
||
<section id="is-the-logistic-activation-function-sigmoid-our-choice">
|
||
<h2>Is the Logistic activation function (Sigmoid) our choice?<a class="headerlink" href="#is-the-logistic-activation-function-sigmoid-our-choice" title="Link to this heading">#</a></h2>
|
||
<p>Although this unfortunate behavior has been empirically observed for
|
||
quite a while (it was one of the reasons why deep neural networks were
|
||
mostly abandoned for a long time), it is only around 2010 that
|
||
significant progress was made in understanding it.</p>
|
||
<p>A paper titled <a class="reference external" href="http://proceedings.mlr.press/v9/glorot10a.html">Understanding the Difficulty of Training Deep
|
||
Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio</a> found that
|
||
the problems with the popular logistic
|
||
sigmoid activation function and the weight initialization technique
|
||
that was most popular at the time, namely random initialization using
|
||
a normal distribution with a mean of 0 and a standard deviation of
|
||
1.</p>
|
||
</section>
|
||
<section id="logistic-function-as-the-root-of-problems">
|
||
<h2>Logistic function as the root of problems<a class="headerlink" href="#logistic-function-as-the-root-of-problems" title="Link to this heading">#</a></h2>
|
||
<p>They showed that with this activation function and this
|
||
initialization scheme, the variance of the outputs of each layer is
|
||
much greater than the variance of its inputs. Going forward in the
|
||
network, the variance keeps increasing after each layer until the
|
||
activation function saturates at the top layers. This is actually made
|
||
worse by the fact that the logistic function has a mean of 0.5, not 0
|
||
(the hyperbolic tangent function has a mean of 0 and behaves slightly
|
||
better than the logistic function in deep networks).</p>
|
||
</section>
|
||
<section id="the-derivative-of-the-logistic-funtion">
|
||
<h2>The derivative of the Logistic funtion<a class="headerlink" href="#the-derivative-of-the-logistic-funtion" title="Link to this heading">#</a></h2>
|
||
<p>Looking at the logistic activation function, when inputs become large
|
||
(negative or positive), the function saturates at 0 or 1, with a
|
||
derivative extremely close to 0. Thus when backpropagation kicks in,
|
||
it has virtually no gradient to propagate back through the network,
|
||
and what little gradient exists keeps getting diluted as
|
||
backpropagation progresses down through the top layers, so there is
|
||
really nothing left for the lower layers.</p>
|
||
<p>In their paper, Glorot and Bengio propose a way to significantly
|
||
alleviate this problem. We need the signal to flow properly in both
|
||
directions: in the forward direction when making predictions, and in
|
||
the reverse direction when backpropagating gradients. We don’t want
|
||
the signal to die out, nor do we want it to explode and saturate. For
|
||
the signal to flow properly, the authors argue that we need the
|
||
variance of the outputs of each layer to be equal to the variance of
|
||
its inputs, and we also need the gradients to have equal variance
|
||
before and after flowing through a layer in the reverse direction.</p>
|
||
</section>
|
||
<section id="insights-from-the-paper-by-glorot-and-bengio">
|
||
<h2>Insights from the paper by Glorot and Bengio<a class="headerlink" href="#insights-from-the-paper-by-glorot-and-bengio" title="Link to this heading">#</a></h2>
|
||
<p>One of the insights in the 2010 paper by Glorot and Bengio was that
|
||
the vanishing/exploding gradients problems were in part due to a poor
|
||
choice of activation function. Until then most people had assumed that
|
||
if Nature had chosen to use roughly sigmoid activation functions in
|
||
biological neurons, they must be an excellent choice. But it turns out
|
||
that other activation functions behave much better in deep neural
|
||
networks, in particular the ReLU activation function, mostly because
|
||
it does not saturate for positive values (and also because it is quite
|
||
fast to compute).</p>
|
||
</section>
|
||
<section id="the-relu-function-family">
|
||
<h2>The RELU function family<a class="headerlink" href="#the-relu-function-family" title="Link to this heading">#</a></h2>
|
||
<p>The ReLU activation function suffers from a problem known as the dying
|
||
ReLUs: during training, some neurons effectively die, meaning they
|
||
stop outputting anything other than 0.</p>
|
||
<p>In some cases, you may find that half of your network’s neurons are
|
||
dead, especially if you used a large learning rate. During training,
|
||
if a neuron’s weights get updated such that the weighted sum of the
|
||
neuron’s inputs is negative, it will start outputting 0. When this
|
||
happen, the neuron is unlikely to come back to life since the gradient
|
||
of the ReLU function is 0 when its input is negative.</p>
|
||
</section>
|
||
<section id="elu-function">
|
||
<h2>ELU function<a class="headerlink" href="#elu-function" title="Link to this heading">#</a></h2>
|
||
<p>To solve this problem, nowadays practitioners use a variant of the
|
||
ReLU function, such as the leaky ReLU discussed above or the so-called
|
||
exponential linear unit (ELU) function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right.
|
||
\end{split}\]</div>
|
||
</section>
|
||
<section id="which-activation-function-should-we-use">
|
||
<h2>Which activation function should we use?<a class="headerlink" href="#which-activation-function-should-we-use" title="Link to this heading">#</a></h2>
|
||
<p>In general it seems that the ELU activation function is better than
|
||
the leaky ReLU function (and its variants), which is better than
|
||
ReLU. ReLU performs better than <span class="math notranslate nohighlight">\(\tanh\)</span> which in turn performs better
|
||
than the logistic function.</p>
|
||
<p>If runtime performance is an issue, then you may opt for the leaky
|
||
ReLU function over the ELU function If you don’t want to tweak yet
|
||
another hyperparameter, you may just use the default <span class="math notranslate nohighlight">\(\alpha\)</span> of
|
||
<span class="math notranslate nohighlight">\(0.01\)</span> for the leaky ReLU, and <span class="math notranslate nohighlight">\(1\)</span> for ELU. If you have spare time and
|
||
computing power, you can use cross-validation or bootstrap to evaluate
|
||
other activation functions.</p>
|
||
</section>
|
||
<section id="more-on-activation-functions-output-layers">
|
||
<h2>More on activation functions, output layers<a class="headerlink" href="#more-on-activation-functions-output-layers" title="Link to this heading">#</a></h2>
|
||
<p>In most cases you can use the ReLU activation function in the hidden
|
||
layers (or one of its variants).</p>
|
||
<p>It is a bit faster to compute than other activation functions, and the
|
||
gradient descent optimization does in general not get stuck.</p>
|
||
<p><strong>For the output layer:</strong></p>
|
||
<ul class="simple">
|
||
<li><p>For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).</p></li>
|
||
<li><p>For regression tasks, you can simply use no activation function at all.</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="fine-tuning-neural-network-hyperparameters">
|
||
<h2>Fine-tuning neural network hyperparameters<a class="headerlink" href="#fine-tuning-neural-network-hyperparameters" title="Link to this heading">#</a></h2>
|
||
<p>The flexibility of neural networks is also one of their main
|
||
drawbacks: there are many hyperparameters to tweak. Not only can you
|
||
use any imaginable network topology (how neurons/nodes are
|
||
interconnected), but even in a simple FFNN you can change the number
|
||
of layers, the number of neurons per layer, the type of activation
|
||
function to use in each layer, the weight initialization logic, the
|
||
stochastic gradient optmized and much more. How do you know what
|
||
combination of hyperparameters is the best for your task?</p>
|
||
<ul class="simple">
|
||
<li><p>You can use grid search with cross-validation to find the right hyperparameters.</p></li>
|
||
</ul>
|
||
<p>However,since there are many hyperparameters to tune, and since
|
||
training a neural network on a large dataset takes a lot of time, you
|
||
will only be able to explore a tiny part of the hyperparameter space.</p>
|
||
<ul class="simple">
|
||
<li><p>You can use randomized search.</p></li>
|
||
<li><p>Or use tools like <a class="reference external" href="http://oscar.calldesk.ai/">Oscar</a>, which implements more complex algorithms to help you find a good set of hyperparameters quickly.</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="hidden-layers">
|
||
<h2>Hidden layers<a class="headerlink" href="#hidden-layers" title="Link to this heading">#</a></h2>
|
||
<p>For many problems you can start with just one or two hidden layers and
|
||
it will work just fine. For the MNIST data set discussed below you can easily get a
|
||
high accuracy using just one hidden layer with a few hundred neurons.
|
||
You can reach for this data set above 98% accuracy using two hidden
|
||
layers with the same total amount of neurons, in roughly the same
|
||
amount of training time.</p>
|
||
<p>For more complex problems, you can gradually ramp up the number of
|
||
hidden layers, until you start overfitting the training set. Very
|
||
complex tasks, such as large image classification or speech
|
||
recognition, typically require networks with dozens of layers and they
|
||
need a huge amount of training data. However, you will rarely have to
|
||
train such networks from scratch: it is much more common to reuse
|
||
parts of a pretrained state-of-the-art network that performs a similar
|
||
task.</p>
|
||
</section>
|
||
<section id="batch-normalization">
|
||
<h2>Batch Normalization<a class="headerlink" href="#batch-normalization" title="Link to this heading">#</a></h2>
|
||
<p>Batch Normalization aims to address the vanishing/exploding gradients
|
||
problems, and more generally the problem that the distribution of each
|
||
layer’s inputs changes during training, as the parameters of the
|
||
previous layers change.</p>
|
||
<p>The technique consists of adding an operation in the model just before
|
||
the activation function of each layer, simply zero-centering and
|
||
normalizing the inputs, then scaling and shifting the result using two
|
||
new parameters per layer (one for scaling, the other for shifting). In
|
||
other words, this operation lets the model learn the optimal scale and
|
||
mean of the inputs for each layer. In order to zero-center and
|
||
normalize the inputs, the algorithm needs to estimate the inputs’ mean
|
||
and standard deviation. It does so by evaluating the mean and standard
|
||
deviation of the inputs over the current mini-batch, from this the
|
||
name batch normalization.</p>
|
||
</section>
|
||
<section id="dropout">
|
||
<h2>Dropout<a class="headerlink" href="#dropout" title="Link to this heading">#</a></h2>
|
||
<p>It is a fairly simple algorithm: at every training step, every neuron
|
||
(including the input neurons but excluding the output neurons) has a
|
||
probability <span class="math notranslate nohighlight">\(p\)</span> of being temporarily dropped out, meaning it will be
|
||
entirely ignored during this training step, but it may be active
|
||
during the next step.</p>
|
||
<p>The hyperparameter <span class="math notranslate nohighlight">\(p\)</span> is called the dropout rate, and it is typically
|
||
set to 50%. After training, the neurons are not dropped anymore. It
|
||
is viewed as one of the most popular regularization techniques.</p>
|
||
</section>
|
||
<section id="gradient-clipping">
|
||
<h2>Gradient Clipping<a class="headerlink" href="#gradient-clipping" title="Link to this heading">#</a></h2>
|
||
<p>A popular technique to lessen the exploding gradients problem is to
|
||
simply clip the gradients during backpropagation so that they never
|
||
exceed some threshold (this is mostly useful for recurrent neural
|
||
networks).</p>
|
||
<p>This technique is called Gradient Clipping.</p>
|
||
<p>In general however, Batch
|
||
Normalization is preferred.</p>
|
||
</section>
|
||
<section id="a-top-down-perspective-on-neural-networks">
|
||
<h2>A top-down perspective on Neural networks<a class="headerlink" href="#a-top-down-perspective-on-neural-networks" title="Link to this heading">#</a></h2>
|
||
<p>The first thing we would like to do is divide the data into two or
|
||
three parts. A training set, a validation or dev (development) set,
|
||
and a test set. The test set is the data on which we want to make
|
||
predictions. The dev set is a subset of the training data we use to
|
||
check how well we are doing out-of-sample, after training the model on
|
||
the training dataset. We use the validation error as a proxy for the
|
||
test error in order to make tweaks to our model. It is crucial that we
|
||
do not use any of the test data to train the algorithm. This is a
|
||
cardinal sin in ML. Then:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Estimate optimal error rate</p></li>
|
||
<li><p>Minimize underfitting (bias) on training data set.</p></li>
|
||
<li><p>Make sure you are not overfitting.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="more-top-down-perspectives">
|
||
<h2>More top-down perspectives<a class="headerlink" href="#more-top-down-perspectives" title="Link to this heading">#</a></h2>
|
||
<p>If the validation and test sets are drawn from the same distributions,
|
||
then a good performance on the validation set should lead to similarly
|
||
good performance on the test set.</p>
|
||
<p>However, sometimes
|
||
the training data and test data differ in subtle ways because, for
|
||
example, they are collected using slightly different methods, or
|
||
because it is cheaper to collect data in one way versus another. In
|
||
this case, there can be a mismatch between the training and test
|
||
data. This can lead to the neural network overfitting these small
|
||
differences between the test and training sets, and a poor performance
|
||
on the test set despite having a good performance on the validation
|
||
set. To rectify this, Andrew Ng suggests making two validation or dev
|
||
sets, one constructed from the training data and one constructed from
|
||
the test data. The difference between the performance of the algorithm
|
||
on these two validation sets quantifies the train-test mismatch. This
|
||
can serve as another important diagnostic when using DNNs for
|
||
supervised learning.</p>
|
||
</section>
|
||
<section id="limitations-of-supervised-learning-with-deep-networks">
|
||
<h2>Limitations of supervised learning with deep networks<a class="headerlink" href="#limitations-of-supervised-learning-with-deep-networks" title="Link to this heading">#</a></h2>
|
||
<p>Like all statistical methods, supervised learning using neural
|
||
networks has important limitations. This is especially important when
|
||
one seeks to apply these methods, especially to physics problems. Like
|
||
all tools, DNNs are not a universal solution. Often, the same or
|
||
better performance on a task can be achieved by using a few
|
||
hand-engineered features (or even a collection of random
|
||
features).</p>
|
||
</section>
|
||
<section id="limitations-of-nns">
|
||
<h2>Limitations of NNs<a class="headerlink" href="#limitations-of-nns" title="Link to this heading">#</a></h2>
|
||
<p>Here we list some of the important limitations of supervised neural network based models.</p>
|
||
<ul class="simple">
|
||
<li><p><strong>Need labeled data</strong>. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).</p></li>
|
||
<li><p><strong>Supervised neural networks are extremely data intensive.</strong> DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="homogeneous-data">
|
||
<h2>Homogeneous data<a class="headerlink" href="#homogeneous-data" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p><strong>Homogeneous data.</strong> Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="more-limitations">
|
||
<h2>More limitations<a class="headerlink" href="#more-limitations" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p><strong>Many problems are not about prediction.</strong> In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a <em>wrong</em> model. The model might or might not be useful for understanding the underlying science.</p></li>
|
||
</ul>
|
||
<p>Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.</p>
|
||
</section>
|
||
<section id="setting-up-a-multi-layer-perceptron-model-for-classification">
|
||
<h2>Setting up a Multi-layer perceptron model for classification<a class="headerlink" href="#setting-up-a-multi-layer-perceptron-model-for-classification" title="Link to this heading">#</a></h2>
|
||
<p>We are now gong to develop an example based on the MNIST data
|
||
base. This is a classification problem and we need to use our
|
||
cross-entropy function we discussed in connection with logistic
|
||
regression. The cross-entropy defines our cost function for the
|
||
classificaton problems with neural networks.</p>
|
||
<p>In binary classification with two classes <span class="math notranslate nohighlight">\((0, 1)\)</span> we define the
|
||
logistic/sigmoid function as the probability that a particular input
|
||
is in class <span class="math notranslate nohighlight">\(0\)</span> or <span class="math notranslate nohighlight">\(1\)</span>. This is possible because the logistic
|
||
function takes any input from the real numbers and inputs a number
|
||
between 0 and 1, and can therefore be interpreted as a probability. It
|
||
also has other nice properties, such as a derivative that is simple to
|
||
calculate.</p>
|
||
<p>For an input <span class="math notranslate nohighlight">\(\boldsymbol{a}\)</span> from the hidden layer, the probability that the input <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>
|
||
is in class 0 or 1 is just. We let <span class="math notranslate nohighlight">\(\theta\)</span> represent the unknown weights and biases to be adjusted by our equations). The variable <span class="math notranslate nohighlight">\(x\)</span>
|
||
represents our activation values <span class="math notranslate nohighlight">\(z\)</span>. We have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp{(- \boldsymbol{x}})} ,
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(y \in \{0, 1\}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> represents the weights and biases
|
||
of our network.</p>
|
||
</section>
|
||
<section id="defining-the-cost-function">
|
||
<h2>Defining the cost function<a class="headerlink" href="#defining-the-cost-function" title="Link to this heading">#</a></h2>
|
||
<p>Our cost function is given as (see the Logistic regression lectures)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
|
||
y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .
|
||
\]</div>
|
||
<p>This last equality means that we can interpret our <em>cost</em> function as a sum over the <em>loss</em> function
|
||
for each point in the dataset <span class="math notranslate nohighlight">\(\mathcal{L}_i(\boldsymbol{\theta})\)</span>.<br />
|
||
The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
|
||
than maximizing a negative number.</p>
|
||
<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:</p>
|
||
<p><span class="math notranslate nohighlight">\(y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\)</span> and</p>
|
||
<p><span class="math notranslate nohighlight">\(y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\)</span></p>
|
||
<p>i.e. a binary bit string of length <span class="math notranslate nohighlight">\(C\)</span>, where <span class="math notranslate nohighlight">\(C = 10\)</span> is the number of classes in the MNIST dataset (numbers from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(9\)</span>)..</p>
|
||
<p>If <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> is the <span class="math notranslate nohighlight">\(i\)</span>-th input (image), <span class="math notranslate nohighlight">\(y_{ic}\)</span> refers to the <span class="math notranslate nohighlight">\(c\)</span>-th component of the <span class="math notranslate nohighlight">\(i\)</span>-th
|
||
output vector <span class="math notranslate nohighlight">\(\boldsymbol{y}_i\)</span>.<br />
|
||
The probability of <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> being in class <span class="math notranslate nohighlight">\(c\)</span> will be given by the softmax function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
|
||
{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,
|
||
\]</div>
|
||
<p>which reduces to the logistic function in the binary case.<br />
|
||
The likelihood of this <span class="math notranslate nohighlight">\(C\)</span>-class classifier
|
||
is now given as:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
|
||
\]</div>
|
||
<p>Again we take the negative log-likelihood to define our cost function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\theta}) = - \log{P(\mathcal{D} \mid \boldsymbol{\theta})}.
|
||
\]</div>
|
||
<p>See the logistic regression lectures for a full definition of the cost function.</p>
|
||
<p>The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!</p>
|
||
</section>
|
||
<section id="example-binary-classification-problem">
|
||
<h2>Example: binary classification problem<a class="headerlink" href="#example-binary-classification-problem" title="Link to this heading">#</a></h2>
|
||
<p>As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters <span class="math notranslate nohighlight">\(\beta\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\boldsymbol{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\boldsymbol{\beta})}\right),
|
||
\]</div>
|
||
<p>where we had defined the logistic (sigmoid) function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(y_i =1\vert x_i,\boldsymbol{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(y_i =0\vert x_i,\boldsymbol{\beta})=1-p(y_i =1\vert x_i,\boldsymbol{\beta}).
|
||
\]</div>
|
||
<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> were defined using a minimization method like gradient descent or Newton-Raphson’s method.</p>
|
||
<p>Now we replace <span class="math notranslate nohighlight">\(x_i\)</span> with the activation <span class="math notranslate nohighlight">\(z_i^l\)</span> for a given layer <span class="math notranslate nohighlight">\(l\)</span> and the outputs as <span class="math notranslate nohighlight">\(y_i=a_i^l=f(z_i^l)\)</span>, with <span class="math notranslate nohighlight">\(z_i^l\)</span> now being a function of the weights <span class="math notranslate nohighlight">\(w_{ij}^l\)</span> and biases <span class="math notranslate nohighlight">\(b_i^l\)</span>.
|
||
We have then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}},
|
||
\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,
|
||
\]</div>
|
||
<p>where the superscript <span class="math notranslate nohighlight">\(l-1\)</span> indicates that these are the outputs from layer <span class="math notranslate nohighlight">\(l-1\)</span>.
|
||
Our cost function at the final layer <span class="math notranslate nohighlight">\(l=L\)</span> is now</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right),
|
||
\]</div>
|
||
<p>where we have defined the targets <span class="math notranslate nohighlight">\(t_i\)</span>. The derivatives of the cost function with respect to the output <span class="math notranslate nohighlight">\(a_i^L\)</span> are then easily calculated and we get</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial \mathcal{C}(\boldsymbol{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.
|
||
\]</div>
|
||
<p>In case we use another activation function than the logistic one, we need to evaluate other derivatives.</p>
|
||
</section>
|
||
<section id="the-softmax-function">
|
||
<h2>The Softmax function<a class="headerlink" href="#the-softmax-function" title="Link to this heading">#</a></h2>
|
||
<p>In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation <span class="math notranslate nohighlight">\(z_i^l\)</span>, that is we need</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
|
||
\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}.
|
||
\]</div>
|
||
<p>For the Softmax function we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.
|
||
\]</div>
|
||
<p>Its derivative with respect to <span class="math notranslate nohighlight">\(z_j^l\)</span> gives</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right),
|
||
\]</div>
|
||
<p>which in case of the simply binary model reduces to having <span class="math notranslate nohighlight">\(i=j\)</span>.</p>
|
||
</section>
|
||
<section id="developing-a-code-for-doing-neural-networks-with-back-propagation">
|
||
<h2>Developing a code for doing neural networks with back propagation<a class="headerlink" href="#developing-a-code-for-doing-neural-networks-with-back-propagation" title="Link to this heading">#</a></h2>
|
||
<p>One can identify a set of key steps when using neural networks to solve supervised learning problems:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Collect and pre-process data</p></li>
|
||
<li><p>Define model and architecture</p></li>
|
||
<li><p>Choose cost function and optimizer</p></li>
|
||
<li><p>Train the model</p></li>
|
||
<li><p>Evaluate model performance on test data</p></li>
|
||
<li><p>Adjust hyperparameters (if necessary, network architecture)</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="collect-and-pre-process-data">
|
||
<h2>Collect and pre-process data<a class="headerlink" href="#collect-and-pre-process-data" title="Link to this heading">#</a></h2>
|
||
<p>Here we will be using the MNIST dataset, which is readily available through the <strong>scikit-learn</strong>
|
||
package. You may also find it for example <a class="reference external" href="http://yann.lecun.com/exdb/mnist/">here</a>.<br />
|
||
The <em>MNIST</em> (Modified National Institute of Standards and Technology) database is a large database
|
||
of handwritten digits that is commonly used for training various image processing systems.<br />
|
||
The MNIST dataset consists of 70 000 images of size <span class="math notranslate nohighlight">\(28\times 28\)</span> pixels, each labeled from 0 to 9.<br />
|
||
The scikit-learn dataset we will use consists of a selection of 1797 images of size <span class="math notranslate nohighlight">\(8\times 8\)</span> collected and processed from this database.</p>
|
||
<p>To feed data into a feed-forward neural network we need to represent
|
||
the inputs as a design/feature matrix <span class="math notranslate nohighlight">\(X = (n_{inputs}, n_{features})\)</span>. Each
|
||
row represents an <em>input</em>, in this case a handwritten digit, and
|
||
each column represents a <em>feature</em>, in this case a pixel. The
|
||
correct answers, also known as <em>labels</em> or <em>targets</em> are
|
||
represented as a 1D array of integers
|
||
<span class="math notranslate nohighlight">\(Y = (n_{inputs}) = (5, 3, 1, 8,...)\)</span>.</p>
|
||
<p>As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from
|
||
measurements of height (in m)<br />
|
||
and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split} X = \begin{bmatrix}
|
||
1.85 & 81\\
|
||
1.71 & 65\\
|
||
1.95 & 103\\
|
||
1.55 & 42\\
|
||
1.63 & 56
|
||
\end{bmatrix} ,\end{split}\]</div>
|
||
<p>and the targets would be:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ Y = (23.7, 22.2, 27.1, 17.5, 21.1) \]</div>
|
||
<p>Since each input image is a 2D matrix, we need to flatten the image
|
||
(i.e. “unravel” the 2D matrix into a 1D array) to turn the data into a
|
||
design/feature matrix. This means we lose all spatial information in the
|
||
image, such as locality and translational invariance. More complicated
|
||
architectures such as Convolutional Neural Networks can take advantage
|
||
of such information, and are most commonly applied when analyzing
|
||
images.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># import necessary packages
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn import datasets
|
||
|
||
|
||
# ensure the same random numbers appear every time
|
||
np.random.seed(0)
|
||
|
||
# display images in notebook
|
||
%matplotlib inline
|
||
plt.rcParams['figure.figsize'] = (12,12)
|
||
|
||
|
||
# download MNIST dataset
|
||
digits = datasets.load_digits()
|
||
|
||
# define inputs and labels
|
||
inputs = digits.images
|
||
labels = digits.target
|
||
|
||
print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
|
||
print("labels = (n_inputs) = " + str(labels.shape))
|
||
|
||
|
||
# flatten the image
|
||
# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
|
||
n_inputs = len(inputs)
|
||
inputs = inputs.reshape(n_inputs, -1)
|
||
print("X = (n_inputs, n_features) = " + str(inputs.shape))
|
||
|
||
|
||
# choose some random images to display
|
||
indices = np.arange(n_inputs)
|
||
random_indices = np.random.choice(indices, size=5)
|
||
|
||
for i, image in enumerate(digits.images[random_indices]):
|
||
plt.subplot(1, 5, i+1)
|
||
plt.axis('off')
|
||
plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
|
||
plt.title("Label: %d" % digits.target[random_indices[i]])
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="train-and-test-datasets">
|
||
<h2>Train and test datasets<a class="headerlink" href="#train-and-test-datasets" title="Link to this heading">#</a></h2>
|
||
<p>Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.</p>
|
||
<p>We will reserve <span class="math notranslate nohighlight">\(80 \%\)</span> of our dataset for training and <span class="math notranslate nohighlight">\(20 \%\)</span> for testing.</p>
|
||
<p>It is important that the train and test datasets are drawn randomly from our dataset, to ensure
|
||
no bias in the sampling.<br />
|
||
Say you are taking measurements of weather data to predict the weather in the coming 5 days.
|
||
You don’t want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data
|
||
collected from 12.00 to 24.00.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.model_selection import train_test_split
|
||
|
||
# one-liner from scikit-learn library
|
||
train_size = 0.8
|
||
test_size = 1 - train_size
|
||
X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
|
||
test_size=test_size)
|
||
|
||
# equivalently in numpy
|
||
def train_test_split_numpy(inputs, labels, train_size, test_size):
|
||
n_inputs = len(inputs)
|
||
inputs_shuffled = inputs.copy()
|
||
labels_shuffled = labels.copy()
|
||
|
||
np.random.shuffle(inputs_shuffled)
|
||
np.random.shuffle(labels_shuffled)
|
||
|
||
train_end = int(n_inputs*train_size)
|
||
X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
|
||
Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
|
||
|
||
return X_train, X_test, Y_train, Y_test
|
||
|
||
#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
|
||
|
||
print("Number of training images: " + str(len(X_train)))
|
||
print("Number of test images: " + str(len(X_test)))
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="define-model-and-architecture">
|
||
<h2>Define model and architecture<a class="headerlink" href="#define-model-and-architecture" title="Link to this heading">#</a></h2>
|
||
<p>Our simple feed-forward neural network will consist of an <em>input</em> layer, a single <em>hidden</em> layer and an <em>output</em> layer. The activation <span class="math notranslate nohighlight">\(y\)</span> of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z = \sum_{i=1}^n w_i a_i ,\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[ y = f(z) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f\)</span> is the activation function, <span class="math notranslate nohighlight">\(a_i\)</span> represents input from neuron <span class="math notranslate nohighlight">\(i\)</span> in the preceding layer
|
||
and <span class="math notranslate nohighlight">\(w_i\)</span> is the weight to input <span class="math notranslate nohighlight">\(i\)</span>.<br />
|
||
The activation of the neurons in the input layer is just the features (e.g. a pixel value).</p>
|
||
<p>The simplest activation function for a neuron is the <em>Heaviside</em> function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split} f(z) =
|
||
\begin{cases}
|
||
1, & z > 0\\
|
||
0, & \text{otherwise}
|
||
\end{cases}
|
||
\end{split}\]</div>
|
||
<p>A feed-forward neural network with this activation is known as a <em>perceptron</em>.<br />
|
||
For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.<br />
|
||
This activation can be generalized to <span class="math notranslate nohighlight">\(k\)</span> classes (using e.g. the <em>one-against-all</em> strategy),
|
||
and we call these architectures <em>multiclass perceptrons</em>.</p>
|
||
<p>However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and<br />
|
||
Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.</p>
|
||
<p>Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).<br />
|
||
We will be using the sigmoid function <span class="math notranslate nohighlight">\(\sigma(x)\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,\]</div>
|
||
<p>which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.</p>
|
||
</section>
|
||
<section id="layers">
|
||
<h2>Layers<a class="headerlink" href="#layers" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Input</p></li>
|
||
</ul>
|
||
<p>Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.</p>
|
||
<ul class="simple">
|
||
<li><p>Hidden layer</p></li>
|
||
</ul>
|
||
<p>We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.<br />
|
||
Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.</p>
|
||
<ul class="simple">
|
||
<li><p>Output</p></li>
|
||
</ul>
|
||
<p>If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
|
||
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.</p>
|
||
<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>
|
||
<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 <span class="math notranslate nohighlight">\(j = 0,1,...,9\)</span>. The activation of each output neuron <span class="math notranslate nohighlight">\(j\)</span> will be according to the <em>softmax</em> function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ P(\text{class $j$} \mid \text{input $\boldsymbol{a}$}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
|
||
{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,\]</div>
|
||
<p>i.e. each neuron <span class="math notranslate nohighlight">\(j\)</span> outputs the probability of being in class <span class="math notranslate nohighlight">\(j\)</span> given an input from the hidden layer <span class="math notranslate nohighlight">\(\boldsymbol{a}\)</span>, with <span class="math notranslate nohighlight">\(\boldsymbol{w}_j\)</span> the weights of neuron <span class="math notranslate nohighlight">\(j\)</span> to the inputs.<br />
|
||
The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.<br />
|
||
The exponent is just the weighted sum of inputs as before:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.\]</div>
|
||
<p>Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
|
||
weights to the output layer.</p>
|
||
</section>
|
||
<section id="weights-and-biases">
|
||
<h2>Weights and biases<a class="headerlink" href="#weights-and-biases" title="Link to this heading">#</a></h2>
|
||
<p>Typically weights are initialized with small values distributed around zero, drawn from a uniform
|
||
or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.</p>
|
||
<p>Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
|
||
of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron <span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(b_j\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.\]</div>
|
||
<p>The bias weights <span class="math notranslate nohighlight">\(\boldsymbol{b}\)</span> are often initialized to zero, but a small value like <span class="math notranslate nohighlight">\(0.01\)</span> ensures all neurons have some output which can be backpropagated in the first training cycle.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># building our neural network
|
||
|
||
n_inputs, n_features = X_train.shape
|
||
n_hidden_neurons = 50
|
||
n_categories = 10
|
||
|
||
# we make the weights normally distributed using numpy.random.randn
|
||
|
||
# weights and bias in the hidden layer
|
||
hidden_weights = np.random.randn(n_features, n_hidden_neurons)
|
||
hidden_bias = np.zeros(n_hidden_neurons) + 0.01
|
||
|
||
# weights and bias in the output layer
|
||
output_weights = np.random.randn(n_hidden_neurons, n_categories)
|
||
output_bias = np.zeros(n_categories) + 0.01
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="feed-forward-pass">
|
||
<h2>Feed-forward pass<a class="headerlink" href="#feed-forward-pass" title="Link to this heading">#</a></h2>
|
||
<p>Denote <span class="math notranslate nohighlight">\(F\)</span> the number of features, <span class="math notranslate nohighlight">\(H\)</span> the number of hidden neurons and <span class="math notranslate nohighlight">\(C\)</span> the number of categories.<br />
|
||
For each input image we calculate a weighted sum of input features (pixel values) to each neuron <span class="math notranslate nohighlight">\(j\)</span> in the hidden layer <span class="math notranslate nohighlight">\(l\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},\]</div>
|
||
<p>this is then passed through our activation function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ a_{j}^{l} = f(z_{j}^{l}) .\]</div>
|
||
<p>We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron <span class="math notranslate nohighlight">\(j\)</span> in the output layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.\]</div>
|
||
<p>Finally we calculate the output of neuron <span class="math notranslate nohighlight">\(j\)</span> in the output layer using the softmax function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
|
||
{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .\]</div>
|
||
</section>
|
||
<section id="matrix-multiplications">
|
||
<h2>Matrix multiplications<a class="headerlink" href="#matrix-multiplications" title="Link to this heading">#</a></h2>
|
||
<p>Since our data has the dimensions <span class="math notranslate nohighlight">\(X = (n_{inputs}, n_{features})\)</span> and our weights to the hidden
|
||
layer have the dimensions<br />
|
||
<span class="math notranslate nohighlight">\(W_{hidden} = (n_{features}, n_{hidden})\)</span>,
|
||
we can easily feed the network all our training data in one go by taking the matrix product</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ X W^{h} = (n_{inputs}, n_{hidden}),\]</div>
|
||
<p>and obtain a matrix that holds the weighted sum of inputs to the hidden layer
|
||
for each input image and each hidden neuron.<br />
|
||
We also add the bias to obtain a matrix of weighted sums to the hidden layer <span class="math notranslate nohighlight">\(Z^{h}\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \boldsymbol{z}^{l} = \boldsymbol{X} \boldsymbol{W}^{l} + \boldsymbol{b}^{l} ,\]</div>
|
||
<p>meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.<br />
|
||
This is then passed through the activation:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \boldsymbol{a}^{l} = f(\boldsymbol{z}^l) .\]</div>
|
||
<p>This is fed to the output layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \boldsymbol{z}^{L} = \boldsymbol{a}^{L} \boldsymbol{W}^{L} + \boldsymbol{b}^{L} .\]</div>
|
||
<p>Finally we receive our output values for each image and each category by passing it through the softmax function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ output = softmax (\boldsymbol{z}^{L}) = (n_{inputs}, n_{categories}) .\]</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># setup the feed-forward pass, subscript h = hidden layer
|
||
|
||
def sigmoid(x):
|
||
return 1/(1 + np.exp(-x))
|
||
|
||
def feed_forward(X):
|
||
# weighted sum of inputs to the hidden layer
|
||
z_h = np.matmul(X, hidden_weights) + hidden_bias
|
||
# activation in the hidden layer
|
||
a_h = sigmoid(z_h)
|
||
|
||
# weighted sum of inputs to the output layer
|
||
z_o = np.matmul(a_h, output_weights) + output_bias
|
||
# softmax output
|
||
# axis 0 holds each input and axis 1 the probabilities of each category
|
||
exp_term = np.exp(z_o)
|
||
probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
|
||
return probabilities
|
||
|
||
probabilities = feed_forward(X_train)
|
||
print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
|
||
print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
|
||
print("probabilities sum up to: " + str(probabilities[0].sum()))
|
||
print()
|
||
|
||
# we obtain a prediction by taking the class with the highest likelihood
|
||
def predict(X):
|
||
probabilities = feed_forward(X)
|
||
return np.argmax(probabilities, axis=1)
|
||
|
||
predictions = predict(X_train)
|
||
print("predictions = (n_inputs) = " + str(predictions.shape))
|
||
print("prediction for image 0: " + str(predictions[0]))
|
||
print("correct label for image 0: " + str(Y_train[0]))
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="choose-cost-function-and-optimizer">
|
||
<h2>Choose cost function and optimizer<a class="headerlink" href="#choose-cost-function-and-optimizer" title="Link to this heading">#</a></h2>
|
||
<p>To measure how well our neural network is doing we need to introduce a cost function.<br />
|
||
We will call the function that gives the error of a single sample output the <em>loss</em> function, and the function
|
||
that gives the total error of our network across all samples the <em>cost</em> function.
|
||
A typical choice for multiclass classification is the <em>cross-entropy</em> loss, also known as the negative log likelihood.</p>
|
||
<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\]</div>
|
||
<p>i.e. a binary bit string of length <span class="math notranslate nohighlight">\(C\)</span>, where <span class="math notranslate nohighlight">\(C = 10\)</span> is the number of classes in the MNIST dataset.</p>
|
||
<p>Let <span class="math notranslate nohighlight">\(y_{ic}\)</span> denote the <span class="math notranslate nohighlight">\(c\)</span>-th component of the <span class="math notranslate nohighlight">\(i\)</span>-th one-hot vector.<br />
|
||
We define the cost function <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> as a sum over the cross-entropy loss for each point <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> in the dataset.</p>
|
||
<p>In the one-hot representation only one of the terms in the loss function is non-zero, namely the
|
||
probability of the correct category <span class="math notranslate nohighlight">\(c'\)</span><br />
|
||
(i.e. the category <span class="math notranslate nohighlight">\(c'\)</span> such that <span class="math notranslate nohighlight">\(y_{ic'} = 1\)</span>). This means that the cross entropy loss only punishes you for how wrong
|
||
you got the correct label. The probability of category <span class="math notranslate nohighlight">\(c\)</span> is given by the softmax function. The vector <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> represents the parameters of our network, i.e. all the weights and biases.</p>
|
||
</section>
|
||
<section id="optimizing-the-cost-function">
|
||
<h2>Optimizing the cost function<a class="headerlink" href="#optimizing-the-cost-function" title="Link to this heading">#</a></h2>
|
||
<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
|
||
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.<br />
|
||
Each parameter <span class="math notranslate nohighlight">\(\theta\)</span> is iteratively adjusted according to the rule</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\eta\)</span> is known as the <em>learning rate</em>, which controls how big a step we take towards the minimum.<br />
|
||
This update can be repeated for any number of iterations, or until we are satisfied with the result.</p>
|
||
<p>A simple and effective improvement is a variant called <em>Batch Gradient Descent</em>.<br />
|
||
Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
|
||
on a subset of the data called a <em>minibatch</em>.<br />
|
||
If there are <span class="math notranslate nohighlight">\(N\)</span> data points and we have a minibatch size of <span class="math notranslate nohighlight">\(M\)</span>, the total number of batches
|
||
is <span class="math notranslate nohighlight">\(N/M\)</span>.<br />
|
||
We denote each minibatch <span class="math notranslate nohighlight">\(B_k\)</span>, with <span class="math notranslate nohighlight">\(k = 1, 2,...,N/M\)</span>. The gradient then becomes:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
|
||
\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,\]</div>
|
||
<p>i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.</p>
|
||
<p>This has two important benefits:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.</p></li>
|
||
<li><p>It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.</p></li>
|
||
</ol>
|
||
<p>The various optmization methods, with codes and algorithms, are discussed in our lectures on <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html">Gradient descent approaches</a>.</p>
|
||
</section>
|
||
<section id="regularization">
|
||
<h2>Regularization<a class="headerlink" href="#regularization" title="Link to this heading">#</a></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>
|
||
<p>We will measure the size of the weights using the so called <em>L2-norm</em>, meaning our cost function becomes:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
|
||
\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2
|
||
= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,\]</div>
|
||
<p>i.e. we sum up all the weights squared. The factor <span class="math notranslate nohighlight">\(\lambda\)</span> is known as a regularization parameter.</p>
|
||
<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 <span class="math notranslate nohighlight">\((64 + 1)\times 50=3250\)</span> weights in
|
||
the hidden layer and <span class="math notranslate nohighlight">\((50 + 1)\times 10=510\)</span> weights to the output
|
||
layer (<span class="math notranslate nohighlight">\(+1\)</span> 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>
|
||
</section>
|
||
<section id="matrix-multiplication">
|
||
<h2>Matrix multiplication<a class="headerlink" href="#matrix-multiplication" title="Link to this heading">#</a></h2>
|
||
<p>To more efficently train our network these equations are implemented using matrix operations.<br />
|
||
The error in the output layer is calculated simply as, with <span class="math notranslate nohighlight">\(\boldsymbol{t}\)</span> being our targets,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .\]</div>
|
||
<p>The gradient for the output weights is calculated as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla W_{L} = \boldsymbol{a}^T \delta_L = (n_{hidden}, n_{categories}) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{a} = (n_{inputs}, n_{hidden})\)</span>. This simply means that we are summing up the gradients for each input.<br />
|
||
Since we are going backwards we have to transpose the activation matrix.</p>
|
||
<p>The gradient with respect to the output bias is then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .\]</div>
|
||
<p>The error in the hidden layer is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f'(a_{h})\)</span> is the derivative of the activation in the hidden layer. The matrix products mean
|
||
that we are summing up the products for each neuron in the output layer. The symbol <span class="math notranslate nohighlight">\(\circ\)</span> denotes
|
||
the <em>Hadamard product</em>, meaning element-wise multiplication.</p>
|
||
<p>This again gives us the gradients in the hidden layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .\]</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># to categorical turns our integer vector into a onehot representation
|
||
from sklearn.metrics import accuracy_score
|
||
|
||
# one-hot in numpy
|
||
def to_categorical_numpy(integer_vector):
|
||
n_inputs = len(integer_vector)
|
||
n_categories = np.max(integer_vector) + 1
|
||
onehot_vector = np.zeros((n_inputs, n_categories))
|
||
onehot_vector[range(n_inputs), integer_vector] = 1
|
||
|
||
return onehot_vector
|
||
|
||
#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
|
||
Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
|
||
|
||
def feed_forward_train(X):
|
||
# weighted sum of inputs to the hidden layer
|
||
z_h = np.matmul(X, hidden_weights) + hidden_bias
|
||
# activation in the hidden layer
|
||
a_h = sigmoid(z_h)
|
||
|
||
# weighted sum of inputs to the output layer
|
||
z_o = np.matmul(a_h, output_weights) + output_bias
|
||
# softmax output
|
||
# axis 0 holds each input and axis 1 the probabilities of each category
|
||
exp_term = np.exp(z_o)
|
||
probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
|
||
# for backpropagation need activations in hidden and output layers
|
||
return a_h, probabilities
|
||
|
||
def backpropagation(X, Y):
|
||
a_h, probabilities = feed_forward_train(X)
|
||
|
||
# error in the output layer
|
||
error_output = probabilities - Y
|
||
# error in the hidden layer
|
||
error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
|
||
|
||
# gradients for the output layer
|
||
output_weights_gradient = np.matmul(a_h.T, error_output)
|
||
output_bias_gradient = np.sum(error_output, axis=0)
|
||
|
||
# gradient for the hidden layer
|
||
hidden_weights_gradient = np.matmul(X.T, error_hidden)
|
||
hidden_bias_gradient = np.sum(error_hidden, axis=0)
|
||
|
||
return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
|
||
|
||
print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
|
||
|
||
eta = 0.01
|
||
lmbd = 0.01
|
||
for i in range(1000):
|
||
# calculate gradients
|
||
dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
|
||
|
||
# regularization term gradients
|
||
dWo += lmbd * output_weights
|
||
dWh += lmbd * hidden_weights
|
||
|
||
# update weights and biases
|
||
output_weights -= eta * dWo
|
||
output_bias -= eta * dBo
|
||
hidden_weights -= eta * dWh
|
||
hidden_bias -= eta * dBh
|
||
|
||
print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="improving-performance">
|
||
<h2>Improving performance<a class="headerlink" href="#improving-performance" title="Link to this heading">#</a></h2>
|
||
<p>As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.<br />
|
||
In order to obtain a network that does something useful, we will have to do a bit more work.</p>
|
||
<p>The choice of <em>hyperparameters</em> such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a <em>grid-search</em> is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates <span class="math notranslate nohighlight">\(\eta = 10^{-6}, 10^{-5},...,10^{-1}\)</span> with different regularization parameters <span class="math notranslate nohighlight">\(\lambda = 10^{-6},...,10^{-0}\)</span>.</p>
|
||
<p>Next, we haven’t implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an <em>iteration</em>, and a full training period
|
||
going through the entire dataset (<span class="math notranslate nohighlight">\(n/M\)</span> batches) an <em>epoch</em>.</p>
|
||
<p>If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.<br />
|
||
Andrew Ng goes through some of these considerations in this <a class="reference external" href="https://youtu.be/F1ka6a13S9I">video</a>. You can find a summary of the video <a class="reference external" href="https://kevinzakka.github.io/2016/09/26/applying-deep-learning/">here</a>.</p>
|
||
</section>
|
||
<section id="full-object-oriented-implementation">
|
||
<h2>Full object-oriented implementation<a class="headerlink" href="#full-object-oriented-implementation" title="Link to this heading">#</a></h2>
|
||
<p>It is very natural to think of the network as an object, with specific instances of the network
|
||
being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>class NeuralNetwork:
|
||
def __init__(
|
||
self,
|
||
X_data,
|
||
Y_data,
|
||
n_hidden_neurons=50,
|
||
n_categories=10,
|
||
epochs=10,
|
||
batch_size=100,
|
||
eta=0.1,
|
||
lmbd=0.0):
|
||
|
||
self.X_data_full = X_data
|
||
self.Y_data_full = Y_data
|
||
|
||
self.n_inputs = X_data.shape[0]
|
||
self.n_features = X_data.shape[1]
|
||
self.n_hidden_neurons = n_hidden_neurons
|
||
self.n_categories = n_categories
|
||
|
||
self.epochs = epochs
|
||
self.batch_size = batch_size
|
||
self.iterations = self.n_inputs // self.batch_size
|
||
self.eta = eta
|
||
self.lmbd = lmbd
|
||
|
||
self.create_biases_and_weights()
|
||
|
||
def create_biases_and_weights(self):
|
||
self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
|
||
self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
|
||
|
||
self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
|
||
self.output_bias = np.zeros(self.n_categories) + 0.01
|
||
|
||
def feed_forward(self):
|
||
# feed-forward for training
|
||
self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
|
||
self.a_h = sigmoid(self.z_h)
|
||
|
||
self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
|
||
|
||
exp_term = np.exp(self.z_o)
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
|
||
def feed_forward_out(self, X):
|
||
# feed-forward for output
|
||
z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
|
||
a_h = sigmoid(z_h)
|
||
|
||
z_o = np.matmul(a_h, self.output_weights) + self.output_bias
|
||
|
||
exp_term = np.exp(z_o)
|
||
probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
return probabilities
|
||
|
||
def backpropagation(self):
|
||
error_output = self.probabilities - self.Y_data
|
||
error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
|
||
|
||
self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
|
||
self.output_bias_gradient = np.sum(error_output, axis=0)
|
||
|
||
self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
|
||
self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
|
||
|
||
if self.lmbd > 0.0:
|
||
self.output_weights_gradient += self.lmbd * self.output_weights
|
||
self.hidden_weights_gradient += self.lmbd * self.hidden_weights
|
||
|
||
self.output_weights -= self.eta * self.output_weights_gradient
|
||
self.output_bias -= self.eta * self.output_bias_gradient
|
||
self.hidden_weights -= self.eta * self.hidden_weights_gradient
|
||
self.hidden_bias -= self.eta * self.hidden_bias_gradient
|
||
|
||
def predict(self, X):
|
||
probabilities = self.feed_forward_out(X)
|
||
return np.argmax(probabilities, axis=1)
|
||
|
||
def predict_probabilities(self, X):
|
||
probabilities = self.feed_forward_out(X)
|
||
return probabilities
|
||
|
||
def train(self):
|
||
data_indices = np.arange(self.n_inputs)
|
||
|
||
for i in range(self.epochs):
|
||
for j in range(self.iterations):
|
||
# pick datapoints with replacement
|
||
chosen_datapoints = np.random.choice(
|
||
data_indices, size=self.batch_size, replace=False
|
||
)
|
||
|
||
# minibatch training data
|
||
self.X_data = self.X_data_full[chosen_datapoints]
|
||
self.Y_data = self.Y_data_full[chosen_datapoints]
|
||
|
||
self.feed_forward()
|
||
self.backpropagation()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="evaluate-model-performance-on-test-data">
|
||
<h2>Evaluate model performance on test data<a class="headerlink" href="#evaluate-model-performance-on-test-data" title="Link to this heading">#</a></h2>
|
||
<p>To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.<br />
|
||
We measure the performance of the network using the <em>accuracy</em> score.<br />
|
||
The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of <span class="math notranslate nohighlight">\(1\)</span>.</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(I\)</span> is the indicator function, <span class="math notranslate nohighlight">\(1\)</span> if <span class="math notranslate nohighlight">\(\tilde{y}_i = y_i\)</span> and <span class="math notranslate nohighlight">\(0\)</span> otherwise.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>epochs = 100
|
||
batch_size = 100
|
||
|
||
dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
|
||
n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
|
||
dnn.train()
|
||
test_predict = dnn.predict(X_test)
|
||
|
||
# accuracy score from scikit library
|
||
print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
|
||
|
||
# equivalent in numpy
|
||
def accuracy_score_numpy(Y_test, Y_pred):
|
||
return np.sum(Y_test == Y_pred) / len(Y_test)
|
||
|
||
#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="adjust-hyperparameters">
|
||
<h2>Adjust hyperparameters<a class="headerlink" href="#adjust-hyperparameters" title="Link to this heading">#</a></h2>
|
||
<p>We now perform a grid search to find the optimal hyperparameters for the network.<br />
|
||
Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around <span class="math notranslate nohighlight">\(98\%\)</span> (<span class="math notranslate nohighlight">\(2\%\)</span> error rate).</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>eta_vals = np.logspace(-5, 1, 7)
|
||
lmbd_vals = np.logspace(-5, 1, 7)
|
||
# store the models for later use
|
||
DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
||
|
||
# grid search
|
||
for i, eta in enumerate(eta_vals):
|
||
for j, lmbd in enumerate(lmbd_vals):
|
||
dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
|
||
n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
|
||
dnn.train()
|
||
|
||
DNN_numpy[i][j] = dnn
|
||
|
||
test_predict = dnn.predict(X_test)
|
||
|
||
print("Learning rate = ", eta)
|
||
print("Lambda = ", lmbd)
|
||
print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
|
||
print()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="visualization">
|
||
<h2>Visualization<a class="headerlink" href="#visualization" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># visual representation of grid search
|
||
# uses seaborn heatmap, you can also do this with matplotlib imshow
|
||
import seaborn as sns
|
||
|
||
sns.set()
|
||
|
||
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||
|
||
for i in range(len(eta_vals)):
|
||
for j in range(len(lmbd_vals)):
|
||
dnn = DNN_numpy[i][j]
|
||
|
||
train_pred = dnn.predict(X_train)
|
||
test_pred = dnn.predict(X_test)
|
||
|
||
train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
|
||
test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
|
||
|
||
|
||
fig, ax = plt.subplots(figsize = (10, 10))
|
||
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
||
ax.set_title("Training Accuracy")
|
||
ax.set_ylabel("$\eta$")
|
||
ax.set_xlabel("$\lambda$")
|
||
plt.show()
|
||
|
||
fig, ax = plt.subplots(figsize = (10, 10))
|
||
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
|
||
ax.set_title("Test Accuracy")
|
||
ax.set_ylabel("$\eta$")
|
||
ax.set_xlabel("$\lambda$")
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="scikit-learn-implementation">
|
||
<h2>scikit-learn implementation<a class="headerlink" href="#scikit-learn-implementation" title="Link to this heading">#</a></h2>
|
||
<p><strong>scikit-learn</strong> focuses more
|
||
on traditional machine learning methods, such as regression,
|
||
clustering, decision trees, etc. As such, it has only two types of
|
||
neural networks: Multi Layer Perceptron outputting continuous values,
|
||
<em>MPLRegressor</em>, and Multi Layer Perceptron outputting labels,
|
||
<em>MLPClassifier</em>. We will see how simple it is to use these classes.</p>
|
||
<p><strong>scikit-learn</strong> implements a few improvements from our neural network,
|
||
such as early stopping, a varying learning rate, different
|
||
optimization methods, etc. We would therefore expect a better
|
||
performance overall.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.neural_network import MLPClassifier
|
||
# store models for later use
|
||
DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
||
|
||
for i, eta in enumerate(eta_vals):
|
||
for j, lmbd in enumerate(lmbd_vals):
|
||
dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
|
||
alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
|
||
dnn.fit(X_train, Y_train)
|
||
|
||
DNN_scikit[i][j] = dnn
|
||
|
||
print("Learning rate = ", eta)
|
||
print("Lambda = ", lmbd)
|
||
print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
|
||
print()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="id1">
|
||
<h2>Visualization<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># optional
|
||
# visual representation of grid search
|
||
# uses seaborn heatmap, could probably do this in matplotlib
|
||
import seaborn as sns
|
||
|
||
sns.set()
|
||
|
||
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||
|
||
for i in range(len(eta_vals)):
|
||
for j in range(len(lmbd_vals)):
|
||
dnn = DNN_scikit[i][j]
|
||
|
||
train_pred = dnn.predict(X_train)
|
||
test_pred = dnn.predict(X_test)
|
||
|
||
train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
|
||
test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
|
||
|
||
|
||
fig, ax = plt.subplots(figsize = (10, 10))
|
||
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
||
ax.set_title("Training Accuracy")
|
||
ax.set_ylabel("$\eta$")
|
||
ax.set_xlabel("$\lambda$")
|
||
plt.show()
|
||
|
||
fig, ax = plt.subplots(figsize = (10, 10))
|
||
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
|
||
ax.set_title("Test Accuracy")
|
||
ax.set_ylabel("$\eta$")
|
||
ax.set_xlabel("$\lambda$")
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="building-neural-networks-in-tensorflow-and-keras">
|
||
<h2>Building neural networks in Tensorflow and Keras<a class="headerlink" href="#building-neural-networks-in-tensorflow-and-keras" title="Link to this heading">#</a></h2>
|
||
<p>Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
|
||
and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
|
||
and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.</p>
|
||
<p>In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
|
||
clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
|
||
NumPy arrays.</p>
|
||
</section>
|
||
<section id="tensorflow">
|
||
<h2>Tensorflow<a class="headerlink" href="#tensorflow" title="Link to this heading">#</a></h2>
|
||
<p>Tensorflow is an open source library machine learning library
|
||
developed by the Google Brain team for internal use. It was released
|
||
under the Apache 2.0 open source license in November 9, 2015.</p>
|
||
<p>Tensorflow is a computational framework that allows you to construct
|
||
machine learning models at different levels of abstraction, from
|
||
high-level, object-oriented APIs like Keras, down to the C++ kernels
|
||
that Tensorflow is built upon. The higher levels of abstraction are
|
||
simpler to use, but less flexible, and our choice of implementation
|
||
should reflect the problems we are trying to solve.</p>
|
||
<p><a class="reference external" href="https://www.tensorflow.org/guide/graphs">Tensorflow uses</a> so-called graphs to represent your computation
|
||
in terms of the dependencies between individual operations, such that you first build a Tensorflow <em>graph</em>
|
||
to represent your model, and then create a Tensorflow <em>session</em> to run the graph.</p>
|
||
<p>In this guide we will analyze the same data as we did in our NumPy and
|
||
scikit-learn tutorial, gathered from the MNIST database of images. We
|
||
will give an introduction to the lower level Python Application
|
||
Program Interfaces (APIs), and see how we use them to build our graph.
|
||
Then we will build (effectively) the same graph in Keras, to see just
|
||
how simple solving a machine learning problem can be.</p>
|
||
<p>To install tensorflow on Unix/Linux systems, use pip as</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>pip3 install tensorflow
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>and/or if you use <strong>anaconda</strong>, just write (or install from the graphical user interface)
|
||
(current release of CPU-only TensorFlow)</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>conda create -n tf tensorflow
|
||
conda activate tf
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>To install the current release of GPU TensorFlow</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>conda create -n tf-gpu tensorflow-gpu
|
||
conda activate tf-gpu
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="using-keras">
|
||
<h2>Using Keras<a class="headerlink" href="#using-keras" title="Link to this heading">#</a></h2>
|
||
<p>Keras is a high level <a class="reference external" href="https://en.wikipedia.org/wiki/Application_programming_interface">neural network</a>
|
||
that supports Tensorflow, CTNK and Theano as backends.<br />
|
||
If you have Anaconda installed you may run the following command</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>conda install keras
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>You can look up the <a class="reference external" href="https://keras.io/">instructions here</a> for more information.</p>
|
||
<p>We will to a large extent use <strong>keras</strong> in this course.</p>
|
||
</section>
|
||
<section id="id2">
|
||
<h2>Collect and pre-process data<a class="headerlink" href="#id2" title="Link to this heading">#</a></h2>
|
||
<p>Let us look again at the MINST data set.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># import necessary packages
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
import tensorflow as tf
|
||
from sklearn import datasets
|
||
|
||
|
||
# ensure the same random numbers appear every time
|
||
np.random.seed(0)
|
||
|
||
# display images in notebook
|
||
%matplotlib inline
|
||
plt.rcParams['figure.figsize'] = (12,12)
|
||
|
||
|
||
# download MNIST dataset
|
||
digits = datasets.load_digits()
|
||
|
||
# define inputs and labels
|
||
inputs = digits.images
|
||
labels = digits.target
|
||
|
||
print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
|
||
print("labels = (n_inputs) = " + str(labels.shape))
|
||
|
||
|
||
# flatten the image
|
||
# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
|
||
n_inputs = len(inputs)
|
||
inputs = inputs.reshape(n_inputs, -1)
|
||
print("X = (n_inputs, n_features) = " + str(inputs.shape))
|
||
|
||
|
||
# choose some random images to display
|
||
indices = np.arange(n_inputs)
|
||
random_indices = np.random.choice(indices, size=5)
|
||
|
||
for i, image in enumerate(digits.images[random_indices]):
|
||
plt.subplot(1, 5, i+1)
|
||
plt.axis('off')
|
||
plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
|
||
plt.title("Label: %d" % digits.target[random_indices[i]])
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from tensorflow.keras.layers import Input
|
||
from tensorflow.keras.models import Sequential #This allows appending layers to existing models
|
||
from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
|
||
from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
|
||
from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
|
||
from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
|
||
|
||
from sklearn.model_selection import train_test_split
|
||
|
||
# one-hot representation of labels
|
||
labels = to_categorical(labels)
|
||
|
||
# split into train and test data
|
||
train_size = 0.8
|
||
test_size = 1 - train_size
|
||
X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
|
||
test_size=test_size)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>
|
||
epochs = 100
|
||
batch_size = 100
|
||
n_neurons_layer1 = 100
|
||
n_neurons_layer2 = 50
|
||
n_categories = 10
|
||
eta_vals = np.logspace(-5, 1, 7)
|
||
lmbd_vals = np.logspace(-5, 1, 7)
|
||
def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
|
||
model = Sequential()
|
||
model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
|
||
model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
|
||
model.add(Dense(n_categories, activation='softmax'))
|
||
|
||
sgd = optimizers.SGD(lr=eta)
|
||
model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
|
||
|
||
return model
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
||
|
||
for i, eta in enumerate(eta_vals):
|
||
for j, lmbd in enumerate(lmbd_vals):
|
||
DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
|
||
eta=eta, lmbd=lmbd)
|
||
DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
|
||
scores = DNN.evaluate(X_test, Y_test)
|
||
|
||
DNN_keras[i][j] = DNN
|
||
|
||
print("Learning rate = ", eta)
|
||
print("Lambda = ", lmbd)
|
||
print("Test accuracy: %.3f" % scores[1])
|
||
print()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># optional
|
||
# visual representation of grid search
|
||
# uses seaborn heatmap, could probably do this in matplotlib
|
||
import seaborn as sns
|
||
|
||
sns.set()
|
||
|
||
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||
|
||
for i in range(len(eta_vals)):
|
||
for j in range(len(lmbd_vals)):
|
||
DNN = DNN_keras[i][j]
|
||
|
||
train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
|
||
test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
|
||
|
||
|
||
fig, ax = plt.subplots(figsize = (10, 10))
|
||
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
||
ax.set_title("Training Accuracy")
|
||
ax.set_ylabel("$\eta$")
|
||
ax.set_xlabel("$\lambda$")
|
||
plt.show()
|
||
|
||
fig, ax = plt.subplots(figsize = (10, 10))
|
||
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
|
||
ax.set_title("Test Accuracy")
|
||
ax.set_ylabel("$\eta$")
|
||
ax.set_xlabel("$\lambda$")
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="building-a-neural-network-code">
|
||
<h2>Building a neural network code<a class="headerlink" href="#building-a-neural-network-code" title="Link to this heading">#</a></h2>
|
||
<p>Here we present a flexible object oriented codebase
|
||
for a feed forward neural network, along with a demonstration of how
|
||
to use it. Before we get into the details of the neural network, we
|
||
will first present some implementations of various schedulers, cost
|
||
functions and activation functions that can be used together with the
|
||
neural network.</p>
|
||
<p>The codes here were developed by Eric Reber and Gregor Kajda during spring 2023.</p>
|
||
<section id="learning-rate-methods">
|
||
<h3>Learning rate methods<a class="headerlink" href="#learning-rate-methods" title="Link to this heading">#</a></h3>
|
||
<p>The code below shows object oriented implementations of the Constant,
|
||
Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All
|
||
of the classes belong to the shared abstract Scheduler class, and
|
||
share the update_change() and reset() methods allowing for any of the
|
||
schedulers to be seamlessly used during the training stage, as will
|
||
later be shown in the fit() method of the neural
|
||
network. Update_change() only has one parameter, the gradient
|
||
(<span class="math notranslate nohighlight">\(δ^l_ja^{l−1}_k\)</span>), and returns the change which will be subtracted
|
||
from the weights. The reset() function takes no parameters, and resets
|
||
the desired variables. For Constant and Momentum, reset does nothing.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
|
||
|
||
class Scheduler:
|
||
"""
|
||
Abstract class for Schedulers
|
||
"""
|
||
|
||
def __init__(self, eta):
|
||
self.eta = eta
|
||
|
||
# should be overwritten
|
||
def update_change(self, gradient):
|
||
raise NotImplementedError
|
||
|
||
# overwritten if needed
|
||
def reset(self):
|
||
pass
|
||
|
||
|
||
class Constant(Scheduler):
|
||
def __init__(self, eta):
|
||
super().__init__(eta)
|
||
|
||
def update_change(self, gradient):
|
||
return self.eta * gradient
|
||
|
||
def reset(self):
|
||
pass
|
||
|
||
|
||
class Momentum(Scheduler):
|
||
def __init__(self, eta: float, momentum: float):
|
||
super().__init__(eta)
|
||
self.momentum = momentum
|
||
self.change = 0
|
||
|
||
def update_change(self, gradient):
|
||
self.change = self.momentum * self.change + self.eta * gradient
|
||
return self.change
|
||
|
||
def reset(self):
|
||
pass
|
||
|
||
|
||
class Adagrad(Scheduler):
|
||
def __init__(self, eta):
|
||
super().__init__(eta)
|
||
self.G_t = None
|
||
|
||
def update_change(self, gradient):
|
||
delta = 1e-8 # avoid division ny zero
|
||
|
||
if self.G_t is None:
|
||
self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))
|
||
|
||
self.G_t += gradient @ gradient.T
|
||
|
||
G_t_inverse = 1 / (
|
||
delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
|
||
)
|
||
return self.eta * gradient * G_t_inverse
|
||
|
||
def reset(self):
|
||
self.G_t = None
|
||
|
||
|
||
class AdagradMomentum(Scheduler):
|
||
def __init__(self, eta, momentum):
|
||
super().__init__(eta)
|
||
self.G_t = None
|
||
self.momentum = momentum
|
||
self.change = 0
|
||
|
||
def update_change(self, gradient):
|
||
delta = 1e-8 # avoid division ny zero
|
||
|
||
if self.G_t is None:
|
||
self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))
|
||
|
||
self.G_t += gradient @ gradient.T
|
||
|
||
G_t_inverse = 1 / (
|
||
delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
|
||
)
|
||
self.change = self.change * self.momentum + self.eta * gradient * G_t_inverse
|
||
return self.change
|
||
|
||
def reset(self):
|
||
self.G_t = None
|
||
|
||
|
||
class RMS_prop(Scheduler):
|
||
def __init__(self, eta, rho):
|
||
super().__init__(eta)
|
||
self.rho = rho
|
||
self.second = 0.0
|
||
|
||
def update_change(self, gradient):
|
||
delta = 1e-8 # avoid division ny zero
|
||
self.second = self.rho * self.second + (1 - self.rho) * gradient * gradient
|
||
return self.eta * gradient / (np.sqrt(self.second + delta))
|
||
|
||
def reset(self):
|
||
self.second = 0.0
|
||
|
||
|
||
class Adam(Scheduler):
|
||
def __init__(self, eta, rho, rho2):
|
||
super().__init__(eta)
|
||
self.rho = rho
|
||
self.rho2 = rho2
|
||
self.moment = 0
|
||
self.second = 0
|
||
self.n_epochs = 1
|
||
|
||
def update_change(self, gradient):
|
||
delta = 1e-8 # avoid division ny zero
|
||
|
||
self.moment = self.rho * self.moment + (1 - self.rho) * gradient
|
||
self.second = self.rho2 * self.second + (1 - self.rho2) * gradient * gradient
|
||
|
||
moment_corrected = self.moment / (1 - self.rho**self.n_epochs)
|
||
second_corrected = self.second / (1 - self.rho2**self.n_epochs)
|
||
|
||
return self.eta * moment_corrected / (np.sqrt(second_corrected + delta))
|
||
|
||
def reset(self):
|
||
self.n_epochs += 1
|
||
self.moment = 0
|
||
self.second = 0
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="usage-of-the-above-learning-rate-schedulers">
|
||
<h3>Usage of the above learning rate schedulers<a class="headerlink" href="#usage-of-the-above-learning-rate-schedulers" title="Link to this heading">#</a></h3>
|
||
<p>To initalize a scheduler, simply create the object and pass in the
|
||
necessary parameters such as the learning rate and the momentum as
|
||
shown below. As the Scheduler class is an abstract class it should not
|
||
called directly, and will raise an error upon usage.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)
|
||
adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here is a small example for how a segment of code using schedulers
|
||
could look. Switching out the schedulers is simple.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>weights = np.ones((3,3))
|
||
print(f"Before scheduler:\n{weights=}")
|
||
|
||
epochs = 10
|
||
for e in range(epochs):
|
||
gradient = np.random.rand(3, 3)
|
||
change = adam_scheduler.update_change(gradient)
|
||
weights = weights - change
|
||
adam_scheduler.reset()
|
||
|
||
print(f"\nAfter scheduler:\n{weights=}")
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="cost-functions">
|
||
<h3>Cost functions<a class="headerlink" href="#cost-functions" title="Link to this heading">#</a></h3>
|
||
<p>Here we discuss cost functions that can be used when creating the
|
||
neural network. Every cost function takes the target vector as its
|
||
parameter, and returns a function valued only at <span class="math notranslate nohighlight">\(x\)</span> such that it may
|
||
easily be differentiated.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
|
||
|
||
def CostOLS(target):
|
||
|
||
def func(X):
|
||
return (1.0 / target.shape[0]) * np.sum((target - X) ** 2)
|
||
|
||
return func
|
||
|
||
|
||
def CostLogReg(target):
|
||
|
||
def func(X):
|
||
|
||
return -(1.0 / target.shape[0]) * np.sum(
|
||
(target * np.log(X + 10e-10)) + ((1 - target) * np.log(1 - X + 10e-10))
|
||
)
|
||
|
||
return func
|
||
|
||
|
||
def CostCrossEntropy(target):
|
||
|
||
def func(X):
|
||
return -(1.0 / target.size) * np.sum(target * np.log(X + 10e-10))
|
||
|
||
return func
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Below we give a short example of how these cost function may be used
|
||
to obtain results if you wish to test them out on your own using
|
||
AutoGrad’s automatics differentiation.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from autograd import grad
|
||
|
||
target = np.array([[1, 2, 3]]).T
|
||
a = np.array([[4, 5, 6]]).T
|
||
|
||
cost_func = CostCrossEntropy
|
||
cost_func_derivative = grad(cost_func(target))
|
||
|
||
valued_at_a = cost_func_derivative(a)
|
||
print(f"Derivative of cost function {cost_func.__name__} valued at a:\n{valued_at_a}")
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="id3">
|
||
<h3>Activation functions<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
|
||
<p>Finally, before we look at the neural network, we will look at the
|
||
activation functions which can be specified between the hidden layers
|
||
and as the output function. Each function can be valued for any given
|
||
vector or matrix X, and can be differentiated via derivate().</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
|
||
from autograd import elementwise_grad
|
||
|
||
def identity(X):
|
||
return X
|
||
|
||
|
||
def sigmoid(X):
|
||
try:
|
||
return 1.0 / (1 + np.exp(-X))
|
||
except FloatingPointError:
|
||
return np.where(X > np.zeros(X.shape), np.ones(X.shape), np.zeros(X.shape))
|
||
|
||
|
||
def softmax(X):
|
||
X = X - np.max(X, axis=-1, keepdims=True)
|
||
delta = 10e-10
|
||
return np.exp(X) / (np.sum(np.exp(X), axis=-1, keepdims=True) + delta)
|
||
|
||
|
||
def RELU(X):
|
||
return np.where(X > np.zeros(X.shape), X, np.zeros(X.shape))
|
||
|
||
|
||
def LRELU(X):
|
||
delta = 10e-4
|
||
return np.where(X > np.zeros(X.shape), X, delta * X)
|
||
|
||
|
||
def derivate(func):
|
||
if func.__name__ == "RELU":
|
||
|
||
def func(X):
|
||
return np.where(X > 0, 1, 0)
|
||
|
||
return func
|
||
|
||
elif func.__name__ == "LRELU":
|
||
|
||
def func(X):
|
||
delta = 10e-4
|
||
return np.where(X > 0, 1, delta)
|
||
|
||
return func
|
||
|
||
else:
|
||
return elementwise_grad(func)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Below follows a short demonstration of how to use an activation
|
||
function. The derivative of the activation function will be important
|
||
when calculating the output delta term during backpropagation. Note
|
||
that derivate() can also be used for cost functions for a more
|
||
generalized approach.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>z = np.array([[4, 5, 6]]).T
|
||
print(f"Input to activation function:\n{z}")
|
||
|
||
act_func = sigmoid
|
||
a = act_func(z)
|
||
print(f"\nOutput from {act_func.__name__} activation function:\n{a}")
|
||
|
||
act_func_derivative = derivate(act_func)
|
||
valued_at_z = act_func_derivative(a)
|
||
print(f"\nDerivative of {act_func.__name__} activation function valued at z:\n{valued_at_z}")
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="the-neural-network">
|
||
<h3>The Neural Network<a class="headerlink" href="#the-neural-network" title="Link to this heading">#</a></h3>
|
||
<p>Now that we have gotten a good understanding of the implementation of
|
||
some important components, we can take a look at an object oriented
|
||
implementation of a feed forward neural network. The feed forward
|
||
neural network has been implemented as a class named FFNN, which can
|
||
be initiated as a regressor or classifier dependant on the choice of
|
||
cost function. The FFNN can have any number of input nodes, hidden
|
||
layers with any amount of hidden nodes, and any amount of output nodes
|
||
meaning it can perform multiclass classification as well as binary
|
||
classification and regression problems. Although there is a lot of
|
||
code present, it makes for an easy to use and generalizeable interface
|
||
for creating many types of neural networks as will be demonstrated
|
||
below.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import math
|
||
import autograd.numpy as np
|
||
import sys
|
||
import warnings
|
||
from autograd import grad, elementwise_grad
|
||
from random import random, seed
|
||
from copy import deepcopy, copy
|
||
from typing import Tuple, Callable
|
||
from sklearn.utils import resample
|
||
|
||
warnings.simplefilter("error")
|
||
|
||
|
||
class FFNN:
|
||
"""
|
||
Description:
|
||
------------
|
||
Feed Forward Neural Network with interface enabling flexible design of a
|
||
nerual networks architecture and the specification of activation function
|
||
in the hidden layers and output layer respectively. This model can be used
|
||
for both regression and classification problems, depending on the output function.
|
||
|
||
Attributes:
|
||
------------
|
||
I dimensions (tuple[int]): A list of positive integers, which specifies the
|
||
number of nodes in each of the networks layers. The first integer in the array
|
||
defines the number of nodes in the input layer, the second integer defines number
|
||
of nodes in the first hidden layer and so on until the last number, which
|
||
specifies the number of nodes in the output layer.
|
||
II hidden_func (Callable): The activation function for the hidden layers
|
||
III output_func (Callable): The activation function for the output layer
|
||
IV cost_func (Callable): Our cost function
|
||
V seed (int): Sets random seed, makes results reproducible
|
||
"""
|
||
|
||
def __init__(
|
||
self,
|
||
dimensions: tuple[int],
|
||
hidden_func: Callable = sigmoid,
|
||
output_func: Callable = lambda x: x,
|
||
cost_func: Callable = CostOLS,
|
||
seed: int = None,
|
||
):
|
||
self.dimensions = dimensions
|
||
self.hidden_func = hidden_func
|
||
self.output_func = output_func
|
||
self.cost_func = cost_func
|
||
self.seed = seed
|
||
self.weights = list()
|
||
self.schedulers_weight = list()
|
||
self.schedulers_bias = list()
|
||
self.a_matrices = list()
|
||
self.z_matrices = list()
|
||
self.classification = None
|
||
|
||
self.reset_weights()
|
||
self._set_classification()
|
||
|
||
def fit(
|
||
self,
|
||
X: np.ndarray,
|
||
t: np.ndarray,
|
||
scheduler: Scheduler,
|
||
batches: int = 1,
|
||
epochs: int = 100,
|
||
lam: float = 0,
|
||
X_val: np.ndarray = None,
|
||
t_val: np.ndarray = None,
|
||
):
|
||
"""
|
||
Description:
|
||
------------
|
||
This function performs the training the neural network by performing the feedforward and backpropagation
|
||
algorithm to update the networks weights.
|
||
|
||
Parameters:
|
||
------------
|
||
I X (np.ndarray) : training data
|
||
II t (np.ndarray) : target data
|
||
III scheduler (Scheduler) : specified scheduler (algorithm for optimization of gradient descent)
|
||
IV scheduler_args (list[int]) : list of all arguments necessary for scheduler
|
||
|
||
Optional Parameters:
|
||
------------
|
||
V batches (int) : number of batches the datasets are split into, default equal to 1
|
||
VI epochs (int) : number of iterations used to train the network, default equal to 100
|
||
VII lam (float) : regularization hyperparameter lambda
|
||
VIII X_val (np.ndarray) : validation set
|
||
IX t_val (np.ndarray) : validation target set
|
||
|
||
Returns:
|
||
------------
|
||
I scores (dict) : A dictionary containing the performance metrics of the model.
|
||
The number of the metrics depends on the parameters passed to the fit-function.
|
||
|
||
"""
|
||
|
||
# setup
|
||
if self.seed is not None:
|
||
np.random.seed(self.seed)
|
||
|
||
val_set = False
|
||
if X_val is not None and t_val is not None:
|
||
val_set = True
|
||
|
||
# creating arrays for score metrics
|
||
train_errors = np.empty(epochs)
|
||
train_errors.fill(np.nan)
|
||
val_errors = np.empty(epochs)
|
||
val_errors.fill(np.nan)
|
||
|
||
train_accs = np.empty(epochs)
|
||
train_accs.fill(np.nan)
|
||
val_accs = np.empty(epochs)
|
||
val_accs.fill(np.nan)
|
||
|
||
self.schedulers_weight = list()
|
||
self.schedulers_bias = list()
|
||
|
||
batch_size = X.shape[0] // batches
|
||
|
||
X, t = resample(X, t)
|
||
|
||
# this function returns a function valued only at X
|
||
cost_function_train = self.cost_func(t)
|
||
if val_set:
|
||
cost_function_val = self.cost_func(t_val)
|
||
|
||
# create schedulers for each weight matrix
|
||
for i in range(len(self.weights)):
|
||
self.schedulers_weight.append(copy(scheduler))
|
||
self.schedulers_bias.append(copy(scheduler))
|
||
|
||
print(f"{scheduler.__class__.__name__}: Eta={scheduler.eta}, Lambda={lam}")
|
||
|
||
try:
|
||
for e in range(epochs):
|
||
for i in range(batches):
|
||
# allows for minibatch gradient descent
|
||
if i == batches - 1:
|
||
# If the for loop has reached the last batch, take all thats left
|
||
X_batch = X[i * batch_size :, :]
|
||
t_batch = t[i * batch_size :, :]
|
||
else:
|
||
X_batch = X[i * batch_size : (i + 1) * batch_size, :]
|
||
t_batch = t[i * batch_size : (i + 1) * batch_size, :]
|
||
|
||
self._feedforward(X_batch)
|
||
self._backpropagate(X_batch, t_batch, lam)
|
||
|
||
# reset schedulers for each epoch (some schedulers pass in this call)
|
||
for scheduler in self.schedulers_weight:
|
||
scheduler.reset()
|
||
|
||
for scheduler in self.schedulers_bias:
|
||
scheduler.reset()
|
||
|
||
# computing performance metrics
|
||
pred_train = self.predict(X)
|
||
train_error = cost_function_train(pred_train)
|
||
|
||
train_errors[e] = train_error
|
||
if val_set:
|
||
|
||
pred_val = self.predict(X_val)
|
||
val_error = cost_function_val(pred_val)
|
||
val_errors[e] = val_error
|
||
|
||
if self.classification:
|
||
train_acc = self._accuracy(self.predict(X), t)
|
||
train_accs[e] = train_acc
|
||
if val_set:
|
||
val_acc = self._accuracy(pred_val, t_val)
|
||
val_accs[e] = val_acc
|
||
|
||
# printing progress bar
|
||
progression = e / epochs
|
||
print_length = self._progress_bar(
|
||
progression,
|
||
train_error=train_errors[e],
|
||
train_acc=train_accs[e],
|
||
val_error=val_errors[e],
|
||
val_acc=val_accs[e],
|
||
)
|
||
except KeyboardInterrupt:
|
||
# allows for stopping training at any point and seeing the result
|
||
pass
|
||
|
||
# visualization of training progression (similiar to tensorflow progression bar)
|
||
sys.stdout.write("\r" + " " * print_length)
|
||
sys.stdout.flush()
|
||
self._progress_bar(
|
||
1,
|
||
train_error=train_errors[e],
|
||
train_acc=train_accs[e],
|
||
val_error=val_errors[e],
|
||
val_acc=val_accs[e],
|
||
)
|
||
sys.stdout.write("")
|
||
|
||
# return performance metrics for the entire run
|
||
scores = dict()
|
||
|
||
scores["train_errors"] = train_errors
|
||
|
||
if val_set:
|
||
scores["val_errors"] = val_errors
|
||
|
||
if self.classification:
|
||
scores["train_accs"] = train_accs
|
||
|
||
if val_set:
|
||
scores["val_accs"] = val_accs
|
||
|
||
return scores
|
||
|
||
def predict(self, X: np.ndarray, *, threshold=0.5):
|
||
"""
|
||
Description:
|
||
------------
|
||
Performs prediction after training of the network has been finished.
|
||
|
||
Parameters:
|
||
------------
|
||
I X (np.ndarray): The design matrix, with n rows of p features each
|
||
|
||
Optional Parameters:
|
||
------------
|
||
II threshold (float) : sets minimal value for a prediction to be predicted as the positive class
|
||
in classification problems
|
||
|
||
Returns:
|
||
------------
|
||
I z (np.ndarray): A prediction vector (row) for each row in our design matrix
|
||
This vector is thresholded if regression=False, meaning that classification results
|
||
in a vector of 1s and 0s, while regressions in an array of decimal numbers
|
||
|
||
"""
|
||
|
||
predict = self._feedforward(X)
|
||
|
||
if self.classification:
|
||
return np.where(predict > threshold, 1, 0)
|
||
else:
|
||
return predict
|
||
|
||
def reset_weights(self):
|
||
"""
|
||
Description:
|
||
------------
|
||
Resets/Reinitializes the weights in order to train the network for a new problem.
|
||
|
||
"""
|
||
if self.seed is not None:
|
||
np.random.seed(self.seed)
|
||
|
||
self.weights = list()
|
||
for i in range(len(self.dimensions) - 1):
|
||
weight_array = np.random.randn(
|
||
self.dimensions[i] + 1, self.dimensions[i + 1]
|
||
)
|
||
weight_array[0, :] = np.random.randn(self.dimensions[i + 1]) * 0.01
|
||
|
||
self.weights.append(weight_array)
|
||
|
||
def _feedforward(self, X: np.ndarray):
|
||
"""
|
||
Description:
|
||
------------
|
||
Calculates the activation of each layer starting at the input and ending at the output.
|
||
Each following activation is calculated from a weighted sum of each of the preceeding
|
||
activations (except in the case of the input layer).
|
||
|
||
Parameters:
|
||
------------
|
||
I X (np.ndarray): The design matrix, with n rows of p features each
|
||
|
||
Returns:
|
||
------------
|
||
I z (np.ndarray): A prediction vector (row) for each row in our design matrix
|
||
"""
|
||
|
||
# reset matrices
|
||
self.a_matrices = list()
|
||
self.z_matrices = list()
|
||
|
||
# if X is just a vector, make it into a matrix
|
||
if len(X.shape) == 1:
|
||
X = X.reshape((1, X.shape[0]))
|
||
|
||
# Add a coloumn of zeros as the first coloumn of the design matrix, in order
|
||
# to add bias to our data
|
||
bias = np.ones((X.shape[0], 1)) * 0.01
|
||
X = np.hstack([bias, X])
|
||
|
||
# a^0, the nodes in the input layer (one a^0 for each row in X - where the
|
||
# exponent indicates layer number).
|
||
a = X
|
||
self.a_matrices.append(a)
|
||
self.z_matrices.append(a)
|
||
|
||
# The feed forward algorithm
|
||
for i in range(len(self.weights)):
|
||
if i < len(self.weights) - 1:
|
||
z = a @ self.weights[i]
|
||
self.z_matrices.append(z)
|
||
a = self.hidden_func(z)
|
||
# bias column again added to the data here
|
||
bias = np.ones((a.shape[0], 1)) * 0.01
|
||
a = np.hstack([bias, a])
|
||
self.a_matrices.append(a)
|
||
else:
|
||
try:
|
||
# a^L, the nodes in our output layers
|
||
z = a @ self.weights[i]
|
||
a = self.output_func(z)
|
||
self.a_matrices.append(a)
|
||
self.z_matrices.append(z)
|
||
except Exception as OverflowError:
|
||
print(
|
||
"OverflowError in fit() in FFNN\nHOW TO DEBUG ERROR: Consider lowering your learning rate or scheduler specific parameters such as momentum, or check if your input values need scaling"
|
||
)
|
||
|
||
# this will be a^L
|
||
return a
|
||
|
||
def _backpropagate(self, X, t, lam):
|
||
"""
|
||
Description:
|
||
------------
|
||
Performs the backpropagation algorithm. In other words, this method
|
||
calculates the gradient of all the layers starting at the
|
||
output layer, and moving from right to left accumulates the gradient until
|
||
the input layer is reached. Each layers respective weights are updated while
|
||
the algorithm propagates backwards from the output layer (auto-differentation in reverse mode).
|
||
|
||
Parameters:
|
||
------------
|
||
I X (np.ndarray): The design matrix, with n rows of p features each.
|
||
II t (np.ndarray): The target vector, with n rows of p targets.
|
||
III lam (float32): regularization parameter used to punish the weights in case of overfitting
|
||
|
||
Returns:
|
||
------------
|
||
No return value.
|
||
|
||
"""
|
||
out_derivative = derivate(self.output_func)
|
||
hidden_derivative = derivate(self.hidden_func)
|
||
|
||
for i in range(len(self.weights) - 1, -1, -1):
|
||
# delta terms for output
|
||
if i == len(self.weights) - 1:
|
||
# for multi-class classification
|
||
if (
|
||
self.output_func.__name__ == "softmax"
|
||
):
|
||
delta_matrix = self.a_matrices[i + 1] - t
|
||
# for single class classification
|
||
else:
|
||
cost_func_derivative = grad(self.cost_func(t))
|
||
delta_matrix = out_derivative(
|
||
self.z_matrices[i + 1]
|
||
) * cost_func_derivative(self.a_matrices[i + 1])
|
||
|
||
# delta terms for hidden layer
|
||
else:
|
||
delta_matrix = (
|
||
self.weights[i + 1][1:, :] @ delta_matrix.T
|
||
).T * hidden_derivative(self.z_matrices[i + 1])
|
||
|
||
# calculate gradient
|
||
gradient_weights = self.a_matrices[i][:, 1:].T @ delta_matrix
|
||
gradient_bias = np.sum(delta_matrix, axis=0).reshape(
|
||
1, delta_matrix.shape[1]
|
||
)
|
||
|
||
# regularization term
|
||
gradient_weights += self.weights[i][1:, :] * lam
|
||
|
||
# use scheduler
|
||
update_matrix = np.vstack(
|
||
[
|
||
self.schedulers_bias[i].update_change(gradient_bias),
|
||
self.schedulers_weight[i].update_change(gradient_weights),
|
||
]
|
||
)
|
||
|
||
# update weights and bias
|
||
self.weights[i] -= update_matrix
|
||
|
||
def _accuracy(self, prediction: np.ndarray, target: np.ndarray):
|
||
"""
|
||
Description:
|
||
------------
|
||
Calculates accuracy of given prediction to target
|
||
|
||
Parameters:
|
||
------------
|
||
I prediction (np.ndarray): vector of predicitons output network
|
||
(1s and 0s in case of classification, and real numbers in case of regression)
|
||
II target (np.ndarray): vector of true values (What the network ideally should predict)
|
||
|
||
Returns:
|
||
------------
|
||
A floating point number representing the percentage of correctly classified instances.
|
||
"""
|
||
assert prediction.size == target.size
|
||
return np.average((target == prediction))
|
||
def _set_classification(self):
|
||
"""
|
||
Description:
|
||
------------
|
||
Decides if FFNN acts as classifier (True) og regressor (False),
|
||
sets self.classification during init()
|
||
"""
|
||
self.classification = False
|
||
if (
|
||
self.cost_func.__name__ == "CostLogReg"
|
||
or self.cost_func.__name__ == "CostCrossEntropy"
|
||
):
|
||
self.classification = True
|
||
|
||
def _progress_bar(self, progression, **kwargs):
|
||
"""
|
||
Description:
|
||
------------
|
||
Displays progress of training
|
||
"""
|
||
print_length = 40
|
||
num_equals = int(progression * print_length)
|
||
num_not = print_length - num_equals
|
||
arrow = ">" if num_equals > 0 else ""
|
||
bar = "[" + "=" * (num_equals - 1) + arrow + "-" * num_not + "]"
|
||
perc_print = self._format(progression * 100, decimals=5)
|
||
line = f" {bar} {perc_print}% "
|
||
|
||
for key in kwargs:
|
||
if not np.isnan(kwargs[key]):
|
||
value = self._format(kwargs[key], decimals=4)
|
||
line += f"| {key}: {value} "
|
||
sys.stdout.write("\r" + line)
|
||
sys.stdout.flush()
|
||
return len(line)
|
||
|
||
def _format(self, value, decimals=4):
|
||
"""
|
||
Description:
|
||
------------
|
||
Formats decimal numbers for progress bar
|
||
"""
|
||
if value > 0:
|
||
v = value
|
||
elif value < 0:
|
||
v = -10 * value
|
||
else:
|
||
v = 1
|
||
n = 1 + math.floor(math.log10(v))
|
||
if n >= decimals - 1:
|
||
return str(round(value))
|
||
return f"{value:.{decimals-n-1}f}"
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Before we make a model, we will quickly generate a dataset we can use
|
||
for our linear regression problem as shown below</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
|
||
from sklearn.model_selection import train_test_split
|
||
|
||
def SkrankeFunction(x, y):
|
||
return np.ravel(0 + 1*x + 2*y + 3*x**2 + 4*x*y + 5*y**2)
|
||
|
||
def create_X(x, y, n):
|
||
if len(x.shape) > 1:
|
||
x = np.ravel(x)
|
||
y = np.ravel(y)
|
||
|
||
N = len(x)
|
||
l = int((n + 1) * (n + 2) / 2) # Number of elements in beta
|
||
X = np.ones((N, l))
|
||
|
||
for i in range(1, n + 1):
|
||
q = int((i) * (i + 1) / 2)
|
||
for k in range(i + 1):
|
||
X[:, q + k] = (x ** (i - k)) * (y**k)
|
||
|
||
return X
|
||
|
||
step=0.5
|
||
x = np.arange(0, 1, step)
|
||
y = np.arange(0, 1, step)
|
||
x, y = np.meshgrid(x, y)
|
||
target = SkrankeFunction(x, y)
|
||
target = target.reshape(target.shape[0], 1)
|
||
|
||
poly_degree=3
|
||
X = create_X(x, y, poly_degree)
|
||
|
||
X_train, X_test, t_train, t_test = train_test_split(X, target)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Now that we have our dataset ready for the regression, we can create
|
||
our regressor. Note that with the seed parameter, we can make sure our
|
||
results stay the same every time we run the neural network. For
|
||
inititialization, we simply specify the dimensions (we wish the amount
|
||
of input nodes to be equal to the datapoints, and the output to
|
||
predict one value).</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>input_nodes = X_train.shape[1]
|
||
output_nodes = 1
|
||
|
||
linear_regression = FFNN((input_nodes, output_nodes), output_func=identity, cost_func=CostOLS, seed=2023)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We then fit our model with our training data using the scheduler of our choice.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights
|
||
|
||
scheduler = Constant(eta=1e-3)
|
||
scores = linear_regression.fit(X_train, t_train, scheduler)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Due to the progress bar we can see the MSE (train_error) throughout
|
||
the FFNN’s training. Note that the fit() function has some optional
|
||
parameters with defualt arguments. For example, the regularization
|
||
hyperparameter can be left ignored if not needed, and equally the FFNN
|
||
will by default run for 100 epochs. These can easily be changed, such
|
||
as for example:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights
|
||
|
||
scores = linear_regression.fit(X_train, t_train, scheduler, lam=1e-4, epochs=1000)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We see that given more epochs to train on, the regressor reaches a lower MSE.</p>
|
||
<p>Let us then switch to a binary classification. We use a binary
|
||
classification dataset, and follow a similar setup to the regression
|
||
case.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_breast_cancer
|
||
from sklearn.preprocessing import MinMaxScaler
|
||
|
||
wisconsin = load_breast_cancer()
|
||
X = wisconsin.data
|
||
target = wisconsin.target
|
||
target = target.reshape(target.shape[0], 1)
|
||
|
||
X_train, X_val, t_train, t_val = train_test_split(X, target)
|
||
|
||
scaler = MinMaxScaler()
|
||
scaler.fit(X_train)
|
||
X_train = scaler.transform(X_train)
|
||
X_val = scaler.transform(X_val)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>input_nodes = X_train.shape[1]
|
||
output_nodes = 1
|
||
|
||
logistic_regression = FFNN((input_nodes, output_nodes), output_func=sigmoid, cost_func=CostLogReg, seed=2023)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We will now make use of our validation data by passing it into our fit function as a keyword argument</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>logistic_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights
|
||
|
||
scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
|
||
scores = logistic_regression.fit(X_train, t_train, scheduler, epochs=1000, X_val=X_val, t_val=t_val)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Finally, we will create a neural network with 2 hidden layers with activation functions.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>input_nodes = X_train.shape[1]
|
||
hidden_nodes1 = 100
|
||
hidden_nodes2 = 30
|
||
output_nodes = 1
|
||
|
||
dims = (input_nodes, hidden_nodes1, hidden_nodes2, output_nodes)
|
||
|
||
neural_network = FFNN(dims, hidden_func=RELU, output_func=sigmoid, cost_func=CostLogReg, seed=2023)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>neural_network.reset_weights() # reset weights such that previous runs or reruns don't affect the weights
|
||
|
||
scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999)
|
||
scores = neural_network.fit(X_train, t_train, scheduler, epochs=1000, X_val=X_val, t_val=t_val)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="multiclass-classification">
|
||
<h3>Multiclass classification<a class="headerlink" href="#multiclass-classification" title="Link to this heading">#</a></h3>
|
||
<p>Finally, we will demonstrate the use case of multiclass classification
|
||
using our FFNN with the famous MNIST dataset, which contain images of
|
||
digits between the range of 0 to 9.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_digits
|
||
|
||
def onehot(target: np.ndarray):
|
||
onehot = np.zeros((target.size, target.max() + 1))
|
||
onehot[np.arange(target.size), target] = 1
|
||
return onehot
|
||
|
||
digits = load_digits()
|
||
|
||
X = digits.data
|
||
target = digits.target
|
||
target = onehot(target)
|
||
|
||
input_nodes = 64
|
||
hidden_nodes1 = 100
|
||
hidden_nodes2 = 30
|
||
output_nodes = 10
|
||
|
||
dims = (input_nodes, hidden_nodes1, hidden_nodes2, output_nodes)
|
||
|
||
multiclass = FFNN(dims, hidden_func=LRELU, output_func=softmax, cost_func=CostCrossEntropy)
|
||
|
||
multiclass.reset_weights() # reset weights such that previous runs or reruns don't affect the weights
|
||
|
||
scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999)
|
||
scores = multiclass.fit(X, target, scheduler, epochs=1000)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
<section id="testing-the-xor-gate-and-other-gates">
|
||
<h2>Testing the XOR gate and other gates<a class="headerlink" href="#testing-the-xor-gate-and-other-gates" title="Link to this heading">#</a></h2>
|
||
<p>Let us now use our code to test the XOR gate.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)
|
||
|
||
# The XOR gate
|
||
yXOR = np.array( [[ 0], [1] ,[1], [0]])
|
||
|
||
input_nodes = X.shape[1]
|
||
output_nodes = 1
|
||
|
||
logistic_regression = FFNN((input_nodes, output_nodes), output_func=sigmoid, cost_func=CostLogReg, seed=2023)
|
||
logistic_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights
|
||
scheduler = Adam(eta=1e-1, rho=0.9, rho2=0.999)
|
||
scores = logistic_regression.fit(X, yXOR, scheduler, epochs=1000)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Not bad, but the results depend strongly on the learning reate. Try different learning rates.</p>
|
||
</section>
|
||
</section>
|
||
|
||
<script type="text/x-thebe-config">
|
||
{
|
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requestKernel: true,
|
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binderOptions: {
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||
<script>kernelName = 'python3'</script>
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||
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|
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|
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<i class="fa-solid fa-list"></i> Contents
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<nav class="bd-toc-nav page-toc">
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<ul class="visible nav section-nav flex-column">
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-october-13-2025">Lecture October 13, 2025</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#readings-and-videos">Readings and videos</a></li>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#material-for-the-lab-sessions-on-tuesday-and-wednesday">Material for the lab sessions on Tuesday and Wednesday</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-material-writing-a-code-which-implements-a-feed-forward-neural-network">Lecture material: Writing a code which implements a feed-forward neural network</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematics-of-deep-learning">Mathematics of deep learning</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reminder-on-books-with-hands-on-material-and-codes">Reminder on books with hands-on material and codes</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reading-recommendations">Reading recommendations</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reminder-from-last-week-first-network-example-simple-percepetron-with-one-input">Reminder from last week: First network example, simple percepetron with one input</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-simple-neural-network-with-no-hidden-layer">Layout of a simple neural network with no hidden layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#optimizing-the-parameters">Optimizing the parameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adding-a-hidden-layer">Adding a hidden layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-simple-neural-network-with-one-hidden-layer">Layout of a simple neural network with one hidden layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-derivatives">The derivatives</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#important-observations">Important observations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-training">The training</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-example">Code example</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-neural-network-and-the-back-propagation-equations">Simple neural network and the back propagation equations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-with-two-hidden-noeds-and-one-output-node">Layout of a simple neural network with two input nodes, one hidden layer with two hidden noeds and one output node</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-ouput-layer">The ouput layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compact-expressions">Compact expressions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#output-layer">Output layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#explicit-derivatives">Explicit derivatives</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-of-the-hidden-layer">Derivatives of the hidden layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-expression">Final expression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#completing-the-list">Completing the list</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-expressions-for-the-biases-of-the-hidden-layer">Final expressions for the biases of the hidden layer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-expressions">Gradient expressions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-equations-for-a-neural-network">Setting up the equations for a neural network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-a-neural-network-with-three-hidden-layers-last-layer-l-l-4-first-layer-l-0">Layout of a neural network with three hidden layers (last layer = <span class="math notranslate nohighlight">\(l=L=4\)</span>, first layer <span class="math notranslate nohighlight">\(l=0\)</span>)</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#definitions">Definitions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inputs-to-the-activation-function">Inputs to the activation function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layout-of-input-to-first-hidden-layer-l-1-from-input-layer-l-0">Layout of input to first hidden layer <span class="math notranslate nohighlight">\(l=1\)</span> from input layer <span class="math notranslate nohighlight">\(l=0\)</span></a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-and-the-chain-rule">Derivatives and the chain rule</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-of-the-cost-function">Derivative of the cost function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-back-propagation-equations-for-a-neural-network">The back propagation equations for a neural network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analyzing-the-last-results">Analyzing the last results</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-considerations">More considerations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-in-terms-of-z-j-l">Derivatives in terms of <span class="math notranslate nohighlight">\(z_j^L\)</span></a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bringing-it-together">Bringing it together</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-back-propagating-equation">Final back propagating equation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-the-chain-rule-and-summing-over-all-k-entries">Using the chain rule and summing over all <span class="math notranslate nohighlight">\(k\)</span> entries</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-1">Setting up the back propagation algorithm, part 1</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-2">Setting up the back propagation algorithm, part 2</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-3">Setting up the Back propagation algorithm, part 3</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#updating-the-gradients">Updating the gradients</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions">Activation functions</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions-logistic-and-hyperbolic-ones">Activation functions, Logistic and Hyperbolic ones</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#relevance">Relevance</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#vanishing-gradients">Vanishing gradients</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exploding-gradients">Exploding gradients</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#is-the-logistic-activation-function-sigmoid-our-choice">Is the Logistic activation function (Sigmoid) our choice?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logistic-function-as-the-root-of-problems">Logistic function as the root of problems</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-derivative-of-the-logistic-funtion">The derivative of the Logistic funtion</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#insights-from-the-paper-by-glorot-and-bengio">Insights from the paper by Glorot and Bengio</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-relu-function-family">The RELU function family</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#elu-function">ELU function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-activation-function-should-we-use">Which activation function should we use?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-on-activation-functions-output-layers">More on activation functions, output layers</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fine-tuning-neural-network-hyperparameters">Fine-tuning neural network hyperparameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#hidden-layers">Hidden layers</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#batch-normalization">Batch Normalization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dropout">Dropout</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-clipping">Gradient Clipping</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-top-down-perspective-on-neural-networks">A top-down perspective on Neural networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-top-down-perspectives">More top-down perspectives</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#limitations-of-supervised-learning-with-deep-networks">Limitations of supervised learning with deep networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#limitations-of-nns">Limitations of NNs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#homogeneous-data">Homogeneous data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-limitations">More limitations</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-a-multi-layer-perceptron-model-for-classification">Setting up a Multi-layer perceptron model for classification</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-cost-function">Defining the cost function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-binary-classification-problem">Example: binary classification problem</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-softmax-function">The Softmax function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#developing-a-code-for-doing-neural-networks-with-back-propagation">Developing a code for doing neural networks with back propagation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#collect-and-pre-process-data">Collect and pre-process data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#train-and-test-datasets">Train and test datasets</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-model-and-architecture">Define model and architecture</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#layers">Layers</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weights-and-biases">Weights and biases</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#feed-forward-pass">Feed-forward pass</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-multiplications">Matrix multiplications</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choose-cost-function-and-optimizer">Choose cost function and optimizer</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#optimizing-the-cost-function">Optimizing the cost function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regularization">Regularization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-multiplication">Matrix multiplication</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#improving-performance">Improving performance</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#full-object-oriented-implementation">Full object-oriented implementation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluate-model-performance-on-test-data">Evaluate model performance on test data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adjust-hyperparameters">Adjust hyperparameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#visualization">Visualization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#scikit-learn-implementation">scikit-learn implementation</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Visualization</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-neural-networks-in-tensorflow-and-keras">Building neural networks in Tensorflow and Keras</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tensorflow">Tensorflow</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-keras">Using Keras</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Collect and pre-process data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-neural-network-code">Building a neural network code</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-rate-methods">Learning rate methods</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-the-above-learning-rate-schedulers">Usage of the above learning rate schedulers</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-functions">Cost functions</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Activation functions</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-neural-network">The Neural Network</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#multiclass-classification">Multiclass classification</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#testing-the-xor-gate-and-other-gates">Testing the XOR gate and other gates</a></li>
|
||
</ul>
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