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Applied Data Analysis and Machine Learning
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Review of Statistics with Resampling Techniques and Linear Algebra
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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Decision Trees, Ensemble Methods and Boosting
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Dimensionality Reduction
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Deep Learning Methods
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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Exercises week 35
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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Exercises week 36
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Week 36: Linear Regression and Statistical interpretations
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Week 37: Statistical interpretations and Resampling Methods
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Week 39: Optimization and Gradient Methods
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Week 40: Gradient descent methods (continued) and start Neural networks
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Exercises week 41
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Week 41 Neural networks and constructing a neural network code
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#plan-for-week-41-october-7-11">
Plan for week 41, October 7-11
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#material-for-the-lecture-on-monday-october-7-2024">
Material for the lecture on Monday October 7, 2024
</a>
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<a class="reference internal nav-link" href="#material-for-the-active-learning-sessions-on-tuesday-and-wednesday">
Material for the active learning sessions on Tuesday and Wednesday
</a>
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#lecture-monday-october-7">
Lecture Monday October 7
</a>
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#introduction-to-neural-networks">
Introduction to Neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#artificial-neurons">
Artificial neurons
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#neural-network-types">
Neural network types
</a>
</li>
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<a class="reference internal nav-link" href="#feed-forward-neural-networks">
Feed-forward neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#convolutional-neural-network">
Convolutional Neural Network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#recurrent-neural-networks">
Recurrent neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#other-types-of-networks">
Other types of networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#multilayer-perceptrons">
Multilayer perceptrons
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-multilayer-perceptrons">
Why multilayer perceptrons?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model">
Illustration of a single perceptron model and a multi-perceptron model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#examples-of-xor-or-and-and-gates">
Examples of XOR, OR and AND gates
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#does-logistic-regression-do-a-better-job">
Does Logistic Regression do a better Job?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adding-neural-networks">
Adding Neural Networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#mathematical-model">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id2">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id3">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id4">
Mathematical model
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrix-vector-notation">
Matrix-vector notation
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrix-vector-notation-and-activation">
Matrix-vector notation and activation
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#activation-functions">
Activation functions
</a>
</li>
<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>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#relevance">
Relevance
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-multilayer-perceptron-mlp">
The multilayer perceptron (MLP)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#from-one-to-many-layers-the-universal-approximation-theorem">
From one to many layers, the universal approximation theorem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">
Deriving the back propagation code for a multilayer perceptron model
</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="#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="#bringing-it-together-first-back-propagation-equation">
Bringing it together, first back propagation equation
</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="#setting-up-the-back-propagation-algorithm">
Setting up the Back propagation algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id5">
Setting up the Back propagation algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id6">
Setting up the Back propagation algorithm
</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="#id7">
Visualization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#testing-our-code-for-the-xor-or-and-and-gates">
Testing our code for the XOR, OR and AND gates
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-and-and-xor-gates">
The AND and XOR Gates
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#representing-the-data-sets">
Representing the Data Sets
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#setting-up-the-neural-network">
Setting up the Neural Network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-code-using-scikit-learn">
The Code using Scikit-Learn
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
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<h1>Week 41 Neural networks and constructing a neural network code</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
<div>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#plan-for-week-41-october-7-11">
Plan for week 41, October 7-11
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#material-for-the-lecture-on-monday-october-7-2024">
Material for the lecture on Monday October 7, 2024
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#material-for-the-active-learning-sessions-on-tuesday-and-wednesday">
Material for the active learning sessions on Tuesday and Wednesday
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#lecture-monday-october-7">
Lecture Monday October 7
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#introduction-to-neural-networks">
Introduction to Neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#artificial-neurons">
Artificial neurons
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#neural-network-types">
Neural network types
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#feed-forward-neural-networks">
Feed-forward neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#convolutional-neural-network">
Convolutional Neural Network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#recurrent-neural-networks">
Recurrent neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#other-types-of-networks">
Other types of networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#multilayer-perceptrons">
Multilayer perceptrons
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-multilayer-perceptrons">
Why multilayer perceptrons?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model">
Illustration of a single perceptron model and a multi-perceptron model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#examples-of-xor-or-and-and-gates">
Examples of XOR, OR and AND gates
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#does-logistic-regression-do-a-better-job">
Does Logistic Regression do a better Job?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adding-neural-networks">
Adding Neural Networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#mathematical-model">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id2">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id3">
Mathematical model
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id4">
Mathematical model
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrix-vector-notation">
Matrix-vector notation
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrix-vector-notation-and-activation">
Matrix-vector notation and activation
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#activation-functions">
Activation functions
</a>
</li>
<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>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#relevance">
Relevance
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-multilayer-perceptron-mlp">
The multilayer perceptron (MLP)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#from-one-to-many-layers-the-universal-approximation-theorem">
From one to many layers, the universal approximation theorem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">
Deriving the back propagation code for a multilayer perceptron model
</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="#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="#bringing-it-together-first-back-propagation-equation">
Bringing it together, first back propagation equation
</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="#setting-up-the-back-propagation-algorithm">
Setting up the Back propagation algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id5">
Setting up the Back propagation algorithm
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id6">
Setting up the Back propagation algorithm
</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="#id7">
Visualization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#testing-our-code-for-the-xor-or-and-and-gates">
Testing our code for the XOR, OR and AND gates
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-and-and-xor-gates">
The AND and XOR Gates
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#representing-the-data-sets">
Representing the Data Sets
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#setting-up-the-neural-network">
Setting up the Neural Network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-code-using-scikit-learn">
The Code using Scikit-Learn
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div>
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
doconce format html week41.do.txt --no_mako -->
<!-- dom:TITLE: Week 41 Neural networks and constructing a neural network code --><div class="tex2jax_ignore mathjax_ignore section" id="week-41-neural-networks-and-constructing-a-neural-network-code">
<h1>Week 41 Neural networks and constructing a neural network code<a class="headerlink" href="#week-41-neural-networks-and-constructing-a-neural-network-code" title="Permalink to this headline"></a></h1>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo</p>
<p>Date: <strong>Week 41</strong></p>
<div class="section" id="plan-for-week-41-october-7-11">
<h2>Plan for week 41, October 7-11<a class="headerlink" href="#plan-for-week-41-october-7-11" title="Permalink to this headline"></a></h2>
</div>
<div class="section" id="material-for-the-lecture-on-monday-october-7-2024">
<h2>Material for the lecture on Monday October 7, 2024<a class="headerlink" href="#material-for-the-lecture-on-monday-october-7-2024" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model.</p></li>
<li><p>Building our own Feed-forward Neural Network</p></li>
</ol>
<!-- * [Video of lecture notes](https://youtu.be/5-RRTO9uDvI) -->
<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesOctober7.pdf) -->
<p><strong>Readings and Videos:</strong></p>
<ol class="simple">
<li><p>These lecture notes</p></li>
<li><p>Rashcka et al chapter 11</p></li>
<li><p>For neural networks we recommend Goodfellow et al chapter 6.</p></li>
</ol>
<p>a. Neural Networks demystified at <a class="reference external" href="https://www.youtube.com/watch?v=bxe2T-V8XRs&amp;amp;list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&amp;amp;ab_channel=WelchLabs">https://www.youtube.com/watch?v=bxe2T-V8XRs&amp;amp;list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&amp;amp;ab_channel=WelchLabs</a></p>
<ol class="simple">
<li><p>Building Neural Networks from scratch at <a class="reference external" href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&amp;amp;ab_channel=sentdex">https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&amp;amp;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>We 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>
</div>
<div class="section" id="material-for-the-active-learning-sessions-on-tuesday-and-wednesday">
<h2>Material for the active learning sessions on Tuesday and Wednesday<a class="headerlink" href="#material-for-the-active-learning-sessions-on-tuesday-and-wednesday" title="Permalink to this headline"></a></h2>
<ul class="simple">
<li><p>Exercise on writing your own stochastic gradient and gradient descent codes. This exercise continues next week with studies of automatic differentiation</p></li>
<li><p>One lecture at the beginning of each session on the material from weeks 39 and 40 and how to write your own gradient descent code</p></li>
<li><p>Discussion of project 2</p></li>
<li><p>Your task before the sessions: revisit the material from weeks 39 and 40 and in particular the material from week 40 on stochastic gradient descent</p></li>
</ul>
</div>
<div class="section" id="lecture-monday-october-7">
<h2>Lecture Monday October 7<a class="headerlink" href="#lecture-monday-october-7" title="Permalink to this headline"></a></h2>
</div>
<div class="section" id="introduction-to-neural-networks">
<h2>Introduction to Neural networks<a class="headerlink" href="#introduction-to-neural-networks" title="Permalink to this headline"></a></h2>
<p>Artificial neural networks are computational systems that can learn to
perform tasks by considering examples, generally without being
programmed with any task-specific rules. It is supposed to mimic a
biological system, wherein neurons interact by sending signals in the
form of mathematical functions between layers. All layers can contain
an arbitrary number of neurons, and each connection is represented by
a weight variable.</p>
</div>
<div class="section" id="artificial-neurons">
<h2>Artificial neurons<a class="headerlink" href="#artificial-neurons" title="Permalink to this headline"></a></h2>
<p>The field of artificial neural networks has a long history of
development, and is closely connected with the advancement of computer
science and computers in general. A model of artificial neurons was
first developed by McCulloch and Pitts in 1943 to study signal
processing in the brain and has later been refined by others. The
general idea is to mimic neural networks in the human brain, which is
composed of billions of neurons that communicate with each other by
sending electrical signals. Each neuron accumulates its incoming
signals, which must exceed an activation threshold to yield an
output. If the threshold is not overcome, the neuron remains inactive,
i.e. has zero output.</p>
<p>This behaviour has inspired a simple mathematical model for an artificial neuron.</p>
<!-- Equation labels as ordinary links -->
<div id="artificialNeuron"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
\label{artificialNeuron} \tag{1}
\end{equation}
\]</div>
<p>Here, the output <span class="math notranslate nohighlight">\(y\)</span> of the neuron is the value of its activation function, which have as input
a weighted sum of signals <span class="math notranslate nohighlight">\(x_i, \dots ,x_n\)</span> received by <span class="math notranslate nohighlight">\(n\)</span> other neurons.</p>
<p>Conceptually, it is helpful to divide neural networks into four
categories:</p>
<ol class="simple">
<li><p>general purpose neural networks for supervised learning,</p></li>
<li><p>neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),</p></li>
<li><p>neural networks for sequential data such as Recurrent Neural Networks (RNNs), and</p></li>
<li><p>neural networks for unsupervised learning such as Deep Boltzmann Machines.</p></li>
</ol>
<p>In natural science, DNNs and CNNs have already found numerous
applications. In statistical physics, they have been applied to detect
phase transitions in 2D Ising and Potts models, lattice gauge
theories, and different phases of polymers, or solving the
Navier-Stokes equation in weather forecasting. Deep learning has also
found interesting applications in quantum physics. Various quantum
phase transitions can be detected and studied using DNNs and CNNs,
topological phases, and even non-equilibrium many-body
localization. Representing quantum states as DNNs quantum state
tomography are among some of the impressive achievements to reveal the
potential of DNNs to facilitate the study of quantum systems.</p>
<p>In quantum information theory, it has been shown that one can perform
gate decompositions with the help of neural.</p>
<p>The applications are not limited to the natural sciences. There is a
plethora of applications in essentially all disciplines, from the
humanities to life science and medicine.</p>
</div>
<div class="section" id="neural-network-types">
<h2>Neural network types<a class="headerlink" href="#neural-network-types" title="Permalink to this headline"></a></h2>
<p>An artificial neural network (ANN), is a computational model that
consists of layers of connected neurons, or nodes or units. We will
refer to these interchangeably as units or nodes, and sometimes as
neurons.</p>
<p>It is supposed to mimic a biological nervous system by letting each
neuron interact with other neurons by sending signals in the form of
mathematical functions between layers. A wide variety of different
ANNs have been developed, but most of them consist of an input layer,
an output layer and eventual layers in-between, called <em>hidden
layers</em>. All layers can contain an arbitrary number of nodes, and each
connection between two nodes is associated with a weight variable.</p>
<p>Neural networks (also called neural nets) are neural-inspired
nonlinear models for supervised learning. As we will see, neural nets
can be viewed as natural, more powerful extensions of supervised
learning methods such as linear and logistic regression and soft-max
methods we discussed earlier.</p>
</div>
<div class="section" id="feed-forward-neural-networks">
<h2>Feed-forward neural networks<a class="headerlink" href="#feed-forward-neural-networks" title="Permalink to this headline"></a></h2>
<p>The feed-forward neural network (FFNN) was the first and simplest type
of ANNs that were devised. In this network, the information moves in
only one direction: forward through the layers.</p>
<p>Nodes are represented by circles, while the arrows display the
connections between the nodes, including the direction of information
flow. Additionally, each arrow corresponds to a weight variable
(figure to come). We observe that each node in a layer is connected
to <em>all</em> nodes in the subsequent layer, making this a so-called
<em>fully-connected</em> FFNN.</p>
</div>
<div class="section" id="convolutional-neural-network">
<h2>Convolutional Neural Network<a class="headerlink" href="#convolutional-neural-network" title="Permalink to this headline"></a></h2>
<p>A different variant of FFNNs are <em>convolutional neural networks</em>
(CNNs), which have a connectivity pattern inspired by the animal
visual cortex. Individual neurons in the visual cortex only respond to
stimuli from small sub-regions of the visual field, called a receptive
field. This makes the neurons well-suited to exploit the strong
spatially local correlation present in natural images. The response of
each neuron can be approximated mathematically as a convolution
operation. (figure to come)</p>
<p>Convolutional neural networks emulate the behaviour of neurons in the
visual cortex by enforcing a <em>local</em> connectivity pattern between
nodes of adjacent layers: Each node in a convolutional layer is
connected only to a subset of the nodes in the previous layer, in
contrast to the fully-connected FFNN. Often, CNNs consist of several
convolutional layers that learn local features of the input, with a
fully-connected layer at the end, which gathers all the local data and
produces the outputs. They have wide applications in image and video
recognition.</p>
</div>
<div class="section" id="recurrent-neural-networks">
<h2>Recurrent neural networks<a class="headerlink" href="#recurrent-neural-networks" title="Permalink to this headline"></a></h2>
<p>So far we have only mentioned ANNs where information flows in one
direction: forward. <em>Recurrent neural networks</em> on the other hand,
have connections between nodes that form directed <em>cycles</em>. This
creates a form of internal memory which are able to capture
information on what has been calculated before; the output is
dependent on the previous computations. Recurrent NNs make use of
sequential information by performing the same task for every element
in a sequence, where each element depends on previous elements. An
example of such information is sentences, making recurrent NNs
especially well-suited for handwriting and speech recognition.</p>
</div>
<div class="section" id="other-types-of-networks">
<h2>Other types of networks<a class="headerlink" href="#other-types-of-networks" title="Permalink to this headline"></a></h2>
<p>There are many other kinds of ANNs that have been developed. One type
that is specifically designed for interpolation in multidimensional
space is the radial basis function (RBF) network. RBFs are typically
made up of three layers: an input layer, a hidden layer with
non-linear radial symmetric activation functions and a linear output
layer (‘’linear’’ here means that each node in the output layer has a
linear activation function). The layers are normally fully-connected
and there are no cycles, thus RBFs can be viewed as a type of
fully-connected FFNN. They are however usually treated as a separate
type of NN due the unusual activation functions.</p>
</div>
<div class="section" id="multilayer-perceptrons">
<h2>Multilayer perceptrons<a class="headerlink" href="#multilayer-perceptrons" title="Permalink to this headline"></a></h2>
<p>One uses often so-called fully-connected feed-forward neural networks
with three or more layers (an input layer, one or more hidden layers
and an output layer) consisting of neurons that have non-linear
activation functions.</p>
<p>Such networks are often called <em>multilayer perceptrons</em> (MLPs).</p>
</div>
<div class="section" id="why-multilayer-perceptrons">
<h2>Why multilayer perceptrons?<a class="headerlink" href="#why-multilayer-perceptrons" title="Permalink to this headline"></a></h2>
<p>According to the <em>Universal approximation theorem</em>, a feed-forward
neural network with just a single hidden layer containing a finite
number of neurons can approximate a continuous multidimensional
function to arbitrary accuracy, assuming the activation function for
the hidden layer is a <strong>non-constant, bounded and
monotonically-increasing continuous function</strong>.</p>
<p>Note that the requirements on the activation function only applies to
the hidden layer, the output nodes are always assumed to be linear, so
as to not restrict the range of output values.</p>
</div>
<div class="section" id="illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model">
<h2>Illustration of a single perceptron model and a multi-perceptron model<a class="headerlink" href="#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" title="Permalink to this headline"></a></h2>
<!-- dom:FIGURE: [figures/nns.png, width=600 frac=0.8] In a) we show a single perceptron model while in b) we dispay a network with two hidden layers, an input layer and an output layer. -->
<!-- begin figure -->
<p><img src="figures/nns.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: In a) we show a single perceptron model while in b) we dispay a network with two hidden layers, an input layer and an output layer.</i></p></p>
<!-- end figure --></div>
<div class="section" id="examples-of-xor-or-and-and-gates">
<h2>Examples of XOR, OR and AND gates<a class="headerlink" href="#examples-of-xor-or-and-and-gates" title="Permalink to this headline"></a></h2>
<p>Let us first try to fit various gates using standard linear
regression. The gates we are thinking of are the classical XOR, OR and
AND gates, well-known elements in computer science. The tables here
show how we can set up the inputs <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> in order to yield a
specific target <span class="math notranslate nohighlight">\(y_i\)</span>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd">Simple code that tests XOR, OR and AND gates with linear regression</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># Design matrix</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span><span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The X.TX matrix:</span><span class="si">{</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">X</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">Xinv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The invers of X.TX matrix:</span><span class="si">{</span><span class="n">Xinv</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># The XOR gate </span>
<span class="n">yXOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="n">ThetaXOR</span> <span class="o">=</span> <span class="n">Xinv</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">yXOR</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The values of theta for the XOR gate:</span><span class="si">{</span><span class="n">ThetaXOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The linear regression prediction for the XOR gate:</span><span class="si">{</span><span class="n">X</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">ThetaXOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># The OR gate </span>
<span class="n">yOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">ThetaOR</span> <span class="o">=</span> <span class="n">Xinv</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">yOR</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The values of theta for the OR gate:</span><span class="si">{</span><span class="n">ThetaOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The linear regression prediction for the OR gate:</span><span class="si">{</span><span class="n">X</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">ThetaOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># The OR gate </span>
<span class="n">yAND</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span> <span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">ThetaAND</span> <span class="o">=</span> <span class="n">Xinv</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">yAND</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The values of theta for the AND gate:</span><span class="si">{</span><span class="n">ThetaAND</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The linear regression prediction for the AND gate:</span><span class="si">{</span><span class="n">X</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">ThetaAND</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The X.TX matrix:[[4. 2. 2.]
[2. 2. 1.]
[2. 1. 2.]]
The invers of X.TX matrix:[[ 7.50000000e-01 -5.00000000e-01 -5.00000000e-01]
[-5.00000000e-01 1.00000000e+00 -2.27693602e-16]
[-5.00000000e-01 9.94484047e-17 1.00000000e+00]]
The values of theta for the XOR gate:[ 5.00000000e-01 -2.22044605e-16 -1.11022302e-16]
The linear regression prediction for the XOR gate:[0.5 0.5 0.5 0.5]
The values of theta for the OR gate:[0.25 0.5 0.5 ]
The linear regression prediction for the OR gate:[0.25 0.75 0.75 1.25]
The values of theta for the AND gate:[-0.25 0.5 0.5 ]
The linear regression prediction for the AND gate:[-0.25 0.25 0.25 0.75]
</pre></div>
</div>
</div>
</div>
<p>What is happening here?</p>
</div>
<div class="section" id="does-logistic-regression-do-a-better-job">
<h2>Does Logistic Regression do a better Job?<a class="headerlink" href="#does-logistic-regression-do-a-better-job" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd">Simple code that tests XOR and OR gates with linear regression</span>
<span class="sd">and logistic regression</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># Design matrix</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span><span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The X.TX matrix:</span><span class="si">{</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">X</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">Xinv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The invers of X.TX matrix:</span><span class="si">{</span><span class="n">Xinv</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># The XOR gate </span>
<span class="n">yXOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="n">ThetaXOR</span> <span class="o">=</span> <span class="n">Xinv</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">yXOR</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The values of theta for the XOR gate:</span><span class="si">{</span><span class="n">ThetaXOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The linear regression prediction for the XOR gate:</span><span class="si">{</span><span class="n">X</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">ThetaXOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># The OR gate </span>
<span class="n">yOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">ThetaOR</span> <span class="o">=</span> <span class="n">Xinv</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">yOR</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The values of theta for the OR gate:</span><span class="si">{</span><span class="n">ThetaOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The linear regression prediction for the OR gate:</span><span class="si">{</span><span class="n">X</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">ThetaOR</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># The OR gate </span>
<span class="n">yAND</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span> <span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">ThetaAND</span> <span class="o">=</span> <span class="n">Xinv</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">yAND</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The values of theta for the AND gate:</span><span class="si">{</span><span class="n">ThetaAND</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The linear regression prediction for the AND gate:</span><span class="si">{</span><span class="n">X</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">ThetaAND</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># Now we change to logistic regression</span>
<span class="c1"># Logistic Regression</span>
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">()</span>
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yOR</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Logistic Regression for OR gate: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">yOR</span><span class="p">)))</span>
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yXOR</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Logistic Regression for XOR gate: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">yXOR</span><span class="p">)))</span>
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yAND</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test set accuracy with Logistic Regression for AND gate: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">yAND</span><span class="p">)))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The X.TX matrix:[[4. 2. 2.]
[2. 2. 1.]
[2. 1. 2.]]
The invers of X.TX matrix:[[ 7.50000000e-01 -5.00000000e-01 -5.00000000e-01]
[-5.00000000e-01 1.00000000e+00 -2.27693602e-16]
[-5.00000000e-01 9.94484047e-17 1.00000000e+00]]
The values of theta for the XOR gate:[ 5.00000000e-01 -2.22044605e-16 -1.11022302e-16]
The linear regression prediction for the XOR gate:[0.5 0.5 0.5 0.5]
The values of theta for the OR gate:[0.25 0.5 0.5 ]
The linear regression prediction for the OR gate:[0.25 0.75 0.75 1.25]
The values of theta for the AND gate:[-0.25 0.5 0.5 ]
The linear regression prediction for the AND gate:[-0.25 0.25 0.25 0.75]
Test set accuracy with Logistic Regression for OR gate: 0.75
Test set accuracy with Logistic Regression for XOR gate: 0.50
Test set accuracy with Logistic Regression for AND gate: 0.75
</pre></div>
</div>
</div>
</div>
<p>Not exactly impressive, but somewhat better.</p>
</div>
<div class="section" id="adding-neural-networks">
<h2>Adding Neural Networks<a class="headerlink" href="#adding-neural-networks" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># and now neural networks with Scikit-Learn and the XOR</span>
<span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPClassifier</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_classification</span>
<span class="n">X</span><span class="p">,</span> <span class="n">yXOR</span> <span class="o">=</span> <span class="n">make_classification</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">FFNN</span> <span class="o">=</span> <span class="n">MLPClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">300</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yXOR</span><span class="p">)</span>
<span class="n">FFNN</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Test set accuracy with Feed Forward Neural Network for XOR gate:</span><span class="si">{</span><span class="n">FFNN</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="w"> </span><span class="n">yXOR</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Test set accuracy with Feed Forward Neural Network for XOR gate:1.0
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="mathematical-model">
<h2>Mathematical model<a class="headerlink" href="#mathematical-model" title="Permalink to this headline"></a></h2>
<p>The output <span class="math notranslate nohighlight">\(y\)</span> is produced via the activation function <span class="math notranslate nohighlight">\(f\)</span></p>
<div class="math notranslate nohighlight">
\[
y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
\]</div>
<p>This function receives <span class="math notranslate nohighlight">\(x_i\)</span> as inputs.
Here the activation <span class="math notranslate nohighlight">\(z=(\sum_{i=1}^n w_ix_i+b_i)\)</span>.
In an FFNN of such neurons, the <em>inputs</em> <span class="math notranslate nohighlight">\(x_i\)</span> are the <em>outputs</em> of
the neurons in the preceding layer. Furthermore, an MLP is
fully-connected, which means that each neuron receives a weighted sum
of the outputs of <em>all</em> neurons in the previous layer.</p>
</div>
<div class="section" id="id1">
<h2>Mathematical model<a class="headerlink" href="#id1" title="Permalink to this headline"></a></h2>
<p>First, for each node <span class="math notranslate nohighlight">\(i\)</span> in the first hidden layer, we calculate a weighted sum <span class="math notranslate nohighlight">\(z_i^1\)</span> of the input coordinates <span class="math notranslate nohighlight">\(x_j\)</span>,</p>
<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
\label{_auto1} \tag{2}
\end{equation}
\]</div>
<p>Here <span class="math notranslate nohighlight">\(b_i\)</span> is the so-called bias which is normally needed in
case of zero activation weights or inputs. How to fix the biases and
the weights will be discussed below. The value of <span class="math notranslate nohighlight">\(z_i^1\)</span> is the
argument to the activation function <span class="math notranslate nohighlight">\(f_i\)</span> of each node <span class="math notranslate nohighlight">\(i\)</span>, The
variable <span class="math notranslate nohighlight">\(M\)</span> stands for all possible inputs to a given node <span class="math notranslate nohighlight">\(i\)</span> in the
first layer. We define the output <span class="math notranslate nohighlight">\(y_i^1\)</span> of all neurons in layer 1 as</p>
<!-- Equation labels as ordinary links -->
<div id="outputLayer1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1} \tag{3}
\end{equation}
\]</div>
<p>where we assume that all nodes in the same layer have identical
activation functions, hence the notation <span class="math notranslate nohighlight">\(f\)</span>. In general, we could assume in the more general case that different layers have different activation functions.
In this case we would identify these functions with a superscript <span class="math notranslate nohighlight">\(l\)</span> for the <span class="math notranslate nohighlight">\(l\)</span>-th layer,</p>
<!-- Equation labels as ordinary links -->
<div id="generalLayer"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer} \tag{4}
\end{equation}
\]</div>
<p>where <span class="math notranslate nohighlight">\(N_l\)</span> is the number of nodes in layer <span class="math notranslate nohighlight">\(l\)</span>. When the output of
all the nodes in the first hidden layer are computed, the values of
the subsequent layer can be calculated and so forth until the output
is obtained.</p>
</div>
<div class="section" id="id2">
<h2>Mathematical model<a class="headerlink" href="#id2" title="Permalink to this headline"></a></h2>
<p>The output of neuron <span class="math notranslate nohighlight">\(i\)</span> in layer 2 is thus,</p>
<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^2 = f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto2} \tag{5}
\end{equation}
\]</div>
<!-- Equation labels as ordinary links -->
<div id="outputLayer2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2} \tag{6}
\end{equation}
\]</div>
<p>where we have substituted <span class="math notranslate nohighlight">\(y_k^1\)</span> with the inputs <span class="math notranslate nohighlight">\(x_k\)</span>. Finally, the ANN output reads</p>
<!-- Equation labels as ordinary links -->
<div id="_auto3"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^3 = f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
\label{_auto3} \tag{7}
\end{equation}
\]</div>
<!-- Equation labels as ordinary links -->
<div id="_auto4"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto4} \tag{8}
\end{equation}
\]</div>
</div>
<div class="section" id="id3">
<h2>Mathematical model<a class="headerlink" href="#id3" title="Permalink to this headline"></a></h2>
<p>We can generalize this expression to an MLP with <span class="math notranslate nohighlight">\(l\)</span> hidden
layers. The complete functional form is,</p>
<!-- Equation labels as ordinary links -->
<div id="completeNN"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right]
\label{completeNN} \tag{9}
\end{equation}
\]</div>
<p>which illustrates a basic property of MLPs: The only independent
variables are the input values <span class="math notranslate nohighlight">\(x_n\)</span>.</p>
</div>
<div class="section" id="id4">
<h2>Mathematical model<a class="headerlink" href="#id4" title="Permalink to this headline"></a></h2>
<p>This confirms that an MLP, despite its quite convoluted mathematical
form, is nothing more than an analytic function, specifically a
mapping of real-valued vectors <span class="math notranslate nohighlight">\(\hat{x} \in \mathbb{R}^n \rightarrow
\hat{y} \in \mathbb{R}^m\)</span>.</p>
<p>Furthermore, the flexibility and universality of an MLP can be
illustrated by realizing that the expression is essentially a nested
sum of scaled activation functions of the form</p>
<!-- Equation labels as ordinary links -->
<div id="_auto5"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
f(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto5} \tag{10}
\end{equation}
\]</div>
<p>where the parameters <span class="math notranslate nohighlight">\(c_i\)</span> are weights and biases. By adjusting these
parameters, the activation functions can be shifted up and down or
left and right, change slope or be rescaled which is the key to the
flexibility of a neural network.</p>
<div class="section" id="matrix-vector-notation">
<h3>Matrix-vector notation<a class="headerlink" href="#matrix-vector-notation" title="Permalink to this headline"></a></h3>
<p>We can introduce a more convenient notation for the activations in an A NN.</p>
<p>Additionally, we can represent the biases and activations
as layer-wise column vectors <span class="math notranslate nohighlight">\(\hat{b}_l\)</span> and <span class="math notranslate nohighlight">\(\hat{y}_l\)</span>, so that the <span class="math notranslate nohighlight">\(i\)</span>-th element of each vector
is the bias <span class="math notranslate nohighlight">\(b_i^l\)</span> and activation <span class="math notranslate nohighlight">\(y_i^l\)</span> of node <span class="math notranslate nohighlight">\(i\)</span> in layer <span class="math notranslate nohighlight">\(l\)</span> respectively.</p>
<p>We have that <span class="math notranslate nohighlight">\(\mathrm{W}_l\)</span> is an <span class="math notranslate nohighlight">\(N_{l-1} \times N_l\)</span> matrix, while <span class="math notranslate nohighlight">\(\hat{b}_l\)</span> and <span class="math notranslate nohighlight">\(\hat{y}_l\)</span> are <span class="math notranslate nohighlight">\(N_l \times 1\)</span> column vectors.
With this notation, the sum becomes a matrix-vector multiplication, and we can write
the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as</p>
<!-- Equation labels as ordinary links -->
<div id="_auto6"></div>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{equation}
\hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &amp;w^2_{12} &amp;w^2_{13} \\
w^2_{21} &amp;w^2_{22} &amp;w^2_{23} \\
w^2_{31} &amp;w^2_{32} &amp;w^2_{33} \\
\end{array} \right] \cdot
\left[\begin{array}{c}
y^1_1 \\
y^1_2 \\
y^1_3 \\
\end{array}\right] +
\left[\begin{array}{c}
b^2_1 \\
b^2_2 \\
b^2_3 \\
\end{array}\right]\right).
\label{_auto6} \tag{11}
\end{equation}
\end{split}\]</div>
</div>
<div class="section" id="matrix-vector-notation-and-activation">
<h3>Matrix-vector notation and activation<a class="headerlink" href="#matrix-vector-notation-and-activation" title="Permalink to this headline"></a></h3>
<p>The activation of node <span class="math notranslate nohighlight">\(i\)</span> in layer 2 is</p>
<!-- Equation labels as ordinary links -->
<div id="_auto7"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
\label{_auto7} \tag{12}
\end{equation}
\]</div>
<p>This is not just a convenient and compact notation, but also a useful
and intuitive way to think about MLPs: The output is calculated by a
series of matrix-vector multiplications and vector additions that are
used as input to the activation functions. For each operation
<span class="math notranslate nohighlight">\(\mathrm{W}_l \hat{y}_{l-1}\)</span> we move forward one layer.</p>
</div>
<div class="section" id="activation-functions">
<h3>Activation functions<a class="headerlink" href="#activation-functions" title="Permalink to this headline"></a></h3>
<p>A property that characterizes a neural network, other than its
connectivity, is the choice of activation function(s). As described
in, the following restrictions are imposed on an activation function
for a 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>
</div>
<div class="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="Permalink to this headline"></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">
\[
f(x) = \frac{1}{1 + e^{-x}},
\]</div>
<p>and the <em>hyperbolic tangent</em> function</p>
<div class="math notranslate nohighlight">
\[
f(x) = \tanh(x)
\]</div>
</div>
<div class="section" id="relevance">
<h3>Relevance<a class="headerlink" href="#relevance" title="Permalink to this headline"></a></h3>
<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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a </span>
<span class="sd">function that takes any real number, z, and outputs a number (0,1).</span>
<span class="sd">It is useful in neural networks for assigning weights on a relative scale.</span>
<span class="sd">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
<span class="kn">import</span> <span class="nn">numpy</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">math</span> <span class="k">as</span> <span class="nn">mt</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">.1</span><span class="p">)</span>
<span class="n">sigma_fn</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">numpy</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)))</span>
<span class="n">sigma</span> <span class="o">=</span> <span class="n">sigma_fn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;z&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;sigmoid function&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="sd">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mf">.02</span><span class="p">)</span>
<span class="n">step_fn</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mf">1.0</span> <span class="k">if</span> <span class="n">z</span> <span class="o">&gt;=</span> <span class="mf">0.0</span> <span class="k">else</span> <span class="mf">0.0</span><span class="p">)</span>
<span class="n">step</span> <span class="o">=</span> <span class="n">step_fn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;z&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;step function&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="sd">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;z&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;sine function&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="sd">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
<span class="sd">unit&quot;&quot;&quot;</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">.1</span><span class="p">)</span>
<span class="n">zero</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">max</span><span class="p">([</span><span class="n">zero</span><span class="p">,</span> <span class="n">z</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;z&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Rectified linear unit&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/week41_59_0.png" src="_images/week41_59_0.png" />
<img alt="_images/week41_59_1.png" src="_images/week41_59_1.png" />
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</div>
</div>
</div>
<div class="section" id="the-multilayer-perceptron-mlp">
<h2>The multilayer perceptron (MLP)<a class="headerlink" href="#the-multilayer-perceptron-mlp" title="Permalink to this headline"></a></h2>
<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
<ol class="simple">
<li><p>A neural network with one or more layers of nodes between the input and the output nodes.</p></li>
<li><p>The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</p></li>
<li><p>The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</p></li>
</ol>
<p>As a convention it is normal to call a network with one layer of input units, one layer of hidden
units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.</p>
<p>For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
Hereafter we will call the various entities of a layer for nodes.
There are also no connections within a single layer.</p>
<p>The number of input nodes does not need to equal the number of output
nodes. This applies also to the hidden layers. Each layer may have its
own number of nodes and activation functions.</p>
<p>The hidden layers have their name from the fact that they are not
linked to observables and as we will see below when we define the
so-called activation <span class="math notranslate nohighlight">\(\hat{z}\)</span>, we can think of this as a basis
expansion of the original inputs <span class="math notranslate nohighlight">\(\hat{x}\)</span>. The difference however
between neural networks and say linear regression is that now these
basis functions (which will correspond to the weights in the network)
are learned from data. This results in an important difference between
neural networks and deep learning approaches on one side and methods
like logistic regression or linear regression and their modifications on the other side.</p>
</div>
<div class="section" id="from-one-to-many-layers-the-universal-approximation-theorem">
<h2>From one to many layers, the universal approximation theorem<a class="headerlink" href="#from-one-to-many-layers-the-universal-approximation-theorem" title="Permalink to this headline"></a></h2>
<p>A neural network with only one layer, what we called the simple
perceptron, is best suited if we have a standard binary model with
clear (linear) boundaries between the outcomes. As such it could
equally well be replaced by standard linear regression or logistic
regression. Networks with one or more hidden layers approximate
systems with more complex boundaries.</p>
<p>As stated earlier,
an important theorem in studies of neural networks, restated without
proof here, is the <a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&amp;rep=rep1&amp;type=pdf">universal approximation
theorem</a>.</p>
<p>It states that a feed-forward network with a single hidden layer
containing a finite number of neurons can approximate continuous
functions on compact subsets of real functions. The theorem thus
states that simple neural networks can represent a wide variety of
interesting functions when given appropriate parameters. It is the
multilayer feedforward architecture itself which gives neural networks
the potential of being universal approximators.</p>
</div>
<div class="section" id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">
<h2>Deriving the back propagation code for a multilayer perceptron model<a class="headerlink" href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" title="Permalink to this headline"></a></h2>
<p>As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
The unknowwn quantities are our weights <span class="math notranslate nohighlight">\(w_{ij}\)</span> and we need to find an algorithm for changing them so that our errors are as small as possible.
This leads us to the famous <a class="reference external" href="https://www.nature.com/articles/323533a0">back propagation algorithm</a>.</p>
<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?</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">
\[
{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
\]</div>
<p>where the <span class="math notranslate nohighlight">\(t_i\)</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">\(\hat{x}\)</span> are given by <span class="math notranslate nohighlight">\(y_i\)</span>. Below we will demonstrate
how the basic equations arising from the back propagation algorithm
can be modified in order to study classification problems with <span class="math notranslate nohighlight">\(K\)</span>
classes.</p>
</div>
<div class="section" id="definitions">
<h2>Definitions<a class="headerlink" href="#definitions" title="Permalink to this headline"></a></h2>
<p>With our definition of the targets <span class="math notranslate nohighlight">\(\hat{t}\)</span>, the outputs of the
network <span class="math notranslate nohighlight">\(\hat{y}\)</span> and the inputs <span class="math notranslate nohighlight">\(\hat{x}\)</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">\(\hat{a}^{l-1}\)</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 here 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">
\[
\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
\]</div>
<p>With the activation values <span class="math notranslate nohighlight">\(\hat{z}^l\)</span> we can in turn define the
output of layer <span class="math notranslate nohighlight">\(l\)</span> as <span class="math notranslate nohighlight">\(\hat{a}^l = f(\hat{z}^l)\)</span> where <span class="math notranslate nohighlight">\(f\)</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">\(f\)</span> for all layers
and their nodes. It means we have</p>
<div class="math notranslate nohighlight">
\[
a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
\]</div>
</div>
<div class="section" id="derivatives-and-the-chain-rule">
<h2>Derivatives and the chain rule<a class="headerlink" href="#derivatives-and-the-chain-rule" title="Permalink to this headline"></a></h2>
<p>From the definition of the activation <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">
\[
\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
\]</div>
</div>
<div class="section" id="derivative-of-the-cost-function">
<h2>Derivative of the cost function<a class="headerlink" href="#derivative-of-the-cost-function" title="Permalink to this headline"></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">
\[
{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
\]</div>
<p>The derivative of this function with respect to the weights is</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
\]</div>
<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
\]</div>
</div>
<div class="section" id="bringing-it-together-first-back-propagation-equation">
<h2>Bringing it together, first back propagation equation<a class="headerlink" href="#bringing-it-together-first-back-propagation-equation" title="Permalink to this headline"></a></h2>
<p>We have thus</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
\]</div>
<p>Defining</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
\]</div>
<p>and using the Hadamard product of two vectors we can write this as</p>
<div class="math notranslate nohighlight">
\[
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
\]</div>
<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>
<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">\(f'(z^L_j)\)</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">
\[
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
\]</div>
</div>
<div class="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="Permalink to this headline"></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>
</div>
<div class="section" id="bringing-it-together">
<h2>Bringing it together<a class="headerlink" href="#bringing-it-together" title="Permalink to this headline"></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>
<p><strong>The starting equations.</strong></p>
<!-- Equation labels as ordinary links -->
<div id="_auto8"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
\label{_auto8} \tag{13}
\end{equation}
\]</div>
<p>and</p>
<!-- Equation labels as ordinary links -->
<div id="_auto9"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
\label{_auto9} \tag{14}
\end{equation}
\]</div>
<p>and</p>
<!-- Equation labels as ordinary links -->
<div id="_auto10"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
\label{_auto10} \tag{15}
\end{equation}
\]</div>
<p>An interesting consequence of the above equations is that when the
activation <span class="math notranslate nohighlight">\(a_k^{L-1}\)</span> is small, the gradient term, that is the
derivative of the cost function with respect to the weights, will also
tend to be small. We say then that the weight learns slowly, meaning
that it changes slowly when we minimize the weights via say gradient
descent. In this case we say the system learns slowly.</p>
<p>Another interesting feature is that is when the activation function,
represented by the sigmoid function here, is rather flat when we move towards
its end values <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span> (see the above Python codes). In these
cases, the derivatives of the activation function will also be close
to zero, meaning again that the gradients will be small and the
network learns slowly again.</p>
<p>We need a fourth equation and we are set. We are going to propagate
backwards in order to the determine the weights and biases. In order
to do so we need to represent the error in the layer before the final
one <span class="math notranslate nohighlight">\(L-1\)</span> in terms of the errors in the final output layer.</p>
</div>
<div class="section" id="final-back-propagating-equation">
<h2>Final back propagating equation<a class="headerlink" href="#final-back-propagating-equation" title="Permalink to this headline"></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>. Using the chain rule and summing over all <span class="math notranslate nohighlight">\(k\)</span> entries we have</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">
\[
\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
\]</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>
</div>
<div class="section" id="setting-up-the-back-propagation-algorithm">
<h2>Setting up the Back propagation algorithm<a class="headerlink" href="#setting-up-the-back-propagation-algorithm" title="Permalink to this headline"></a></h2>
<p>The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
<p>First, we set up the input data <span class="math notranslate nohighlight">\(\hat{x}\)</span> and the activations
<span class="math notranslate nohighlight">\(\hat{z}_1\)</span> of the input layer and compute the activation function and
the pertinent outputs <span class="math notranslate nohighlight">\(\hat{a}^1\)</span>.</p>
<p>Secondly, we perform then the feed forward till we reach the output
layer and compute all <span class="math notranslate nohighlight">\(\hat{z}_l\)</span> of the input layer and compute the
activation function and the pertinent outputs <span class="math notranslate nohighlight">\(\hat{a}^l\)</span> for
<span class="math notranslate nohighlight">\(l=2,3,\dots,L\)</span>.</p>
<p>Thereafter we compute the ouput error <span class="math notranslate nohighlight">\(\hat{\delta}^L\)</span> by computing all</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
\]</div>
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
\]</div>
<p>Finally, we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> and update the weights and biases according to the rules</p>
<div class="math notranslate nohighlight">
\[
w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
\]</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>The parameter <span class="math notranslate nohighlight">\(\eta\)</span> is the learning parameter discussed in connection with the gradient descent methods.
Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.</p>
</div>
<div class="section" id="id5">
<h2>Setting up the Back propagation algorithm<a class="headerlink" href="#id5" title="Permalink to this headline"></a></h2>
<p>The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
<p>First, 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>Secondly, 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">\(l=2,3,\dots,L\)</span>.</p>
<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">
\[
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
\]</div>
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
\]</div>
<p>Finally, we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> and update the weights and biases according to the rules</p>
<div class="math notranslate nohighlight">
\[
w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
\]</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>The parameter <span class="math notranslate nohighlight">\(\eta\)</span> is the learning parameter discussed in connection with the gradient descent methods.
Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.</p>
</div>
<div class="section" id="id6">
<h2>Setting up the Back propagation algorithm<a class="headerlink" href="#id6" title="Permalink to this headline"></a></h2>
<p>The four equations derived discussed above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.</p>
<p>First, 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>Secondly, 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">\(l=2,3,\dots,L\)</span>.</p>
<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">
\[
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
\]</div>
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
\]</div>
<p>Finally, we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> and update the weights and biases according to the rules</p>
<div class="math notranslate nohighlight">
\[
w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
\]</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>The parameter <span class="math notranslate nohighlight">\(\eta\)</span> is the learning parameter discussed in connection with the gradient descent methods.
Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.</p>
</div>
<div class="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="Permalink to this headline"></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>
</div>
<div class="section" id="defining-the-cost-function">
<h2>Defining the cost function<a class="headerlink" href="#defining-the-cost-function" title="Permalink to this headline"></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>
</div>
<div class="section" id="example-binary-classification-problem">
<h2>Example: binary classification problem<a class="headerlink" href="#example-binary-classification-problem" title="Permalink to this headline"></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>
</div>
<div class="section" id="the-softmax-function">
<h2>The Softmax function<a class="headerlink" href="#the-softmax-function" title="Permalink to this headline"></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>
</div>
<div class="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="Permalink to this headline"></a></h2>
<p>One can identify a set of key steps when using neural networks to solve supervised learning problems:</p>
<ol class="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>
</div>
<div class="section" id="collect-and-pre-process-data">
<h2>Collect and pre-process data<a class="headerlink" href="#collect-and-pre-process-data" title="Permalink to this headline"></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">
\[\begin{split} X = \begin{bmatrix}
1.85 &amp; 81\\
1.71 &amp; 65\\
1.95 &amp; 103\\
1.55 &amp; 42\\
1.63 &amp; 56
\end{bmatrix} ,\end{split}\]</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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import necessary packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="c1"># ensure the same random numbers appear every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># display images in notebook</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
<span class="c1"># download MNIST dataset</span>
<span class="n">digits</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">()</span>
<span class="c1"># define inputs and labels</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">images</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;inputs = (n_inputs, pixel_width, pixel_height) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;labels = (n_inputs) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="c1"># flatten the image</span>
<span class="c1"># the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;X = (n_inputs, n_features) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="c1"># choose some random images to display</span>
<span class="n">indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">)</span>
<span class="n">random_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">image</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">digits</span><span class="o">.</span><span class="n">images</span><span class="p">[</span><span class="n">random_indices</span><span class="p">]):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Label: </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">random_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>inputs = (n_inputs, pixel_width, pixel_height) = (1797, 8, 8)
labels = (n_inputs) = (1797,)
X = (n_inputs, n_features) = (1797, 64)
</pre></div>
</div>
<img alt="_images/week41_176_1.png" src="_images/week41_176_1.png" />
</div>
</div>
</div>
<div class="section" id="train-and-test-datasets">
<h2>Train and test datasets<a class="headerlink" href="#train-and-test-datasets" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="c1"># one-liner from scikit-learn library</span>
<span class="n">train_size</span> <span class="o">=</span> <span class="mf">0.8</span>
<span class="n">test_size</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">train_size</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span>
<span class="n">test_size</span><span class="o">=</span><span class="n">test_size</span><span class="p">)</span>
<span class="c1"># equivalently in numpy</span>
<span class="k">def</span> <span class="nf">train_test_split_numpy</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="p">,</span> <span class="n">test_size</span><span class="p">):</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
<span class="n">inputs_shuffled</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">labels_shuffled</span> <span class="o">=</span> <span class="n">labels</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">inputs_shuffled</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">labels_shuffled</span><span class="p">)</span>
<span class="n">train_end</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n_inputs</span><span class="o">*</span><span class="n">train_size</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span> <span class="o">=</span> <span class="n">inputs_shuffled</span><span class="p">[:</span><span class="n">train_end</span><span class="p">],</span> <span class="n">inputs_shuffled</span><span class="p">[</span><span class="n">train_end</span><span class="p">:]</span>
<span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">labels_shuffled</span><span class="p">[:</span><span class="n">train_end</span><span class="p">],</span> <span class="n">labels_shuffled</span><span class="p">[</span><span class="n">train_end</span><span class="p">:]</span>
<span class="k">return</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span>
<span class="c1">#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Number of training images: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X_train</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Number of test images: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X_test</span><span class="p">)))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Number of training images: 1437
Number of test images: 360
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="define-model-and-architecture">
<h2>Define model and architecture<a class="headerlink" href="#define-model-and-architecture" title="Permalink to this headline"></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">
\[\begin{split} f(z) =
\begin{cases}
1, &amp; z &gt; 0\\
0, &amp; \text{otherwise}
\end{cases}
\end{split}\]</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>
</div>
<div class="section" id="layers">
<h2>Layers<a class="headerlink" href="#layers" title="Permalink to this headline"></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>
</div>
<div class="section" id="weights-and-biases">
<h2>Weights and biases<a class="headerlink" href="#weights-and-biases" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># building our neural network</span>
<span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">shape</span>
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">10</span>
<span class="c1"># we make the weights normally distributed using numpy.random.randn</span>
<span class="c1"># weights and bias in the hidden layer</span>
<span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_hidden_neurons</span><span class="p">)</span>
<span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="c1"># weights and bias in the output layer</span>
<span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">)</span>
<span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="feed-forward-pass">
<h2>Feed-forward pass<a class="headerlink" href="#feed-forward-pass" title="Permalink to this headline"></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>
</div>
<div class="section" id="matrix-multiplications">
<h2>Matrix multiplications<a class="headerlink" href="#matrix-multiplications" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup the feed-forward pass, subscript h = hidden layer</span>
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="c1"># weighted sum of inputs to the hidden layer</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
<span class="c1"># activation in the hidden layer</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="c1"># weighted sum of inputs to the output layer</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
<span class="c1"># softmax output</span>
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;probabilities = (n_inputs, n_categories) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;probability that image 0 is in category 0,1,2,...,9 = </span><span class="se">\n</span><span class="s2">&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;probabilities sum up to: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sum</span><span class="p">()))</span>
<span class="nb">print</span><span class="p">()</span>
<span class="c1"># we obtain a prediction by taking the class with the highest likelihood</span>
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">predictions</span> <span class="o">=</span> <span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;predictions = (n_inputs) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">predictions</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;prediction for image 0: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">predictions</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;correct label for image 0: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">Y_train</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>probabilities = (n_inputs, n_categories) = (1437, 10)
probability that image 0 is in category 0,1,2,...,9 =
[5.41511965e-04 2.17174962e-03 8.84355903e-03 1.44970586e-03
1.10378326e-04 5.08318298e-09 2.03256632e-04 1.92507116e-03
9.84443254e-01 3.11507992e-04]
probabilities sum up to: 1.0
predictions = (n_inputs) = (1437,)
prediction for image 0: 8
correct label for image 0: 6
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="choose-cost-function-and-optimizer">
<h2>Choose cost function and optimizer<a class="headerlink" href="#choose-cost-function-and-optimizer" title="Permalink to this headline"></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>
</div>
<div class="section" id="optimizing-the-cost-function">
<h2>Optimizing the cost function<a class="headerlink" href="#optimizing-the-cost-function" title="Permalink to this headline"></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="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>
</div>
<div class="section" id="regularization">
<h2>Regularization<a class="headerlink" href="#regularization" title="Permalink to this headline"></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>
</div>
<div class="section" id="matrix-multiplication">
<h2>Matrix multiplication<a class="headerlink" href="#matrix-multiplication" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># to categorical turns our integer vector into a onehot representation</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
<span class="c1"># one-hot in numpy</span>
<span class="k">def</span> <span class="nf">to_categorical_numpy</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">):</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">)</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">onehot_vector</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">))</span>
<span class="n">onehot_vector</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">),</span> <span class="n">integer_vector</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">onehot_vector</span>
<span class="c1">#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)</span>
<span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">Y_test_onehot</span> <span class="o">=</span> <span class="n">to_categorical_numpy</span><span class="p">(</span><span class="n">Y_train</span><span class="p">),</span> <span class="n">to_categorical_numpy</span><span class="p">(</span><span class="n">Y_test</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">feed_forward_train</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="c1"># weighted sum of inputs to the hidden layer</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
<span class="c1"># activation in the hidden layer</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="c1"># weighted sum of inputs to the output layer</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
<span class="c1"># softmax output</span>
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># for backpropagation need activations in hidden and output layers</span>
<span class="k">return</span> <span class="n">a_h</span><span class="p">,</span> <span class="n">probabilities</span>
<span class="k">def</span> <span class="nf">backpropagation</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
<span class="n">a_h</span><span class="p">,</span> <span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward_train</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="c1"># error in the output layer</span>
<span class="n">error_output</span> <span class="o">=</span> <span class="n">probabilities</span> <span class="o">-</span> <span class="n">Y</span>
<span class="c1"># error in the hidden layer</span>
<span class="n">error_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">output_weights</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="n">a_h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a_h</span><span class="p">)</span>
<span class="c1"># gradients for the output layer</span>
<span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
<span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># gradient for the hidden layer</span>
<span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
<span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">return</span> <span class="n">output_weights_gradient</span><span class="p">,</span> <span class="n">output_bias_gradient</span><span class="p">,</span> <span class="n">hidden_weights_gradient</span><span class="p">,</span> <span class="n">hidden_bias_gradient</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Old accuracy on training data: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">Y_train</span><span class="p">)))</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="n">lmbd</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
<span class="c1"># calculate gradients</span>
<span class="n">dWo</span><span class="p">,</span> <span class="n">dBo</span><span class="p">,</span> <span class="n">dWh</span><span class="p">,</span> <span class="n">dBh</span> <span class="o">=</span> <span class="n">backpropagation</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">)</span>
<span class="c1"># regularization term gradients</span>
<span class="n">dWo</span> <span class="o">+=</span> <span class="n">lmbd</span> <span class="o">*</span> <span class="n">output_weights</span>
<span class="n">dWh</span> <span class="o">+=</span> <span class="n">lmbd</span> <span class="o">*</span> <span class="n">hidden_weights</span>
<span class="c1"># update weights and biases</span>
<span class="n">output_weights</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dWo</span>
<span class="n">output_bias</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dBo</span>
<span class="n">hidden_weights</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dWh</span>
<span class="n">hidden_bias</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dBh</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;New accuracy on training data: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">Y_train</span><span class="p">)))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Old accuracy on training data: 0.1440501043841336
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>New accuracy on training data: 0.09951287404314545
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="improving-performance">
<h2>Improving performance<a class="headerlink" href="#improving-performance" title="Permalink to this headline"></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>
</div>
<div class="section" id="full-object-oriented-implementation">
<h2>Full object-oriented implementation<a class="headerlink" href="#full-object-oriented-implementation" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NeuralNetwork</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
<span class="bp">self</span><span class="p">,</span>
<span class="n">X_data</span><span class="p">,</span>
<span class="n">Y_data</span><span class="p">,</span>
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span>
<span class="n">n_categories</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">epochs</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">eta</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
<span class="n">lmbd</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span> <span class="o">=</span> <span class="n">X_data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span> <span class="o">=</span> <span class="n">Y_data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span> <span class="o">=</span> <span class="n">X_data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_features</span> <span class="o">=</span> <span class="n">X_data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="n">n_hidden_neurons</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span> <span class="o">=</span> <span class="n">n_categories</span>
<span class="bp">self</span><span class="o">.</span><span class="n">epochs</span> <span class="o">=</span> <span class="n">epochs</span>
<span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span> <span class="o">=</span> <span class="n">batch_size</span>
<span class="bp">self</span><span class="o">.</span><span class="n">iterations</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span> <span class="o">//</span> <span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span>
<span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">=</span> <span class="n">eta</span>
<span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">=</span> <span class="n">lmbd</span>
<span class="bp">self</span><span class="o">.</span><span class="n">create_biases_and_weights</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">create_biases_and_weights</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_features</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="c1"># feed-forward for training</span>
<span class="bp">self</span><span class="o">.</span><span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
<span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_h</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_o</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">feed_forward_out</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="c1"># feed-forward for output</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="k">def</span> <span class="nf">backpropagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">error_output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">probabilities</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span>
<span class="n">error_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">&gt;</span> <span class="mf">0.0</span><span class="p">:</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span>
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward_out</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">predict_probabilities</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward_out</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">data_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">epochs</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">iterations</span><span class="p">):</span>
<span class="c1"># pick datapoints with replacement</span>
<span class="n">chosen_datapoints</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span>
<span class="n">data_indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span>
<span class="p">)</span>
<span class="c1"># minibatch training data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">X_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">feed_forward</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">backpropagation</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="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="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
<span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># accuracy score from scikit library</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on test set: &quot;</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_predict</span><span class="p">))</span>
<span class="c1"># equivalent in numpy</span>
<span class="k">def</span> <span class="nf">accuracy_score_numpy</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">Y_pred</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">Y_test</span> <span class="o">==</span> <span class="n">Y_pred</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">Y_test</span><span class="p">)</span>
<span class="c1">#print(&quot;Accuracy score on test set: &quot;, accuracy_score_numpy(Y_test, test_predict))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Accuracy score on test set: 0.9444444444444444
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="adjust-hyperparameters">
<h2>Adjust hyperparameters<a class="headerlink" href="#adjust-hyperparameters" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="c1"># store the models for later use</span>
<span class="n">DNN_numpy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="c1"># grid search</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
<span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Learning rate = &quot;</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Lambda = &quot;</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on test set: &quot;</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_predict</span><span class="p">))</span>
<span class="nb">print</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 1e-05
Accuracy score on test set: 0.11666666666666667
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.0001
Accuracy score on test set: 0.20833333333333334
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.001
Accuracy score on test set: 0.12222222222222222
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.01
Accuracy score on test set: 0.14722222222222223
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.1
Accuracy score on test set: 0.17777777777777778
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 1.0
Accuracy score on test set: 0.16111111111111112
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 10.0
Accuracy score on test set: 0.20277777777777778
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 1e-05
Accuracy score on test set: 0.5305555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.0001
Accuracy score on test set: 0.5944444444444444
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.001
Accuracy score on test set: 0.5888888888888889
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.01
Accuracy score on test set: 0.6111111111111112
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.1
Accuracy score on test set: 0.5222222222222223
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 1.0
Accuracy score on test set: 0.5555555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 10.0
Accuracy score on test set: 0.8055555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 1e-05
Accuracy score on test set: 0.85
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 0.0001
Accuracy score on test set: 0.85
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 0.001
Accuracy score on test set: 0.875
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 0.01
Accuracy score on test set: 0.8666666666666667
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 0.1
Accuracy score on test set: 0.8638888888888889
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 1.0
Accuracy score on test set: 0.9555555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 10.0
Accuracy score on test set: 0.925
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 1e-05
Accuracy score on test set: 0.9472222222222222
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 0.0001
Accuracy score on test set: 0.9277777777777778
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 0.001
Accuracy score on test set: 0.9472222222222222
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 0.01
Accuracy score on test set: 0.9305555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 0.1
Accuracy score on test set: 0.9555555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 1.0
Accuracy score on test set: 0.7694444444444445
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
Lambda = 10.0
Accuracy score on test set: 0.19166666666666668
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 1e-05
Accuracy score on test set: 0.10555555555555556
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 0.0001
Accuracy score on test set: 0.08611111111111111
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 0.001
Accuracy score on test set: 0.10555555555555556
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 0.01
Accuracy score on test set: 0.08888888888888889
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 0.1
Accuracy score on test set: 0.08611111111111111
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 1.0
Accuracy score on test set: 0.08888888888888889
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
Lambda = 10.0
Accuracy score on test set: 0.09166666666666666
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 1e-05
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 0.0001
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 0.001
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 0.01
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 0.1
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 1.0
Accuracy score on test set: 0.10555555555555556
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
Lambda = 10.0
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
Lambda = 1e-05
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
Lambda = 0.0001
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
Lambda = 0.001
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
Lambda = 0.01
Accuracy score on test set: 0.07777777777777778
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:43: RuntimeWarning: overflow encountered in exp
exp_term = np.exp(self.z_o)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_12588/1630775253.py:44: RuntimeWarning: invalid value encountered in true_divide
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">12</span><span class="p">],</span> <span class="n">line</span> <span class="mi">11</span>
<span class="g g-Whitespace"> </span><span class="mi">8</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">9</span> <span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">10</span> <span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
<span class="ne">---&gt; </span><span class="mi">11</span> <span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
<span class="g g-Whitespace"> </span><span class="mi">13</span> <span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="g g-Whitespace"> </span><span class="mi">15</span> <span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="nn">Cell In[10], line 99,</span> in <span class="ni">NeuralNetwork.train</span><span class="nt">(self)</span>
<span class="g g-Whitespace"> </span><span class="mi">96</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="g g-Whitespace"> </span><span class="mi">98</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward</span><span class="p">()</span>
<span class="ne">---&gt; </span><span class="mi">99</span> <span class="bp">self</span><span class="o">.</span><span class="n">backpropagation</span><span class="p">()</span>
<span class="nn">Cell In[10], line 64,</span> in <span class="ni">NeuralNetwork.backpropagation</span><span class="nt">(self)</span>
<span class="g g-Whitespace"> </span><span class="mi">61</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">62</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="ne">---&gt; </span><span class="mi">64</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">65</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">67</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">&gt;</span> <span class="mf">0.0</span><span class="p">:</span>
<span class="ne">KeyboardInterrupt</span>:
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="visualization">
<h2>Visualization<a class="headerlink" href="#visualization" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># visual representation of grid search</span>
<span class="c1"># uses seaborn heatmap, you can also do this with matplotlib imshow</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
<span class="n">train_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">train_pred</span><span class="p">)</span>
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Test Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="scikit-learn-implementation">
<h2>scikit-learn implementation<a class="headerlink" href="#scikit-learn-implementation" title="Permalink to this headline"></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-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPClassifier</span>
<span class="c1"># store models for later use</span>
<span class="n">DNN_scikit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">MLPClassifier</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">),</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;logistic&#39;</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">learning_rate_init</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="n">epochs</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Learning rate = &quot;</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Lambda = &quot;</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on test set: &quot;</span><span class="p">,</span> <span class="n">dnn</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">))</span>
<span class="nb">print</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id7">
<h2>Visualization<a class="headerlink" href="#id7" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># optional</span>
<span class="c1"># visual representation of grid search</span>
<span class="c1"># uses seaborn heatmap, could probably do this in matplotlib</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
<span class="n">train_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">train_pred</span><span class="p">)</span>
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Test Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="testing-our-code-for-the-xor-or-and-and-gates">
<h2>Testing our code for the XOR, OR and AND gates<a class="headerlink" href="#testing-our-code-for-the-xor-or-and-and-gates" title="Permalink to this headline"></a></h2>
<p>Last week we discussed three different types of gates, the so-called
XOR, the OR and the AND gates. Their inputs and outputs can be
summarized using the following tables, first for the OR gate with
inputs <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> and outputs <span class="math notranslate nohighlight">\(y\)</span>:</p>
<table class="dotable" border="1">
<thead>
<tr><th align="center">$x_1$</th> <th align="center">$x_2$</th> <th align="center">$y$</th> </tr>
</thead>
<tbody>
<tr><td align="center"> 0 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
<tr><td align="center"> 0 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
<tr><td align="center"> 1 </td> <td align="center"> 0 </td> <td align="center"> 1 </td> </tr>
<tr><td align="center"> 1 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
</tbody>
</table></div>
<div class="section" id="the-and-and-xor-gates">
<h2>The AND and XOR Gates<a class="headerlink" href="#the-and-and-xor-gates" title="Permalink to this headline"></a></h2>
<p>The AND gate is defined as</p>
<table class="dotable" border="1">
<thead>
<tr><th align="center">$x_1$</th> <th align="center">$x_2$</th> <th align="center">$y$</th> </tr>
</thead>
<tbody>
<tr><td align="center"> 0 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
<tr><td align="center"> 0 </td> <td align="center"> 1 </td> <td align="center"> 0 </td> </tr>
<tr><td align="center"> 1 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
<tr><td align="center"> 1 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
</tbody>
</table>
<p>And finally we have the XOR gate</p>
<table class="dotable" border="1">
<thead>
<tr><th align="center">$x_1$</th> <th align="center">$x_2$</th> <th align="center">$y$</th> </tr>
</thead>
<tbody>
<tr><td align="center"> 0 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
<tr><td align="center"> 0 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
<tr><td align="center"> 1 </td> <td align="center"> 0 </td> <td align="center"> 1 </td> </tr>
<tr><td align="center"> 1 </td> <td align="center"> 1 </td> <td align="center"> 0 </td> </tr>
</tbody>
</table></div>
<div class="section" id="representing-the-data-sets">
<h2>Representing the Data Sets<a class="headerlink" href="#representing-the-data-sets" title="Permalink to this headline"></a></h2>
<p>Our design matrix is defined by the input values <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>. Since we have four possible outputs, our design matrix reads</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{X}=\begin{bmatrix} 0 &amp; 0 \\
0 &amp; 1 \\
1 &amp; 0 \\
1 &amp; 1 \end{bmatrix},
\end{split}\]</div>
<p>while the vector of outputs is <span class="math notranslate nohighlight">\(\boldsymbol{y}^T=[0,1,1,0]\)</span> for the XOR gate, <span class="math notranslate nohighlight">\(\boldsymbol{y}^T=[0,0,0,1]\)</span> for the AND gate and <span class="math notranslate nohighlight">\(\boldsymbol{y}^T=[0,1,1,1]\)</span> for the OR gate.</p>
</div>
<div class="section" id="setting-up-the-neural-network">
<h2>Setting up the Neural Network<a class="headerlink" href="#setting-up-the-neural-network" title="Permalink to this headline"></a></h2>
<p>We define first our design matrix and the various output vectors for the different gates.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd">Simple code that tests XOR, OR and AND gates with linear regression</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="c1"># import necessary packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="c1"># weighted sum of inputs to the hidden layer</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
<span class="c1"># activation in the hidden layer</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="c1"># weighted sum of inputs to the output layer</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
<span class="c1"># softmax output</span>
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="c1"># we obtain a prediction by taking the class with the highest likelihood</span>
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># ensure the same random numbers appear every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># Design matrix</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span><span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="c1"># The XOR gate</span>
<span class="n">yXOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="c1"># The OR gate</span>
<span class="n">yOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="c1"># The AND gate</span>
<span class="n">yAND</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span> <span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="c1"># Defining the neural network</span>
<span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c1"># we make the weights normally distributed using numpy.random.randn</span>
<span class="c1"># weights and bias in the hidden layer</span>
<span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_hidden_neurons</span><span class="p">)</span>
<span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="c1"># weights and bias in the output layer</span>
<span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">)</span>
<span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">probabilities</span><span class="p">)</span>
<span class="n">predictions</span> <span class="o">=</span> <span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">predictions</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Not an impressive result, but this was our first forward pass with randomly assigned weights. Let us now add the full network with the back-propagation algorithm discussed above.</p>
</div>
<div class="section" id="the-code-using-scikit-learn">
<h2>The Code using Scikit-Learn<a class="headerlink" href="#the-code-using-scikit-learn" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import necessary packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPClassifier</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="c1"># ensure the same random numbers appear every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># Design matrix</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span><span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="c1"># The XOR gate</span>
<span class="n">yXOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="c1"># The OR gate</span>
<span class="n">yOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="c1"># The AND gate</span>
<span class="n">yAND</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span> <span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="c1"># Defining the neural network</span>
<span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="c1"># store models for later use</span>
<span class="n">DNN_scikit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">MLPClassifier</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">),</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;logistic&#39;</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">learning_rate_init</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="n">epochs</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yXOR</span><span class="p">)</span>
<span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Learning rate = &quot;</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Lambda = &quot;</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on data set: &quot;</span><span class="p">,</span> <span class="n">dnn</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yXOR</span><span class="p">))</span>
<span class="nb">print</span><span class="p">()</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">yXOR</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Test Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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