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<title>Week 40: Gradient descent methods (continued) and start Neural networks &#8212; Applied Data Analysis and Machine Learning</title>
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Applied Data Analysis and Machine Learning
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About the course
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Textbooks
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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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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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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7. Optimization, the central part of any Machine Learning algortithm
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8. Support Vector Machines, overarching aims
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Decision Trees, Ensemble Methods and Boosting
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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Dimensionality Reduction
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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17. Recurrent neural networks: Overarching view
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Weekly material, notes and exercises
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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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Exercises week 37
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Week 37: Statistical interpretations and Resampling Methods
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Exercises week 38
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Week 38: Logistic Regression and Optimization
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Exercises week 39
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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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Exercises week 42
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Week 42 Constructing a Neural Network code with examples
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Projects
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Project 1 on Machine Learning, deadline October 7 (midnight), 2024
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Project 2 on Machine Learning, deadline November 4 (Midnight)
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#plans-for-week-40">
Plans for week 40
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#lecture-monday-september-30-2024">
Lecture Monday September 30, 2024
</a>
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#suggested-readings-and-videos">
Suggested readings and videos
</a>
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<a class="reference internal nav-link" href="#lab-sessions-tuesday-and-wednesday">
Lab sessions Tuesday and Wednesday
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#summary-from-last-week-using-gradient-descent-methods-limitations">
Summary from last week, using gradient descent methods, limitations
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-implementation-of-gd-for-ols-ridge-and-lasso">
Simple implementation of GD for OLS, Ridge and Lasso
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#but-none-of-these-can-compete-with-newton-s-method">
But none of these can compete with Newton’s method
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-descent-and-logistic-regression">
Gradient descent and Logistic regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#overview-video-on-stochastic-gradient-descent">
Overview video on Stochastic Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#batches-and-mini-batches">
Batches and mini-batches
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#stochastic-gradient-descent-sgd">
Stochastic Gradient Descent (SGD)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#stochastic-gradient-descent">
Stochastic Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#computation-of-gradients">
Computation of gradients
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#sgd-example">
SGD example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-gradient-step">
The gradient step
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-example-code">
Simple example code
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#when-do-we-stop">
When do we stop?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#slightly-different-approach">
Slightly different approach
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#time-decay-rate">
Time decay rate
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#code-with-a-number-of-minibatches-which-varies">
Code with a Number of Minibatches which varies
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#replace-or-not">
Replace or not
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#momentum-based-gd">
Momentum based GD
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-on-momentum-based-approaches">
More on momentum based approaches
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#momentum-parameter">
Momentum parameter
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#second-moment-of-the-gradient">
Second moment of the gradient
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rms-prop">
RMS prop
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adam-optimizer">
ADAM optimizer
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#algorithms-and-codes-for-adagrad-rmsprop-and-adam">
Algorithms and codes for Adagrad, RMSprop and Adam
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adagrad-algorithm-taken-from-goodfellow-et-al">
AdaGrad algorithm, taken from Goodfellow et al
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rmsprop-algorithm-taken-from-goodfellow-et-al">
RMSProp algorithm, taken from Goodfellow et al
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adam-algorithm-taken-from-goodfellow-et-al">
ADAM algorithm, taken from Goodfellow et al
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#practical-tips">
Practical tips
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#automatic-differentiation">
Automatic differentiation
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-autograd">
Using autograd
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#autograd-with-more-complicated-functions">
Autograd with more complicated functions
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-complicated-functions-using-the-elements-of-their-arguments-directly">
More complicated functions using the elements of their arguments directly
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#functions-using-mathematical-functions-from-numpy">
Functions using mathematical functions from Numpy
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-autograd">
More autograd
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#and-with-loops">
And with loops
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-recursion">
Using recursion
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-autograd-with-ols">
Using Autograd with OLS
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#same-code-but-now-with-momentum-gradient-descent">
Same code but now with momentum gradient descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#including-stochastic-gradient-descent-with-autograd">
Including Stochastic Gradient Descent with Autograd
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
Same code but now with momentum gradient descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#similar-second-order-function-now-problem-but-now-with-adagrad">
Similar (second order function now) problem but now with AdaGrad
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent">
RMSprop for adaptive learning rate with Stochastic Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#and-finally-adam">
And finally ADAM
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#and-logistic-regression">
And Logistic Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#introducing-jax">
Introducing JAX
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#getting-started-with-jax-note-the-way-we-import-numpy">
Getting started with Jax, note the way we import numpy
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-warm-up-example">
A warm-up example
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-more-advanced-example">
A more advanced example
</a>
</li>
</ul>
</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="#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>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id5">
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>
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<h1>Week 40: Gradient descent methods (continued) and start Neural networks</h1>
<!-- Table of contents -->
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<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="#plans-for-week-40">
Plans for week 40
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#lecture-monday-september-30-2024">
Lecture Monday September 30, 2024
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#suggested-readings-and-videos">
Suggested readings and videos
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#lab-sessions-tuesday-and-wednesday">
Lab sessions Tuesday and Wednesday
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#summary-from-last-week-using-gradient-descent-methods-limitations">
Summary from last week, using gradient descent methods, limitations
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-implementation-of-gd-for-ols-ridge-and-lasso">
Simple implementation of GD for OLS, Ridge and Lasso
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#but-none-of-these-can-compete-with-newton-s-method">
But none of these can compete with Newton’s method
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-descent-and-logistic-regression">
Gradient descent and Logistic regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#overview-video-on-stochastic-gradient-descent">
Overview video on Stochastic Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#batches-and-mini-batches">
Batches and mini-batches
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#stochastic-gradient-descent-sgd">
Stochastic Gradient Descent (SGD)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#stochastic-gradient-descent">
Stochastic Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#computation-of-gradients">
Computation of gradients
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#sgd-example">
SGD example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-gradient-step">
The gradient step
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-example-code">
Simple example code
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#when-do-we-stop">
When do we stop?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#slightly-different-approach">
Slightly different approach
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#time-decay-rate">
Time decay rate
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#code-with-a-number-of-minibatches-which-varies">
Code with a Number of Minibatches which varies
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#replace-or-not">
Replace or not
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#momentum-based-gd">
Momentum based GD
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-on-momentum-based-approaches">
More on momentum based approaches
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#momentum-parameter">
Momentum parameter
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#second-moment-of-the-gradient">
Second moment of the gradient
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rms-prop">
RMS prop
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adam-optimizer">
ADAM optimizer
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#algorithms-and-codes-for-adagrad-rmsprop-and-adam">
Algorithms and codes for Adagrad, RMSprop and Adam
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adagrad-algorithm-taken-from-goodfellow-et-al">
AdaGrad algorithm, taken from Goodfellow et al
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rmsprop-algorithm-taken-from-goodfellow-et-al">
RMSProp algorithm, taken from Goodfellow et al
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adam-algorithm-taken-from-goodfellow-et-al">
ADAM algorithm, taken from Goodfellow et al
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#practical-tips">
Practical tips
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#automatic-differentiation">
Automatic differentiation
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-autograd">
Using autograd
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#autograd-with-more-complicated-functions">
Autograd with more complicated functions
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-complicated-functions-using-the-elements-of-their-arguments-directly">
More complicated functions using the elements of their arguments directly
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#functions-using-mathematical-functions-from-numpy">
Functions using mathematical functions from Numpy
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-autograd">
More autograd
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#and-with-loops">
And with loops
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-recursion">
Using recursion
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#using-autograd-with-ols">
Using Autograd with OLS
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#same-code-but-now-with-momentum-gradient-descent">
Same code but now with momentum gradient descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#including-stochastic-gradient-descent-with-autograd">
Including Stochastic Gradient Descent with Autograd
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
Same code but now with momentum gradient descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#similar-second-order-function-now-problem-but-now-with-adagrad">
Similar (second order function now) problem but now with AdaGrad
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent">
RMSprop for adaptive learning rate with Stochastic Gradient Descent
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#and-finally-adam">
And finally ADAM
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#and-logistic-regression">
And Logistic Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#introducing-jax">
Introducing JAX
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#getting-started-with-jax-note-the-way-we-import-numpy">
Getting started with Jax, note the way we import numpy
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-warm-up-example">
A warm-up example
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-more-advanced-example">
A more advanced example
</a>
</li>
</ul>
</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="#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>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id5">
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>
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</li>
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</div>
<div>
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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<!-- dom:TITLE: Week 40: Gradient descent methods (continued) and start Neural networks --><div class="tex2jax_ignore mathjax_ignore section" id="week-40-gradient-descent-methods-continued-and-start-neural-networks">
<h1>Week 40: Gradient descent methods (continued) and start Neural networks<a class="headerlink" href="#week-40-gradient-descent-methods-continued-and-start-neural-networks" title="Permalink to this headline"></a></h1>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</p>
<p>Date: <strong>September 30-October 4, 2024</strong></p>
<div class="section" id="plans-for-week-40">
<h2>Plans for week 40<a class="headerlink" href="#plans-for-week-40" title="Permalink to this headline"></a></h2>
</div>
<div class="section" id="lecture-monday-september-30-2024">
<h2>Lecture Monday September 30, 2024<a class="headerlink" href="#lecture-monday-september-30-2024" title="Permalink to this headline"></a></h2>
<ol class="simple">
<li><p>Stochastic Gradient descent with examples and automatic differentiation</p></li>
<li><p>If we get time, we start with the basics of Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model</p></li>
<li><p><a class="reference external" href="https://youtu.be/jdJoOrCIdII">Video of lecture</a></p></li>
<li><p>Whiteboard notes at <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember30.pdf">https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember30.pdf</a></p></li>
</ol>
</div>
<div class="section" id="suggested-readings-and-videos">
<h2>Suggested readings and videos<a class="headerlink" href="#suggested-readings-and-videos" title="Permalink to this headline"></a></h2>
<p><strong>Readings and Videos:</strong></p>
<ol class="simple">
<li><p>The lecture notes for week 40 (these notes)</p></li>
<li><p>For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well.</p></li>
<li><p>For neural networks we recommend Goodfellow et al chapter 6 and Raschka et al chapter 2 (contains also material about gradient descent) and chapter 11 (we will use this next week)</p></li>
<li><p>Video on gradient descent at <a class="reference external" href="https://www.youtube.com/watch?v=sDv4f4s2SB8">https://www.youtube.com/watch?v=sDv4f4s2SB8</a></p></li>
<li><p>Video on stochastic gradient descent at <a class="reference external" href="https://www.youtube.com/watch?v=vMh0zPT0tLI">https://www.youtube.com/watch?v=vMh0zPT0tLI</a></p></li>
<li><p>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></li>
<li><p>Building Neural Networks from scratch at URL:<a class="reference external" href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&amp;ab_channel=sentdex">https://www.youtube.com/watch?v=Wo5dMEP_BbI&amp;list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&amp;ab_channel=sentdex</a>”</p></li>
</ol>
</div>
<div class="section" id="lab-sessions-tuesday-and-wednesday">
<h2>Lab sessions Tuesday and Wednesday<a class="headerlink" href="#lab-sessions-tuesday-and-wednesday" title="Permalink to this headline"></a></h2>
<p><strong>Material for the active learning sessions on Tuesday and Wednesday.</strong></p>
<ul class="simple">
<li><p>Work on project 1 and discussions on how to structure your report</p></li>
<li><p>No weekly exercises for week 40, project work only</p></li>
<li><p>Video on how to write scientific reports recorded during one of the lab sessions at <a class="reference external" href="https://youtu.be/tVW1ZDmZnwM">https://youtu.be/tVW1ZDmZnwM</a></p></li>
<li><p>A general guideline can be found at <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md">https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md</a>.</p></li>
</ul>
</div>
<div class="section" id="summary-from-last-week-using-gradient-descent-methods-limitations">
<h2>Summary from last week, using gradient descent methods, limitations<a class="headerlink" href="#summary-from-last-week-using-gradient-descent-methods-limitations" title="Permalink to this headline"></a></h2>
<ul class="simple">
<li><p><strong>Gradient descent (GD) finds local minima of our function</strong>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</p></li>
<li><p><strong>GD is sensitive to initial conditions</strong>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</p></li>
<li><p><strong>Gradients are computationally expensive to calculate for large datasets</strong>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, <span class="math notranslate nohighlight">\(E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2\)</span>; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> <span class="math notranslate nohighlight">\(n\)</span> data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called “mini batches”. This has the added benefit of introducing stochasticity into our algorithm.</p></li>
<li><p><strong>GD is very sensitive to choices of learning rates</strong>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</p></li>
<li><p><strong>GD treats all directions in parameter space uniformly.</strong> Another major drawback of GD is that unlike Newton’s method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</p></li>
<li><p>GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</p></li>
</ul>
</div>
<div class="section" id="simple-implementation-of-gd-for-ols-ridge-and-lasso">
<h2>Simple implementation of GD for OLS, Ridge and Lasso<a class="headerlink" href="#simple-implementation-of-gd-for-ols-ridge-and-lasso" title="Permalink to this headline"></a></h2>
<p>Last week we studied both several gradient methods. With and without an update of the learning.
We summarize some of these here for the methods we hvae studied in project one, without the inclusion of momentum.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># the number of datapoints with a 2nd-order polynomial</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span>
<span class="c1"># Design matrix including the intercept</span>
<span class="c1"># No scaling of data of and all data used for training </span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">]</span>
<span class="c1"># Learning rate and number of iterations</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.05</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">100</span>
<span class="c1"># OLS part</span>
<span class="n">beta_OLS</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">gradient</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="mi">3</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradient</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</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="p">(</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta_OLS</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
<span class="n">beta_OLS</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradient</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Parameters for OLS using gradient descent&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta_OLS</span><span class="p">)</span>
<span class="c1">#Ridge and Lasso parameter Lambda</span>
<span class="n">Lambda</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="n">Id</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="n">Lambda</span><span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</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="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="c1"># Gradient descent with Ridge</span>
<span class="n">beta_Ridge</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">gradient</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="mi">3</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">/</span><span class="n">n</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="p">(</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta_Ridge</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">Lambda</span><span class="o">*</span><span class="n">beta_Ridge</span>
<span class="n">beta_Ridge</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Parameters for Ridge using gradient descent&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta_Ridge</span><span class="p">)</span>
<span class="c1"># Gradient descent with Lasso</span>
<span class="n">beta_Lasso</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">gradient</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="mi">3</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">/</span><span class="n">n</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="p">(</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta_Lasso</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="n">Lambda</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">beta_Lasso</span><span class="p">)</span>
<span class="n">beta_Lasso</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Parameters for Lasso using gradient descent&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta_Lasso</span><span class="p">)</span>
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Parameters for OLS using gradient descent
[[3.8184887 ]
[3.47851966]
[4.77551387]]
Parameters for Ridge using gradient descent
[[3.92021197]
[3.11388017]
[4.9458396 ]]
Parameters for Lasso using gradient descent
[[3.87323528]
[3.3008836 ]
[4.86284277]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="but-none-of-these-can-compete-with-newton-s-method">
<h2>But none of these can compete with Newton’s method<a class="headerlink" href="#but-none-of-these-can-compete-with-newton-s-method" title="Permalink to this headline"></a></h2>
<p>Note that we here have introduced automatic differentiation</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Using Newton&#39;s method</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">beta</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</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">y</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">beta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta_linreg</span><span class="p">)</span>
<span class="c1"># Hessian matrix</span>
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">XT_X</span>
<span class="c1"># Note that here the Hessian does not depend on the parameters beta</span>
<span class="n">invH</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">H</span><span class="p">)</span>
<span class="n">beta</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">5</span>
<span class="c1"># define the gradient</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="n">training_gradient</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
<span class="n">beta</span> <span class="o">-=</span> <span class="n">invH</span> <span class="o">@</span> <span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span><span class="n">gradients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">gradients</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;beta from own Newton code&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
[[4.]
[3.]
[5.]]
0 [-31.91417133] [-44.48017159]
1 [3.48805429e-13] [4.67477123e-13]
2 [6.75015599e-16] [1.30675215e-15]
3 [-1.17239551e-15] [-1.78477759e-15]
4 [6.75015599e-16] [1.30675215e-15]
beta from own Newton code
[[4.]
[3.]
[5.]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="gradient-descent-and-logistic-regression">
<h2>Gradient descent and Logistic regression<a class="headerlink" href="#gradient-descent-and-logistic-regression" title="Permalink to this headline"></a></h2>
<p>Finally, we complete these examples by adding a simple code for
Logistic regression. Note the more general approach with a class for
the method. Here we use a so-called <strong>AND</strong> gate for our data set.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="k">class</span> <span class="nc">LogisticRegression</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">learning_rate</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">num_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</span> <span class="o">=</span> <span class="n">learning_rate</span>
<span class="bp">self</span><span class="o">.</span><span class="n">num_iterations</span> <span class="o">=</span> <span class="n">num_iterations</span>
<span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</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">z</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">GDfit</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">y</span><span class="p">):</span>
<span class="n">n_data</span><span class="p">,</span> <span class="n">num_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</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">num_features</span><span class="p">)</span>
<span class="k">for</span> <span class="n">_</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">num_iterations</span><span class="p">):</span>
<span class="n">linear_model</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span>
<span class="n">y_predicted</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">linear_model</span><span class="p">)</span>
<span class="c1"># Gradient calculation</span>
<span class="n">gradient</span> <span class="o">=</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="p">(</span><span class="n">y_predicted</span> <span class="o">-</span> <span class="n">y</span><span class="p">))</span><span class="o">/</span><span class="n">n_data</span>
<span class="c1"># Update beta_logreg</span>
<span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</span><span class="o">*</span><span class="n">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">linear_model</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta_logreg</span>
<span class="n">y_predicted</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">linear_model</span><span class="p">)</span>
<span class="k">return</span> <span class="p">[</span><span class="mi">1</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="mf">0.5</span> <span class="k">else</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y_predicted</span><span class="p">]</span>
<span class="c1"># Example usage</span>
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">&quot;__main__&quot;</span><span class="p">:</span>
<span class="c1"># Sample data</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="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="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="n">y</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="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"># This is an AND gate</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">num_iterations</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">GDfit</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">predictions</span> <span class="o">=</span> <span class="n">model</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="s2">&quot;Predictions:&quot;</span><span class="p">,</span> <span class="n">predictions</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>Predictions: [1, 1, 1, 1]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="overview-video-on-stochastic-gradient-descent">
<h2>Overview video on Stochastic Gradient Descent<a class="headerlink" href="#overview-video-on-stochastic-gradient-descent" title="Permalink to this headline"></a></h2>
<p><a class="reference external" href="https://www.youtube.com/watch?v=vMh0zPT0tLI&amp;ab_channel=StatQuestwithJoshStarmer">What is Stochastic Gradient Descent</a>
There are several reasons for using stochastic gradient descent. Some of these are:</p>
<ol class="simple">
<li><p>Efficiency: Updates weights more frequently using a single or a small batch of samples, which speeds up convergence.</p></li>
<li><p>Hopefully avoid Local Minima</p></li>
<li><p>Memory Usage: Requires less memory compared to computing gradients for the entire dataset.</p></li>
</ol>
</div>
<div class="section" id="batches-and-mini-batches">
<h2>Batches and mini-batches<a class="headerlink" href="#batches-and-mini-batches" title="Permalink to this headline"></a></h2>
<p>In gradient descent we compute the cost function and its gradient for all data points we have.</p>
<p>In large-scale applications such as the <a class="reference external" href="https://www.image-net.org/challenges/LSVRC/">ILSVRC challenge</a>, the
training data can have on order of millions of examples. Hence, it
seems wasteful to compute the full cost function over the entire
training set in order to perform only a single parameter update. A
very common approach to addressing this challenge is to compute the
gradient over batches of the training data. For example, a typical batch could contain some thousand examples from
an entire training set of several millions. This batch is then used to
perform a parameter update.</p>
</div>
<div class="section" id="stochastic-gradient-descent-sgd">
<h2>Stochastic Gradient Descent (SGD)<a class="headerlink" href="#stochastic-gradient-descent-sgd" title="Permalink to this headline"></a></h2>
<p>In stochastic gradient descent, the extreme case is the case where we
have only one batch, that is we include the whole data set.</p>
<p>This process is called Stochastic Gradient
Descent (SGD) (or also sometimes on-line gradient descent). This is
relatively less common to see because in practice due to vectorized
code optimizations it can be computationally much more efficient to
evaluate the gradient for 100 examples, than the gradient for one
example 100 times. Even though SGD technically refers to using a
single example at a time to evaluate the gradient, you will hear
people use the term SGD even when referring to mini-batch gradient
descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD
for “Batch gradient descent” are rare to see), where it is usually
assumed that mini-batches are used. The size of the mini-batch is a
hyperparameter but it is not very common to cross-validate or bootstrap it. It is
usually based on memory constraints (if any), or set to some value,
e.g. 32, 64 or 128. We use powers of 2 in practice because many
vectorized operation implementations work faster when their inputs are
sized in powers of 2.</p>
<p>In our notes with SGD we mean stochastic gradient descent with mini-batches.</p>
</div>
<div class="section" id="stochastic-gradient-descent">
<h2>Stochastic Gradient Descent<a class="headerlink" href="#stochastic-gradient-descent" title="Permalink to this headline"></a></h2>
<p>Stochastic gradient descent (SGD) and variants thereof address some of
the shortcomings of the Gradient descent method discussed above.</p>
<p>The underlying idea of SGD comes from the observation that the cost
function, which we want to minimize, can almost always be written as a
sum over <span class="math notranslate nohighlight">\(n\)</span> data points <span class="math notranslate nohighlight">\(\{\mathbf{x}_i\}_{i=1}^n\)</span>,</p>
<div class="math notranslate nohighlight">
\[
C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
\mathbf{\beta}).
\]</div>
</div>
<div class="section" id="computation-of-gradients">
<h2>Computation of gradients<a class="headerlink" href="#computation-of-gradients" title="Permalink to this headline"></a></h2>
<p>This in turn means that the gradient can be
computed as a sum over <span class="math notranslate nohighlight">\(i\)</span>-gradients</p>
<div class="math notranslate nohighlight">
\[
\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
\mathbf{\beta}).
\]</div>
<p>Stochasticity/randomness is introduced by only taking the
gradient on a subset of the data called minibatches. If there are <span class="math notranslate nohighlight">\(n\)</span>
data points and the size of each minibatch is <span class="math notranslate nohighlight">\(M\)</span>, there will be <span class="math notranslate nohighlight">\(n/M\)</span>
minibatches. We denote these minibatches by <span class="math notranslate nohighlight">\(B_k\)</span> where
<span class="math notranslate nohighlight">\(k=1,\cdots,n/M\)</span>.</p>
</div>
<div class="section" id="sgd-example">
<h2>SGD example<a class="headerlink" href="#sgd-example" title="Permalink to this headline"></a></h2>
<p>As an example, suppose we have <span class="math notranslate nohighlight">\(10\)</span> data points <span class="math notranslate nohighlight">\((\mathbf{x}_1,\cdots, \mathbf{x}_{10})\)</span>
and we choose to have <span class="math notranslate nohighlight">\(M=5\)</span> minibathces,
then each minibatch contains two data points. In particular we have
<span class="math notranslate nohighlight">\(B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
(\mathbf{x}_9,\mathbf{x}_{10})\)</span>. Note that if you choose <span class="math notranslate nohighlight">\(M=1\)</span> you
have only a single batch with all data points and on the other extreme,
you may choose <span class="math notranslate nohighlight">\(M=n\)</span> resulting in a minibatch for each datapoint, i.e
<span class="math notranslate nohighlight">\(B_k = \mathbf{x}_k\)</span>.</p>
<p>The idea is now to approximate the gradient by replacing the sum over
all data points with a sum over the data points in one the minibatches
picked at random in each gradient descent step</p>
<div class="math notranslate nohighlight">
\[
\nabla_{\beta}
C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
c_i(\mathbf{x}_i, \mathbf{\beta}).
\]</div>
</div>
<div class="section" id="the-gradient-step">
<h2>The gradient step<a class="headerlink" href="#the-gradient-step" title="Permalink to this headline"></a></h2>
<p>Thus a gradient descent step now looks like</p>
<div class="math notranslate nohighlight">
\[
\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
\mathbf{\beta})
\]</div>
<p>where <span class="math notranslate nohighlight">\(k\)</span> is picked at random with equal
probability from <span class="math notranslate nohighlight">\([1,n/M]\)</span>. An iteration over the number of
minibathces (n/M) is commonly referred to as an epoch. Thus it is
typical to choose a number of epochs and for each epoch iterate over
the number of minibatches, as exemplified in the code below.</p>
</div>
<div class="section" id="simple-example-code">
<h2>Simple example code<a class="headerlink" href="#simple-example-code" 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="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1">#100 datapoints </span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">10</span> <span class="c1">#number of epochs</span>
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n_epochs</span><span class="o">+</span><span class="mi">1</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="n">m</span><span class="p">):</span>
<span class="n">k</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="c1">#Pick the k-th minibatch at random</span>
<span class="c1">#Compute the gradient using the data in minibatch Bk</span>
<span class="c1">#Compute new suggestion for </span>
<span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
</pre></div>
</div>
</div>
</div>
<p>Taking the gradient only on a subset of the data has two important
benefits. First, it introduces randomness which decreases the chance
that our opmization scheme gets stuck in a local minima. Second, if
the size of the minibatches are small relative to the number of
datapoints (<span class="math notranslate nohighlight">\(M &lt; n\)</span>), the computation of the gradient is much
cheaper since we sum over the datapoints in the <span class="math notranslate nohighlight">\(k-th\)</span> minibatch and not
all <span class="math notranslate nohighlight">\(n\)</span> datapoints.</p>
</div>
<div class="section" id="when-do-we-stop">
<h2>When do we stop?<a class="headerlink" href="#when-do-we-stop" title="Permalink to this headline"></a></h2>
<p>A natural question is when do we stop the search for a new minimum?
One possibility is to compute the full gradient after a given number
of epochs and check if the norm of the gradient is smaller than some
threshold and stop if true. However, the condition that the gradient
is zero is valid also for local minima, so this would only tell us
that we are close to a local/global minimum. However, we could also
evaluate the cost function at this point, store the result and
continue the search. If the test kicks in at a later stage we can
compare the values of the cost function and keep the <span class="math notranslate nohighlight">\(\beta\)</span> that
gave the lowest value.</p>
</div>
<div class="section" id="slightly-different-approach">
<h2>Slightly different approach<a class="headerlink" href="#slightly-different-approach" title="Permalink to this headline"></a></h2>
<p>Another approach is to let the step length <span class="math notranslate nohighlight">\(\gamma_j\)</span> depend on the
number of epochs in such a way that it becomes very small after a
reasonable time such that we do not move at all. Such approaches are
also called scaling. There are many such ways to <a class="reference external" href="https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1">scale the learning
rate</a>
and <a class="reference external" href="https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf">discussions here</a>. See
also
<a class="reference external" href="https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1">https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1</a>
for a discussion of different scaling functions for the learning rate.</p>
</div>
<div class="section" id="time-decay-rate">
<h2>Time decay rate<a class="headerlink" href="#time-decay-rate" title="Permalink to this headline"></a></h2>
<p>As an example, let <span class="math notranslate nohighlight">\(e = 0,1,2,3,\cdots\)</span> denote the current epoch and let <span class="math notranslate nohighlight">\(t_0, t_1 &gt; 0\)</span> be two fixed numbers. Furthermore, let <span class="math notranslate nohighlight">\(t = e \cdot m + i\)</span> where <span class="math notranslate nohighlight">\(m\)</span> is the number of minibatches and <span class="math notranslate nohighlight">\(i=0,\cdots,m-1\)</span>. Then the function $<span class="math notranslate nohighlight">\(\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} \)</span><span class="math notranslate nohighlight">\( goes to zero as the number of epochs gets large. I.e. we start with a step length \)</span>\gamma_j (0; t_0, t_1) = t_0/t_1<span class="math notranslate nohighlight">\( which decays in *time* \)</span>t$.</p>
<p>In this way we can fix the number of epochs, compute <span class="math notranslate nohighlight">\(\beta\)</span> and
evaluate the cost function at the end. Repeating the computation will
give a different result since the scheme is random by design. Then we
pick the final <span class="math notranslate nohighlight">\(\beta\)</span> that gives the lowest value of the cost
function.</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">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="k">def</span> <span class="nf">step_length</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">t0</span><span class="p">,</span><span class="n">t1</span><span class="p">):</span>
<span class="k">return</span> <span class="n">t0</span><span class="o">/</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">t1</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1">#100 datapoints </span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">500</span> <span class="c1">#number of epochs</span>
<span class="n">t0</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">t1</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">gamma_j</span> <span class="o">=</span> <span class="n">t0</span><span class="o">/</span><span class="n">t1</span>
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n_epochs</span><span class="o">+</span><span class="mi">1</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="n">m</span><span class="p">):</span>
<span class="n">k</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="c1">#Pick the k-th minibatch at random</span>
<span class="c1">#Compute the gradient using the data in minibatch Bk</span>
<span class="c1">#Compute new suggestion for beta</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">epoch</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">i</span>
<span class="n">gamma_j</span> <span class="o">=</span> <span class="n">step_length</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">t0</span><span class="p">,</span><span class="n">t1</span><span class="p">)</span>
<span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;gamma_j after </span><span class="si">%d</span><span class="s2"> epochs: </span><span class="si">%g</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n_epochs</span><span class="p">,</span><span class="n">gamma_j</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>gamma_j after 500 epochs: 9.97108e-05
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="code-with-a-number-of-minibatches-which-varies">
<h2>Code with a Number of Minibatches which varies<a class="headerlink" href="#code-with-a-number-of-minibatches-which-varies" title="Permalink to this headline"></a></h2>
<p>In the code here we vary the number of mini-batches.</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="o">%</span><span class="k">matplotlib</span> inline
<span class="c1"># Importing various packages</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</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</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">inv</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="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Hessian matrix</span>
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">XT_X</span>
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</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">eig</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">1.0</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">EigValues</span><span class="p">)</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">/</span><span class="n">n</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="p">((</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own gd&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">xnew</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="mi">0</span><span class="p">],[</span><span class="mi">2</span><span class="p">]])</span>
<span class="n">Xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">xnew</span><span class="p">]</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">ypredict2</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="n">t0</span><span class="p">,</span> <span class="n">t1</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">50</span>
<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="k">return</span> <span class="n">t0</span><span class="o">/</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">t1</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">):</span>
<span class="c1"># Can you figure out a better way of setting up the contributions to each batch?</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="n">m</span><span class="p">):</span>
<span class="n">random_index</span> <span class="o">=</span> <span class="n">M</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span> <span class="n">xi</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">((</span><span class="n">xi</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">-</span><span class="n">yi</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="n">learning_schedule</span><span class="p">(</span><span class="n">epoch</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">i</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span> <span class="o">-</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own sdg&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">&quot;r-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict2</span><span class="p">,</span> <span class="s2">&quot;b-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</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="s1">&#39;ro&#39;</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="mi">0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$y$&#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="sa">r</span><span class="s1">&#39;Random numbers &#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">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
[[3.90300704]
[3.16913489]]
Eigenvalues of Hessian Matrix:[0.2964378 4.12443871]
theta from own gd
[[3.90300704]
[3.16913489]]
theta from own sdg
[[3.93272428]
[3.16328315]]
</pre></div>
</div>
<img alt="_images/week40_34_1.png" src="_images/week40_34_1.png" />
</div>
</div>
</div>
<div class="section" id="replace-or-not">
<h2>Replace or not<a class="headerlink" href="#replace-or-not" title="Permalink to this headline"></a></h2>
<p>In the above code, we have use replacement in setting up the
mini-batches. The discussion
<a class="reference external" href="https://sebastianraschka.com/faq/docs/sgd-methods.html">here</a> may be
useful.</p>
</div>
<div class="section" id="momentum-based-gd">
<h2>Momentum based GD<a class="headerlink" href="#momentum-based-gd" title="Permalink to this headline"></a></h2>
<p>The stochastic gradient descent (SGD) is almost always used with a
<em>momentum</em> or inertia term that serves as a memory of the direction we
are moving in parameter space. This is typically implemented as
follows</p>
<div class="math notranslate nohighlight">
\[
\mathbf{v}_{t}=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber
\]</div>
<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\boldsymbol{\theta}_{t+1}= \boldsymbol{\theta}_t -\mathbf{v}_{t},
\label{_auto1} \tag{1}
\end{equation}
\]</div>
<p>where we have introduced a momentum parameter <span class="math notranslate nohighlight">\(\gamma\)</span>, with
<span class="math notranslate nohighlight">\(0\le\gamma\le 1\)</span>, and for brevity we dropped the explicit notation to
indicate the gradient is to be taken over a different mini-batch at
each step. We call this algorithm gradient descent with momentum
(GDM). From these equations, it is clear that <span class="math notranslate nohighlight">\(\mathbf{v}_t\)</span> is a
running average of recently encountered gradients and
<span class="math notranslate nohighlight">\((1-\gamma)^{-1}\)</span> sets the characteristic time scale for the memory
used in the averaging procedure. Consistent with this, when
<span class="math notranslate nohighlight">\(\gamma=0\)</span>, this just reduces down to ordinary SGD as discussed
earlier. An equivalent way of writing the updates is</p>
<div class="math notranslate nohighlight">
\[
\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
\]</div>
<p>where we have defined <span class="math notranslate nohighlight">\(\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}\)</span>.</p>
</div>
<div class="section" id="more-on-momentum-based-approaches">
<h2>More on momentum based approaches<a class="headerlink" href="#more-on-momentum-based-approaches" title="Permalink to this headline"></a></h2>
<p>Let us try to get more intuition from these equations. It is helpful
to consider a simple physical analogy with a particle of mass <span class="math notranslate nohighlight">\(m\)</span>
moving in a viscous medium with drag coefficient <span class="math notranslate nohighlight">\(\mu\)</span> and potential
<span class="math notranslate nohighlight">\(E(\mathbf{w})\)</span>. If we denote the particle’s position by <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>,
then its motion is described by</p>
<div class="math notranslate nohighlight">
\[
m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
\]</div>
<p>We can discretize this equation in the usual way to get</p>
<div class="math notranslate nohighlight">
\[
m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
\]</div>
<p>Rearranging this equation, we can rewrite this as</p>
<div class="math notranslate nohighlight">
\[
\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
\]</div>
</div>
<div class="section" id="momentum-parameter">
<h2>Momentum parameter<a class="headerlink" href="#momentum-parameter" title="Permalink to this headline"></a></h2>
<p>Notice that this equation is identical to previous one if we identify
the position of the particle, <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>, with the parameters
<span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>. This allows us to identify the momentum
parameter and learning rate with the mass of the particle and the
viscous drag as:</p>
<div class="math notranslate nohighlight">
\[
\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
\]</div>
<p>Thus, as the name suggests, the momentum parameter is proportional to
the mass of the particle and effectively provides inertia.
Furthermore, in the large viscosity/small learning rate limit, our
memory time scales as <span class="math notranslate nohighlight">\((1-\gamma)^{-1} \approx m/(\mu \Delta t)\)</span>.</p>
<p>Why is momentum useful? SGD momentum helps the gradient descent
algorithm gain speed in directions with persistent but small gradients
even in the presence of stochasticity, while suppressing oscillations
in high-curvature directions. This becomes especially important in
situations where the landscape is shallow and flat in some directions
and narrow and steep in others. It has been argued that first-order
methods (with appropriate initial conditions) can perform comparable
to more expensive second order methods, especially in the context of
complex deep learning models.</p>
<p>These beneficial properties of momentum can sometimes become even more
pronounced by using a slight modification of the classical momentum
algorithm called Nesterov Accelerated Gradient (NAG).</p>
<p>In the NAG algorithm, rather than calculating the gradient at the
current parameters, <span class="math notranslate nohighlight">\(\nabla_\theta E(\boldsymbol{\theta}_t)\)</span>, one
calculates the gradient at the expected value of the parameters given
our current momentum, <span class="math notranslate nohighlight">\(\nabla_\theta E(\boldsymbol{\theta}_t +\gamma
\mathbf{v}_{t-1})\)</span>. This yields the NAG update rule</p>
<div class="math notranslate nohighlight">
\[
\mathbf{v}_{t}=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber
\]</div>
<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\boldsymbol{\theta}_{t+1}= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
\label{_auto2} \tag{2}
\end{equation}
\]</div>
<p>One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of <span class="math notranslate nohighlight">\(\gamma\)</span>.</p>
</div>
<div class="section" id="second-moment-of-the-gradient">
<h2>Second moment of the gradient<a class="headerlink" href="#second-moment-of-the-gradient" title="Permalink to this headline"></a></h2>
<p>In stochastic gradient descent, with and without momentum, we still
have to specify a schedule for tuning the learning rates <span class="math notranslate nohighlight">\(\eta_t\)</span>
as a function of time. As discussed in the context of Newton’s
method, this presents a number of dilemmas. The learning rate is
limited by the steepest direction which can change depending on the
current position in the landscape. To circumvent this problem, ideally
our algorithm would keep track of curvature and take large steps in
shallow, flat directions and small steps in steep, narrow directions.
Second-order methods accomplish this by calculating or approximating
the Hessian and normalizing the learning rate by the
curvature. However, this is very computationally expensive for
extremely large models. Ideally, we would like to be able to
adaptively change the step size to match the landscape without paying
the steep computational price of calculating or approximating
Hessians.</p>
<p>During the last decade a number of methods have been introduced that accomplish
this by tracking not only the gradient, but also the second moment of
the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
<a class="reference external" href="https://arxiv.org/abs/1412.6980">ADAM</a>.</p>
</div>
<div class="section" id="rms-prop">
<h2>RMS prop<a class="headerlink" href="#rms-prop" title="Permalink to this headline"></a></h2>
<p>In RMS prop, in addition to keeping a running average of the first
moment of the gradient, we also keep track of the second moment
denoted by <span class="math notranslate nohighlight">\(\mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2]\)</span>. The update rule
for RMS prop is given by</p>
<!-- Equation labels as ordinary links -->
<div id="_auto3"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\mathbf{g}_t = \nabla_\theta E(\boldsymbol{\theta})
\label{_auto3} \tag{3}
\end{equation}
\]</div>
<div class="math notranslate nohighlight">
\[
\mathbf{s}_t =\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber
\]</div>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\theta}_{t+1}=\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
\]</div>
<p>where <span class="math notranslate nohighlight">\(\beta\)</span> controls the averaging time of the second moment and is
typically taken to be about <span class="math notranslate nohighlight">\(\beta=0.9\)</span>, <span class="math notranslate nohighlight">\(\eta_t\)</span> is a learning rate
typically chosen to be <span class="math notranslate nohighlight">\(10^{-3}\)</span>, and <span class="math notranslate nohighlight">\(\epsilon\sim 10^{-8} \)</span> is a
small regularization constant to prevent divergences. Multiplication
and division by vectors is understood as an element-wise operation. It
is clear from this formula that the learning rate is reduced in
directions where the norm of the gradient is consistently large. This
greatly speeds up the convergence by allowing us to use a larger
learning rate for flat directions.</p>
</div>
<div class="section" id="adam-optimizer">
<h2><a class="reference external" href="https://arxiv.org/abs/1412.6980">ADAM optimizer</a><a class="headerlink" href="#adam-optimizer" title="Permalink to this headline"></a></h2>
<p>A related algorithm is the ADAM optimizer. In
<a class="reference external" href="https://arxiv.org/abs/1412.6980">ADAM</a>, we keep a running average of
both the first and second moment of the gradient and use this
information to adaptively change the learning rate for different
parameters. The method isefficient when working with large
problems involving lots data and/or parameters. It is a combination of the
gradient descent with momentum algorithm and the RMSprop algorithm
discussed above.</p>
<p>In addition to keeping a running average of the first and
second moments of the gradient
(i.e. <span class="math notranslate nohighlight">\(\mathbf{m}_t=\mathbb{E}[\mathbf{g}_t]\)</span> and
<span class="math notranslate nohighlight">\(\mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t]\)</span>, respectively), ADAM
performs an additional bias correction to account for the fact that we
are estimating the first two moments of the gradient using a running
average (denoted by the hats in the update rule below). The update
rule for ADAM is given by (where multiplication and division are once
again understood to be element-wise operations below)</p>
<!-- Equation labels as ordinary links -->
<div id="_auto4"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\mathbf{g}_t = \nabla_\theta E(\boldsymbol{\theta})
\label{_auto4} \tag{4}
\end{equation}
\]</div>
<div class="math notranslate nohighlight">
\[
\mathbf{m}_t = \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber
\]</div>
<div class="math notranslate nohighlight">
\[
\mathbf{s}_t =\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber
\]</div>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\mathbf{m}}_t={\mathbf{m}_t \over 1-\beta_1^t} \nonumber
\]</div>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\mathbf{s}}_t ={\mathbf{s}_t \over1-\beta_2^t} \nonumber
\]</div>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\theta}_{t+1}=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber
\]</div>
<!-- Equation labels as ordinary links -->
<div id="_auto5"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\label{_auto5} \tag{5}
\end{equation}
\]</div>
<p>where <span class="math notranslate nohighlight">\(\beta_1\)</span> and <span class="math notranslate nohighlight">\(\beta_2\)</span> set the memory lifetime of the first and
second moment and are typically taken to be <span class="math notranslate nohighlight">\(0.9\)</span> and <span class="math notranslate nohighlight">\(0.99\)</span>
respectively, and <span class="math notranslate nohighlight">\(\eta\)</span> and <span class="math notranslate nohighlight">\(\epsilon\)</span> are identical to RMSprop.</p>
<p>Like in RMSprop, the effective step size of a parameter depends on the
magnitude of its gradient squared. To understand this better, let us
rewrite this expression in terms of the variance
<span class="math notranslate nohighlight">\(\boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t -
(\boldsymbol{\mathbf{m}}_t)^2\)</span>. Consider a single parameter <span class="math notranslate nohighlight">\(\theta_t\)</span>. The
update rule for this parameter is given by</p>
<div class="math notranslate nohighlight">
\[
\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
\]</div>
</div>
<div class="section" id="algorithms-and-codes-for-adagrad-rmsprop-and-adam">
<h2>Algorithms and codes for Adagrad, RMSprop and Adam<a class="headerlink" href="#algorithms-and-codes-for-adagrad-rmsprop-and-adam" title="Permalink to this headline"></a></h2>
<p>The algorithms we have implemented are well described in the text by <a class="reference external" href="https://www.deeplearningbook.org/contents/optimization.html">Goodfellow, Bengio and Courville, chapter 8</a>.</p>
<p>The codes which implement these algorithms are discussed after our presentation of automatic differentiation.</p>
</div>
<div class="section" id="adagrad-algorithm-taken-from-goodfellow-et-al">
<h2>AdaGrad algorithm, taken from <a class="reference external" href="https://www.deeplearningbook.org/contents/optimization.html">Goodfellow et al</a><a class="headerlink" href="#adagrad-algorithm-taken-from-goodfellow-et-al" title="Permalink to this headline"></a></h2>
<!-- dom:FIGURE: [figures/adagrad.png, width=600 frac=0.8] -->
<!-- begin figure -->
<p><img src="figures/adagrad.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
<!-- end figure --></div>
<div class="section" id="rmsprop-algorithm-taken-from-goodfellow-et-al">
<h2>RMSProp algorithm, taken from <a class="reference external" href="https://www.deeplearningbook.org/contents/optimization.html">Goodfellow et al</a><a class="headerlink" href="#rmsprop-algorithm-taken-from-goodfellow-et-al" title="Permalink to this headline"></a></h2>
<!-- dom:FIGURE: [figures/rmsprop.png, width=600 frac=0.8] -->
<!-- begin figure -->
<p><img src="figures/rmsprop.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
<!-- end figure --></div>
<div class="section" id="adam-algorithm-taken-from-goodfellow-et-al">
<h2>ADAM algorithm, taken from <a class="reference external" href="https://www.deeplearningbook.org/contents/optimization.html">Goodfellow et al</a><a class="headerlink" href="#adam-algorithm-taken-from-goodfellow-et-al" title="Permalink to this headline"></a></h2>
<!-- dom:FIGURE: [figures/adam.png, width=600 frac=0.8] -->
<!-- begin figure -->
<p><img src="figures/adam.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
<!-- end figure --></div>
<div class="section" id="practical-tips">
<h2>Practical tips<a class="headerlink" href="#practical-tips" title="Permalink to this headline"></a></h2>
<ul class="simple">
<li><p><strong>Randomize the data when making mini-batches</strong>. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.</p></li>
<li><p><strong>Transform your inputs</strong>. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.</p></li>
<li><p><strong>Monitor the out-of-sample performance.</strong> Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This <em>early stopping</em> significantly improves performance in many settings.</p></li>
<li><p><strong>Adaptive optimization methods don’t always have good generalization.</strong> Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.</p></li>
</ul>
<p>Geron’s text, see chapter 11, has several interesting discussions.</p>
</div>
<div class="section" id="automatic-differentiation">
<h2>Automatic differentiation<a class="headerlink" href="#automatic-differentiation" title="Permalink to this headline"></a></h2>
<p><a class="reference external" href="https://en.wikipedia.org/wiki/Automatic_differentiation">Automatic differentiation (AD)</a>,
also called algorithmic
differentiation or computational differentiation,is a set of
techniques to numerically evaluate the derivative of a function
specified by a computer program. AD exploits the fact that every
computer program, no matter how complicated, executes a sequence of
elementary arithmetic operations (addition, subtraction,
multiplication, division, etc.) and elementary functions (exp, log,
sin, cos, etc.). By applying the chain rule repeatedly to these
operations, derivatives of arbitrary order can be computed
automatically, accurately to working precision, and using at most a
small constant factor more arithmetic operations than the original
program.</p>
<p>Automatic differentiation is neither:</p>
<ul class="simple">
<li><p>Symbolic differentiation, nor</p></li>
<li><p>Numerical differentiation (the method of finite differences).</p></li>
</ul>
<p>Symbolic differentiation can lead to inefficient code and faces the
difficulty of converting a computer program into a single expression,
while numerical differentiation can introduce round-off errors in the
discretization process and cancellation</p>
<p>Python has tools for so-called <strong>automatic differentiation</strong>.
Consider the following example</p>
<div class="math notranslate nohighlight">
\[
f(x) = \sin\left(2\pi x + x^2\right)
\]</div>
<p>which has the following derivative</p>
<div class="math notranslate nohighlight">
\[
f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
\]</div>
<p>Using <strong>autograd</strong> we have</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">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># To do elementwise differentiation:</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">elementwise_grad</span> <span class="k">as</span> <span class="n">egrad</span>
<span class="c1"># To plot:</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">def</span> <span class="nf">f</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">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">f_grad_analytic</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">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
<span class="c1"># Do the comparison:</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</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">1000</span><span class="p">)</span>
<span class="n">f_grad</span> <span class="o">=</span> <span class="n">egrad</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
<span class="n">computed</span> <span class="o">=</span> <span class="n">f_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">analytic</span> <span class="o">=</span> <span class="n">f_grad_analytic</span><span class="p">(</span><span class="n">x</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="s1">&#39;Derivative computed from Autograd compared with the analytical derivative&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">computed</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;autograd&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">analytic</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">&#39;analytic&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</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="nb">print</span><span class="p">(</span><span class="s2">&quot;The max absolute difference is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">computed</span> <span class="o">-</span> <span class="n">analytic</span><span class="p">))))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/week40_80_0.png" src="_images/week40_80_0.png" />
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The max absolute difference is: 1.77636e-15
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="using-autograd">
<h2>Using autograd<a class="headerlink" href="#using-autograd" title="Permalink to this headline"></a></h2>
<p>Here we
experiment with what kind of functions Autograd is capable
of finding the gradient of. The following Python functions are just
meant to illustrate what Autograd can do, but please feel free to
experiment with other, possibly more complicated, functions as well.</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">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">f1_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f1</span><span class="p">)</span>
<span class="c1"># Remember to send in float as argument to the computed gradient from Autograd!</span>
<span class="n">a</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="c1"># See the evaluated gradient at a using autograd:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The gradient of f1 evaluated at a = </span><span class="si">%g</span><span class="s2"> using autograd is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">f1_grad</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
<span class="c1"># Compare with the analytical derivative, that is f1&#39;(x) = 3*x**2 </span>
<span class="n">grad_analytical</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The gradient of f1 evaluated at a = </span><span class="si">%g</span><span class="s2"> by finding the analytic expression is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">grad_analytical</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 gradient of f1 evaluated at a = 1 using autograd is: 3
The gradient of f1 evaluated at a = 1 by finding the analytic expression is: 3
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="autograd-with-more-complicated-functions">
<h2>Autograd with more complicated functions<a class="headerlink" href="#autograd-with-more-complicated-functions" title="Permalink to this headline"></a></h2>
<p>To differentiate with respect to two (or more) arguments of a Python
function, Autograd need to know at which variable the function if
being differentiated with respect to.</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">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f2</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">3</span><span class="o">*</span><span class="n">x1</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x2</span><span class="o">*</span><span class="p">(</span><span class="n">x1</span> <span class="o">-</span> <span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="c1"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
<span class="n">f2_grad_x1</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
<span class="n">f2_grad_x2</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">x1</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">x2</span> <span class="o">=</span> <span class="mf">3.0</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Evaluating at x1 = </span><span class="si">%g</span><span class="s2">, x2 = </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;-&quot;</span><span class="o">*</span><span class="mi">30</span><span class="p">)</span>
<span class="c1"># Compare with the analytical derivatives:</span>
<span class="c1"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
<span class="n">f2_grad_x1_analytical</span> <span class="o">=</span> <span class="mi">9</span><span class="o">*</span><span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x2</span>
<span class="c1"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
<span class="n">f2_grad_x2_analytical</span> <span class="o">=</span> <span class="n">x1</span> <span class="o">-</span> <span class="mi">5</span>
<span class="c1"># See the evaluated derivations:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The derivative of f2 w.r.t x1: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x1</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The analytical derivative of f2 w.r.t x1: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x1</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The derivative of f2 w.r.t x2: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x2</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The analytical derivative of f2 w.r.t x2: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x2</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</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>Evaluating at x1 = 1, x2 = 3
------------------------------
The derivative of f2 w.r.t x1: 12
The analytical derivative of f2 w.r.t x1: 12
The derivative of f2 w.r.t x2: -4
The analytical derivative of f2 w.r.t x2: -4
</pre></div>
</div>
</div>
</div>
<p>Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.</p>
</div>
<div class="section" id="more-complicated-functions-using-the-elements-of-their-arguments-directly">
<h2>More complicated functions using the elements of their arguments directly<a class="headerlink" href="#more-complicated-functions-using-the-elements-of-their-arguments-directly" 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="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f3</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> <span class="c1"># Assumes x is an array of length 5 or higher</span>
<span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">+</span> <span class="mi">11</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span>
<span class="n">f3_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f3</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="c1"># Print the computed gradient:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The computed gradient of f3 is: &quot;</span><span class="p">,</span> <span class="n">f3_grad</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="c1"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
<span class="n">f3_grad_analytical</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="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">22</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]])</span>
<span class="c1"># Print the analytical gradient:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The analytical gradient of f3 is: &quot;</span><span class="p">,</span> <span class="n">f3_grad_analytical</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The computed gradient of f3 is: [ 2. 3. 5. 7. 88.]
The analytical gradient of f3 is: [ 2. 3. 5. 7. 88.]
</pre></div>
</div>
</div>
</div>
<p>Note that in this case, when sending an array as input argument, the
output from Autograd is another array. This is the true gradient of
the function, as opposed to the function in the previous example. By
using arrays to represent the variables, the output from Autograd
might be easier to work with, as the output is closer to what one
could expect form a gradient-evaluting function.</p>
</div>
<div class="section" id="functions-using-mathematical-functions-from-numpy">
<h2>Functions using mathematical functions from Numpy<a class="headerlink" href="#functions-using-mathematical-functions-from-numpy" 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="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f4</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">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
<span class="n">f4_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f4</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="mf">2.7</span>
<span class="c1"># Print the computed derivative:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The computed derivative of f4 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f4_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
<span class="c1"># The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi</span>
<span class="n">f4_grad_analytical</span> <span class="o">=</span> <span class="n">x</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span>
<span class="c1"># Print the analytical gradient:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The analytical gradient of f4 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f4_grad_analytical</span><span class="p">))</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The computed derivative of f4 at x = 2.7 is: 13.8759
The analytical gradient of f4 at x = 2.7 is: 13.8759
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="more-autograd">
<h2>More autograd<a class="headerlink" href="#more-autograd" 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="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f5</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">f5_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f5</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="mf">2.7</span>
<span class="c1"># Print the computed derivative:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The computed derivative of f5 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f5_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The computed derivative of f5 at x = 2.7 is: 5.4
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="and-with-loops">
<h2>And with loops<a class="headerlink" href="#and-with-loops" 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="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f6_for</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">val</span> <span class="o">=</span> <span class="mi">0</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">10</span><span class="p">):</span>
<span class="n">val</span> <span class="o">=</span> <span class="n">val</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span>
<span class="k">return</span> <span class="n">val</span>
<span class="k">def</span> <span class="nf">f6_while</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="n">val</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">10</span><span class="p">:</span>
<span class="n">val</span> <span class="o">=</span> <span class="n">val</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span>
<span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">val</span>
<span class="n">f6_for_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f6_for</span><span class="p">)</span>
<span class="n">f6_while_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f6_while</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="mf">0.5</span>
<span class="c1"># Print the computed derivaties of f6_for and f6_while</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The computed derivative of f6_for at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f6_for_grad</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="s2">&quot;The computed derivative of f6_while at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f6_while_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The computed derivative of f6_for at x = 0.5 is: 3.95703
The computed derivative of f6_while at x = 0.5 is: 3.95703
</pre></div>
</div>
</div>
</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="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="c1"># Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9</span>
<span class="c1"># The analytical derivative is: sum(i*x**(i-1)) </span>
<span class="n">f6_grad_analytical</span> <span class="o">=</span> <span class="mi">0</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">10</span><span class="p">):</span>
<span class="n">f6_grad_analytical</span> <span class="o">+=</span> <span class="n">i</span><span class="o">*</span><span class="n">x</span><span class="o">**</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="nb">print</span><span class="p">(</span><span class="s2">&quot;The analytical derivative of f6 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f6_grad_analytical</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 analytical derivative of f6 at x = 0.5 is: 3.95703
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="using-recursion">
<h2>Using recursion<a class="headerlink" href="#using-recursion" 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="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">f7</span><span class="p">(</span><span class="n">n</span><span class="p">):</span> <span class="c1"># Assume that n is an integer</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">n</span><span class="o">*</span><span class="n">f7</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">f7_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f7</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mf">2.0</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The computed derivative of f7 at n = </span><span class="si">%d</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">f7_grad</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
<span class="c1"># The function f7 is an implementation of the factorial of n.</span>
<span class="c1"># By using the product rule, one can find that the derivative is:</span>
<span class="n">f7_grad_analytical</span> <span class="o">=</span> <span class="mi">0</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">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<span class="n">tmp</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<span class="k">if</span> <span class="n">k</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
<span class="n">tmp</span> <span class="o">*=</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
<span class="n">f7_grad_analytical</span> <span class="o">+=</span> <span class="n">tmp</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The analytical derivative of f7 at n = </span><span class="si">%d</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">&quot;</span><span class="o">%</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">f7_grad_analytical</span><span class="p">))</span>
</pre></div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The computed derivative of f7 at n = 2 is: 1
The analytical derivative of f7 at n = 2 is: 1
</pre></div>
</div>
</div>
</div>
<p>Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.</p>
</div>
<div class="section" id="using-autograd-with-ols">
<h2>Using Autograd with OLS<a class="headerlink" href="#using-autograd-with-ols" title="Permalink to this headline"></a></h2>
<p>We conclude the part on optmization by showing how we can make codes
for linear regression and logistic regression using <strong>autograd</strong>. The
first example shows results with ordinary leats squares.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Using Autograd to calculate gradients for OLS</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">beta</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</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">y</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</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</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Hessian matrix</span>
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">XT_X</span>
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</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">eig</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">1.0</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">EigValues</span><span class="p">)</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="c1"># define the gradient</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="n">training_gradient</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own gd&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">xnew</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="mi">0</span><span class="p">],[</span><span class="mi">2</span><span class="p">]])</span>
<span class="n">Xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">xnew</span><span class="p">]</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">ypredict2</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">&quot;r-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict2</span><span class="p">,</span> <span class="s2">&quot;b-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</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="s1">&#39;ro&#39;</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="mi">0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$y$&#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="sa">r</span><span class="s1">&#39;Random numbers &#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>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
[[4.13791264]
[2.92552059]]
Eigenvalues of Hessian Matrix:[0.27874136 4.16226023]
theta from own gd
[[4.13791264]
[2.92552059]]
</pre></div>
</div>
<img alt="_images/week40_100_1.png" src="_images/week40_100_1.png" />
</div>
</div>
</div>
<div class="section" id="same-code-but-now-with-momentum-gradient-descent">
<h2>Same code but now with momentum gradient descent<a class="headerlink" href="#same-code-but-now-with-momentum-gradient-descent" title="Permalink to this headline"></a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Using Autograd to calculate gradients for OLS</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">beta</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</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">y</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="c1">#+np.random.randn(n,1)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Hessian matrix</span>
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">XT_X</span>
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</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">eig</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">1.0</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">EigValues</span><span class="p">)</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">30</span>
<span class="c1"># define the gradient</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="n">training_gradient</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span><span class="n">gradients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">gradients</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own gd&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="c1"># Now improve with momentum gradient descent</span>
<span class="n">change</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">delta_momentum</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="c1"># calculate gradient</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="n">training_gradient</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="c1"># calculate update</span>
<span class="n">new_change</span> <span class="o">=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span><span class="o">+</span><span class="n">delta_momentum</span><span class="o">*</span><span class="n">change</span>
<span class="c1"># take a step</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">new_change</span>
<span class="c1"># save the change</span>
<span class="n">change</span> <span class="o">=</span> <span class="n">new_change</span>
<span class="nb">print</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span><span class="n">gradients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">gradients</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own gd wth momentum&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
[[4.]
[3.]]
Eigenvalues of Hessian Matrix:[0.32606365 3.80499859]
0 [-10.98955596] [-10.8332972]
1 [-0.26421616] [0.25444295]
2 [-0.24157456] [0.23263884]
3 [-0.22087319] [0.2127032]
4 [-0.20194579] [0.19447592]
5 [-0.18464035] [0.1778106]
6 [-0.16881787] [0.16257339]
7 [-0.15435127] [0.1486419]
8 [-0.14112437] [0.13590426]
9 [-0.12903093] [0.12425815]
10 [-0.11797382] [0.11361003]
11 [-0.10786423] [0.10387439]
12 [-0.09862097] [0.09497303]
13 [-0.09016979] [0.08683446]
14 [-0.08244283] [0.07939331]
15 [-0.07537801] [0.07258982]
16 [-0.06891861] [0.06636934]
17 [-0.06301273] [0.06068192]
18 [-0.05761295] [0.05548187]
19 [-0.05267589] [0.05072744]
20 [-0.04816191] [0.04638043]
21 [-0.04403475] [0.04240593]
22 [-0.04026126] [0.03877201]
23 [-0.03681113] [0.0354495]
24 [-0.03365665] [0.03241171]
25 [-0.0307725] [0.02963424]
26 [-0.02813549] [0.02709478]
27 [-0.02572446] [0.02477293]
28 [-0.02352005] [0.02265005]
29 [-0.02150453] [0.02070909]
theta from own gd
[[3.93969971]
[3.05806981]]
0 [-0.01966173] [0.01893445]
1 [-0.01797685] [0.0173119]
2 [-0.01593089] [0.01534161]
3 [-0.01395192] [0.01343585]
4 [-0.01216264] [0.01171276]
5 [-0.0105836] [0.01019212]
6 [-0.00920294] [0.00886253]
7 [-0.00800011] [0.00770419]
8 [-0.00695371] [0.00669649]
9 [-0.0060439] [0.00582034]
10 [-0.00525303] [0.00505872]
11 [-0.00456562] [0.00439674]
12 [-0.00396815] [0.00382137]
13 [-0.00344887] [0.0033213]
14 [-0.00299754] [0.00288666]
15 [-0.00260527] [0.0025089]
16 [-0.00226433] [0.00218058]
17 [-0.00196801] [0.00189522]
18 [-0.00171047] [0.0016472]
19 [-0.00148663] [0.00143164]
20 [-0.00129209] [0.00124429]
21 [-0.001123] [0.00108146]
22 [-0.00097604] [0.00093994]
23 [-0.00084831] [0.00081693]
24 [-0.0007373] [0.00071003]
25 [-0.00064081] [0.00061711]
26 [-0.00055695] [0.00053635]
27 [-0.00048407] [0.00046616]
28 [-0.00042072] [0.00040516]
29 [-0.00036566] [0.00035214]
theta from own gd wth momentum
[[3.99902531]
[3.00093864]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="including-stochastic-gradient-descent-with-autograd">
<h2>Including Stochastic Gradient Descent with Autograd<a class="headerlink" href="#including-stochastic-gradient-descent-with-autograd" title="Permalink to this headline"></a></h2>
<p>In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using <strong>autograd</strong>.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Using Autograd to calculate gradients using SGD</span>
<span class="c1"># OLS example</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="c1"># Note change from previous example</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</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</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</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</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Hessian matrix</span>
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">XT_X</span>
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</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">eig</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">1.0</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">EigValues</span><span class="p">)</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own gd&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">xnew</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="mi">0</span><span class="p">],[</span><span class="mi">2</span><span class="p">]])</span>
<span class="n">Xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">xnew</span><span class="p">]</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">ypredict2</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">&quot;r-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict2</span><span class="p">,</span> <span class="s2">&quot;b-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</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="s1">&#39;ro&#39;</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="mi">0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$y$&#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="sa">r</span><span class="s1">&#39;Random numbers &#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="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="n">t0</span><span class="p">,</span> <span class="n">t1</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">50</span>
<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="k">return</span> <span class="n">t0</span><span class="o">/</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">t1</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">):</span>
<span class="c1"># Can you figure out a better way of setting up the contributions to each batch?</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="n">m</span><span class="p">):</span>
<span class="n">random_index</span> <span class="o">=</span> <span class="n">M</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="n">learning_schedule</span><span class="p">(</span><span class="n">epoch</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">i</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span> <span class="o">-</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own sdg&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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>Own inversion
[[4.11762444]
[3.04098313]]
Eigenvalues of Hessian Matrix:[0.29738252 4.51279273]
theta from own gd
[[4.11762444]
[3.04098313]]
</pre></div>
</div>
<img alt="_images/week40_104_1.png" src="_images/week40_104_1.png" />
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>theta from own sdg
[[4.07058967]
[3.024004 ]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id1">
<h2>Same code but now with momentum gradient descent<a class="headerlink" href="#id1" title="Permalink to this headline"></a></h2>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Using Autograd to calculate gradients using SGD</span>
<span class="c1"># OLS example</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="c1"># Note change from previous example</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</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</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</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</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Hessian matrix</span>
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">XT_X</span>
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</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">eig</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">1.0</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">EigValues</span><span class="p">)</span>
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">100</span>
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own gd&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="n">t0</span><span class="p">,</span> <span class="n">t1</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">50</span>
<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
<span class="k">return</span> <span class="n">t0</span><span class="o">/</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">t1</span><span class="p">)</span>
<span class="n">theta</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="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">change</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">delta_momentum</span> <span class="o">=</span> <span class="mf">0.3</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</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="n">m</span><span class="p">):</span>
<span class="n">random_index</span> <span class="o">=</span> <span class="n">M</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">eta</span> <span class="o">=</span> <span class="n">learning_schedule</span><span class="p">(</span><span class="n">epoch</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">i</span><span class="p">)</span>
<span class="c1"># calculate update</span>
<span class="n">new_change</span> <span class="o">=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span><span class="o">+</span><span class="n">delta_momentum</span><span class="o">*</span><span class="n">change</span>
<span class="c1"># take a step</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">new_change</span>
<span class="c1"># save the change</span>
<span class="n">change</span> <span class="o">=</span> <span class="n">new_change</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own sdg with momentum&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
[[4.4252885 ]
[2.70944365]]
Eigenvalues of Hessian Matrix:[0.28973035 4.32089655]
theta from own gd
[[4.42394588]
[2.71059619]]
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>theta from own sdg with momentum
[[4.44845593]
[2.72577807]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="similar-second-order-function-now-problem-but-now-with-adagrad">
<h2>Similar (second order function now) problem but now with AdaGrad<a class="headerlink" href="#similar-second-order-function-now-problem-but-now-with-adagrad" 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"># Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent</span>
<span class="c1"># OLS example</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="c1"># Note change from previous example</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</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</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">x</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># Define parameters for Stochastic Gradient Descent</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="c1"># Guess for unknown parameters theta</span>
<span class="n">theta</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># Value for learning rate</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="c1"># Including AdaGrad parameter to avoid possible division by zero</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">):</span>
<span class="n">Giter</span> <span class="o">=</span> <span class="mf">0.0</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="n">m</span><span class="p">):</span>
<span class="n">random_index</span> <span class="o">=</span> <span class="n">M</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">Giter</span> <span class="o">+=</span> <span class="n">gradients</span><span class="o">*</span><span class="n">gradients</span>
<span class="n">update</span> <span class="o">=</span> <span class="n">gradients</span><span class="o">*</span><span class="n">eta</span><span class="o">/</span><span class="p">(</span><span class="n">delta</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">Giter</span><span class="p">))</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">update</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own AdaGrad&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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>Own inversion
[[2.]
[3.]
[4.]]
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>theta from own AdaGrad
[[2.00036797]
[2.99817613]
[4.00177485]]
</pre></div>
</div>
</div>
</div>
<p>Running this code we note an almost perfect agreement with the results from matrix inversion.</p>
</div>
<div class="section" id="rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent">
<h2>RMSprop for adaptive learning rate with Stochastic Gradient Descent<a class="headerlink" href="#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" 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"># Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent</span>
<span class="c1"># OLS example</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="c1"># Note change from previous example</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</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</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">x</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="c1"># +np.random.randn(n,1)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># Define parameters for Stochastic Gradient Descent</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="c1"># Guess for unknown parameters theta</span>
<span class="n">theta</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># Value for learning rate</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="c1"># Value for parameter rho</span>
<span class="n">rho</span> <span class="o">=</span> <span class="mf">0.99</span>
<span class="c1"># Including AdaGrad parameter to avoid possible division by zero</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">):</span>
<span class="n">Giter</span> <span class="o">=</span> <span class="mf">0.0</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="n">m</span><span class="p">):</span>
<span class="n">random_index</span> <span class="o">=</span> <span class="n">M</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="c1"># Accumulated gradient</span>
<span class="c1"># Scaling with rho the new and the previous results</span>
<span class="n">Giter</span> <span class="o">=</span> <span class="p">(</span><span class="n">rho</span><span class="o">*</span><span class="n">Giter</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">rho</span><span class="p">)</span><span class="o">*</span><span class="n">gradients</span><span class="o">*</span><span class="n">gradients</span><span class="p">)</span>
<span class="c1"># Taking the diagonal only and inverting</span>
<span class="n">update</span> <span class="o">=</span> <span class="n">gradients</span><span class="o">*</span><span class="n">eta</span><span class="o">/</span><span class="p">(</span><span class="n">delta</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">Giter</span><span class="p">))</span>
<span class="c1"># Hadamard product</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">update</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own RMSprop&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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>Own inversion
[[2.]
[3.]
[4.]]
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>theta from own RMSprop
[[2.00119865]
[3.01346635]
[3.99284588]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="and-finally-adam">
<h2>And finally <a class="reference external" href="https://arxiv.org/pdf/1412.6980.pdf">ADAM</a><a class="headerlink" href="#and-finally-adam" title="Permalink to this headline"></a></h2>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent</span>
<span class="c1"># OLS example</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="c1"># Note change from previous example</span>
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</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</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">x</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">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="c1"># +np.random.randn(n,1)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">]</span>
<span class="n">XT_X</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">X</span>
<span class="n">theta_linreg</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">XT_X</span><span class="p">)</span> <span class="o">@</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">y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Own inversion&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># Define parameters for Stochastic Gradient Descent</span>
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
<span class="c1"># Guess for unknown parameters theta</span>
<span class="n">theta</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="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># Value for learning rate</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="c1"># Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980</span>
<span class="n">beta1</span> <span class="o">=</span> <span class="mf">0.9</span>
<span class="n">beta2</span> <span class="o">=</span> <span class="mf">0.999</span>
<span class="c1"># Including AdaGrad parameter to avoid possible division by zero</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-7</span>
<span class="nb">iter</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">):</span>
<span class="n">first_moment</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">second_moment</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="nb">iter</span> <span class="o">+=</span> <span class="mi">1</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="n">m</span><span class="p">):</span>
<span class="n">random_index</span> <span class="o">=</span> <span class="n">M</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">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="n">M</span><span class="p">]</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="c1"># Computing moments first</span>
<span class="n">first_moment</span> <span class="o">=</span> <span class="n">beta1</span><span class="o">*</span><span class="n">first_moment</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">beta1</span><span class="p">)</span><span class="o">*</span><span class="n">gradients</span>
<span class="n">second_moment</span> <span class="o">=</span> <span class="n">beta2</span><span class="o">*</span><span class="n">second_moment</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">beta2</span><span class="p">)</span><span class="o">*</span><span class="n">gradients</span><span class="o">*</span><span class="n">gradients</span>
<span class="n">first_term</span> <span class="o">=</span> <span class="n">first_moment</span><span class="o">/</span><span class="p">(</span><span class="mf">1.0</span><span class="o">-</span><span class="n">beta1</span><span class="o">**</span><span class="nb">iter</span><span class="p">)</span>
<span class="n">second_term</span> <span class="o">=</span> <span class="n">second_moment</span><span class="o">/</span><span class="p">(</span><span class="mf">1.0</span><span class="o">-</span><span class="n">beta2</span><span class="o">**</span><span class="nb">iter</span><span class="p">)</span>
<span class="c1"># Scaling with rho the new and the previous results</span>
<span class="n">update</span> <span class="o">=</span> <span class="n">eta</span><span class="o">*</span><span class="n">first_term</span><span class="o">/</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">second_term</span><span class="p">)</span><span class="o">+</span><span class="n">delta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">-=</span> <span class="n">update</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;theta from own ADAM&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
[[2.]
[3.]
[4.]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>theta from own ADAM
[[1.99997244]
[3.00018876]
[3.99983332]]
</pre></div>
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</div>
</div>
<div class="section" id="and-logistic-regression">
<h2>And Logistic Regression<a class="headerlink" href="#and-logistic-regression" title="Permalink to this headline"></a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</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="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">x</span> <span class="o">/</span> <span class="mf">2.</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">logistic_predictions</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span> <span class="n">inputs</span><span class="p">):</span>
<span class="c1"># Outputs probability of a label being true according to logistic model.</span>
<span class="k">return</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">weights</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">training_loss</span><span class="p">(</span><span class="n">weights</span><span class="p">):</span>
<span class="c1"># Training loss is the negative log-likelihood of the training labels.</span>
<span class="n">preds</span> <span class="o">=</span> <span class="n">logistic_predictions</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span> <span class="n">inputs</span><span class="p">)</span>
<span class="n">label_probabilities</span> <span class="o">=</span> <span class="n">preds</span> <span class="o">*</span> <span class="n">targets</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">preds</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">targets</span><span class="p">)</span>
<span class="k">return</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">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">label_probabilities</span><span class="p">))</span>
<span class="c1"># Build a toy dataset.</span>
<span class="n">inputs</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="mf">0.52</span><span class="p">,</span> <span class="mf">1.12</span><span class="p">,</span> <span class="mf">0.77</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.88</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.08</span><span class="p">,</span> <span class="mf">0.15</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.52</span><span class="p">,</span> <span class="mf">0.06</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.30</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.74</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.49</span><span class="p">,</span> <span class="mf">1.39</span><span class="p">]])</span>
<span class="n">targets</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="kc">True</span><span class="p">,</span> <span class="kc">True</span><span class="p">,</span> <span class="kc">False</span><span class="p">,</span> <span class="kc">True</span><span class="p">])</span>
<span class="c1"># Define a function that returns gradients of training loss using Autograd.</span>
<span class="n">training_gradient_fun</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">training_loss</span><span class="p">)</span>
<span class="c1"># Optimize weights using gradient descent.</span>
<span class="n">weights</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="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Initial loss:&quot;</span><span class="p">,</span> <span class="n">training_loss</span><span class="p">(</span><span class="n">weights</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="mi">100</span><span class="p">):</span>
<span class="n">weights</span> <span class="o">-=</span> <span class="n">training_gradient_fun</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.01</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Trained loss:&quot;</span><span class="p">,</span> <span class="n">training_loss</span><span class="p">(</span><span class="n">weights</span><span class="p">))</span>
</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Initial loss: 2.772588722239781
Trained loss: 1.067270675787016
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</div>
<div class="section" id="introducing-jax">
<h2>Introducing <a class="reference external" href="https://jax.readthedocs.io/en/latest/">JAX</a><a class="headerlink" href="#introducing-jax" title="Permalink to this headline"></a></h2>
<p>Presently, instead of using <strong>autograd</strong>, we recommend using <a class="reference external" href="https://jax.readthedocs.io/en/latest/">JAX</a></p>
<p><strong>JAX</strong> is Autograd and <a class="reference external" href="https://www.tensorflow.org/xla">XLA (Accelerated Linear Algebra))</a>,
brought together for high-performance numerical computing and machine learning research.
It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.</p>
<div class="section" id="getting-started-with-jax-note-the-way-we-import-numpy">
<h3>Getting started with Jax, note the way we import numpy<a class="headerlink" href="#getting-started-with-jax-note-the-way-we-import-numpy" title="Permalink to this headline"></a></h3>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">jax</span>
<span class="kn">import</span> <span class="nn">jax.numpy</span> <span class="k">as</span> <span class="nn">jnp</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">jax</span> <span class="kn">import</span> <span class="n">grad</span> <span class="k">as</span> <span class="n">jax_grad</span>
</pre></div>
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</div>
<div class="section" id="a-warm-up-example">
<h3>A warm-up example<a class="headerlink" href="#a-warm-up-example" title="Permalink to this headline"></a></h3>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
<span class="k">def</span> <span class="nf">analytical_gradient</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span>
<span class="k">def</span> <span class="nf">gradient_descent</span><span class="p">(</span><span class="n">starting_point</span><span class="p">,</span> <span class="n">learning_rate</span><span class="p">,</span> <span class="n">num_iterations</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s2">&quot;analytical&quot;</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">starting_point</span>
<span class="n">trajectory_x</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span>
<span class="n">trajectory_y</span> <span class="o">=</span> <span class="p">[</span><span class="n">function</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span>
<span class="k">if</span> <span class="n">solver</span> <span class="o">==</span> <span class="s2">&quot;analytical&quot;</span><span class="p">:</span>
<span class="n">grad</span> <span class="o">=</span> <span class="n">analytical_gradient</span>
<span class="k">elif</span> <span class="n">solver</span> <span class="o">==</span> <span class="s2">&quot;jax&quot;</span><span class="p">:</span>
<span class="n">grad</span> <span class="o">=</span> <span class="n">jax_grad</span><span class="p">(</span><span class="n">function</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">float64</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">learning_rate</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">float64</span><span class="p">(</span><span class="n">learning_rate</span><span class="p">)</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iterations</span><span class="p">):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">learning_rate</span> <span class="o">*</span> <span class="n">grad</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">trajectory_x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">trajectory_y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">function</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="k">return</span> <span class="n">trajectory_x</span><span class="p">,</span> <span class="n">trajectory_y</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</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="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">function</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;f(x)&quot;</span><span class="p">)</span>
<span class="n">descent_x</span><span class="p">,</span> <span class="n">descent_y</span> <span class="o">=</span> <span class="n">gradient_descent</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s2">&quot;analytical&quot;</span><span class="p">)</span>
<span class="n">jax_descend_x</span><span class="p">,</span> <span class="n">jax_descend_y</span> <span class="o">=</span> <span class="n">gradient_descent</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s2">&quot;jax&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">descent_x</span><span class="p">,</span> <span class="n">descent_y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Gradient descent&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">jax_descend_x</span><span class="p">,</span> <span class="n">jax_descend_y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;JAX&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/jax/_src/numpy/lax_numpy.py:173: UserWarning: Explicitly requested dtype float64 requested in asarray is not available, and will be truncated to dtype float32. To enable more dtypes, set the jax_enable_x64 configuration option or the JAX_ENABLE_X64 shell environment variable. See https://github.com/google/jax#current-gotchas for more.
return asarray(x, dtype=self.dtype)
</pre></div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[&lt;matplotlib.lines.Line2D at 0x10cc52cd0&gt;]
</pre></div>
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<img alt="_images/week40_120_2.png" src="_images/week40_120_2.png" />
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<div class="section" id="a-more-advanced-example">
<h3>A more advanced example<a class="headerlink" href="#a-more-advanced-example" title="Permalink to this headline"></a></h3>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">backend</span> <span class="o">=</span> <span class="n">np</span>
<span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="n">backend</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">analytical_gradient</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">backend</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">backend</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</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="mi">100</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">function</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;f(x)&quot;</span><span class="p">)</span>
<span class="n">descent_x</span><span class="p">,</span> <span class="n">descent_y</span> <span class="o">=</span> <span class="n">gradient_descent</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mi">300</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s2">&quot;analytical&quot;</span><span class="p">)</span>
<span class="c1"># Change the backend to JAX</span>
<span class="n">backend</span> <span class="o">=</span> <span class="n">jnp</span>
<span class="n">jax_descend_x</span><span class="p">,</span> <span class="n">jax_descend_y</span> <span class="o">=</span> <span class="n">gradient_descent</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mi">300</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s2">&quot;jax&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">descent_x</span><span class="p">,</span> <span class="n">descent_y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Gradient descent&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;v&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">jax_descend_x</span><span class="p">,</span> <span class="n">jax_descend_y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;JAX&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.collections.PathCollection at 0x10fcfaeb0&gt;
</pre></div>
</div>
<img alt="_images/week40_122_1.png" src="_images/week40_122_1.png" />
</div>
</div>
</div>
</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">
\[
\begin{equation}
y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
\label{artificialNeuron} \tag{6}
\end{equation}
\]</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">
<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 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">
<div class="cell_input docutils container">
<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="id2">
<h2>Mathematical model<a class="headerlink" href="#id2" 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="_auto6"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
\label{_auto6} \tag{7}
\end{equation}
\]</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">
\[
\begin{equation}
y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1} \tag{8}
\end{equation}
\]</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">
\[
\begin{equation}
y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer} \tag{9}
\end{equation}
\]</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="id3">
<h2>Mathematical model<a class="headerlink" href="#id3" 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="_auto7"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^2 = f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto7} \tag{10}
\end{equation}
\]</div>
<!-- Equation labels as ordinary links -->
<div id="outputLayer2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2} \tag{11}
\end{equation}
\]</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="_auto8"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^3 = f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
\label{_auto8} \tag{12}
\end{equation}
\]</div>
<!-- Equation labels as ordinary links -->
<div id="_auto9"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto9} \tag{13}
\end{equation}
\]</div>
</div>
<div class="section" id="id4">
<h2>Mathematical model<a class="headerlink" href="#id4" 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">
\[
\begin{equation}
y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right]
\label{completeNN} \tag{14}
\end{equation}
\]</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="id5">
<h2>Mathematical model<a class="headerlink" href="#id5" 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="_auto10"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
f(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto10} \tag{15}
\end{equation}
\]</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="_auto11"></div>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{equation}
\hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &amp;w^2_{12} &amp;w^2_{13} \\
w^2_{21} &amp;w^2_{22} &amp;w^2_{23} \\
w^2_{31} &amp;w^2_{32} &amp;w^2_{33} \\
\end{array} \right] \cdot
\left[\begin{array}{c}
y^1_1 \\
y^1_2 \\
y^1_3 \\
\end{array}\right] +
\left[\begin{array}{c}
b^2_1 \\
b^2_2 \\
b^2_3 \\
\end{array}\right]\right).
\label{_auto11} \tag{16}
\end{equation}
\end{split}\]</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="_auto12"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
\label{_auto12} \tag{17}
\end{equation}
\]</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>
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