lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
06b09eb681a311aec756811f6f464148168d0bcb
FYS-STK4155/doc/LectureNotes/_build/html/chapter10.html
T
Morten Hjorth-Jensen 06b09eb681 update
2023-11-21 06:16:54 +01:00

2509 lines
219 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>14. Building a Feed Forward Neural Network &#8212; Applied Data Analysis and Machine Learning</title>
<link href="_static/css/theme.css" rel="stylesheet">
<link href="_static/css/index.ff1ffe594081f20da1ef19478df9384b.css" rel="stylesheet">
<link rel="stylesheet"
href="_static/vendor/fontawesome/5.13.0/css/all.min.css">
<link rel="preload" as="font" type="font/woff2" crossorigin
href="_static/vendor/fontawesome/5.13.0/webfonts/fa-solid-900.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin
href="_static/vendor/fontawesome/5.13.0/webfonts/fa-brands-400.woff2">
<link rel="stylesheet" type="text/css" href="_static/pygments.css" />
<link rel="stylesheet" type="text/css" href="_static/sphinx-book-theme.css?digest=c3fdc42140077d1ad13ad2f1588a4309" />
<link rel="stylesheet" type="text/css" href="_static/togglebutton.css" />
<link rel="stylesheet" type="text/css" href="_static/copybutton.css" />
<link rel="stylesheet" type="text/css" href="_static/mystnb.css" />
<link rel="stylesheet" type="text/css" href="_static/sphinx-thebe.css" />
<link rel="stylesheet" type="text/css" href="_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css" />
<link rel="stylesheet" type="text/css" href="_static/panels-variables.06eb56fa6e07937060861dad626602ad.css" />
<link rel="preload" as="script" href="_static/js/index.be7d3bbb2ef33a8344ce.js">
<script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
<script src="_static/jquery.js"></script>
<script src="_static/underscore.js"></script>
<script src="_static/doctools.js"></script>
<script src="_static/clipboard.min.js"></script>
<script src="_static/copybutton.js"></script>
<script>let toggleHintShow = 'Click to show';</script>
<script>let toggleHintHide = 'Click to hide';</script>
<script>let toggleOpenOnPrint = 'true';</script>
<script src="_static/togglebutton.js"></script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown, .tag_hide_input div.cell_input, .tag_hide-input div.cell_input, .tag_hide_output div.cell_output, .tag_hide-output div.cell_output, .tag_hide_cell.cell, .tag_hide-cell.cell';</script>
<script src="_static/sphinx-book-theme.d59cb220de22ca1c485ebbdc042f0030.js"></script>
<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"
const thebe_selector = ".thebe,.cell"
const thebe_selector_input = "pre"
const thebe_selector_output = ".output, .cell_output"
</script>
<script async="async" src="_static/sphinx-thebe.js"></script>
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
<script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
<link rel="next" title="15. Solving Differential Equations with Deep Learning" href="chapter11.html" />
<link rel="prev" title="13. Neural networks" href="chapter9.html" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="docsearch:language" content="None">
<!-- Google Analytics -->
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="80">
<div class="container-fluid" id="banner"></div>
<div class="container-xl">
<div class="row">
<div class="col-12 col-md-3 bd-sidebar site-navigation show" id="site-navigation">
<div class="navbar-brand-box">
<a class="navbar-brand text-wrap" href="index.html">
<!-- `logo` is deprecated in Sphinx 4.0, so remove this when we stop supporting 3 -->
<img src="_static/logo.png" class="logo" alt="logo">
<h1 class="site-logo" id="site-title">Applied Data Analysis and Machine Learning</h1>
</a>
</div><form class="bd-search d-flex align-items-center" action="search.html" method="get">
<i class="icon fas fa-search"></i>
<input type="search" class="form-control" name="q" id="search-input" placeholder="Search this book..." aria-label="Search this book..." autocomplete="off" >
</form><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
<div class="bd-toc-item active">
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="intro.html">
Applied Data Analysis and Machine Learning
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
About the course
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="schedule.html">
Teaching schedule with links to material
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="teachers.html">
Teachers and Grading
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="textbooks.html">
Textbooks
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Review of Statistics with Resampling Techniques and Linear Algebra
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="statistics.html">
1. Elements of Probability Theory and Statistical Data Analysis
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="linalg.html">
2. Linear Algebra, Handling of Arrays and more Python Features
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
From Regression to Support Vector Machines
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter1.html">
3. Linear Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter2.html">
4. Ridge and Lasso Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter3.html">
5. Resampling Methods
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter4.html">
6. Logistic Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapteroptimization.html">
7. Optimization, the central part of any Machine Learning algortithm
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter5.html">
8. Support Vector Machines, overarching aims
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Decision Trees, Ensemble Methods and Boosting
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter6.html">
9. Decision trees, overarching aims
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter7.html">
10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Dimensionality Reduction
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter8.html">
11. Basic ideas of the Principal Component Analysis (PCA)
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="clustering.html">
12. Clustering and Unsupervised Learning
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Deep Learning Methods
</span>
</p>
<ul class="current nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter9.html">
13. Neural networks
</a>
</li>
<li class="toctree-l1 current active">
<a class="current reference internal" href="#">
14. Building a Feed Forward Neural Network
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter11.html">
15. Solving Differential Equations with Deep Learning
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter12.html">
16. Convolutional Neural Networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter13.html">
17. Recurrent neural networks: Overarching view
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Weekly material, notes and exercises
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek34.html">
Exercises week 34
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week34.html">
Week 34: Introduction to the course, Logistics and Practicalities
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek35.html">
Exercises week 35
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week35.html">
Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek36.html">
Exercises week 36
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week36.html">
Week 36: Statistical interpretation of Linear Regression and Resampling techniques
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek37.html">
Exercises week 37
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week37.html">
Week 37: Statistical interpretations and Resampling Methods
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek38.html">
Exercises week 38
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week38.html">
Week 38: Logistic Regression and Optimization
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek39.html">
Exercises week 39
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week39.html">
Week 39: Optimization and Gradient Methods
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week40.html">
Week 40: Gradient descent methods (continued) and start Neural networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek41.html">
Exercises week 41
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week41.html">
Week 41 Neural networks and constructing a neural network code
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek42.html">
Exercises week 42
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week42.html">
Week 42 Constructing a Neural Network code with introduction to Tensor flow
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="exercisesweek43.html">
Exercises weeks 43 and 44
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week43.html">
Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week44.html">
Week 44, Convolutional Neural Networks (CNN)
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week45.html">
Week 45, Recurrent Neural Networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week46.html">
Week 46: Decision Trees, Ensemble methods and Random Forests
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="week47.html">
Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course
</a>
</li>
</ul>
<p aria-level="2" class="caption" role="heading">
<span class="caption-text">
Projects
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="project1.html">
Project 1 on Machine Learning, deadline October 9 (midnight), 2023
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="project2.html">
Project 2 on Machine Learning, deadline November 17 (Midnight)
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="project3.html">
Project 3 on Machine Learning, deadline December 18 (midnight), 2023
</a>
</li>
</ul>
</div>
</nav> <!-- To handle the deprecated key -->
<div class="navbar_extra_footer">
Powered by <a href="https://jupyterbook.org">Jupyter Book</a>
</div>
</div>
<main class="col py-md-3 pl-md-4 bd-content overflow-auto" role="main">
<div class="topbar container-xl fixed-top">
<div class="topbar-contents row">
<div class="col-12 col-md-3 bd-topbar-whitespace site-navigation show"></div>
<div class="col pl-md-4 topbar-main">
<button id="navbar-toggler" class="navbar-toggler ml-0" type="button" data-toggle="collapse"
data-toggle="tooltip" data-placement="bottom" data-target=".site-navigation" aria-controls="navbar-menu"
aria-expanded="true" aria-label="Toggle navigation" aria-controls="site-navigation"
title="Toggle navigation" data-toggle="tooltip" data-placement="left">
<i class="fas fa-bars"></i>
<i class="fas fa-arrow-left"></i>
<i class="fas fa-arrow-up"></i>
</button>
<div class="dropdown-buttons-trigger">
<button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn" aria-label="Download this page"><i
class="fas fa-download"></i></button>
<div class="dropdown-buttons">
<!-- ipynb file if we had a myst markdown file -->
<!-- Download raw file -->
<a class="dropdown-buttons" href="_sources/chapter10.ipynb"><button type="button"
class="btn btn-secondary topbarbtn" title="Download source file" data-toggle="tooltip"
data-placement="left">.ipynb</button></a>
<!-- Download PDF via print -->
<button type="button" id="download-print" class="btn btn-secondary topbarbtn" title="Print to PDF"
onclick="printPdf(this)" data-toggle="tooltip" data-placement="left">.pdf</button>
</div>
</div>
<!-- Source interaction buttons -->
<!-- Full screen (wrap in <a> to have style consistency -->
<a class="full-screen-button"><button type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip"
data-placement="bottom" onclick="toggleFullScreen()" aria-label="Fullscreen mode"
title="Fullscreen mode"><i
class="fas fa-expand"></i></button></a>
<!-- Launch buttons -->
</div>
<!-- Table of contents -->
<div class="d-none d-md-block col-md-2 bd-toc show noprint">
<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> Contents
</div>
<nav id="bd-toc-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="#defining-the-cost-function">
14.1. Defining the cost function
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-binary-classification-problem">
14.1.1. Example: binary classification problem
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-softmax-function">
14.1.2. The Softmax function
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#developing-a-code-for-doing-neural-networks-with-back-propagation">
14.2. Developing a code for doing neural networks with back propagation
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#collect-and-pre-process-data">
14.2.1. Collect and pre-process data
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#train-and-test-datasets">
14.2.2. Train and test datasets
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#define-model-and-architecture">
14.2.3. Define model and architecture
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#layers">
14.2.4. Layers
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#feed-forward-pass">
14.2.5. Feed-forward pass
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#choose-cost-function-and-optimizer">
14.2.6. Choose cost function and optimizer
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#optimizing-the-cost-function">
14.2.7. Optimizing the cost function
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#regularization">
14.2.8. Regularization
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrix-multiplication">
14.2.9. Matrix multiplication
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#improving-performance">
14.3. Improving performance
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#evaluate-model-performance-on-test-data">
14.4. Evaluate model performance on test data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adjust-hyperparameters">
14.5. Adjust hyperparameters
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#visualization">
14.6. Visualization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#scikit-learn-implementation">
14.7. scikit-learn implementation
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
14.8. Visualization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#building-neural-networks-in-tensorflow-and-keras">
14.9. Building neural networks in Tensorflow and Keras
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-breast-cancer-data-now-with-keras">
14.10. The Breast Cancer Data, now with Keras
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#fine-tuning-neural-network-hyperparameters">
14.11. Fine-tuning neural network hyperparameters
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#which-activation-function-should-i-use">
14.12. Which activation function should I use?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-relu-function-family">
14.13. The RELU function family
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#batch-normalization">
14.14. Batch Normalization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#dropout">
14.15. Dropout
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-clipping">
14.16. Gradient Clipping
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-top-down-perspective-on-neural-networks">
14.17. A top-down perspective on Neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#limitations-of-supervised-learning-with-deep-networks">
14.18. Limitations of supervised learning with deep networks
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div id="main-content" class="row">
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
<!-- Table of contents that is only displayed when printing the page -->
<div id="jb-print-docs-body" class="onlyprint">
<h1>Building a Feed Forward Neural Network</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
<div>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#defining-the-cost-function">
14.1. Defining the cost function
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-binary-classification-problem">
14.1.1. Example: binary classification problem
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-softmax-function">
14.1.2. The Softmax function
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#developing-a-code-for-doing-neural-networks-with-back-propagation">
14.2. Developing a code for doing neural networks with back propagation
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#collect-and-pre-process-data">
14.2.1. Collect and pre-process data
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#train-and-test-datasets">
14.2.2. Train and test datasets
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#define-model-and-architecture">
14.2.3. Define model and architecture
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#layers">
14.2.4. Layers
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#feed-forward-pass">
14.2.5. Feed-forward pass
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#choose-cost-function-and-optimizer">
14.2.6. Choose cost function and optimizer
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#optimizing-the-cost-function">
14.2.7. Optimizing the cost function
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#regularization">
14.2.8. Regularization
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrix-multiplication">
14.2.9. Matrix multiplication
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#improving-performance">
14.3. Improving performance
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#evaluate-model-performance-on-test-data">
14.4. Evaluate model performance on test data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#adjust-hyperparameters">
14.5. Adjust hyperparameters
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#visualization">
14.6. Visualization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#scikit-learn-implementation">
14.7. scikit-learn implementation
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
14.8. Visualization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#building-neural-networks-in-tensorflow-and-keras">
14.9. Building neural networks in Tensorflow and Keras
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-breast-cancer-data-now-with-keras">
14.10. The Breast Cancer Data, now with Keras
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#fine-tuning-neural-network-hyperparameters">
14.11. Fine-tuning neural network hyperparameters
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#which-activation-function-should-i-use">
14.12. Which activation function should I use?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-relu-function-family">
14.13. The RELU function family
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#batch-normalization">
14.14. Batch Normalization
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#dropout">
14.15. Dropout
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#gradient-clipping">
14.16. Gradient Clipping
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-top-down-perspective-on-neural-networks">
14.17. A top-down perspective on Neural networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#limitations-of-supervised-learning-with-deep-networks">
14.18. Limitations of supervised learning with deep networks
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div>
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
doconce format html chapter10.do.txt --no_mako --><div class="tex2jax_ignore mathjax_ignore section" id="building-a-feed-forward-neural-network">
<h1><span class="section-number">14. </span>Building a Feed Forward Neural Network<a class="headerlink" href="#building-a-feed-forward-neural-network" title="Permalink to this headline"></a></h1>
<p>We are now gong to develop an example based on the MNIST data
base. This is a classification problem and we need to use our
cross-entropy function we discussed in connection with logistic
regression. The cross-entropy defines our cost function for the
classificaton problems with neural networks.</p>
<p>In binary classification with two classes <span class="math notranslate nohighlight">\((0, 1)\)</span> we define the
logistic/sigmoid function as the probability that a particular input
is in class <span class="math notranslate nohighlight">\(0\)</span> or <span class="math notranslate nohighlight">\(1\)</span>. This is possible because the logistic
function takes any input from the real numbers and inputs a number
between 0 and 1, and can therefore be interpreted as a probability. It
also has other nice properties, such as a derivative that is simple to
calculate.</p>
<p>For an input <span class="math notranslate nohighlight">\(\boldsymbol{a}\)</span> from the hidden layer, the probability that the input <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>
is in class 0 or 1 is just. We let <span class="math notranslate nohighlight">\(\theta\)</span> represent the unknown weights and biases to be adjusted by our equations). The variable <span class="math notranslate nohighlight">\(x\)</span>
represents our activation values <span class="math notranslate nohighlight">\(z\)</span>. We have</p>
<div class="math notranslate nohighlight">
\[
P(y = 0 \mid \hat{x}, \hat{\theta}) = \frac{1}{1 + \exp{(- \hat{x}})} ,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
P(y = 1 \mid \hat{x}, \hat{\theta}) = 1 - P(y = 0 \mid \hat{x}, \hat{\theta}) ,
\]</div>
<p>where <span class="math notranslate nohighlight">\(y \in \{0, 1\}\)</span> and <span class="math notranslate nohighlight">\(\hat{\theta}\)</span> represents the weights and biases
of our network.</p>
<div class="section" id="defining-the-cost-function">
<h2><span class="section-number">14.1. </span>Defining the cost function<a class="headerlink" href="#defining-the-cost-function" title="Permalink to this headline"></a></h2>
<p>Our cost function is given as (see the Logistic regression lectures)</p>
<div class="math notranslate nohighlight">
\[
\mathcal{C}(\hat{\theta}) = - \ln P(\mathcal{D} \mid \hat{\theta}) = - \sum_{i=1}^n
y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\hat{\theta}) .
\]</div>
<p>This last equality means that we can interpret our <em>cost</em> function as a sum over the <em>loss</em> function
for each point in the dataset <span class="math notranslate nohighlight">\(\mathcal{L}_i(\hat{\theta})\)</span>.<br />
The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
than maximizing a negative number.</p>
<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:</p>
<p><span class="math notranslate nohighlight">\(y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\)</span> and</p>
<p><span class="math notranslate nohighlight">\(y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\)</span></p>
<p>i.e. a binary bit string of length <span class="math notranslate nohighlight">\(C\)</span>, where <span class="math notranslate nohighlight">\(C = 10\)</span> is the number of classes in the MNIST dataset (numbers from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(9\)</span>)..</p>
<p>If <span class="math notranslate nohighlight">\(\hat{x}_i\)</span> is the <span class="math notranslate nohighlight">\(i\)</span>-th input (image), <span class="math notranslate nohighlight">\(y_{ic}\)</span> refers to the <span class="math notranslate nohighlight">\(c\)</span>-th component of the <span class="math notranslate nohighlight">\(i\)</span>-th
output vector <span class="math notranslate nohighlight">\(\hat{y}_i\)</span>.<br />
The probability of <span class="math notranslate nohighlight">\(\hat{x}_i\)</span> being in class <span class="math notranslate nohighlight">\(c\)</span> will be given by the softmax function:</p>
<div class="math notranslate nohighlight">
\[
P(y_{ic} = 1 \mid \hat{x}_i, \hat{\theta}) = \frac{\exp{((\hat{a}_i^{hidden})^T \hat{w}_c)}}
{\sum_{c'=0}^{C-1} \exp{((\hat{a}_i^{hidden})^T \hat{w}_{c'})}} ,
\]</div>
<p>which reduces to the logistic function in the binary case.<br />
The likelihood of this <span class="math notranslate nohighlight">\(C\)</span>-class classifier
is now given as:</p>
<div class="math notranslate nohighlight">
\[
P(\mathcal{D} \mid \hat{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
\]</div>
<p>Again we take the negative log-likelihood to define our cost function:</p>
<div class="math notranslate nohighlight">
\[
\mathcal{C}(\hat{\theta}) = - \log{P(\mathcal{D} \mid \hat{\theta})}.
\]</div>
<p>See the logistic regression lectures for a full definition of the cost function.</p>
<p>The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!</p>
<div class="section" id="example-binary-classification-problem">
<h3><span class="section-number">14.1.1. </span>Example: binary classification problem<a class="headerlink" href="#example-binary-classification-problem" title="Permalink to this headline"></a></h3>
<p>As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters <span class="math notranslate nohighlight">\(\beta\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\mathcal{C}(\hat{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\hat{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\hat{\beta})}\right),
\]</div>
<p>where we had defined the logistic (sigmoid) function</p>
<div class="math notranslate nohighlight">
\[
p(y_i =1\vert x_i,\hat{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}},
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
p(y_i =0\vert x_i,\hat{\beta})=1-p(y_i =1\vert x_i,\hat{\beta}).
\]</div>
<p>The parameters <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> were defined using a minimization method like gradient descent or Newton-Raphson’s method.</p>
<p>Now we replace <span class="math notranslate nohighlight">\(x_i\)</span> with the activation <span class="math notranslate nohighlight">\(z_i^l\)</span> for a given layer <span class="math notranslate nohighlight">\(l\)</span> and the outputs as <span class="math notranslate nohighlight">\(y_i=a_i^l=f(z_i^l)\)</span>, with <span class="math notranslate nohighlight">\(z_i^l\)</span> now being a function of the weights <span class="math notranslate nohighlight">\(w_{ij}^l\)</span> and biases <span class="math notranslate nohighlight">\(b_i^l\)</span>.
We have then</p>
<div class="math notranslate nohighlight">
\[
a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}},
\]</div>
<p>with</p>
<div class="math notranslate nohighlight">
\[
z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,
\]</div>
<p>where the superscript <span class="math notranslate nohighlight">\(l-1\)</span> indicates that these are the outputs from layer <span class="math notranslate nohighlight">\(l-1\)</span>.
Our cost function at the final layer <span class="math notranslate nohighlight">\(l=L\)</span> is now</p>
<div class="math notranslate nohighlight">
\[
\mathcal{C}(\hat{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right),
\]</div>
<p>where we have defined the targets <span class="math notranslate nohighlight">\(t_i\)</span>. The derivatives of the cost function with respect to the output <span class="math notranslate nohighlight">\(a_i^L\)</span> are then easily calculated and we get</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \mathcal{C}(\hat{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.
\]</div>
<p>In case we use another activation function than the logistic one, we need to evaluate other derivatives.</p>
</div>
<div class="section" id="the-softmax-function">
<h3><span class="section-number">14.1.2. </span>The Softmax function<a class="headerlink" href="#the-softmax-function" title="Permalink to this headline"></a></h3>
<p>In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation <span class="math notranslate nohighlight">\(z_i^l\)</span>, that is we need</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}.
\]</div>
<p>For the Softmax function we have</p>
<div class="math notranslate nohighlight">
\[
f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.
\]</div>
<p>Its derivative with respect to <span class="math notranslate nohighlight">\(z_j^l\)</span> gives</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right),
\]</div>
<p>which in case of the simply binary model reduces to having <span class="math notranslate nohighlight">\(i=j\)</span>.</p>
</div>
</div>
<div class="section" id="developing-a-code-for-doing-neural-networks-with-back-propagation">
<h2><span class="section-number">14.2. </span>Developing a code for doing neural networks with back propagation<a class="headerlink" href="#developing-a-code-for-doing-neural-networks-with-back-propagation" title="Permalink to this headline"></a></h2>
<p>One can identify a set of key steps when using neural networks to solve supervised learning problems:</p>
<ol class="simple">
<li><p>Collect and pre-process data</p></li>
<li><p>Define model and architecture</p></li>
<li><p>Choose cost function and optimizer</p></li>
<li><p>Train the model</p></li>
<li><p>Evaluate model performance on test data</p></li>
<li><p>Adjust hyperparameters (if necessary, network architecture)</p></li>
</ol>
<div class="section" id="collect-and-pre-process-data">
<h3><span class="section-number">14.2.1. </span>Collect and pre-process data<a class="headerlink" href="#collect-and-pre-process-data" title="Permalink to this headline"></a></h3>
<p>Here we will be using the MNIST dataset, which is readily available through the <strong>scikit-learn</strong>
package. You may also find it for example <a class="reference external" href="http://yann.lecun.com/exdb/mnist/">here</a>.<br />
The <em>MNIST</em> (Modified National Institute of Standards and Technology) database is a large database
of handwritten digits that is commonly used for training various image processing systems.<br />
The MNIST dataset consists of 70 000 images of size <span class="math notranslate nohighlight">\(28\times 28\)</span> pixels, each labeled from 0 to 9.<br />
The scikit-learn dataset we will use consists of a selection of 1797 images of size <span class="math notranslate nohighlight">\(8\times 8\)</span> collected and processed from this database.</p>
<p>To feed data into a feed-forward neural network we need to represent
the inputs as a design/feature matrix <span class="math notranslate nohighlight">\(X = (n_{inputs}, n_{features})\)</span>. Each
row represents an <em>input</em>, in this case a handwritten digit, and
each column represents a <em>feature</em>, in this case a pixel. The
correct answers, also known as <em>labels</em> or <em>targets</em> are
represented as a 1D array of integers
<span class="math notranslate nohighlight">\(Y = (n_{inputs}) = (5, 3, 1, 8,...)\)</span>.</p>
<p>As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from
measurements of height (in m)<br />
and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:</p>
<div class="math notranslate nohighlight">
\[\begin{split} X = \begin{bmatrix}
1.85 &amp; 81\\
1.71 &amp; 65\\
1.95 &amp; 103\\
1.55 &amp; 42\\
1.63 &amp; 56
\end{bmatrix} ,\end{split}\]</div>
<p>and the targets would be:</p>
<div class="math notranslate nohighlight">
\[ Y = (23.7, 22.2, 27.1, 17.5, 21.1) \]</div>
<p>Since each input image is a 2D matrix, we need to flatten the image
(i.e. “unravel” the 2D matrix into a 1D array) to turn the data into a
design/feature matrix. This means we lose all spatial information in the
image, such as locality and translational invariance. More complicated
architectures such as Convolutional Neural Networks can take advantage
of such information, and are most commonly applied when analyzing
images.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="c1"># import necessary packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="c1"># ensure the same random numbers appear every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># display images in notebook</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
<span class="c1"># download MNIST dataset</span>
<span class="n">digits</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">()</span>
<span class="c1"># define inputs and labels</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">images</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;inputs = (n_inputs, pixel_width, pixel_height) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;labels = (n_inputs) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="c1"># flatten the image</span>
<span class="c1"># the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;X = (n_inputs, n_features) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="c1"># choose some random images to display</span>
<span class="n">indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">)</span>
<span class="n">random_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">image</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">digits</span><span class="o">.</span><span class="n">images</span><span class="p">[</span><span class="n">random_indices</span><span class="p">]):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Label: </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">random_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>inputs = (n_inputs, pixel_width, pixel_height) = (1797, 8, 8)
labels = (n_inputs) = (1797,)
X = (n_inputs, n_features) = (1797, 64)
</pre></div>
</div>
<img alt="_images/chapter10_39_1.png" src="_images/chapter10_39_1.png" />
</div>
</div>
</div>
<div class="section" id="train-and-test-datasets">
<h3><span class="section-number">14.2.2. </span>Train and test datasets<a class="headerlink" href="#train-and-test-datasets" title="Permalink to this headline"></a></h3>
<p>Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.</p>
<p>We will reserve <span class="math notranslate nohighlight">\(80 \%\)</span> of our dataset for training and <span class="math notranslate nohighlight">\(20 \%\)</span> for testing.</p>
<p>It is important that the train and test datasets are drawn randomly from our dataset, to ensure
no bias in the sampling.<br />
Say you are taking measurements of weather data to predict the weather in the coming 5 days.
You don’t want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data
collected from 12.00 to 24.00.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="c1"># one-liner from scikit-learn library</span>
<span class="n">train_size</span> <span class="o">=</span> <span class="mf">0.8</span>
<span class="n">test_size</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">train_size</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span>
<span class="n">test_size</span><span class="o">=</span><span class="n">test_size</span><span class="p">)</span>
<span class="c1"># equivalently in numpy</span>
<span class="k">def</span> <span class="nf">train_test_split_numpy</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="p">,</span> <span class="n">test_size</span><span class="p">):</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
<span class="n">inputs_shuffled</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">labels_shuffled</span> <span class="o">=</span> <span class="n">labels</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">inputs_shuffled</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">labels_shuffled</span><span class="p">)</span>
<span class="n">train_end</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n_inputs</span><span class="o">*</span><span class="n">train_size</span><span class="p">)</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span> <span class="o">=</span> <span class="n">inputs_shuffled</span><span class="p">[:</span><span class="n">train_end</span><span class="p">],</span> <span class="n">inputs_shuffled</span><span class="p">[</span><span class="n">train_end</span><span class="p">:]</span>
<span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">labels_shuffled</span><span class="p">[:</span><span class="n">train_end</span><span class="p">],</span> <span class="n">labels_shuffled</span><span class="p">[</span><span class="n">train_end</span><span class="p">:]</span>
<span class="k">return</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span>
<span class="c1">#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Number of training images: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X_train</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Number of test images: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X_test</span><span class="p">)))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Number of training images: 1437
Number of test images: 360
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="define-model-and-architecture">
<h3><span class="section-number">14.2.3. </span>Define model and architecture<a class="headerlink" href="#define-model-and-architecture" title="Permalink to this headline"></a></h3>
<p>Our simple feed-forward neural network will consist of an <em>input</em> layer, a single <em>hidden</em> layer and an <em>output</em> layer. The activation <span class="math notranslate nohighlight">\(y\)</span> of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have</p>
<div class="math notranslate nohighlight">
\[ z = \sum_{i=1}^n w_i a_i ,\]</div>
<div class="math notranslate nohighlight">
\[ y = f(z) ,\]</div>
<p>where <span class="math notranslate nohighlight">\(f\)</span> is the activation function, <span class="math notranslate nohighlight">\(a_i\)</span> represents input from neuron <span class="math notranslate nohighlight">\(i\)</span> in the preceding layer
and <span class="math notranslate nohighlight">\(w_i\)</span> is the weight to input <span class="math notranslate nohighlight">\(i\)</span>.<br />
The activation of the neurons in the input layer is just the features (e.g. a pixel value).</p>
<p>The simplest activation function for a neuron is the <em>Heaviside</em> function:</p>
<div class="math notranslate nohighlight">
\[\begin{split} f(z) =
\begin{cases}
1, &amp; z &gt; 0\\
0, &amp; \text{otherwise}
\end{cases}
\end{split}\]</div>
<p>A feed-forward neural network with this activation is known as a <em>perceptron</em>.<br />
For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.<br />
This activation can be generalized to <span class="math notranslate nohighlight">\(k\)</span> classes (using e.g. the <em>one-against-all</em> strategy),
and we call these architectures <em>multiclass perceptrons</em>.</p>
<p>However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and<br />
Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.</p>
<p>Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).<br />
We will be using the sigmoid function <span class="math notranslate nohighlight">\(\sigma(x)\)</span>:</p>
<div class="math notranslate nohighlight">
\[ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,\]</div>
<p>which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.</p>
</div>
<div class="section" id="layers">
<h3><span class="section-number">14.2.4. </span>Layers<a class="headerlink" href="#layers" title="Permalink to this headline"></a></h3>
<ul class="simple">
<li><p>Input</p></li>
</ul>
<p>Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.</p>
<ul class="simple">
<li><p>Hidden layer</p></li>
</ul>
<p>We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.<br />
Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.</p>
<ul class="simple">
<li><p>Output</p></li>
</ul>
<p>If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
which could output 0 or 1 according to the Heaviside function. This would be an example of a <em>hard</em> classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a <em>soft</em> classifier, which outputs the probability of being in class 0 or 1.</p>
<p>For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.</p>
<p>Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons <span class="math notranslate nohighlight">\(j = 0,1,...,9\)</span>. The activation of each output neuron <span class="math notranslate nohighlight">\(j\)</span> will be according to the <em>softmax</em> function:</p>
<div class="math notranslate nohighlight">
\[ P(\text{class $j$} \mid \text{input $\hat{a}$}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}}
{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,\]</div>
<p>i.e. each neuron <span class="math notranslate nohighlight">\(j\)</span> outputs the probability of being in class <span class="math notranslate nohighlight">\(j\)</span> given an input from the hidden layer <span class="math notranslate nohighlight">\(\hat{a}\)</span>, with <span class="math notranslate nohighlight">\(\hat{w}_j\)</span> the weights of neuron <span class="math notranslate nohighlight">\(j\)</span> to the inputs.<br />
The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.<br />
The exponent is just the weighted sum of inputs as before:</p>
<div class="math notranslate nohighlight">
\[ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.\]</div>
<p>Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
weights to the output layer.</p>
<p>Typically weights are initialized with small values distributed around zero, drawn from a uniform
or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.</p>
<p>Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron <span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(b_j\)</span>:</p>
<div class="math notranslate nohighlight">
\[ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.\]</div>
<p>The bias weights <span class="math notranslate nohighlight">\(\hat{b}\)</span> are often initialized to zero, but a small value like <span class="math notranslate nohighlight">\(0.01\)</span> ensures all neurons have some output which can be backpropagated in the first training cycle.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># building our neural network</span>
<span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">shape</span>
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">10</span>
<span class="c1"># we make the weights normally distributed using numpy.random.randn</span>
<span class="c1"># weights and bias in the hidden layer</span>
<span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_hidden_neurons</span><span class="p">)</span>
<span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="c1"># weights and bias in the output layer</span>
<span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">)</span>
<span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="feed-forward-pass">
<h3><span class="section-number">14.2.5. </span>Feed-forward pass<a class="headerlink" href="#feed-forward-pass" title="Permalink to this headline"></a></h3>
<p>Denote <span class="math notranslate nohighlight">\(F\)</span> the number of features, <span class="math notranslate nohighlight">\(H\)</span> the number of hidden neurons and <span class="math notranslate nohighlight">\(C\)</span> the number of categories.<br />
For each input image we calculate a weighted sum of input features (pixel values) to each neuron <span class="math notranslate nohighlight">\(j\)</span> in the hidden layer <span class="math notranslate nohighlight">\(l\)</span>:</p>
<div class="math notranslate nohighlight">
\[ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},\]</div>
<p>this is then passed through our activation function</p>
<div class="math notranslate nohighlight">
\[ a_{j}^{l} = f(z_{j}^{l}) .\]</div>
<p>We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron <span class="math notranslate nohighlight">\(j\)</span> in the output layer:</p>
<div class="math notranslate nohighlight">
\[ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.\]</div>
<p>Finally we calculate the output of neuron <span class="math notranslate nohighlight">\(j\)</span> in the output layer using the softmax function:</p>
<div class="math notranslate nohighlight">
\[ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .\]</div>
<p>Since our data has the dimensions <span class="math notranslate nohighlight">\(X = (n_{inputs}, n_{features})\)</span> and our weights to the hidden
layer have the dimensions<br />
<span class="math notranslate nohighlight">\(W_{hidden} = (n_{features}, n_{hidden})\)</span>,
we can easily feed the network all our training data in one go by taking the matrix product</p>
<div class="math notranslate nohighlight">
\[ X W^{h} = (n_{inputs}, n_{hidden}),\]</div>
<p>and obtain a matrix that holds the weighted sum of inputs to the hidden layer
for each input image and each hidden neuron.<br />
We also add the bias to obtain a matrix of weighted sums to the hidden layer <span class="math notranslate nohighlight">\(Z^{h}\)</span>:</p>
<div class="math notranslate nohighlight">
\[ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,\]</div>
<p>meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.<br />
This is then passed through the activation:</p>
<div class="math notranslate nohighlight">
\[ \hat{a}^{l} = f(\hat{z}^l) .\]</div>
<p>This is fed to the output layer:</p>
<div class="math notranslate nohighlight">
\[ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .\]</div>
<p>Finally we receive our output values for each image and each category by passing it through the softmax function:</p>
<div class="math notranslate nohighlight">
\[ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .\]</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup the feed-forward pass, subscript h = hidden layer</span>
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="c1"># weighted sum of inputs to the hidden layer</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
<span class="c1"># activation in the hidden layer</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="c1"># weighted sum of inputs to the output layer</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
<span class="c1"># softmax output</span>
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;probabilities = (n_inputs, n_categories) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;probability that image 0 is in category 0,1,2,...,9 = </span><span class="se">\n</span><span class="s2">&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;probabilities sum up to: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sum</span><span class="p">()))</span>
<span class="nb">print</span><span class="p">()</span>
<span class="c1"># we obtain a prediction by taking the class with the highest likelihood</span>
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">predictions</span> <span class="o">=</span> <span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;predictions = (n_inputs) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">predictions</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;prediction for image 0: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">predictions</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;correct label for image 0: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">Y_train</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>probabilities = (n_inputs, n_categories) = (1437, 10)
probability that image 0 is in category 0,1,2,...,9 =
[5.41511965e-04 2.17174962e-03 8.84355903e-03 1.44970586e-03
1.10378326e-04 5.08318298e-09 2.03256632e-04 1.92507116e-03
9.84443254e-01 3.11507992e-04]
probabilities sum up to: 1.0
predictions = (n_inputs) = (1437,)
prediction for image 0: 8
correct label for image 0: 6
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="choose-cost-function-and-optimizer">
<h3><span class="section-number">14.2.6. </span>Choose cost function and optimizer<a class="headerlink" href="#choose-cost-function-and-optimizer" title="Permalink to this headline"></a></h3>
<p>To measure how well our neural network is doing we need to introduce a cost function.<br />
We will call the function that gives the error of a single sample output the <em>loss</em> function, and the function
that gives the total error of our network across all samples the <em>cost</em> function.
A typical choice for multiclass classification is the <em>cross-entropy</em> loss, also known as the negative log likelihood.</p>
<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:</p>
<div class="math notranslate nohighlight">
\[ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\]</div>
<div class="math notranslate nohighlight">
\[ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\]</div>
<p>i.e. a binary bit string of length <span class="math notranslate nohighlight">\(C\)</span>, where <span class="math notranslate nohighlight">\(C = 10\)</span> is the number of classes in the MNIST dataset.</p>
<p>Let <span class="math notranslate nohighlight">\(y_{ic}\)</span> denote the <span class="math notranslate nohighlight">\(c\)</span>-th component of the <span class="math notranslate nohighlight">\(i\)</span>-th one-hot vector.<br />
We define the cost function <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> as a sum over the cross-entropy loss for each point <span class="math notranslate nohighlight">\(\hat{x}_i\)</span> in the dataset.</p>
<p>In the one-hot representation only one of the terms in the loss function is non-zero, namely the
probability of the correct category <span class="math notranslate nohighlight">\(c'\)</span><br />
(i.e. the category <span class="math notranslate nohighlight">\(c'\)</span> such that <span class="math notranslate nohighlight">\(y_{ic'} = 1\)</span>). This means that the cross entropy loss only punishes you for how wrong
you got the correct label. The probability of category <span class="math notranslate nohighlight">\(c\)</span> is given by the softmax function. The vector <span class="math notranslate nohighlight">\(\hat{\theta}\)</span> represents the parameters of our network, i.e. all the weights and biases.</p>
</div>
<div class="section" id="optimizing-the-cost-function">
<h3><span class="section-number">14.2.7. </span>Optimizing the cost function<a class="headerlink" href="#optimizing-the-cost-function" title="Permalink to this headline"></a></h3>
<p>The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is <em>gradient descent</em> and its generalizations. The idea behind gradient descent
is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a <em>local</em> minimum of the cost function.<br />
Each parameter <span class="math notranslate nohighlight">\(\theta\)</span> is iteratively adjusted according to the rule</p>
<div class="math notranslate nohighlight">
\[ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,\]</div>
<p>where <span class="math notranslate nohighlight">\(\eta\)</span> is known as the <em>learning rate</em>, which controls how big a step we take towards the minimum.<br />
This update can be repeated for any number of iterations, or until we are satisfied with the result.</p>
<p>A simple and effective improvement is a variant called <em>Batch Gradient Descent</em>.<br />
Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
on a subset of the data called a <em>minibatch</em>.<br />
If there are <span class="math notranslate nohighlight">\(N\)</span> data points and we have a minibatch size of <span class="math notranslate nohighlight">\(M\)</span>, the total number of batches
is <span class="math notranslate nohighlight">\(N/M\)</span>.<br />
We denote each minibatch <span class="math notranslate nohighlight">\(B_k\)</span>, with <span class="math notranslate nohighlight">\(k = 1, 2,...,N/M\)</span>. The gradient then becomes:</p>
<div class="math notranslate nohighlight">
\[ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,\]</div>
<p>i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.</p>
<p>This has two important benefits:</p>
<ol class="simple">
<li><p>Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.</p></li>
<li><p>It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.</p></li>
</ol>
<p>The various optmization methods, with codes and algorithms, are discussed in our lectures on <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html">Gradient descent approaches</a>.</p>
</div>
<div class="section" id="regularization">
<h3><span class="section-number">14.2.8. </span>Regularization<a class="headerlink" href="#regularization" title="Permalink to this headline"></a></h3>
<p>It is common to add an extra term to the cost function, proportional
to the size of the weights. This is equivalent to constraining the
size of the weights, so that they do not grow out of control.
Constraining the size of the weights means that the weights cannot
grow arbitrarily large to fit the training data, and in this way
reduces <em>overfitting</em>.</p>
<p>We will measure the size of the weights using the so called <em>L2-norm</em>, meaning our cost function becomes:</p>
<div class="math notranslate nohighlight">
\[ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2
= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,\]</div>
<p>i.e. we sum up all the weights squared. The factor <span class="math notranslate nohighlight">\(\lambda\)</span> is known as a regularization parameter.</p>
<p>In order to train the model, we need to calculate the derivative of
the cost function with respect to every bias and weight in the
network. In total our network has <span class="math notranslate nohighlight">\((64 + 1)\times 50=3250\)</span> weights in
the hidden layer and <span class="math notranslate nohighlight">\((50 + 1)\times 10=510\)</span> weights to the output
layer (<span class="math notranslate nohighlight">\(+1\)</span> for the bias), and the gradient must be calculated for
every parameter. We use the <em>backpropagation</em> algorithm discussed
above. This is a clever use of the chain rule that allows us to
calculate the gradient efficently.</p>
</div>
<div class="section" id="matrix-multiplication">
<h3><span class="section-number">14.2.9. </span>Matrix multiplication<a class="headerlink" href="#matrix-multiplication" title="Permalink to this headline"></a></h3>
<p>To more efficently train our network these equations are implemented using matrix operations.<br />
The error in the output layer is calculated simply as, with <span class="math notranslate nohighlight">\(\hat{t}\)</span> being our targets,</p>
<div class="math notranslate nohighlight">
\[ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .\]</div>
<p>The gradient for the output weights is calculated as</p>
<div class="math notranslate nohighlight">
\[ \nabla W_{L} = \hat{a}^T \delta_L = (n_{hidden}, n_{categories}) ,\]</div>
<p>where <span class="math notranslate nohighlight">\(\hat{a} = (n_{inputs}, n_{hidden})\)</span>. This simply means that we are summing up the gradients for each input.<br />
Since we are going backwards we have to transpose the activation matrix.</p>
<p>The gradient with respect to the output bias is then</p>
<div class="math notranslate nohighlight">
\[ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .\]</div>
<p>The error in the hidden layer is</p>
<div class="math notranslate nohighlight">
\[ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,\]</div>
<p>where <span class="math notranslate nohighlight">\(f'(a_{h})\)</span> is the derivative of the activation in the hidden layer. The matrix products mean
that we are summing up the products for each neuron in the output layer. The symbol <span class="math notranslate nohighlight">\(\circ\)</span> denotes
the <em>Hadamard product</em>, meaning element-wise multiplication.</p>
<p>This again gives us the gradients in the hidden layer:</p>
<div class="math notranslate nohighlight">
\[ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,\]</div>
<div class="math notranslate nohighlight">
\[ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .\]</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># to categorical turns our integer vector into a onehot representation</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
<span class="c1"># one-hot in numpy</span>
<span class="k">def</span> <span class="nf">to_categorical_numpy</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">):</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">)</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">onehot_vector</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">))</span>
<span class="n">onehot_vector</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">),</span> <span class="n">integer_vector</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">onehot_vector</span>
<span class="c1">#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)</span>
<span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">Y_test_onehot</span> <span class="o">=</span> <span class="n">to_categorical_numpy</span><span class="p">(</span><span class="n">Y_train</span><span class="p">),</span> <span class="n">to_categorical_numpy</span><span class="p">(</span><span class="n">Y_test</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">feed_forward_train</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="c1"># weighted sum of inputs to the hidden layer</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
<span class="c1"># activation in the hidden layer</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="c1"># weighted sum of inputs to the output layer</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
<span class="c1"># softmax output</span>
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># for backpropagation need activations in hidden and output layers</span>
<span class="k">return</span> <span class="n">a_h</span><span class="p">,</span> <span class="n">probabilities</span>
<span class="k">def</span> <span class="nf">backpropagation</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
<span class="n">a_h</span><span class="p">,</span> <span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward_train</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="c1"># error in the output layer</span>
<span class="n">error_output</span> <span class="o">=</span> <span class="n">probabilities</span> <span class="o">-</span> <span class="n">Y</span>
<span class="c1"># error in the hidden layer</span>
<span class="n">error_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">output_weights</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="n">a_h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a_h</span><span class="p">)</span>
<span class="c1"># gradients for the output layer</span>
<span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
<span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># gradient for the hidden layer</span>
<span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
<span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">return</span> <span class="n">output_weights_gradient</span><span class="p">,</span> <span class="n">output_bias_gradient</span><span class="p">,</span> <span class="n">hidden_weights_gradient</span><span class="p">,</span> <span class="n">hidden_bias_gradient</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Old accuracy on training data: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">Y_train</span><span class="p">)))</span>
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="n">lmbd</span> <span class="o">=</span> <span class="mf">0.01</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
<span class="c1"># calculate gradients</span>
<span class="n">dWo</span><span class="p">,</span> <span class="n">dBo</span><span class="p">,</span> <span class="n">dWh</span><span class="p">,</span> <span class="n">dBh</span> <span class="o">=</span> <span class="n">backpropagation</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">)</span>
<span class="c1"># regularization term gradients</span>
<span class="n">dWo</span> <span class="o">+=</span> <span class="n">lmbd</span> <span class="o">*</span> <span class="n">output_weights</span>
<span class="n">dWh</span> <span class="o">+=</span> <span class="n">lmbd</span> <span class="o">*</span> <span class="n">hidden_weights</span>
<span class="c1"># update weights and biases</span>
<span class="n">output_weights</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dWo</span>
<span class="n">output_bias</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dBo</span>
<span class="n">hidden_weights</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dWh</span>
<span class="n">hidden_bias</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dBh</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;New accuracy on training data: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">Y_train</span><span class="p">)))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Old accuracy on training data: 0.1440501043841336
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_15302/953065564.py:4: RuntimeWarning: overflow encountered in exp
return 1/(1 + np.exp(-x))
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>New accuracy on training data: 0.09951287404314545
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="section" id="improving-performance">
<h2><span class="section-number">14.3. </span>Improving performance<a class="headerlink" href="#improving-performance" title="Permalink to this headline"></a></h2>
<p>As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.<br />
In order to obtain a network that does something useful, we will have to do a bit more work.</p>
<p>The choice of <em>hyperparameters</em> such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a <em>grid-search</em> is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates <span class="math notranslate nohighlight">\(\eta = 10^{-6}, 10^{-5},...,10^{-1}\)</span> with different regularization parameters <span class="math notranslate nohighlight">\(\lambda = 10^{-6},...,10^{-0}\)</span>.</p>
<p>Next, we haven’t implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an <em>iteration</em>, and a full training period
going through the entire dataset (<span class="math notranslate nohighlight">\(n/M\)</span> batches) an <em>epoch</em>.</p>
<p>If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.<br />
Andrew Ng goes through some of these considerations in this <a class="reference external" href="https://youtu.be/F1ka6a13S9I">video</a>. You can find a summary of the video <a class="reference external" href="https://kevinzakka.github.io/2016/09/26/applying-deep-learning/">here</a>.</p>
<p>It is very natural to think of the network as an object, with specific instances of the network
being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NeuralNetwork</span><span class="p">:</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
<span class="bp">self</span><span class="p">,</span>
<span class="n">X_data</span><span class="p">,</span>
<span class="n">Y_data</span><span class="p">,</span>
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span>
<span class="n">n_categories</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">epochs</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">eta</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
<span class="n">lmbd</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span> <span class="o">=</span> <span class="n">X_data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span> <span class="o">=</span> <span class="n">Y_data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span> <span class="o">=</span> <span class="n">X_data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_features</span> <span class="o">=</span> <span class="n">X_data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="n">n_hidden_neurons</span>
<span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span> <span class="o">=</span> <span class="n">n_categories</span>
<span class="bp">self</span><span class="o">.</span><span class="n">epochs</span> <span class="o">=</span> <span class="n">epochs</span>
<span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span> <span class="o">=</span> <span class="n">batch_size</span>
<span class="bp">self</span><span class="o">.</span><span class="n">iterations</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span> <span class="o">//</span> <span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span>
<span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">=</span> <span class="n">eta</span>
<span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">=</span> <span class="n">lmbd</span>
<span class="bp">self</span><span class="o">.</span><span class="n">create_biases_and_weights</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">create_biases_and_weights</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_features</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="c1"># feed-forward for training</span>
<span class="bp">self</span><span class="o">.</span><span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
<span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_h</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_o</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">feed_forward_out</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="c1"># feed-forward for output</span>
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="k">def</span> <span class="nf">backpropagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">error_output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">probabilities</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span>
<span class="n">error_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">&gt;</span> <span class="mf">0.0</span><span class="p">:</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span>
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span>
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span>
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward_out</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">predict_probabilities</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward_out</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">return</span> <span class="n">probabilities</span>
<span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">data_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">epochs</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">iterations</span><span class="p">):</span>
<span class="c1"># pick datapoints with replacement</span>
<span class="n">chosen_datapoints</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span>
<span class="n">data_indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span>
<span class="p">)</span>
<span class="c1"># minibatch training data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">X_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">feed_forward</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">backpropagation</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="evaluate-model-performance-on-test-data">
<h2><span class="section-number">14.4. </span>Evaluate model performance on test data<a class="headerlink" href="#evaluate-model-performance-on-test-data" title="Permalink to this headline"></a></h2>
<p>To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.<br />
We measure the performance of the network using the <em>accuracy</em> score.<br />
The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of <span class="math notranslate nohighlight">\(1\)</span>.</p>
<div class="math notranslate nohighlight">
\[ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,\]</div>
<p>where <span class="math notranslate nohighlight">\(I\)</span> is the indicator function, <span class="math notranslate nohighlight">\(1\)</span> if <span class="math notranslate nohighlight">\(\hat{y}_i = y_i\)</span> and <span class="math notranslate nohighlight">\(0\)</span> otherwise.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
<span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># accuracy score from scikit library</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on test set: &quot;</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_predict</span><span class="p">))</span>
<span class="c1"># equivalent in numpy</span>
<span class="k">def</span> <span class="nf">accuracy_score_numpy</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">Y_pred</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">Y_test</span> <span class="o">==</span> <span class="n">Y_pred</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">Y_test</span><span class="p">)</span>
<span class="c1">#print(&quot;Accuracy score on test set: &quot;, accuracy_score_numpy(Y_test, test_predict))</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Accuracy score on test set: 0.9444444444444444
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="adjust-hyperparameters">
<h2><span class="section-number">14.5. </span>Adjust hyperparameters<a class="headerlink" href="#adjust-hyperparameters" title="Permalink to this headline"></a></h2>
<p>We now perform a grid search to find the optimal hyperparameters for the network.<br />
Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around <span class="math notranslate nohighlight">\(98\%\)</span> (<span class="math notranslate nohighlight">\(2\%\)</span> error rate).</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="c1"># store the models for later use</span>
<span class="n">DNN_numpy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="c1"># grid search</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
<span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Learning rate = &quot;</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Lambda = &quot;</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on test set: &quot;</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_predict</span><span class="p">))</span>
<span class="nb">print</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 1e-05
Accuracy score on test set: 0.11666666666666667
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.0001
Accuracy score on test set: 0.20833333333333334
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.001
Accuracy score on test set: 0.12222222222222222
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.01
Accuracy score on test set: 0.14722222222222223
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 0.1
Accuracy score on test set: 0.17777777777777778
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 1.0
Accuracy score on test set: 0.16111111111111112
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
Lambda = 10.0
Accuracy score on test set: 0.20277777777777778
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 1e-05
Accuracy score on test set: 0.5305555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.0001
Accuracy score on test set: 0.5944444444444444
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.001
Accuracy score on test set: 0.5888888888888889
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.01
Accuracy score on test set: 0.6111111111111112
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 0.1
Accuracy score on test set: 0.5222222222222223
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 1.0
Accuracy score on test set: 0.5555555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
Lambda = 10.0
Accuracy score on test set: 0.8055555555555556
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 1e-05
Accuracy score on test set: 0.85
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
Lambda = 0.0001
Accuracy score on test set: 0.85
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="nn">Input In [8],</span> in <span class="ni">&lt;cell line: 7&gt;</span><span class="nt">()</span>
<span class="g g-Whitespace"> </span><span class="mi">8</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">9</span> <span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">10</span> <span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
<span class="ne">---&gt; </span><span class="mi">11</span> <span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
<span class="g g-Whitespace"> </span><span class="mi">13</span> <span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="g g-Whitespace"> </span><span class="mi">15</span> <span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="nn">Input In [6],</span> in <span class="ni">NeuralNetwork.train</span><span class="nt">(self)</span>
<span class="g g-Whitespace"> </span><span class="mi">95</span> <span class="bp">self</span><span class="o">.</span><span class="n">X_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="g g-Whitespace"> </span><span class="mi">96</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
<span class="ne">---&gt; </span><span class="mi">98</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward</span><span class="p">()</span>
<span class="g g-Whitespace"> </span><span class="mi">99</span> <span class="bp">self</span><span class="o">.</span><span class="n">backpropagation</span><span class="p">()</span>
<span class="nn">Input In [6],</span> in <span class="ni">NeuralNetwork.feed_forward</span><span class="nt">(self)</span>
<span class="g g-Whitespace"> </span><span class="mi">36</span> <span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">37</span> <span class="c1"># feed-forward for training</span>
<span class="ne">---&gt; </span><span class="mi">38</span> <span class="bp">self</span><span class="o">.</span><span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
<span class="g g-Whitespace"> </span><span class="mi">39</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_h</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">41</span> <span class="bp">self</span><span class="o">.</span><span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
<span class="ne">KeyboardInterrupt</span>:
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="visualization">
<h2><span class="section-number">14.6. </span>Visualization<a class="headerlink" href="#visualization" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># visual representation of grid search</span>
<span class="c1"># uses seaborn heatmap, you can also do this with matplotlib imshow</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
<span class="n">train_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">train_pred</span><span class="p">)</span>
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Test Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="scikit-learn-implementation">
<h2><span class="section-number">14.7. </span>scikit-learn implementation<a class="headerlink" href="#scikit-learn-implementation" title="Permalink to this headline"></a></h2>
<p><strong>scikit-learn</strong> focuses more
on traditional machine learning methods, such as regression,
clustering, decision trees, etc. As such, it has only two types of
neural networks: Multi Layer Perceptron outputting continuous values,
<em>MPLRegressor</em>, and Multi Layer Perceptron outputting labels,
<em>MLPClassifier</em>. We will see how simple it is to use these classes.</p>
<p><strong>scikit-learn</strong> implements a few improvements from our neural network,
such as early stopping, a varying learning rate, different
optimization methods, etc. We would therefore expect a better
performance overall.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPClassifier</span>
<span class="c1"># store models for later use</span>
<span class="n">DNN_scikit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">MLPClassifier</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">),</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;logistic&#39;</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">learning_rate_init</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="n">epochs</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Learning rate = &quot;</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Lambda = &quot;</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Accuracy score on test set: &quot;</span><span class="p">,</span> <span class="n">dnn</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">))</span>
<span class="nb">print</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id1">
<h2><span class="section-number">14.8. </span>Visualization<a class="headerlink" href="#id1" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># optional</span>
<span class="c1"># visual representation of grid search</span>
<span class="c1"># uses seaborn heatmap, could probably do this in matplotlib</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
<span class="n">train_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">train_pred</span><span class="p">)</span>
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Test Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="building-neural-networks-in-tensorflow-and-keras">
<h2><span class="section-number">14.9. </span>Building neural networks in Tensorflow and Keras<a class="headerlink" href="#building-neural-networks-in-tensorflow-and-keras" title="Permalink to this headline"></a></h2>
<p>Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.</p>
<p>In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
NumPy arrays.</p>
<p>Tensorflow is an open source library machine learning library
developed by the Google Brain team for internal use. It was released
under the Apache 2.0 open source license in November 9, 2015.</p>
<p>Tensorflow is a computational framework that allows you to construct
machine learning models at different levels of abstraction, from
high-level, object-oriented APIs like Keras, down to the C++ kernels
that Tensorflow is built upon. The higher levels of abstraction are
simpler to use, but less flexible, and our choice of implementation
should reflect the problems we are trying to solve.</p>
<p><a class="reference external" href="https://www.tensorflow.org/guide/graphs">Tensorflow uses</a> so-called graphs to represent your computation
in terms of the dependencies between individual operations, such that you first build a Tensorflow <em>graph</em>
to represent your model, and then create a Tensorflow <em>session</em> to run the graph.</p>
<p>In this guide we will analyze the same data as we did in our NumPy and
scikit-learn tutorial, gathered from the MNIST database of images. We
will give an introduction to the lower level Python Application
Program Interfaces (APIs), and see how we use them to build our graph.
Then we will build (effectively) the same graph in Keras, to see just
how simple solving a machine learning problem can be.</p>
<p>To install tensorflow on Unix/Linux systems, use pip as <strong>pip3 install tensorflow</strong>
and/or if you use <strong>anaconda</strong>, just write (or install from the graphical user interface)
(current release of CPU-only TensorFlow)</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">create</span> <span class="o">-</span><span class="n">n</span> <span class="n">tf</span> <span class="n">tensorflow</span>
<span class="n">conda</span> <span class="n">activate</span> <span class="n">tf</span>
</pre></div>
</div>
</div>
</div>
<p>To install the current release of GPU TensorFlow</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">create</span> <span class="o">-</span><span class="n">n</span> <span class="n">tf</span><span class="o">-</span><span class="n">gpu</span> <span class="n">tensorflow</span><span class="o">-</span><span class="n">gpu</span>
<span class="n">conda</span> <span class="n">activate</span> <span class="n">tf</span><span class="o">-</span><span class="n">gpu</span>
</pre></div>
</div>
</div>
</div>
<p>Keras is a high level <a class="reference external" href="https://en.wikipedia.org/wiki/Application_programming_interface">neural network</a>
that supports Tensorflow, CTNK and Theano as backends.<br />
If you have Anaconda installed you may run the following command</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">install</span> <span class="n">keras</span>
</pre></div>
</div>
</div>
</div>
<p>You can look up the <a class="reference external" href="https://keras.io/">instructions here</a> for more information.</p>
<p>We will to a large extent use <strong>keras</strong> in this course.</p>
<p>Let us look again at the MINST data set.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import necessary packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="c1"># ensure the same random numbers appear every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># display images in notebook</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
<span class="c1"># download MNIST dataset</span>
<span class="n">digits</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">()</span>
<span class="c1"># define inputs and labels</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">images</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;inputs = (n_inputs, pixel_width, pixel_height) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;labels = (n_inputs) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="c1"># flatten the image</span>
<span class="c1"># the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64</span>
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;X = (n_inputs, n_features) = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="c1"># choose some random images to display</span>
<span class="n">indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">)</span>
<span class="n">random_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">image</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">digits</span><span class="o">.</span><span class="n">images</span><span class="p">[</span><span class="n">random_indices</span><span class="p">]):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Label: </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">random_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Input</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.models</span> <span class="kn">import</span> <span class="n">Sequential</span> <span class="c1">#This allows appending layers to existing models</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Dense</span> <span class="c1">#This allows defining the characteristics of a particular layer</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">optimizers</span> <span class="c1">#This allows using whichever optimiser we want (sgd,adam,RMSprop)</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">regularizers</span> <span class="c1">#This allows using whichever regularizer we want (l1,l2,l1_l2)</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.utils</span> <span class="kn">import</span> <span class="n">to_categorical</span> <span class="c1">#This allows using categorical cross entropy as the cost function</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="c1"># one-hot representation of labels</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">to_categorical</span><span class="p">(</span><span class="n">labels</span><span class="p">)</span>
<span class="c1"># split into train and test data</span>
<span class="n">train_size</span> <span class="o">=</span> <span class="mf">0.8</span>
<span class="n">test_size</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">train_size</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span>
<span class="n">test_size</span><span class="o">=</span><span class="n">test_size</span><span class="p">)</span>
</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="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">n_neurons_layer1</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">n_neurons_layer2</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">create_neural_network_keras</span><span class="p">(</span><span class="n">n_neurons_layer1</span><span class="p">,</span> <span class="n">n_neurons_layer2</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Sequential</span><span class="p">()</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neurons_layer1</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;sigmoid&#39;</span><span class="p">,</span> <span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lmbd</span><span class="p">)))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neurons_layer2</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;sigmoid&#39;</span><span class="p">,</span> <span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lmbd</span><span class="p">)))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_categories</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;softmax&#39;</span><span class="p">))</span>
<span class="n">sgd</span> <span class="o">=</span> <span class="n">optimizers</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">lr</span><span class="o">=</span><span class="n">eta</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">&#39;categorical_crossentropy&#39;</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="n">sgd</span><span class="p">,</span> <span class="n">metrics</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;accuracy&#39;</span><span class="p">])</span>
<span class="k">return</span> <span class="n">model</span>
</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="n">DNN_keras</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">DNN</span> <span class="o">=</span> <span class="n">create_neural_network_keras</span><span class="p">(</span><span class="n">n_neurons_layer1</span><span class="p">,</span> <span class="n">n_neurons_layer2</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">,</span>
<span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">)</span>
<span class="n">DNN</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">scores</span> <span class="o">=</span> <span class="n">DNN</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">)</span>
<span class="n">DNN_keras</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">DNN</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Learning rate = &quot;</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Lambda = &quot;</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test accuracy: </span><span class="si">%.3f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">scores</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="nb">print</span><span class="p">()</span>
</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="c1"># optional</span>
<span class="c1"># visual representation of grid search</span>
<span class="c1"># uses seaborn heatmap, could probably do this in matplotlib</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
<span class="n">DNN</span> <span class="o">=</span> <span class="n">DNN_keras</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">DNN</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">DNN</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Test Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="the-breast-cancer-data-now-with-keras">
<h2><span class="section-number">14.10. </span>The Breast Cancer Data, now with Keras<a class="headerlink" href="#the-breast-cancer-data-now-with-keras" 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">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Input</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.models</span> <span class="kn">import</span> <span class="n">Sequential</span> <span class="c1">#This allows appending layers to existing models</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Dense</span> <span class="c1">#This allows defining the characteristics of a particular layer</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">optimizers</span> <span class="c1">#This allows using whichever optimiser we want (sgd,adam,RMSprop)</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">regularizers</span> <span class="c1">#This allows using whichever regularizer we want (l1,l2,l1_l2)</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.utils</span> <span class="kn">import</span> <span class="n">to_categorical</span> <span class="c1">#This allows using categorical cross entropy as the cost function</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">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span> <span class="k">as</span> <span class="n">splitter</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
<span class="kn">import</span> <span class="nn">pickle</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="sd">&quot;&quot;&quot;Load breast cancer dataset&quot;&quot;&quot;</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c1">#create same seed for random number every time</span>
<span class="n">cancer</span><span class="o">=</span><span class="n">load_breast_cancer</span><span class="p">()</span> <span class="c1">#Download breast cancer dataset</span>
<span class="n">inputs</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span> <span class="c1">#Feature matrix of 569 rows (samples) and 30 columns (parameters)</span>
<span class="n">outputs</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span> <span class="c1">#Label array of 569 rows (0 for benign and 1 for malignant)</span>
<span class="n">labels</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">30</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;The content of the breast cancer dataset is:&#39;</span><span class="p">)</span> <span class="c1">#Print information about the datasets</span>
<span class="nb">print</span><span class="p">(</span><span class="n">labels</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;-------------------------&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;inputs = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;outputs = &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">outputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;labels = &quot;</span><span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
<span class="n">x</span><span class="o">=</span><span class="n">inputs</span> <span class="c1">#Reassign the Feature and Label matrices to other variables</span>
<span class="n">y</span><span class="o">=</span><span class="n">outputs</span>
<span class="c1">#%% </span>
<span class="c1"># Visualisation of dataset (for correlation analysis)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</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">x</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</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;Mean radius&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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;Mean perimeter&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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">plt</span><span class="o">.</span><span class="n">figure</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">x</span><span class="p">[:,</span><span class="mi">5</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">6</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</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;Mean compactness&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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;Mean concavity&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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">plt</span><span class="o">.</span><span class="n">figure</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">x</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</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;Mean radius&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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;Mean texture&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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">plt</span><span class="o">.</span><span class="n">figure</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">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</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;Mean perimeter&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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;Mean compactness&#39;</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#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="c1"># Generate training and testing datasets</span>
<span class="c1">#Select features relevant to classification (texture,perimeter,compactness and symmetery) </span>
<span class="c1">#and add to input matrix</span>
<span class="n">temp1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</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">temp2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</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">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">temp1</span><span class="p">,</span><span class="n">temp2</span><span class="p">))</span>
<span class="n">temp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">5</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">5</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">hstack</span><span class="p">((</span><span class="n">X</span><span class="p">,</span><span class="n">temp</span><span class="p">))</span>
<span class="n">temp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">8</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">8</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">hstack</span><span class="p">((</span><span class="n">X</span><span class="p">,</span><span class="n">temp</span><span class="p">))</span>
<span class="n">X_train</span><span class="p">,</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_train</span><span class="p">,</span><span class="n">y_test</span><span class="o">=</span><span class="n">splitter</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">test_size</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span> <span class="c1">#Split datasets into training and testing</span>
<span class="n">y_train</span><span class="o">=</span><span class="n">to_categorical</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span> <span class="c1">#Convert labels to categorical when using categorical cross entropy</span>
<span class="n">y_test</span><span class="o">=</span><span class="n">to_categorical</span><span class="p">(</span><span class="n">y_test</span><span class="p">)</span>
<span class="k">del</span> <span class="n">temp1</span><span class="p">,</span><span class="n">temp2</span><span class="p">,</span><span class="n">temp</span>
<span class="c1"># %%</span>
<span class="c1"># Define tunable parameters&quot;</span>
<span class="n">eta</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span> <span class="c1">#Define vector of learning rates (parameter to SGD optimiser)</span>
<span class="n">lamda</span><span class="o">=</span><span class="mf">0.01</span> <span class="c1">#Define hyperparameter</span>
<span class="n">n_layers</span><span class="o">=</span><span class="mi">2</span> <span class="c1">#Define number of hidden layers in the model</span>
<span class="n">n_neuron</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span> <span class="c1">#Define number of neurons per layer</span>
<span class="n">epochs</span><span class="o">=</span><span class="mi">100</span> <span class="c1">#Number of reiterations over the input data</span>
<span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span> <span class="c1">#Number of samples per gradient update</span>
<span class="c1"># %%</span>
<span class="sd">&quot;&quot;&quot;Define function to return Deep Neural Network model&quot;&quot;&quot;</span>
<span class="k">def</span> <span class="nf">NN_model</span><span class="p">(</span><span class="n">inputsize</span><span class="p">,</span><span class="n">n_layers</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">lamda</span><span class="p">):</span>
<span class="n">model</span><span class="o">=</span><span class="n">Sequential</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">n_layers</span><span class="p">):</span> <span class="c1">#Run loop to add hidden layers to the model</span>
<span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="o">==</span><span class="mi">0</span><span class="p">):</span> <span class="c1">#First layer requires input dimensions</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">activation</span><span class="o">=</span><span class="s1">&#39;relu&#39;</span><span class="p">,</span><span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lamda</span><span class="p">),</span><span class="n">input_dim</span><span class="o">=</span><span class="n">inputsize</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span> <span class="c1">#Subsequent layers are capable of automatic shape inferencing</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">activation</span><span class="o">=</span><span class="s1">&#39;relu&#39;</span><span class="p">,</span><span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lamda</span><span class="p">)))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">activation</span><span class="o">=</span><span class="s1">&#39;softmax&#39;</span><span class="p">))</span> <span class="c1">#2 outputs - ordered and disordered (softmax for prob)</span>
<span class="n">sgd</span><span class="o">=</span><span class="n">optimizers</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">lr</span><span class="o">=</span><span class="n">eta</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">&#39;categorical_crossentropy&#39;</span><span class="p">,</span><span class="n">optimizer</span><span class="o">=</span><span class="n">sgd</span><span class="p">,</span><span class="n">metrics</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;accuracy&#39;</span><span class="p">])</span>
<span class="k">return</span> <span class="n">model</span>
<span class="n">Train_accuracy</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">),</span><span class="nb">len</span><span class="p">(</span><span class="n">eta</span><span class="p">)))</span> <span class="c1">#Define matrices to store accuracy scores as a function</span>
<span class="n">Test_accuracy</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">),</span><span class="nb">len</span><span class="p">(</span><span class="n">eta</span><span class="p">)))</span> <span class="c1">#of learning rate and number of hidden neurons for </span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">)):</span> <span class="c1">#run loops over hidden neurons and learning rates to calculate </span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta</span><span class="p">)):</span> <span class="c1">#accuracy scores </span>
<span class="n">DNN_model</span><span class="o">=</span><span class="n">NN_model</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">n_layers</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">eta</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="n">lamda</span><span class="p">)</span>
<span class="n">DNN_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">,</span><span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span><span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span><span class="n">verbose</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">Train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="o">=</span><span class="n">DNN_model</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">Test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="o">=</span><span class="n">DNN_model</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">plot_data</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">data</span><span class="p">,</span><span class="n">title</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="c1"># plot results</span>
<span class="n">fontsize</span><span class="o">=</span><span class="mi">16</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">cax</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">&#39;nearest&#39;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">cbar</span><span class="o">=</span><span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">cax</span><span class="p">)</span>
<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;accuracy (%)&#39;</span><span class="p">,</span><span class="n">rotation</span><span class="o">=</span><span class="mi">90</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="n">fontsize</span><span class="p">)</span>
<span class="n">cbar</span><span class="o">.</span><span class="n">set_ticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">.2</span><span class="p">,</span><span class="mf">.4</span><span class="p">,</span><span class="mf">0.6</span><span class="p">,</span><span class="mf">0.8</span><span class="p">,</span><span class="mf">1.0</span><span class="p">])</span>
<span class="n">cbar</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([</span><span class="s1">&#39;0%&#39;</span><span class="p">,</span><span class="s1">&#39;20%&#39;</span><span class="p">,</span><span class="s1">&#39;40%&#39;</span><span class="p">,</span><span class="s1">&#39;60%&#39;</span><span class="p">,</span><span class="s1">&#39;80%&#39;</span><span class="p">,</span><span class="s1">&#39;100%&#39;</span><span class="p">])</span>
<span class="c1"># put text on matrix elements</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x_val</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">y_val</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">))):</span>
<span class="n">c</span> <span class="o">=</span> <span class="s2">&quot;$</span><span class="si">{0:.1f}</span><span class="se">\\</span><span class="s2">%$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="mi">100</span><span class="o">*</span><span class="n">data</span><span class="p">[</span><span class="n">j</span><span class="p">,</span><span class="n">i</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x_val</span><span class="p">,</span> <span class="n">y_val</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">)</span>
<span class="c1"># convert axis vaues to to string labels</span>
<span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">]</span>
<span class="n">y</span><span class="o">=</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y</span><span class="p">]</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([</span><span class="s1">&#39;&#39;</span><span class="p">]</span><span class="o">+</span><span class="n">y</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;$</span><span class="se">\\</span><span class="s1">mathrm{learning</span><span class="se">\\</span><span class="s1"> rate}$&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="n">fontsize</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;$</span><span class="se">\\</span><span class="s1">mathrm{hidden</span><span class="se">\\</span><span class="s1"> neurons}$&#39;</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="n">fontsize</span><span class="p">)</span>
<span class="k">if</span> <span class="n">title</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</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="n">title</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</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">plot_data</span><span class="p">(</span><span class="n">eta</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">Train_accuracy</span><span class="p">,</span> <span class="s1">&#39;training&#39;</span><span class="p">)</span>
<span class="n">plot_data</span><span class="p">(</span><span class="n">eta</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">Test_accuracy</span><span class="p">,</span> <span class="s1">&#39;testing&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="fine-tuning-neural-network-hyperparameters">
<h2><span class="section-number">14.11. </span>Fine-tuning neural network hyperparameters<a class="headerlink" href="#fine-tuning-neural-network-hyperparameters" title="Permalink to this headline"></a></h2>
<p>The flexibility of neural networks is also one of their main
drawbacks: there are many hyperparameters to tweak. Not only can you
use any imaginable network topology (how neurons/nodes are interconnected),
but even in a simple FFNN you can change the number of layers, the
number of neurons per layer, the type of activation function to use in
each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you
know what combination of hyperparameters is the best for your task?</p>
<ul class="simple">
<li><p>You can use grid search with cross-validation to find the right hyperparameters.</p></li>
</ul>
<p>However,since there are many hyperparameters to tune, and since
training a neural network on a large dataset takes a lot of time, you
will only be able to explore a tiny part of the hyperparameter space.</p>
<ul class="simple">
<li><p>You can use randomized search.</p></li>
<li><p>Or use tools like <a class="reference external" href="http://oscar.calldesk.ai/">Oscar</a>, which implements more complex algorithms to help you find a good set of hyperparameters quickly.</p></li>
</ul>
<p>For many problems you can start with just one or two hidden layers and it will work just fine.
For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a
few hundred neurons.
You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of
neurons, in roughly the same amount of training time.</p>
<p>For more complex problems, you can gradually
ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such
as large image classification or speech recognition, typically require networks with dozens of layers
and they need a huge amount
of training data. However, you will rarely have to train such networks from scratch: it is much more
common to reuse parts of a pretrained state-of-the-art network that performs a similar task.</p>
</div>
<div class="section" id="which-activation-function-should-i-use">
<h2><span class="section-number">14.12. </span>Which activation function should I use?<a class="headerlink" href="#which-activation-function-should-i-use" title="Permalink to this headline"></a></h2>
<p>The Back propagation algorithm we derived above works by going from
the output layer to the input layer, propagating the error gradient on
the way. Once the algorithm has computed the gradient of the cost
function with regards to each parameter in the network, it uses these
gradients to update each parameter with a Gradient Descent (GD) step.</p>
<p>Unfortunately for us, the gradients often get smaller and smaller as the
algorithm progresses down to the first hidden layers. As a result, the
GD update leaves the lower layer connection weights
virtually unchanged, and training never converges to a good
solution. This is known in the literature as
<strong>the vanishing gradients problem</strong>.</p>
<p>In other cases, the opposite can happen, namely the the gradients can grow bigger and
bigger. The result is that many of the layers get large updates of the
weights the
algorithm diverges. This is the <strong>exploding gradients problem</strong>, which is
mostly encountered in recurrent neural networks. More generally, deep
neural networks suffer from unstable gradients, different layers may
learn at widely different speeds</p>
<p>Although this unfortunate behavior has been empirically observed for
quite a while (it was one of the reasons why deep neural networks were
mostly abandoned for a long time), it is only around 2010 that
significant progress was made in understanding it.</p>
<p>A paper titled <a class="reference external" href="http://proceedings.mlr.press/v9/glorot10a.html">Understanding the Difficulty of Training Deep
Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio</a> found that
the problems with the popular logistic
sigmoid activation function and the weight initialization technique
that was most popular at the time, namely random initialization using
a normal distribution with a mean of 0 and a standard deviation of
1.</p>
<p>They showed that with this activation function and this
initialization scheme, the variance of the outputs of each layer is
much greater than the variance of its inputs. Going forward in the
network, the variance keeps increasing after each layer until the
activation function saturates at the top layers. This is actually made
worse by the fact that the logistic function has a mean of 0.5, not 0
(the hyperbolic tangent function has a mean of 0 and behaves slightly
better than the logistic function in deep networks).</p>
<p>Looking at the logistic activation function, when inputs become large
(negative or positive), the function saturates at 0 or 1, with a
derivative extremely close to 0. Thus when backpropagation kicks in,
it has virtually no gradient to propagate back through the network,
and what little gradient exists keeps getting diluted as
backpropagation progresses down through the top layers, so there is
really nothing left for the lower layers.</p>
<p>In their paper, Glorot and Bengio propose a way to significantly
alleviate this problem. We need the signal to flow properly in both
directions: in the forward direction when making predictions, and in
the reverse direction when backpropagating gradients. We don’t want
the signal to die out, nor do we want it to explode and saturate. For
the signal to flow properly, the authors argue that we need the
variance of the outputs of each layer to be equal to the variance of
its inputs, and we also need the gradients to have equal variance
before and after flowing through a layer in the reverse direction.</p>
<p>One of the insights in the 2010 paper by Glorot and Bengio was that
the vanishing/exploding gradients problems were in part due to a poor
choice of activation function. Until then most people had assumed that
if Nature had chosen to use roughly sigmoid activation functions in
biological neurons, they must be an excellent choice. But it turns out
that other activation functions behave much better in deep neural
networks, in particular the ReLU activation function, mostly because
it does not saturate for positive values (and also because it is quite
fast to compute).</p>
</div>
<div class="section" id="the-relu-function-family">
<h2><span class="section-number">14.13. </span>The RELU function family<a class="headerlink" href="#the-relu-function-family" title="Permalink to this headline"></a></h2>
<p>The ReLU activation function suffers from a problem known as the dying
ReLUs: during training, some neurons effectively die, meaning they
stop outputting anything other than 0.</p>
<p>In some cases, you may find that half of your network’s neurons are
dead, especially if you used a large learning rate. During training,
if a neuron’s weights get updated such that the weighted sum of the
neuron’s inputs is negative, it will start outputting 0. When this
happen, the neuron is unlikely to come back to life since the gradient
of the ReLU function is 0 when its input is negative.</p>
<p>To solve this problem, nowadays practitioners use a variant of the ReLU
function, such as the leaky ReLU discussed above or the so-called
exponential linear unit (ELU) function</p>
<div class="math notranslate nohighlight">
\[\begin{split}
ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) &amp; z &lt; 0,\\ z &amp; z \ge 0.\end{array}\right.
\end{split}\]</div>
<p>In general it seems that the ELU activation function is better than
the leaky ReLU function (and its variants), which is better than
ReLU. ReLU performs better than <span class="math notranslate nohighlight">\(\tanh\)</span> which in turn performs better
than the logistic function.</p>
<p>If runtime
performance is an issue, then you may opt for the leaky ReLU function over the
ELU function If you don’t
want to tweak yet another hyperparameter, you may just use the default
<span class="math notranslate nohighlight">\(\alpha\)</span> of <span class="math notranslate nohighlight">\(0.01\)</span> for the leaky ReLU, and <span class="math notranslate nohighlight">\(1\)</span> for ELU. If you have
spare time and computing power, you can use cross-validation or
bootstrap to evaluate other activation functions.</p>
<p>In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).</p>
<p>It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.</p>
<p><strong>For the output layer:</strong></p>
<ul class="simple">
<li><p>For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).</p></li>
<li><p>For regression tasks, you can simply use no activation function at all.</p></li>
</ul>
</div>
<div class="section" id="batch-normalization">
<h2><span class="section-number">14.14. </span>Batch Normalization<a class="headerlink" href="#batch-normalization" title="Permalink to this headline"></a></h2>
<p>Batch Normalization
aims to address the vanishing/exploding gradients problems, and more generally the problem that the
distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.</p>
<p>The technique consists of adding an operation in the model just before the activation function of each
layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new
parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model
learn the optimal scale and mean of the inputs for each layer.
In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and
standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current
mini-batch, from this the name batch normalization.</p>
</div>
<div class="section" id="dropout">
<h2><span class="section-number">14.15. </span>Dropout<a class="headerlink" href="#dropout" title="Permalink to this headline"></a></h2>
<p>It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but
excluding the output neurons) has a probability <span class="math notranslate nohighlight">\(p\)</span> of being temporarily dropped out, meaning it will be
entirely ignored during this training step, but it may be active during the next step.</p>
<p>The
hyperparameter <span class="math notranslate nohighlight">\(p\)</span> is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.
It is viewed as one of the most popular regularization techniques.</p>
</div>
<div class="section" id="gradient-clipping">
<h2><span class="section-number">14.16. </span>Gradient Clipping<a class="headerlink" href="#gradient-clipping" title="Permalink to this headline"></a></h2>
<p>A popular technique to lessen the exploding gradients problem is to simply clip the gradients during
backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural
networks).</p>
<p>This technique is called Gradient Clipping.</p>
<p>In general however, Batch
Normalization is preferred.</p>
</div>
<div class="section" id="a-top-down-perspective-on-neural-networks">
<h2><span class="section-number">14.17. </span>A top-down perspective on Neural networks<a class="headerlink" href="#a-top-down-perspective-on-neural-networks" title="Permalink to this headline"></a></h2>
<p>The first thing we would like to do is divide the data into two or three
parts. A training set, a validation or dev (development) set, and a
test set. The test set is the data on which we want to make
predictions. The dev set is a subset of the training data we use to
check how well we are doing out-of-sample, after training the model on
the training dataset. We use the validation error as a proxy for the
test error in order to make tweaks to our model. It is crucial that we
do not use any of the test data to train the algorithm. This is a
cardinal sin in ML. Then:</p>
<ul class="simple">
<li><p>Estimate optimal error rate</p></li>
<li><p>Minimize underfitting (bias) on training data set.</p></li>
<li><p>Make sure you are not overfitting.</p></li>
</ul>
<p>If the validation and test sets are drawn from the same distributions,
then a good performance on the validation set should lead to similarly
good performance on the test set.</p>
<p>However, sometimes
the training data and test data differ in subtle ways because, for
example, they are collected using slightly different methods, or
because it is cheaper to collect data in one way versus another. In
this case, there can be a mismatch between the training and test
data. This can lead to the neural network overfitting these small
differences between the test and training sets, and a poor performance
on the test set despite having a good performance on the validation
set. To rectify this, Andrew Ng suggests making two validation or dev
sets, one constructed from the training data and one constructed from
the test data. The difference between the performance of the algorithm
on these two validation sets quantifies the train-test mismatch. This
can serve as another important diagnostic when using DNNs for
supervised learning.</p>
</div>
<div class="section" id="limitations-of-supervised-learning-with-deep-networks">
<h2><span class="section-number">14.18. </span>Limitations of supervised learning with deep networks<a class="headerlink" href="#limitations-of-supervised-learning-with-deep-networks" title="Permalink to this headline"></a></h2>
<p>Like all statistical methods, supervised learning using neural
networks has important limitations. This is especially important when
one seeks to apply these methods, especially to physics problems. Like
all tools, DNNs are not a universal solution. Often, the same or
better performance on a task can be achieved by using a few
hand-engineered features (or even a collection of random
features).</p>
<p>Here we list some of the important limitations of supervised neural network based models.</p>
<ul class="simple">
<li><p><strong>Need labeled data</strong>. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).</p></li>
<li><p><strong>Supervised neural networks are extremely data intensive.</strong> DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.</p></li>
<li><p><strong>Homogeneous data.</strong> Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.</p></li>
<li><p><strong>Many problems are not about prediction.</strong> In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a <em>wrong</em> model. The model might or might not be useful for understanding the underlying science.</p></li>
</ul>
<p>Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.</p>
</div>
</div>
<script type="text/x-thebe-config">
{
requestKernel: true,
binderOptions: {
repo: "binder-examples/jupyter-stacks-datascience",
ref: "master",
},
codeMirrorConfig: {
theme: "abcdef",
mode: "python"
},
kernelOptions: {
kernelName: "python3",
path: "./."
},
predefinedOutput: true
}
</script>
<script>kernelName = 'python3'</script>
</div>
<!-- Previous / next buttons -->
<div class='prev-next-area'>
<a class='left-prev' id="prev-link" href="chapter9.html" title="previous page">
<i class="fas fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title"><span class="section-number">13. </span>Neural networks</p>
</div>
</a>
<a class='right-next' id="next-link" href="chapter11.html" title="next page">
<div class="prev-next-info">
<p class="prev-next-subtitle">next</p>
<p class="prev-next-title"><span class="section-number">15. </span>Solving Differential Equations with Deep Learning</p>
</div>
<i class="fas fa-angle-right"></i>
</a>
</div>
</div>
</div>
<footer class="footer">
<p>
By Morten Hjorth-Jensen<br/>
&copy; Copyright 2021.<br/>
</p>
</footer>
</main>
</div>
</div>
<script src="_static/js/index.be7d3bbb2ef33a8344ce.js"></script>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink