Morten Hjorth-Jensen
1922 lines
106 KiB
HTML
1922 lines
106 KiB
HTML
<!--
|
|
Automatically generated HTML file from DocOnce source
|
|
(https://github.com/doconce/doconce/)
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Week 39: Optimization and Gradient Methods">
|
|
|
|
<title>Week 39: Optimization and Gradient Methods</title>
|
|
|
|
|
|
<style type="text/css">
|
|
/* bloodish style */
|
|
|
|
body {
|
|
font-family: Helvetica, Verdana, Arial, Sans-serif;
|
|
color: #404040;
|
|
background: #ffffff;
|
|
}
|
|
h1 { font-size: 1.8em; color: #8A0808; }
|
|
h2 { font-size: 1.6em; color: #8A0808; }
|
|
h3 { font-size: 1.4em; color: #8A0808; }
|
|
h4 { color: #8A0808; }
|
|
a { color: #8A0808; text-decoration:none; }
|
|
tt { font-family: "Courier New", Courier; }
|
|
/* pre style removed because it will interfer with pygments */
|
|
p { text-indent: 0px; }
|
|
hr { border: 0; width: 80%; border-bottom: 1px solid #aaa}
|
|
p.caption { width: 80%; font-style: normal; text-align: left; }
|
|
hr.figure { border: 0; width: 80%; border-bottom: 1px solid #aaa}
|
|
.alert-text-small { font-size: 80%; }
|
|
.alert-text-large { font-size: 130%; }
|
|
.alert-text-normal { font-size: 90%; }
|
|
.alert {
|
|
padding:8px 35px 8px 14px; margin-bottom:18px;
|
|
text-shadow:0 1px 0 rgba(255,255,255,0.5);
|
|
border:1px solid #bababa;
|
|
border-radius: 4px;
|
|
-webkit-border-radius: 4px;
|
|
-moz-border-radius: 4px;
|
|
color: #555;
|
|
background-color: #f8f8f8;
|
|
background-position: 10px 5px;
|
|
background-repeat: no-repeat;
|
|
background-size: 38px;
|
|
padding-left: 55px;
|
|
width: 75%;
|
|
}
|
|
.alert-block {padding-top:14px; padding-bottom:14px}
|
|
.alert-block > p, .alert-block > ul {margin-bottom:1em}
|
|
.alert li {margin-top: 1em}
|
|
.alert-block p+p {margin-top:5px}
|
|
.alert-notice { background-image: url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_gray_notice.png); }
|
|
.alert-summary { background-image:url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_gray_summary.png); }
|
|
.alert-warning { background-image: url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_gray_warning.png); }
|
|
.alert-question {background-image:url(https://cdn.rawgit.com/doconce/doconce/master/bundled/html_images/small_gray_question.png); }
|
|
|
|
div { text-align: justify; text-justify: inter-word; }
|
|
</style>
|
|
|
|
|
|
</head>
|
|
|
|
<!-- tocinfo
|
|
{'highest level': 2,
|
|
'sections': [('Plan for week 39', 2, None, 'plan-for-week-39'),
|
|
('Thursday September 30', 2, None, 'thursday-september-30'),
|
|
('Searching for Optimal Regularization Parameters $\\lambda$',
|
|
2,
|
|
None,
|
|
'searching-for-optimal-regularization-parameters-lambda'),
|
|
('Grid Search', 2, None, 'grid-search'),
|
|
('Randomized Grid Search', 2, None, 'randomized-grid-search'),
|
|
('Optimization, the central part of any Machine Learning '
|
|
'algortithm',
|
|
2,
|
|
None,
|
|
'optimization-the-central-part-of-any-machine-learning-algortithm'),
|
|
('Revisiting our Logistic Regression case',
|
|
2,
|
|
None,
|
|
'revisiting-our-logistic-regression-case'),
|
|
('The equations to solve', 2, None, 'the-equations-to-solve'),
|
|
("Solving using Newton-Raphson's method",
|
|
2,
|
|
None,
|
|
'solving-using-newton-raphson-s-method'),
|
|
("Brief reminder on Newton-Raphson's method",
|
|
2,
|
|
None,
|
|
'brief-reminder-on-newton-raphson-s-method'),
|
|
('The equations', 2, None, 'the-equations'),
|
|
('Simple geometric interpretation',
|
|
2,
|
|
None,
|
|
'simple-geometric-interpretation'),
|
|
('Extending to more than one variable',
|
|
2,
|
|
None,
|
|
'extending-to-more-than-one-variable'),
|
|
('Steepest descent', 2, None, 'steepest-descent'),
|
|
('More on Steepest descent', 2, None, 'more-on-steepest-descent'),
|
|
('The ideal', 2, None, 'the-ideal'),
|
|
('The sensitiveness of the gradient descent',
|
|
2,
|
|
None,
|
|
'the-sensitiveness-of-the-gradient-descent'),
|
|
('Convex functions', 2, None, 'convex-functions'),
|
|
('Convex function', 2, None, 'convex-function'),
|
|
('Conditions on convex functions',
|
|
2,
|
|
None,
|
|
'conditions-on-convex-functions'),
|
|
('More on convex functions', 2, None, 'more-on-convex-functions'),
|
|
('Some simple problems', 2, None, 'some-simple-problems'),
|
|
('Standard steepest descent',
|
|
2,
|
|
None,
|
|
'standard-steepest-descent'),
|
|
('Gradient method', 2, None, 'gradient-method'),
|
|
('Steepest descent method', 2, None, 'steepest-descent-method'),
|
|
('Steepest descent method', 2, None, 'steepest-descent-method'),
|
|
('Final expressions', 2, None, 'final-expressions'),
|
|
('Steepest descent example', 2, None, 'steepest-descent-example'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Conjugate gradient method and iterations',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method-and-iterations'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Conjugate gradient method',
|
|
2,
|
|
None,
|
|
'conjugate-gradient-method'),
|
|
('Revisiting our first homework',
|
|
2,
|
|
None,
|
|
'revisiting-our-first-homework'),
|
|
('Gradient descent example', 2, None, 'gradient-descent-example'),
|
|
('The derivative of the cost/loss function',
|
|
2,
|
|
None,
|
|
'the-derivative-of-the-cost-loss-function'),
|
|
('The Hessian matrix', 2, None, 'the-hessian-matrix'),
|
|
('Simple program', 2, None, 'simple-program'),
|
|
('Gradient Descent Example', 2, None, 'gradient-descent-example'),
|
|
('And a corresponding example using _scikit-learn_',
|
|
2,
|
|
None,
|
|
'and-a-corresponding-example-using-_scikit-learn_'),
|
|
('Gradient descent and Ridge',
|
|
2,
|
|
None,
|
|
'gradient-descent-and-ridge'),
|
|
('Program example for gradient descent with Ridge Regression',
|
|
2,
|
|
None,
|
|
'program-example-for-gradient-descent-with-ridge-regression'),
|
|
('Using gradient descent methods, limitations',
|
|
2,
|
|
None,
|
|
'using-gradient-descent-methods-limitations'),
|
|
('To be added', 2, None, 'to-be-added'),
|
|
('Friday October 1', 2, None, 'friday-october-1'),
|
|
('Stochastic Gradient Descent',
|
|
2,
|
|
None,
|
|
'stochastic-gradient-descent'),
|
|
('Computation of gradients', 2, None, 'computation-of-gradients'),
|
|
('SGD example', 2, None, 'sgd-example'),
|
|
('The gradient step', 2, None, 'the-gradient-step'),
|
|
('Simple example code', 2, None, 'simple-example-code'),
|
|
('When do we stop?', 2, None, 'when-do-we-stop'),
|
|
('Slightly different approach',
|
|
2,
|
|
None,
|
|
'slightly-different-approach'),
|
|
('Program for stochastic gradient',
|
|
2,
|
|
None,
|
|
'program-for-stochastic-gradient')]}
|
|
end of tocinfo -->
|
|
|
|
<body>
|
|
|
|
|
|
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
TeX: {
|
|
equationNumbers: { autoNumber: "AMS" },
|
|
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript" async
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
|
</script>
|
|
|
|
|
|
|
|
|
|
<!-- ------------------- main content ---------------------- -->
|
|
|
|
|
|
|
|
<center><h1>Week 39: Optimization and Gradient Methods</h1></center> <!-- document title -->
|
|
|
|
<p>
|
|
<!-- author(s): Morten Hjorth-Jensen -->
|
|
|
|
<center>
|
|
<b>Morten Hjorth-Jensen</b> [1, 2]
|
|
</center>
|
|
|
|
<p>
|
|
<!-- institution(s) -->
|
|
|
|
<center>[1] <b>Department of Physics, University of Oslo</b></center>
|
|
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
|
<br>
|
|
<p>
|
|
<center><h4>Sep 30, 2021</h4></center> <!-- date -->
|
|
<br>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="plan-for-week-39">Plan for week 39 </h2>
|
|
|
|
<ul>
|
|
<li> Thursday: Repetition of Logistic regression equations and classification problems and discussion of Gradient methods. Discussion of project 1 and examples on how to implement Logistic Regression</li>
|
|
<li> Friday: Stochastic Gradient descent with examples and automatic differentiation</li>
|
|
<li> Reading recommendations:</li>
|
|
</ul>
|
|
|
|
See <a href="https://compphysics.github.io/MachineLearning/doc/web/course.html" target="_blank">lecture notes for week 39</a>.
|
|
For a good discussion on gradient methods, see Goodfellow et al section 4.3-4.5 and chapter 8. We will come back to the latter chapter in our discussion of Neural networks as well.
|
|
|
|
<p>
|
|
<b>For project 1, chapter 5 of Goodfellow et al is a good read, in particular sections 5.1-5.55 and 5.7-5.11</b>.
|
|
These sections summarize neatly what we have done till now and point to what is coming with respect to deep learning.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="thursday-september-30">Thursday September 30 </h2>
|
|
|
|
<p>
|
|
<a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/OverarchingAimsWeek39.mp4?vrtx=view-as-webpage" target="_blank">Overview Video, why do we care about gradient methods?</a>
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="searching-for-optimal-regularization-parameters-lambda">Searching for Optimal Regularization Parameters \( \lambda \) </h2>
|
|
|
|
<p>
|
|
In project 1, when using Ridge and Lasso regression, we end up
|
|
searching for the optimal parameter \( \lambda \) which minimizes our
|
|
selected scores (MSE or \( R2 \) values for example). The brute force
|
|
approach, as discussed in the code here for Ridge regression, consists
|
|
in evaluating the MSE as function of different \( \lambda \) values.
|
|
Based on these calculations, one tries then to determine the value of the hyperparameter \( \lambda \)
|
|
which results in optimal scores (for example the smallest MSE or an \( R2=1 \)).
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> linear_model
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
|
|
<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2021</span>)
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n)
|
|
|
|
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree<span style="color: #666666">-1</span>))
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,Maxpolydegree): <span style="color: #408080; font-style: italic">#No intercept column</span>
|
|
X[:,degree<span style="color: #666666">-1</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>(degree)
|
|
|
|
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
|
|
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
|
|
MSERidgePredict <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
|
|
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">2</span>, nlambdas)
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(nlambdas):
|
|
lmb <span style="color: #666666">=</span> lambdas[i]
|
|
RegRidge <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>Ridge(lmb)
|
|
RegRidge<span style="color: #666666">.</span>fit(X_train,y_train)
|
|
ypredictRidge <span style="color: #666666">=</span> RegRidge<span style="color: #666666">.</span>predict(X_test)
|
|
MSERidgePredict[i] <span style="color: #666666">=</span> MSE(y_test,ypredictRidge)
|
|
|
|
<span style="color: #408080; font-style: italic"># Now plot the results</span>
|
|
plt<span style="color: #666666">.</span>figure()
|
|
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), MSERidgePredict, <span style="color: #BA2121">'g--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'MSE SL Ridge Test'</span>)
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MSE'</span>)
|
|
plt<span style="color: #666666">.</span>legend()
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
Here we have performed a rather data greedy calculation as function of the regularization parameter \( \lambda \). There is no resampling here. The latter can easily be added by employing the function <b>RidgeCV</b> instead of just calling the <b>Ridge</b> function. For <b>RidgeCV</b> we need to pass the array of \( \lambda \) values.
|
|
By inspecting the figure we can in turn determine which is the optimal regularization parameter.
|
|
This becomes however less functional in the long run.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="grid-search">Grid Search </h2>
|
|
|
|
<p>
|
|
An alternative is to use the so-called grid search functionality
|
|
included with the library <b>Scikit-Learn</b>, as demonstrated for the same
|
|
example here.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> GridSearchCV
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
|
|
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
|
|
<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2021</span>)
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n)
|
|
|
|
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree<span style="color: #666666">-1</span>))
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,Maxpolydegree): <span style="color: #408080; font-style: italic">#No intercept column</span>
|
|
X[:,degree<span style="color: #666666">-1</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>(degree)
|
|
|
|
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
|
|
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-4</span>, <span style="color: #666666">2</span>, nlambdas)
|
|
<span style="color: #408080; font-style: italic"># create and fit a ridge regression model, testing each alpha</span>
|
|
model <span style="color: #666666">=</span> Ridge()
|
|
gridsearch <span style="color: #666666">=</span> GridSearchCV(estimator<span style="color: #666666">=</span>model, param_grid<span style="color: #666666">=</span><span style="color: #008000">dict</span>(alpha<span style="color: #666666">=</span>lambdas))
|
|
gridsearch<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
<span style="color: #008000">print</span>(gridsearch)
|
|
ypredictRidge <span style="color: #666666">=</span> gridsearch<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #408080; font-style: italic"># summarize the results of the grid search</span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Best estimated lambda-value: </span><span style="color: #BB6688; font-weight: bold">{</span>gridsearch<span style="color: #666666">.</span>best_estimator_<span style="color: #666666">.</span>alpha<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE score: </span><span style="color: #BB6688; font-weight: bold">{</span>MSE(y_test,ypredictRidge)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"R2 score: </span><span style="color: #BB6688; font-weight: bold">{</span>R2(y_test,ypredictRidge)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
</pre></div>
|
|
<p>
|
|
By default the grid search function includes cross validation with
|
|
five folds. The <a href="https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" target="_blank">Scikit-Learn
|
|
documentation</a>
|
|
contains more information on how to set the different parameters.
|
|
|
|
<p>
|
|
If we take out the random noise, running the above codes results in \( \lambda=0 \) yielding the best fit.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="randomized-grid-search">Randomized Grid Search </h2>
|
|
|
|
<p>
|
|
An alternative to the above manual grid set up, is to use a random
|
|
search where the parameters are tuned from a random distribution
|
|
(uniform below) for a fixed number of iterations. A model is
|
|
constructed and evaluated for each combination of chosen parameters.
|
|
We repeat the previous example but now with a random search. Note
|
|
that values of \( \lambda \) are now limited to be within \( x\in
|
|
[0,1] \). This domain may not be the most relevant one for the specific
|
|
case under study.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> GridSearchCV
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">scipy.stats</span> <span style="color: #008000; font-weight: bold">import</span> uniform <span style="color: #008000; font-weight: bold">as</span> randuniform
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> RandomizedSearchCV
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
|
|
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
|
|
<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2021</span>)
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n)
|
|
|
|
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((n,Maxpolydegree<span style="color: #666666">-1</span>))
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,Maxpolydegree): <span style="color: #408080; font-style: italic">#No intercept column</span>
|
|
X[:,degree<span style="color: #666666">-1</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**</span>(degree)
|
|
|
|
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
|
|
|
|
param_grid <span style="color: #666666">=</span> {<span style="color: #BA2121">'alpha'</span>: randuniform()}
|
|
<span style="color: #408080; font-style: italic"># create and fit a ridge regression model, testing each alpha</span>
|
|
model <span style="color: #666666">=</span> Ridge()
|
|
gridsearch <span style="color: #666666">=</span> RandomizedSearchCV(estimator<span style="color: #666666">=</span>model, param_distributions<span style="color: #666666">=</span>param_grid, n_iter<span style="color: #666666">=100</span>)
|
|
gridsearch<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
<span style="color: #008000">print</span>(gridsearch)
|
|
ypredictRidge <span style="color: #666666">=</span> gridsearch<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #408080; font-style: italic"># summarize the results of the grid search</span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Best estimated lambda-value: </span><span style="color: #BB6688; font-weight: bold">{</span>gridsearch<span style="color: #666666">.</span>best_estimator_<span style="color: #666666">.</span>alpha<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"MSE score: </span><span style="color: #BB6688; font-weight: bold">{</span>MSE(y_test,ypredictRidge)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"R2 score: </span><span style="color: #BB6688; font-weight: bold">{</span>R2(y_test,ypredictRidge)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="optimization-the-central-part-of-any-machine-learning-algortithm">Optimization, the central part of any Machine Learning algortithm </h2>
|
|
|
|
<p>
|
|
Almost every problem in machine learning and data science starts with
|
|
a dataset \( X \), a model \( g(\beta) \), which is a function of the
|
|
parameters \( \beta \) and a cost function \( C(X, g(\beta)) \) that allows
|
|
us to judge how well the model \( g(\beta) \) explains the observations
|
|
\( X \). The model is fit by finding the values of \( \beta \) that minimize
|
|
the cost function. Ideally we would be able to solve for \( \beta \)
|
|
analytically, however this is not possible in general and we must use
|
|
some approximative/numerical method to compute the minimum.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="revisiting-our-logistic-regression-case">Revisiting our Logistic Regression case </h2>
|
|
|
|
<p>
|
|
In our discussion on Logistic Regression we studied the
|
|
case of
|
|
two classes, with \( y_i \) either
|
|
\( 0 \) or \( 1 \). Furthermore we assumed also that we have only two
|
|
parameters \( \beta \) in our fitting, that is we
|
|
defined probabilities
|
|
|
|
$$
|
|
\begin{align*}
|
|
p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
|
|
p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
|
|
\end{align*}
|
|
$$
|
|
|
|
where \( \boldsymbol{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-equations-to-solve">The equations to solve </h2>
|
|
|
|
<p>
|
|
Our compact equations used a definition of a vector \( \boldsymbol{y} \) with \( n \)
|
|
elements \( y_i \), an \( n\times p \) matrix \( \boldsymbol{X} \) which contains the
|
|
\( x_i \) values and a vector \( \boldsymbol{p} \) of fitted probabilities
|
|
\( p(y_i\vert x_i,\boldsymbol{\beta}) \). We rewrote in a more compact form
|
|
the first derivative of the cost function as
|
|
|
|
$$
|
|
\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
|
|
$$
|
|
|
|
<p>
|
|
If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements
|
|
\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as
|
|
|
|
$$
|
|
\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
|
|
$$
|
|
|
|
This defines what is called the Hessian matrix.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="solving-using-newton-raphson-s-method">Solving using Newton-Raphson's method </h2>
|
|
|
|
<p>
|
|
If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
|
|
|
|
<p>
|
|
Our iterative scheme is then given by
|
|
|
|
$$
|
|
\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T}\right)^{-1}_{\boldsymbol{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right)_{\boldsymbol{\beta}^{\mathrm{old}}},
|
|
$$
|
|
|
|
or in matrix form as
|
|
|
|
$$
|
|
\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X} \right)^{-1}\times \left(-\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{p}) \right)_{\boldsymbol{\beta}^{\mathrm{old}}}.
|
|
$$
|
|
|
|
The right-hand side is computed with the old values of \( \beta \).
|
|
|
|
<p>
|
|
If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="brief-reminder-on-newton-raphson-s-method">Brief reminder on Newton-Raphson's method </h2>
|
|
|
|
<p>
|
|
Let us quickly remind ourselves how we derive the above method.
|
|
|
|
<p>
|
|
Perhaps the most celebrated of all one-dimensional root-finding
|
|
routines is Newton's method, also called the Newton-Raphson
|
|
method. This method requires the evaluation of both the
|
|
function \( f \) and its derivative \( f' \) at arbitrary points.
|
|
If you can only calculate the derivative
|
|
numerically and/or your function is not of the smooth type, we
|
|
normally discourage the use of this method.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-equations">The equations </h2>
|
|
|
|
<p>
|
|
The Newton-Raphson formula consists geometrically of extending the
|
|
tangent line at a current point until it crosses zero, then setting
|
|
the next guess to the abscissa of that zero-crossing. The mathematics
|
|
behind this method is rather simple. Employing a Taylor expansion for
|
|
\( x \) sufficiently close to the solution \( s \), we have
|
|
|
|
$$
|
|
f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
|
|
\label{eq:taylornr}
|
|
$$
|
|
|
|
<p>
|
|
For small enough values of the function and for well-behaved
|
|
functions, the terms beyond linear are unimportant, hence we obtain
|
|
|
|
$$
|
|
f(x)+(s-x)f'(x)\approx 0,
|
|
$$
|
|
|
|
yielding
|
|
$$
|
|
s\approx x-\frac{f(x)}{f'(x)}.
|
|
$$
|
|
|
|
<p>
|
|
Having in mind an iterative procedure, it is natural to start iterating with
|
|
$$
|
|
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="simple-geometric-interpretation">Simple geometric interpretation </h2>
|
|
|
|
<p>
|
|
The above is Newton-Raphson's method. It has a simple geometric
|
|
interpretation, namely \( x_{n+1} \) is the point where the tangent from
|
|
\( (x_n,f(x_n)) \) crosses the \( x \)-axis. Close to the solution,
|
|
Newton-Raphson converges fast to the desired result. However, if we
|
|
are far from a root, where the higher-order terms in the series are
|
|
important, the Newton-Raphson formula can give grossly inaccurate
|
|
results. For instance, the initial guess for the root might be so far
|
|
from the true root as to let the search interval include a local
|
|
maximum or minimum of the function. If an iteration places a trial
|
|
guess near such a local extremum, so that the first derivative nearly
|
|
vanishes, then Newton-Raphson may fail totally
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="extending-to-more-than-one-variable">Extending to more than one variable </h2>
|
|
|
|
<p>
|
|
Newton's method can be generalized to systems of several non-linear equations
|
|
and variables. Consider the case with two equations
|
|
$$
|
|
\begin{array}{cc} f_1(x_1,x_2) &=0\\
|
|
f_2(x_1,x_2) &=0,\end{array}
|
|
$$
|
|
|
|
which we Taylor expand to obtain
|
|
|
|
$$
|
|
\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
|
|
\partial f_1/\partial x_1+h_2
|
|
\partial f_1/\partial x_2+\dots\\
|
|
0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
|
|
\partial f_2/\partial x_1+h_2
|
|
\partial f_2/\partial x_2+\dots
|
|
\end{array}.
|
|
$$
|
|
|
|
Defining the Jacobian matrix \( {\bf \boldsymbol{J}} \) we have
|
|
$$
|
|
{\bf \boldsymbol{J}}=\left( \begin{array}{cc}
|
|
\partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
|
|
\partial f_2/\partial x_1 &\partial f_2/\partial x_2
|
|
\end{array} \right),
|
|
$$
|
|
|
|
we can rephrase Newton's method as
|
|
$$
|
|
\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
|
|
\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
|
|
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
|
|
$$
|
|
|
|
where we have defined
|
|
$$
|
|
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
|
|
-{\bf \boldsymbol{J}}^{-1}
|
|
\left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
|
|
$$
|
|
|
|
We need thus to compute the inverse of the Jacobian matrix and it
|
|
is to understand that difficulties may
|
|
arise in case \( {\bf \boldsymbol{J}} \) is nearly singular.
|
|
|
|
<p>
|
|
It is rather straightforward to extend the above scheme to systems of
|
|
more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="steepest-descent">Steepest descent </h2>
|
|
|
|
<p>
|
|
The basic idea of gradient descent is
|
|
that a function \( F(\mathbf{x}) \),
|
|
\( \mathbf{x} \equiv (x_1,\cdots,x_n) \), decreases fastest if one goes from \( \bf {x} \) in the
|
|
direction of the negative gradient \( -\nabla F(\mathbf{x}) \).
|
|
|
|
<p>
|
|
It can be shown that if
|
|
$$
|
|
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
|
|
$$
|
|
|
|
with \( \gamma_k > 0 \).
|
|
|
|
<p>
|
|
For \( \gamma_k \) small enough, then \( F(\mathbf{x}_{k+1}) \leq
|
|
F(\mathbf{x}_k) \). This means that for a sufficiently small \( \gamma_k \)
|
|
we are always moving towards smaller function values, i.e a minimum.
|
|
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="more-on-steepest-descent">More on Steepest descent </h2>
|
|
|
|
<p>
|
|
The previous observation is the basis of the method of steepest
|
|
descent, which is also referred to as just gradient descent (GD). One
|
|
starts with an initial guess \( \mathbf{x}_0 \) for a minimum of \( F \) and
|
|
computes new approximations according to
|
|
|
|
$$
|
|
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
|
|
$$
|
|
|
|
<p>
|
|
The parameter \( \gamma_k \) is often referred to as the step length or
|
|
the learning rate within the context of Machine Learning.
|
|
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="the-ideal">The ideal </h2>
|
|
|
|
<p>
|
|
Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
|
|
minimum of the function \( F \). In general we do not know if we are in a
|
|
global or local minimum. In the special case when \( F \) is a convex
|
|
function, all local minima are also global minima, so in this case
|
|
gradient descent can converge to the global solution. The advantage of
|
|
this scheme is that it is conceptually simple and straightforward to
|
|
implement. However the method in this form has some severe
|
|
limitations:
|
|
|
|
<p>
|
|
In machine learing we are often faced with non-convex high dimensional
|
|
cost functions with many local minima. Since GD is deterministic we
|
|
will get stuck in a local minimum, if the method converges, unless we
|
|
have a very good intial guess. This also implies that the scheme is
|
|
sensitive to the chosen initial condition.
|
|
|
|
<p>
|
|
Note that the gradient is a function of \( \mathbf{x} =
|
|
(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.
|
|
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="the-sensitiveness-of-the-gradient-descent">The sensitiveness of the gradient descent </h2>
|
|
|
|
<p>
|
|
The gradient descent method
|
|
is sensitive to the choice of learning rate \( \gamma_k \). This is due
|
|
to the fact that we are only guaranteed that \( F(\mathbf{x}_{k+1}) \leq
|
|
F(\mathbf{x}_k) \) for sufficiently small \( \gamma_k \). The problem is to
|
|
determine an optimal learning rate. If the learning rate is chosen too
|
|
small the method will take a long time to converge and if it is too
|
|
large we can experience erratic behavior.
|
|
|
|
<p>
|
|
Many of these shortcomings can be alleviated by introducing
|
|
randomness. One such method is that of Stochastic Gradient Descent
|
|
(SGD), see below.
|
|
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="convex-functions">Convex functions </h2>
|
|
|
|
<p>
|
|
Ideally we want our cost/loss function to be convex(concave).
|
|
|
|
<p>
|
|
First we give the definition of a convex set: A set \( C \) in
|
|
\( \mathbb{R}^n \) is said to be convex if, for all \( x \) and \( y \) in \( C \) and
|
|
all \( t \in (0,1) \) , the point \( (1 − t)x + ty \) also belongs to
|
|
C. Geometrically this means that every point on the line segment
|
|
connecting \( x \) and \( y \) is in \( C \) as discussed below.
|
|
|
|
<p>
|
|
The convex subsets of \( \mathbb{R} \) are the intervals of
|
|
\( \mathbb{R} \). Examples of convex sets of \( \mathbb{R}^2 \) are the
|
|
regular polygons (triangles, rectangles, pentagons, etc...).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="convex-function">Convex function </h2>
|
|
|
|
<p>
|
|
<b>Convex function</b>: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conditions-on-convex-functions">Conditions on convex functions </h2>
|
|
|
|
<p>
|
|
In the following we state first and second-order conditions which
|
|
ensures convexity of a function \( f \). We write \( D_f \) to denote the
|
|
domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more
|
|
details and proofs we refer to: <a href="http://stanford.edu/boyd/cvxbook/, 2004" target="_blank">S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press</a>.
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>First order condition</b>
|
|
<p>
|
|
Suppose \( f \) is differentiable (i.e \( \nabla f(x) \) is well defined for
|
|
all \( x \) in the domain of \( f \)). Then \( f \) is convex if and only if \( D_f \)
|
|
is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
|
|
for all \( x,y \in D_f \). This condition means that for a convex function
|
|
the first order Taylor expansion (right hand side above) at any point
|
|
a global under estimator of the function. To convince yourself you can
|
|
make a drawing of \( f(x) = x^2+1 \) and draw the tangent line to \( f(x) \) and
|
|
note that it is always below the graph.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>Second order condition</b>
|
|
<p>
|
|
Assume that \( f \) is twice
|
|
differentiable, i.e the Hessian matrix exists at each point in
|
|
\( D_f \). Then \( f \) is convex if and only if \( D_f \) is a convex set and its
|
|
Hessian is positive semi-definite for all \( x\in D_f \). For a
|
|
single-variable function this reduces to \( f''(x) \geq 0 \). Geometrically this means that \( f \) has nonnegative curvature
|
|
everywhere.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="more-on-convex-functions">More on convex functions </h2>
|
|
|
|
<p>
|
|
The next result is of great importance to us and the reason why we are
|
|
going on about convex functions. In machine learning we frequently
|
|
have to minimize a loss/cost function in order to find the best
|
|
parameters for the model we are considering.
|
|
|
|
<p>
|
|
Ideally we want the
|
|
global minimum (for high-dimensional models it is hard to know
|
|
if we have local or global minimum). However, if the cost/loss function
|
|
is convex the following result provides invaluable information:
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>Any minimum is global for convex functions</b>
|
|
<p>
|
|
Consider the problem of finding \( x \in \mathbb{R}^n \) such that \( f(x) \)
|
|
is minimal, where \( f \) is convex and differentiable. Then, any point
|
|
\( x^* \) that satisfies \( \nabla f(x^*) = 0 \) is a global minimum.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="some-simple-problems">Some simple problems </h2>
|
|
|
|
<ol>
|
|
<li> Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.</li>
|
|
<li> Using the second order condition show that the following functions are convex on the specified domain.</li>
|
|
|
|
<ul>
|
|
<li> \( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).</li>
|
|
<li> \( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).</li>
|
|
</ul>
|
|
|
|
<li> Let \( f(x) = x^2 \) and \( g(x) = e^x \). Show that \( f(g(x)) \) and \( g(f(x)) \) is convex for \( x \in \mathbb{R} \). Also show that if \( f(x) \) is any convex function than \( h(x) = e^{f(x)} \) is convex.</li>
|
|
<li> A norm is any function that satisfy the following properties</li>
|
|
|
|
<ul>
|
|
<li> \( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).</li>
|
|
<li> \( f(x+y) \leq f(x) + f(y) \)</li>
|
|
<li> \( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)</li>
|
|
</ul>
|
|
|
|
</ol>
|
|
|
|
Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="standard-steepest-descent">Standard steepest descent </h2>
|
|
|
|
<p>
|
|
Before we proceed, we would like to discuss the approach called the
|
|
<b>standard Steepest descent</b> (different from the above steepest descent discussion), which again leads to us having to be able
|
|
to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
|
|
|
|
<p>
|
|
<a href="https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf" target="_blank">The success of the CG method</a>
|
|
for finding solutions of non-linear problems is based on the theory
|
|
of conjugate gradients for linear systems of equations. It belongs to
|
|
the class of iterative methods for solving problems from linear
|
|
algebra of the type
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}.
|
|
\end{equation*}
|
|
$$
|
|
|
|
<p>
|
|
In the iterative process we end up with a problem like
|
|
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{r}= \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x},
|
|
\end{equation*}
|
|
$$
|
|
|
|
where \( \boldsymbol{r} \) is the so-called residual or error in the iterative process.
|
|
|
|
<p>
|
|
When we have found the exact solution, \( \boldsymbol{r}=0 \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="gradient-method">Gradient method </h2>
|
|
|
|
<p>
|
|
The residual is zero when we reach the minimum of the quadratic equation
|
|
$$
|
|
\begin{equation*}
|
|
P(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T\boldsymbol{b},
|
|
\end{equation*}
|
|
$$
|
|
|
|
<p>
|
|
with the constraint that the matrix \( \boldsymbol{A} \) is positive definite and
|
|
symmetric. This defines also the Hessian and we want it to be positive definite.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="steepest-descent-method">Steepest descent method </h2>
|
|
|
|
<p>
|
|
We denote the initial guess for \( \boldsymbol{x} \) as \( \boldsymbol{x}_0 \).
|
|
We can assume without loss of generality that
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{x}_0=0,
|
|
\end{equation*}
|
|
$$
|
|
|
|
or consider the system
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
|
|
\end{equation*}
|
|
$$
|
|
|
|
instead.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="steepest-descent-method">Steepest descent method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
One can show that the solution \( \boldsymbol{x} \) is also the unique minimizer of the quadratic form
|
|
$$
|
|
\begin{equation*}
|
|
f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
|
|
\end{equation*}
|
|
$$
|
|
|
|
This suggests taking the first basis vector \( \boldsymbol{r}_1 \) (see below for definition)
|
|
to be the gradient of \( f \) at \( \boldsymbol{x}=\boldsymbol{x}_0 \),
|
|
which equals
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
|
|
\end{equation*}
|
|
$$
|
|
|
|
and
|
|
\( \boldsymbol{x}_0=0 \) it is equal \( -\boldsymbol{b} \).
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="final-expressions">Final expressions </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
We can compute the residual iteratively as
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1},
|
|
\end{equation*}
|
|
$$
|
|
|
|
which equals
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{r}_k),
|
|
\end{equation*}
|
|
$$
|
|
|
|
or
|
|
$$
|
|
\begin{equation*}
|
|
(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{r}_k,
|
|
\end{equation*}
|
|
$$
|
|
|
|
which gives
|
|
|
|
$$
|
|
\alpha_k = \frac{\boldsymbol{r}_k^T\boldsymbol{r}_k}{\boldsymbol{r}_k^T\boldsymbol{A}\boldsymbol{r}_k}
|
|
$$
|
|
|
|
leading to the iterative scheme
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{x}_{k+1}=\boldsymbol{x}_k-\alpha_k\boldsymbol{r}_{k},
|
|
\end{equation*}
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="steepest-descent-example">Steepest descent example </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">la</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.optimize</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sopt</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.mplot3d</span> <span style="color: #008000; font-weight: bold">import</span> axes3d
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(x):
|
|
<span style="color: #008000; font-weight: bold">return</span> x[<span style="color: #666666">0</span>]<span style="color: #666666">**2</span> <span style="color: #666666">+</span> <span style="color: #666666">3.0*</span>x[<span style="color: #666666">1</span>]<span style="color: #666666">**2</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">df</span>(x):
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">2*</span>x[<span style="color: #666666">0</span>], <span style="color: #666666">6*</span>x[<span style="color: #666666">1</span>]])
|
|
|
|
fig <span style="color: #666666">=</span> pt<span style="color: #666666">.</span>figure()
|
|
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>gca(projection<span style="color: #666666">=</span><span style="color: #BA2121">"3d"</span>)
|
|
|
|
xmesh, ymesh <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mgrid[<span style="color: #666666">-3</span>:<span style="color: #666666">3</span>:<span style="color: #666666">50</span>j,<span style="color: #666666">-3</span>:<span style="color: #666666">3</span>:<span style="color: #666666">50</span>j]
|
|
fmesh <span style="color: #666666">=</span> f(np<span style="color: #666666">.</span>array([xmesh, ymesh]))
|
|
ax<span style="color: #666666">.</span>plot_surface(xmesh, ymesh, fmesh)
|
|
</pre></div>
|
|
<p>
|
|
And then as countor plot
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>pt<span style="color: #666666">.</span>axis(<span style="color: #BA2121">"equal"</span>)
|
|
pt<span style="color: #666666">.</span>contour(xmesh, ymesh, fmesh)
|
|
guesses <span style="color: #666666">=</span> [np<span style="color: #666666">.</span>array([<span style="color: #666666">2</span>, <span style="color: #666666">2./5</span>])]
|
|
</pre></div>
|
|
<p>
|
|
Find guesses
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>x <span style="color: #666666">=</span> guesses[<span style="color: #666666">-1</span>]
|
|
s <span style="color: #666666">=</span> <span style="color: #666666">-</span>df(x)
|
|
</pre></div>
|
|
<p>
|
|
Run it!
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f1d</span>(alpha):
|
|
<span style="color: #008000; font-weight: bold">return</span> f(x <span style="color: #666666">+</span> alpha<span style="color: #666666">*</span>s)
|
|
|
|
alpha_opt <span style="color: #666666">=</span> sopt<span style="color: #666666">.</span>golden(f1d)
|
|
next_guess <span style="color: #666666">=</span> x <span style="color: #666666">+</span> alpha_opt <span style="color: #666666">*</span> s
|
|
guesses<span style="color: #666666">.</span>append(next_guess)
|
|
<span style="color: #008000">print</span>(next_guess)
|
|
</pre></div>
|
|
<p>
|
|
What happened?
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>pt<span style="color: #666666">.</span>axis(<span style="color: #BA2121">"equal"</span>)
|
|
pt<span style="color: #666666">.</span>contour(xmesh, ymesh, fmesh, <span style="color: #666666">50</span>)
|
|
it_array <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array(guesses)
|
|
pt<span style="color: #666666">.</span>plot(it_array<span style="color: #666666">.</span>T[<span style="color: #666666">0</span>], it_array<span style="color: #666666">.</span>T[<span style="color: #666666">1</span>], <span style="color: #BA2121">"x-"</span>)
|
|
</pre></div>
|
|
<p>
|
|
Note that we did only one iteration here. We can easily add more using our previous guesses.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
In the CG method we define so-called conjugate directions and two vectors
|
|
\( \boldsymbol{s} \) and \( \boldsymbol{t} \)
|
|
are said to be
|
|
conjugate if
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{s}^T\boldsymbol{A}\boldsymbol{t}= 0.
|
|
\end{equation*}
|
|
$$
|
|
|
|
The philosophy of the CG method is to perform searches in various conjugate directions
|
|
of our vectors \( \boldsymbol{x}_i \) obeying the above criterion, namely
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{x}_i^T\boldsymbol{A}\boldsymbol{x}_j= 0.
|
|
\end{equation*}
|
|
$$
|
|
|
|
Two vectors are conjugate if they are orthogonal with respect to
|
|
this inner product. Being conjugate is a symmetric relation: if \( \boldsymbol{s} \) is conjugate to \( \boldsymbol{t} \), then \( \boldsymbol{t} \) is conjugate to \( \boldsymbol{s} \).
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
An example is given by the eigenvectors of the matrix
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{v}_i^T\boldsymbol{A}\boldsymbol{v}_j= \lambda\boldsymbol{v}_i^T\boldsymbol{v}_j,
|
|
\end{equation*}
|
|
$$
|
|
|
|
which is zero unless \( i=j \).
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
Assume now that we have a symmetric positive-definite matrix \( \boldsymbol{A} \) of size
|
|
\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+\alpha_i\boldsymbol{p}_{i}.
|
|
\end{equation*}
|
|
$$
|
|
|
|
We assume that \( \boldsymbol{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
|
|
Then the \( \boldsymbol{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
|
|
$ \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}$ in this basis, namely
|
|
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i \boldsymbol{p}_i.
|
|
\end{equation*}
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
The coefficients are given by
|
|
$$
|
|
\begin{equation*}
|
|
\mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
|
|
\end{equation*}
|
|
$$
|
|
|
|
Multiplying with \( \boldsymbol{p}_k^T \) from the left gives
|
|
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{p}_i= \boldsymbol{p}_k^T \boldsymbol{b},
|
|
\end{equation*}
|
|
$$
|
|
|
|
and we can define the coefficients \( \alpha_k \) as
|
|
|
|
$$
|
|
\begin{equation*}
|
|
\alpha_k = \frac{\boldsymbol{p}_k^T \boldsymbol{b}}{\boldsymbol{p}_k^T \boldsymbol{A} \boldsymbol{p}_k}
|
|
\end{equation*}
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method-and-iterations">Conjugate gradient method and iterations </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
If we choose the conjugate vectors \( \boldsymbol{p}_k \) carefully,
|
|
then we may not need all of them to obtain a good approximation to the solution
|
|
\( \boldsymbol{x} \).
|
|
We want to regard the conjugate gradient method as an iterative method.
|
|
This will us to solve systems where \( n \) is so large that the direct
|
|
method would take too much time.
|
|
|
|
<p>
|
|
We denote the initial guess for \( \boldsymbol{x} \) as \( \boldsymbol{x}_0 \).
|
|
We can assume without loss of generality that
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{x}_0=0,
|
|
\end{equation*}
|
|
$$
|
|
|
|
or consider the system
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
|
|
\end{equation*}
|
|
$$
|
|
|
|
instead.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
One can show that the solution \( \boldsymbol{x} \) is also the unique minimizer of the quadratic form
|
|
$$
|
|
\begin{equation*}
|
|
f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
|
|
\end{equation*}
|
|
$$
|
|
|
|
This suggests taking the first basis vector \( \boldsymbol{p}_1 \)
|
|
to be the gradient of \( f \) at \( \boldsymbol{x}=\boldsymbol{x}_0 \),
|
|
which equals
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
|
|
\end{equation*}
|
|
$$
|
|
|
|
and
|
|
\( \boldsymbol{x}_0=0 \) it is equal \( -\boldsymbol{b} \).
|
|
The other vectors in the basis will be conjugate to the gradient,
|
|
hence the name conjugate gradient method.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
Let \( \boldsymbol{r}_k \) be the residual at the \( k \)-th step:
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{r}_k=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k.
|
|
\end{equation*}
|
|
$$
|
|
|
|
Note that \( \boldsymbol{r}_k \) is the negative gradient of \( f \) at
|
|
\( \boldsymbol{x}=\boldsymbol{x}_k \),
|
|
so the gradient descent method would be to move in the direction \( \boldsymbol{r}_k \).
|
|
Here, we insist that the directions \( \boldsymbol{p}_k \) are conjugate to each other,
|
|
so we take the direction closest to the gradient \( \boldsymbol{r}_k \)
|
|
under the conjugacy constraint.
|
|
This gives the following expression
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{p}_{k+1}=\boldsymbol{r}_k-\frac{\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{r}_k}{\boldsymbol{p}_k^T\boldsymbol{A}\boldsymbol{p}_k} \boldsymbol{p}_k.
|
|
\end{equation*}
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="conjugate-gradient-method">Conjugate gradient method </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
We can also compute the residual iteratively as
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1},
|
|
\end{equation*}
|
|
$$
|
|
|
|
which equals
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{p}_k),
|
|
\end{equation*}
|
|
$$
|
|
|
|
or
|
|
$$
|
|
\begin{equation*}
|
|
(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{p}_k,
|
|
\end{equation*}
|
|
$$
|
|
|
|
which gives
|
|
|
|
$$
|
|
\begin{equation*}
|
|
\boldsymbol{r}_{k+1}=\boldsymbol{r}_k-\boldsymbol{A}\boldsymbol{p}_{k},
|
|
\end{equation*}
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="revisiting-our-first-homework">Revisiting our first homework </h2>
|
|
|
|
<p>
|
|
We will use linear regression as a case study for the gradient descent
|
|
methods. Linear regression is a great test case for the gradient
|
|
descent methods discussed in the lectures since it has several
|
|
desirable properties such as:
|
|
|
|
<ol>
|
|
<li> An analytical solution (recall homework set 1).</li>
|
|
<li> The gradient can be computed analytically.</li>
|
|
<li> The cost function is convex which guarantees that gradient descent converges for small enough learning rates</li>
|
|
</ol>
|
|
|
|
We revisit an example similar to what we had in the first homework set. We had a function of the type
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(m,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(m,<span style="color: #666666">1</span>)
|
|
</pre></div>
|
|
<p>
|
|
with \( x_i \in [0,1] \) is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \).
|
|
The linear regression model is given by
|
|
$$
|
|
h_\beta(x) = \boldsymbol{y} = \beta_0 + \beta_1 x,
|
|
$$
|
|
|
|
such that
|
|
$$
|
|
\boldsymbol{y}_i = \beta_0 + \beta_1 x_i.
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="gradient-descent-example">Gradient descent example </h2>
|
|
|
|
<p>
|
|
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\boldsymbol{y}} = (\boldsymbol{y}_1,\cdots,\boldsymbol{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
|
|
|
|
<p>
|
|
It is convenient to write \( \mathbf{\boldsymbol{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by (we keep the intercept here)
|
|
$$
|
|
X \equiv \begin{bmatrix}
|
|
1 & x_1 \\
|
|
\vdots & \vdots \\
|
|
1 & x_{100} & \\
|
|
\end{bmatrix}.
|
|
$$
|
|
|
|
The cost/loss/risk function is given by (
|
|
$$
|
|
C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right]
|
|
$$
|
|
|
|
and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-derivative-of-the-cost-loss-function">The derivative of the cost/loss function </h2>
|
|
|
|
<p>
|
|
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
|
|
$$
|
|
\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
|
|
\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
|
|
\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}),
|
|
$$
|
|
|
|
where \( X \) is the design matrix defined above.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-hessian-matrix">The Hessian matrix </h2>
|
|
The Hessian matrix of \( C(\beta) \) is given by
|
|
$$
|
|
\boldsymbol{H} \equiv \begin{bmatrix}
|
|
\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
|
|
\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
|
|
\end{bmatrix} = \frac{2}{n}X^T X.
|
|
$$
|
|
|
|
This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="simple-program">Simple program </h2>
|
|
|
|
<p>
|
|
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
|
|
$$
|
|
\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
|
|
$$
|
|
|
|
<p>
|
|
We can use the expression we computed for the gradient and let use a
|
|
\( \beta_0 \) be chosen randomly and let \( \gamma = 0.001 \). Stop iterating
|
|
when \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \). <b>Note that the code below does not include the latter stop criterion</b>.
|
|
|
|
<p>
|
|
And finally we can compare our solution for \( \beta \) with the analytic result given by
|
|
\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="gradient-descent-example">Gradient Descent Example </h2>
|
|
|
|
<p>
|
|
Here our simple example
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.mplot3d</span> <span style="color: #008000; font-weight: bold">import</span> Axes3D
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span> <span style="color: #008000; font-weight: bold">import</span> cm
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib.ticker</span> <span style="color: #008000; font-weight: bold">import</span> LinearLocator, FormatStrFormatter
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sys</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># the number of datapoints</span>
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
|
|
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
|
|
<span style="color: #408080; font-style: italic"># Hessian matrix</span>
|
|
H <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
|
|
<span style="color: #408080; font-style: italic"># Get the eigenvalues</span>
|
|
EigValues, EigVectors <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(H)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"Eigenvalues of Hessian Matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>EigValues<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
|
|
|
beta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
|
|
<span style="color: #008000">print</span>(beta_linreg)
|
|
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
|
|
|
|
eta <span style="color: #666666">=</span> <span style="color: #666666">1.0/</span>np<span style="color: #666666">.</span>max(EigValues)
|
|
Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
|
|
gradient <span style="color: #666666">=</span> (<span style="color: #666666">2.0/</span>n)<span style="color: #666666">*</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> (X <span style="color: #666666">@</span> beta<span style="color: #666666">-</span>y)
|
|
beta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradient
|
|
|
|
<span style="color: #008000">print</span>(beta)
|
|
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
|
|
xbnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
|
|
ypredict <span style="color: #666666">=</span> xbnew<span style="color: #666666">.</span>dot(beta)
|
|
ypredict2 <span style="color: #666666">=</span> xbnew<span style="color: #666666">.</span>dot(beta_linreg)
|
|
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">"r-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">"b-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
|
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Gradient descent example'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="and-a-corresponding-example-using-_scikit-learn_">And a corresponding example using <b>scikit-learn</b> </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDRegressor
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
|
|
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
|
|
beta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
|
|
<span style="color: #008000">print</span>(beta_linreg)
|
|
sgdreg <span style="color: #666666">=</span> SGDRegressor(max_iter <span style="color: #666666">=</span> <span style="color: #666666">50</span>, penalty<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>, eta0<span style="color: #666666">=0.1</span>)
|
|
sgdreg<span style="color: #666666">.</span>fit(x,y<span style="color: #666666">.</span>ravel())
|
|
<span style="color: #008000">print</span>(sgdreg<span style="color: #666666">.</span>intercept_, sgdreg<span style="color: #666666">.</span>coef_)
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split -->
|
|
|
|
<h2 id="gradient-descent-and-ridge">Gradient descent and Ridge </h2>
|
|
|
|
<p>
|
|
We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \),
|
|
$$
|
|
C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0.
|
|
$$
|
|
|
|
<p>
|
|
In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows
|
|
$$
|
|
\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
|
|
\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
|
|
\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
|
|
$$
|
|
|
|
<p>
|
|
We can easily extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by
|
|
$$
|
|
\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="program-example-for-gradient-descent-with-ridge-regression">Program example for gradient descent with Ridge Regression </h2>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.mplot3d</span> <span style="color: #008000; font-weight: bold">import</span> Axes3D
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span> <span style="color: #008000; font-weight: bold">import</span> cm
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib.ticker</span> <span style="color: #008000; font-weight: bold">import</span> LinearLocator, FormatStrFormatter
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sys</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># the number of datapoints</span>
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n,<span style="color: #666666">1</span>)
|
|
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((n,<span style="color: #666666">1</span>)), x]
|
|
XT_X <span style="color: #666666">=</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X
|
|
|
|
<span style="color: #408080; font-style: italic">#Ridge parameter lambda</span>
|
|
lmbda <span style="color: #666666">=</span> <span style="color: #666666">0.001</span>
|
|
Id <span style="color: #666666">=</span> lmbda<span style="color: #666666">*</span> np<span style="color: #666666">.</span>eye(XT_X<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])
|
|
|
|
beta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(XT_X<span style="color: #666666">+</span>Id) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
|
|
<span style="color: #008000">print</span>(beta_linreg)
|
|
<span style="color: #408080; font-style: italic"># Start plain gradient descent</span>
|
|
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
|
|
|
|
eta <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
|
|
Niterations <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
|
|
gradients <span style="color: #666666">=</span> <span style="color: #666666">2.0/</span>n<span style="color: #666666">*</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> (X <span style="color: #666666">@</span> (beta)<span style="color: #666666">-</span>y)<span style="color: #666666">+2*</span>lmbda<span style="color: #666666">*</span>beta
|
|
beta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
|
|
|
|
<span style="color: #008000">print</span>(beta)
|
|
ypredict <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
|
|
ypredict2 <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta_linreg
|
|
plt<span style="color: #666666">.</span>plot(x, ypredict, <span style="color: #BA2121">"r-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, ypredict2, <span style="color: #BA2121">"b-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
|
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Gradient descent example for Ridge'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="using-gradient-descent-methods-limitations">Using gradient descent methods, limitations </h2>
|
|
|
|
<ul>
|
|
<li> <b>Gradient descent (GD) finds local minima of our function</b>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</li>
|
|
<li> <b>GD is sensitive to initial conditions</b>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</li>
|
|
<li> <b>Gradients are computationally expensive to calculate for large datasets</b>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.</li>
|
|
<li> <b>GD is very sensitive to choices of learning rates</b>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</li>
|
|
<li> <b>GD treats all directions in parameter space uniformly.</b> Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</li>
|
|
<li> GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</li>
|
|
</ul>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="to-be-added">To be added </h2>
|
|
|
|
<p>
|
|
We will add here an example which computes the likelihood \( p_i \), sets up the gradient and the Hessian matrix.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="friday-october-1">Friday October 1 </h2>
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="stochastic-gradient-descent">Stochastic Gradient Descent </h2>
|
|
|
|
<p>
|
|
Stochastic gradient descent (SGD) and variants thereof address some of
|
|
the shortcomings of the Gradient descent method discussed above.
|
|
|
|
<p>
|
|
The underlying idea of SGD comes from the observation that the cost
|
|
function, which we want to minimize, can almost always be written as a
|
|
sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \),
|
|
$$
|
|
C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
|
|
\mathbf{\beta}).
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="computation-of-gradients">Computation of gradients </h2>
|
|
|
|
<p>
|
|
This in turn means that the gradient can be
|
|
computed as a sum over \( i \)-gradients
|
|
$$
|
|
\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
|
|
\mathbf{\beta}).
|
|
$$
|
|
|
|
<p>
|
|
Stochasticity/randomness is introduced by only taking the
|
|
gradient on a subset of the data called minibatches. If there are \( n \)
|
|
data points and the size of each minibatch is \( M \), there will be \( n/M \)
|
|
minibatches. We denote these minibatches by \( B_k \) where
|
|
\( k=1,\cdots,n/M \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="sgd-example">SGD example </h2>
|
|
As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \)
|
|
and we choose to have \( M=5 \) minibathces,
|
|
then each minibatch contains two data points. In particular we have
|
|
\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
|
|
(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you
|
|
have only a single batch with all data points and on the other extreme,
|
|
you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e
|
|
\( B_k = \mathbf{x}_k \).
|
|
|
|
<p>
|
|
The idea is now to approximate the gradient by replacing the sum over
|
|
all data points with a sum over the data points in one the minibatches
|
|
picked at random in each gradient descent step
|
|
$$
|
|
\nabla_{\beta}
|
|
C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
|
|
\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
|
|
c_i(\mathbf{x}_i, \mathbf{\beta}).
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-gradient-step">The gradient step </h2>
|
|
|
|
<p>
|
|
Thus a gradient descent step now looks like
|
|
$$
|
|
\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
|
|
\mathbf{\beta})
|
|
$$
|
|
|
|
<p>
|
|
where \( k \) is picked at random with equal
|
|
probability from \( [1,n/M] \). An iteration over the number of
|
|
minibathces (n/M) is commonly referred to as an epoch. Thus it is
|
|
typical to choose a number of epochs and for each epoch iterate over
|
|
the number of minibatches, as exemplified in the code below.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="simple-example-code">Simple example code </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
|
|
M <span style="color: #666666">=</span> <span style="color: #666666">5</span> <span style="color: #408080; font-style: italic">#size of each minibatch</span>
|
|
m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
|
|
n_epochs <span style="color: #666666">=</span> <span style="color: #666666">10</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
|
|
|
|
j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
|
|
<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
|
|
k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
|
|
<span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
|
|
<span style="color: #408080; font-style: italic">#Compute new suggestion for </span>
|
|
j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
|
</pre></div>
|
|
<p>
|
|
Taking the gradient only on a subset of the data has two important
|
|
benefits. First, it introduces randomness which decreases the chance
|
|
that our opmization scheme gets stuck in a local minima. Second, if
|
|
the size of the minibatches are small relative to the number of
|
|
datapoints (\( M < n \)), the computation of the gradient is much
|
|
cheaper since we sum over the datapoints in the \( k-th \) minibatch and not
|
|
all \( n \) datapoints.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="when-do-we-stop">When do we stop? </h2>
|
|
|
|
<p>
|
|
A natural question is when do we stop the search for a new minimum?
|
|
One possibility is to compute the full gradient after a given number
|
|
of epochs and check if the norm of the gradient is smaller than some
|
|
threshold and stop if true. However, the condition that the gradient
|
|
is zero is valid also for local minima, so this would only tell us
|
|
that we are close to a local/global minimum. However, we could also
|
|
evaluate the cost function at this point, store the result and
|
|
continue the search. If the test kicks in at a later stage we can
|
|
compare the values of the cost function and keep the \( \beta \) that
|
|
gave the lowest value.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="slightly-different-approach">Slightly different approach </h2>
|
|
|
|
<p>
|
|
Another approach is to let the step length \( \gamma_j \) depend on the
|
|
number of epochs in such a way that it becomes very small after a
|
|
reasonable time such that we do not move at all.
|
|
|
|
<p>
|
|
As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in <em>time</em> \( t \).
|
|
|
|
<p>
|
|
In this way we can fix the number of epochs, compute \( \beta \) and
|
|
evaluate the cost function at the end. Repeating the computation will
|
|
give a different result since the scheme is random by design. Then we
|
|
pick the final \( \beta \) that gives the lowest value of the cost
|
|
function.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">step_length</span>(t,t0,t1):
|
|
<span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#100 datapoints </span>
|
|
M <span style="color: #666666">=</span> <span style="color: #666666">5</span> <span style="color: #408080; font-style: italic">#size of each minibatch</span>
|
|
m <span style="color: #666666">=</span> <span style="color: #008000">int</span>(n<span style="color: #666666">/</span>M) <span style="color: #408080; font-style: italic">#number of minibatches</span>
|
|
n_epochs <span style="color: #666666">=</span> <span style="color: #666666">500</span> <span style="color: #408080; font-style: italic">#number of epochs</span>
|
|
t0 <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
|
t1 <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
|
|
gamma_j <span style="color: #666666">=</span> t0<span style="color: #666666">/</span>t1
|
|
j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
|
|
<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n_epochs<span style="color: #666666">+1</span>):
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
|
|
k <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m) <span style="color: #408080; font-style: italic">#Pick the k-th minibatch at random</span>
|
|
<span style="color: #408080; font-style: italic">#Compute the gradient using the data in minibatch Bk</span>
|
|
<span style="color: #408080; font-style: italic">#Compute new suggestion for beta</span>
|
|
t <span style="color: #666666">=</span> epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i
|
|
gamma_j <span style="color: #666666">=</span> step_length(t,t0,t1)
|
|
j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
|
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"gamma_j after </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> epochs: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> (n_epochs,gamma_j))
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="program-for-stochastic-gradient">Program for stochastic gradient </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">import</span> exp, sqrt
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDRegressor
|
|
|
|
m <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(m,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(m,<span style="color: #666666">1</span>)
|
|
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((m,<span style="color: #666666">1</span>)), x]
|
|
theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> (X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Own inversion"</span>)
|
|
<span style="color: #008000">print</span>(theta_linreg)
|
|
sgdreg <span style="color: #666666">=</span> SGDRegressor(max_iter <span style="color: #666666">=</span> <span style="color: #666666">50</span>, penalty<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>, eta0<span style="color: #666666">=0.1</span>)
|
|
sgdreg<span style="color: #666666">.</span>fit(x,y<span style="color: #666666">.</span>ravel())
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"sgdreg from scikit"</span>)
|
|
<span style="color: #008000">print</span>(sgdreg<span style="color: #666666">.</span>intercept_, sgdreg<span style="color: #666666">.</span>coef_)
|
|
|
|
|
|
theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
|
|
eta <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
|
|
Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
|
|
gradients <span style="color: #666666">=</span> <span style="color: #666666">2.0/</span>m<span style="color: #666666">*</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> ((X <span style="color: #666666">@</span> theta)<span style="color: #666666">-</span>y)
|
|
theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"theta from own gd"</span>)
|
|
<span style="color: #008000">print</span>(theta)
|
|
|
|
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
|
|
Xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
|
|
ypredict <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta)
|
|
ypredict2 <span style="color: #666666">=</span> Xnew<span style="color: #666666">.</span>dot(theta_linreg)
|
|
|
|
|
|
n_epochs <span style="color: #666666">=</span> <span style="color: #666666">50</span>
|
|
t0, t1 <span style="color: #666666">=</span> <span style="color: #666666">5</span>, <span style="color: #666666">50</span>
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">learning_schedule</span>(t):
|
|
<span style="color: #008000; font-weight: bold">return</span> t0<span style="color: #666666">/</span>(t<span style="color: #666666">+</span>t1)
|
|
|
|
theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> epoch <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_epochs):
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(m):
|
|
random_index <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randint(m)
|
|
xi <span style="color: #666666">=</span> X[random_index:random_index<span style="color: #666666">+1</span>]
|
|
yi <span style="color: #666666">=</span> y[random_index:random_index<span style="color: #666666">+1</span>]
|
|
gradients <span style="color: #666666">=</span> <span style="color: #666666">2</span> <span style="color: #666666">*</span> xi<span style="color: #666666">.</span>T <span style="color: #666666">@</span> ((xi <span style="color: #666666">@</span> theta)<span style="color: #666666">-</span>yi)
|
|
eta <span style="color: #666666">=</span> learning_schedule(epoch<span style="color: #666666">*</span>m<span style="color: #666666">+</span>i)
|
|
theta <span style="color: #666666">=</span> theta <span style="color: #666666">-</span> eta<span style="color: #666666">*</span>gradients
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"theta from own sdg"</span>)
|
|
<span style="color: #008000">print</span>(theta)
|
|
|
|
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">"r-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">"b-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
|
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Random numbers '</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<b>Challenge</b>: try to write a similar code for a Logistic Regression case.
|
|
|
|
<p>
|
|
|
|
<!-- ------------------- end of main content --------------- -->
|
|
|
|
|
|
<center style="font-size:80%">
|
|
<!-- copyright --> © 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
|
|
</center>
|
|
|
|
|
|
</body>
|
|
</html>
|
|
|
|
|