Corrected a typo in GD
This commit is contained in:
mhjensen
@@ -0,0 +1,245 @@
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<a class="navbar-brand" href="Splines-bs.html">Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- ------------------- main content ---------------------- -->
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<div class="jumbotron">
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<center><h1>Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Sep 20, 2018</h4></center> <!-- date -->
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<br>
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<p>
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<p><a href="._Splines-bs001.html" class="btn btn-primary btn-lg">Read »</a></p>
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<li class="active"><a href="._Splines-bs000.html">1</a></li>
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<li><a href="._Splines-bs001.html">2</a></li>
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<li><a href="._Splines-bs002.html">3</a></li>
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<li><a href="._Splines-bs003.html">4</a></li>
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<li><a href="._Splines-bs004.html">5</a></li>
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<li><a href="._Splines-bs005.html">6</a></li>
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<li><a href="._Splines-bs006.html">7</a></li>
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<li><a href="._Splines-bs007.html">8</a></li>
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<li><a href="._Splines-bs008.html">9</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Splines-bs036.html">37</a></li>
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<li><a href="._Splines-bs001.html">»</a></li>
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<!-- ------------------- end of main content --------------- -->
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<!-- include javascript, jQuery *first* -->
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<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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</body>
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</html>
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@@ -0,0 +1,231 @@
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<meta name="description" content="Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods">
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<title>Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods</title>
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<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||||
('The Hessian matrix', 2, None, '___sec15'),
|
||||
('Simple program', 2, None, '___sec16'),
|
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|
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|
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('Computation of gradients', 2, None, '___sec19'),
|
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('SGD example', 2, None, '___sec20'),
|
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('The gradient step', 2, None, '___sec21'),
|
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('Simple example code', 2, None, '___sec22'),
|
||||
('When do we stop?', 2, None, '___sec23'),
|
||||
('Slightly different approach', 2, None, '___sec24'),
|
||||
('Conjugate gradient (CG) method', 2, None, '___sec25'),
|
||||
('Conjugate gradient method', 2, None, '___sec26'),
|
||||
("Conjugate gradient method, Newton's method first",
|
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2,
|
||||
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|
||||
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|
||||
('Conjugate gradient method', 2, None, '___sec28'),
|
||||
('Conjugate gradient method', 2, None, '___sec29'),
|
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('Conjugate gradient method', 2, None, '___sec30'),
|
||||
('Conjugate gradient method', 2, None, '___sec31'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec32'),
|
||||
('Conjugate gradient method', 2, None, '___sec33'),
|
||||
('Conjugate gradient method', 2, None, '___sec34'),
|
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('Conjugate gradient method', 2, None, '___sec35')]}
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<a class="navbar-brand" href="Splines-bs.html">Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods</a>
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<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<a name="part0001"></a>
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<!-- !split -->
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||||
<h2 id="___sec0" class="anchor">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 \( g(\theta) \), which is a function of the
|
||||
parameters \( \theta \) and a cost function \( C(X, g(\theta)) \) that allows
|
||||
us to judge how well the model \( g(\theta) \) explains the observations
|
||||
\( X \). The model is fit by finding the values of \( \theta \) that minimize
|
||||
the cost function. Ideally we would be able to solve for \( \theta \)
|
||||
analytically, however this is not possible in general and we must use
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some approximative/numerical method to compute the minimum.
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- !split -->
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<h2 id="___sec1" class="anchor">Steepest descent </h2>
|
||||
|
||||
<p>
|
||||
The method of steepest descent 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
|
||||
$$
|
||||
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ \gamma_k > 0
|
||||
$$
|
||||
|
||||
<p>
|
||||
for \( \gamma_k \) small enough, then \( F(\mathbf{x}_{k+1}) \leq
|
||||
F(\mathbf{x}_k) \). This means that for a sufficiently small \( \gamma_k \)
|
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we are always moving towards smaller function values, i.e a minimum.
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|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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||||
</li>
|
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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||||
|
||||
<a name="part0003"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec2" class="anchor">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
|
||||
compute 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 in the context of Machine Learning.
|
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
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||||
<h2 id="___sec3" class="anchor">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 minimum. 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.
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
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<a name="part0005"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec4" class="anchor">The sensitiveness of the gradient descent </h2>
|
||||
|
||||
<p>
|
||||
GD 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 to
|
||||
small the method will take a long to converge and if it is to 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
|
||||
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|
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|
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|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
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||||
</ul>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0006"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec5" class="anchor">Gradient Descent Example </h2>
|
||||
We revisit now our simple linear regression example with a linear polynomial.
|
||||
<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>
|
||||
|
||||
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(<span style="color: #666666">100</span>,<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(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
|
||||
xb <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">100</span>,<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(xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb))<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(theta_linreg)
|
||||
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>
|
||||
m <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>m<span style="color: #666666">*</span>xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>dot(theta)<span style="color: #666666">-</span>y)
|
||||
theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
|
||||
|
||||
<span style="color: #008000; font-weight: bold">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>]])
|
||||
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(theta)
|
||||
ypredict2 <span style="color: #666666">=</span> xbnew<span style="color: #666666">.</span>dot(theta_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'Random numbers '</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
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||||
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||||
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||||
<a name="part0007"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec6" class="anchor">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
|
||||
|
||||
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(<span style="color: #666666">100</span>,<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(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
|
||||
xb <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">100</span>,<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(xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb))<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(theta_linreg)
|
||||
sgdreg <span style="color: #666666">=</span> SGDRegressor(n_iter <span style="color: #666666">=</span> <span style="color: #666666">50</span>, penalty<span style="color: #666666">=</span><span style="color: #008000">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; font-weight: bold">print</span>(sgdreg<span style="color: #666666">.</span>intercept_, sgdreg<span style="color: #666666">.</span>coef_)
|
||||
</pre></div>
|
||||
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|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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<a name="part0008"></a>
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||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec7" class="anchor">Convex functions </h2>
|
||||
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
|
||||
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|
||||
C. Geometrically this means that every point on the line segment
|
||||
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|
||||
|
||||
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|
||||
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...).
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<h2 id="___sec8" class="anchor">Convex function </h2>
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<p>
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||||
Convex function: 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.
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- !split -->
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<h2 id="___sec9" class="anchor">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: S. Boyd and L. Vandenberghe. Convex
|
||||
Optimization. Cambridge University Press, <a href="http://stanford.edu/" target="_self"><tt>http://stanford.edu/</tt></a>
|
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boyd/cvxbook/, 2004.
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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>
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
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>
|
||||
</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.
|
||||
|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
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||||
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||||
|
||||
<a name="part0011"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec10" class="anchor">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. Ideally we want the
|
||||
global minimum, however 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="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
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>
|
||||
</div>
|
||||
|
||||
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.
|
||||
|
||||
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|
||||
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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<a name="part0012"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec11" class="anchor">Some simple problems </h2>
|
||||
|
||||
<ol>
|
||||
<li> Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity.</li>
|
||||
</ol>
|
||||
|
||||
Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $$\lambda f(x) + (1-\lambda)f(y) - f(\lambda x + (1-\lambda) y ) \geq 0. $$
|
||||
|
||||
<ol>
|
||||
<li> Using the second order condition show that the following functions are convex on the specified domain.</li>
|
||||
</ol>
|
||||
|
||||
\( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
|
||||
\( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
|
||||
|
||||
<ol>
|
||||
<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>
|
||||
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|
||||
|
||||
\( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).
|
||||
\( f(x+y) \leq f(x) + f(y) \)
|
||||
\( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)
|
||||
Using the definition of convexity, show that a function satisfying the properties above is convex (the third condition is not needed to show this).
|
||||
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||||
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|
||||
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
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|
||||
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||||
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|
||||
<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<a name="part0013"></a>
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<!-- !split -->
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||||
|
||||
<h2 id="___sec12" class="anchor">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 the example from homework set 1 where we had
|
||||
$$
|
||||
y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100
|
||||
$$
|
||||
|
||||
with \( x_i \in [0,1] \) chosen randomly with a uniform distribution. Additionally \( \xi_i \) represents stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \).
|
||||
The linear regression model is given by
|
||||
$$
|
||||
h_\theta(x) = \hat{y} = \theta_0 + \theta_1 x,
|
||||
$$
|
||||
|
||||
such that
|
||||
$$
|
||||
\hat{y}_i = \theta_0 + \theta_1 x_i.
|
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$$
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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||||
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<a name="part0014"></a>
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<!-- !split -->
|
||||
|
||||
<h2 id="___sec13" class="anchor">Gradient descent example </h2>
|
||||
|
||||
<p>
|
||||
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \theta = (\theta_0, \theta_1)^T \)
|
||||
|
||||
<p>
|
||||
t is convenient to write \( \mathbf{\hat{y}} = X\theta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by
|
||||
$$
|
||||
\begin{equation}
|
||||
X \equiv \begin{bmatrix}
|
||||
1 & x_1 \\
|
||||
\vdots & \vdots \\
|
||||
1 & x_{100} & \\
|
||||
\end{bmatrix}.
|
||||
\tag{1}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
The loss function is given by
|
||||
$$
|
||||
C(\theta) = ||X\theta-\mathbf{y}||^2 = ||X\theta||^2 - 2 \mathbf{y}^T X\theta + ||\mathbf{y}||^2 = \sum_{i=1}^{100} (\theta_0 + \theta_1 x_i)^2 - 2 y_i (\theta_0 + \theta_1 x_i) + y_i^2
|
||||
$$
|
||||
|
||||
and we want to find \( \theta \) such that \( C(\theta) \) is minimized.
|
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|
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|
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|
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|
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|
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|
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|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
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||||
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||||
<a name="part0015"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec14" class="anchor">The derivative of the cost/loss function </h2>
|
||||
|
||||
<p>
|
||||
Computing \( \partial C(\theta) / \partial \theta_0 \) and \( \partial C(\theta) / \partial \theta_1 \) we can show that the gradient can be written as
|
||||
$$
|
||||
\nabla_\theta C(\theta) = (\partial C(\theta) / \partial \theta_0, \partial C(\theta) / \partial \theta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\theta_0+\theta_1x_i-y_i\right) \\
|
||||
\sum_{i=1}^{100}\left( x_i (\theta_0+\theta_1x_i)-y_ix_i\right) \\
|
||||
\end{bmatrix} = 2X^T(X\theta - \mathbf{y}),
|
||||
$$
|
||||
|
||||
where \( X \) is the design matrix defined above.
|
||||
|
||||
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|
||||
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{'highest level': 2,
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'sections': [('Optimization, the central part of any Machine Learning '
|
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'algortithm',
|
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2,
|
||||
None,
|
||||
'___sec0'),
|
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('Steepest descent', 2, None, '___sec1'),
|
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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<a name="part0016"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec15" class="anchor">The Hessian matrix </h2>
|
||||
The Hessian matrix of \( C(\theta) \) is given by
|
||||
$$
|
||||
\hat{H} \equiv \begin{bmatrix}
|
||||
\frac{\partial^2 C(\theta)}{\partial \theta_0^2} & \frac{\partial^2 C(\theta)}{\partial \theta_0 \partial \theta_1} \\
|
||||
\frac{\partial^2 C(\theta)}{\partial \theta_0 \partial \theta_1} & \frac{\partial^2 C(\theta)}{\partial \theta_1^2} & \\
|
||||
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|
||||
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|
||||
|
||||
This result implies that \( C(\theta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
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|
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|
||||
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|
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|
||||
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|
||||
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<a name="part0017"></a>
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<!-- !split -->
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<h2 id="___sec16" class="anchor">Simple program </h2>
|
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|
||||
<p>
|
||||
We can now write a program that minimizes \( C(\theta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
|
||||
$$
|
||||
\theta_{k+1} = \theta_k - \gamma \nabla_\theta C(\theta_k), \ k=0,1,\cdots
|
||||
$$
|
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|
||||
<p>
|
||||
We can use the expression we computed for the gradient and let use a
|
||||
\( \theta_0 \) be chosen randomly and let \( \gamma = 0.001 \). Stop iterating
|
||||
when \( ||\nabla_\theta C(\theta_k) || < \epsilon = 10^{-8} \).
|
||||
|
||||
<p>
|
||||
And finally we can compare our solution for \( \theta \) with the analytic result given by
|
||||
\( \theta= (X^TX)^{-1} X^T \mathbf{y} \).
|
||||
<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: #BA2121; font-style: italic">"""</span>
|
||||
<span style="color: #BA2121; font-style: italic">The following setup is just a suggestion, feel free to write it the way you like.</span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic">#Setup problem described in the exercise</span>
|
||||
N <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#Nr of datapoints</span>
|
||||
M <span style="color: #666666">=</span> <span style="color: #666666">2</span> <span style="color: #408080; font-style: italic">#Nr of features</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(N) <span style="color: #408080; font-style: italic">#Uniformly generated x-values in [0,1]</span>
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<span style="color: #666666">**2</span> <span style="color: #666666">+</span> <span style="color: #666666">0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(N)
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones(N),x] <span style="color: #408080; font-style: italic">#Construct design matrix</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic">#Compute theta according to normal equations to compare with GD solution</span>
|
||||
Xt_X_inv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(np<span style="color: #666666">.</span>dot(X<span style="color: #666666">.</span>T,X))
|
||||
Xt_y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>dot(X<span style="color: #666666">.</span>transpose(),y)
|
||||
theta_NE <span style="color: #666666">=</span> np<span style="color: #666666">.</span>dot(Xt_X_inv,Xt_y)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(theta_NE)
|
||||
</pre></div>
|
||||
<p>
|
||||
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|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<a name="part0018"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec17" class="anchor">Gradient descent and Ridge </h2>
|
||||
|
||||
<p>
|
||||
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \theta \),
|
||||
$$
|
||||
C_{\text{ridge}}(\theta) = ||X\theta -\mathbf{y}||^2 + \lambda ||\theta||^2, \ \lambda \geq 0.
|
||||
$$
|
||||
|
||||
<p>
|
||||
In order to minimize \( C_{\text{ridge}}(\theta) \) using GD we only have adjust the gradient as follows
|
||||
$$
|
||||
\nabla_\theta C_{\text{ridge}}(\theta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\theta_0+\theta_1x_i-y_i\right) \\
|
||||
\sum_{i=1}^{100}\left( x_i (\theta_0+\theta_1x_i)-y_ix_i\right) \\
|
||||
\end{bmatrix} + 2\lambda\begin{bmatrix} \theta_0 \\ \theta_1\end{bmatrix} = 2 (X^T(X\theta - \mathbf{y})+\lambda \theta).
|
||||
$$
|
||||
|
||||
<p>
|
||||
We can now extend our program to minimize \( C_{\text{ridge}}(\theta) \) using gradient descent and compare with the analytical solution given by
|
||||
$$
|
||||
\theta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y},
|
||||
$$
|
||||
|
||||
for \( \lambda = {0,1,10,50,100} \) (\( \lambda = 0 \) corresponds to ordinary least squares).
|
||||
We can then compute \( ||\theta_{\text{ridge}}|| \) for each \( \lambda \).
|
||||
|
||||
<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: #BA2121; font-style: italic">"""</span>
|
||||
<span style="color: #BA2121; font-style: italic">The following setup is just a suggestion, feel free to write it the way you like.</span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic">#Setup problem described in the exercise</span>
|
||||
N <span style="color: #666666">=</span> <span style="color: #666666">100</span> <span style="color: #408080; font-style: italic">#Nr of datapoints</span>
|
||||
M <span style="color: #666666">=</span> <span style="color: #666666">2</span> <span style="color: #408080; font-style: italic">#Nr of features</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> <span style="color: #666666">5*</span>x<span style="color: #666666">**2</span> <span style="color: #666666">+</span> <span style="color: #666666">0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(N)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic">#Compute analytic theta for Ridge regression </span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones(N),x]
|
||||
XT_X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>dot(X<span style="color: #666666">.</span>T,X)
|
||||
|
||||
l <span style="color: #666666">=</span> <span style="color: #666666">0.1</span> <span style="color: #408080; font-style: italic">#Ridge parameter lambda</span>
|
||||
Id <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>])
|
||||
|
||||
Z <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>l<span style="color: #666666">*</span>Id)
|
||||
theta_ridge <span style="color: #666666">=</span> np<span style="color: #666666">.</span>dot(Z,np<span style="color: #666666">.</span>dot(X<span style="color: #666666">.</span>T,y))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(theta_ridge)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>norm(theta_ridge)) <span style="color: #408080; font-style: italic">#||theta||</span>
|
||||
</pre></div>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<a name="part0019"></a>
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<!-- !split -->
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|
||||
<h2 id="___sec18" class="anchor">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 \( n \) datapoints \( \{\mathbf{x}_i\}_{i=1}^n \),
|
||||
$$
|
||||
C(\mathbf{\theta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
|
||||
\mathbf{\theta}).
|
||||
$$
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|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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||||
<a name="part0020"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec19" class="anchor">Computation of gradients </h2>
|
||||
|
||||
<p>
|
||||
This in turn means that the gradient can be
|
||||
computed as a sum over \( i \)-gradients
|
||||
$$
|
||||
\nabla_\theta C(\mathbf{\theta}) = \sum_i^n \nabla_\theta c_i(\mathbf{x}_i,
|
||||
\mathbf{\theta}).
|
||||
$$
|
||||
|
||||
<p>
|
||||
Stochasticity/randomness is introduced by only taking the
|
||||
gradient on a subset of the data called minibatches. If there are \( n \)
|
||||
datapoints and the size of each minibatch is \( M \), there will be \( n/M \)
|
||||
minibatches. We denote these minibatches by \( B_k \) where
|
||||
\( k=1,\cdots,n/M \).
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||||
('Gradient descent and Ridge', 2, None, '___sec17'),
|
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('Stochastic Gradient Descent', 2, None, '___sec18'),
|
||||
('Computation of gradients', 2, None, '___sec19'),
|
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('SGD example', 2, None, '___sec20'),
|
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|
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|
||||
('When do we stop?', 2, None, '___sec23'),
|
||||
('Slightly different approach', 2, None, '___sec24'),
|
||||
('Conjugate gradient (CG) method', 2, None, '___sec25'),
|
||||
('Conjugate gradient method', 2, None, '___sec26'),
|
||||
("Conjugate gradient method, Newton's method first",
|
||||
2,
|
||||
None,
|
||||
'___sec27'),
|
||||
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|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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|
||||
</ul>
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<a name="part0021"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec20" class="anchor">SGD example </h2>
|
||||
As an example, suppose we have \( 10 \) datapoints \( ( \mathbf{x}_1,
|
||||
\cdots, \mathbf{x}_{10} ) \) and we choose to have \( M=5 \) minibathces,
|
||||
then each minibatch contains two datapoints. 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 datapoints 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 datapoints with a sum over the datapoints in one the minibatches
|
||||
picked at random in each gradient descent step
|
||||
$$
|
||||
\nabla_\theta
|
||||
C(\mathbf{\theta}) = \sum_{i=1}^n \nabla_\theta c_i(\mathbf{x}_i,
|
||||
\mathbf{\theta}) \rightarrow \sum_{i \in B_k}^n \nabla_\theta
|
||||
c_i(\mathbf{x}_i, \mathbf{\theta}).
|
||||
$$
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|
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
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|
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|
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<a name="part0022"></a>
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<!-- !split -->
|
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|
||||
<h2 id="___sec21" class="anchor">The gradient step </h2>
|
||||
|
||||
<p>
|
||||
Thus a gradient descent step now looks like
|
||||
$$
|
||||
\theta_{j+1} = \theta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\theta c_i(\mathbf{x}_i,
|
||||
\mathbf{\theta})
|
||||
$$
|
||||
|
||||
<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.
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|
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'algortithm',
|
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2,
|
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None,
|
||||
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|
||||
('Steepest descent', 2, None, '___sec1'),
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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('Conjugate gradient method', 2, None, '___sec34'),
|
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('Conjugate gradient method', 2, None, '___sec35')]}
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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</li>
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||||
<a name="part0023"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec22" class="anchor">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 theta</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>
|
||||
<p>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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||||
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<a name="part0024"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec23" class="anchor">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
|
||||
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|
||||
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|
||||
evaluate the cost function at this point, store the result and
|
||||
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|
||||
compare the values of the cost function and keep the \( \theta \) that
|
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gave the lowest value.
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|
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|
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|
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||||
('Conjugate gradient method', 2, None, '___sec31'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec32'),
|
||||
('Conjugate gradient method', 2, None, '___sec33'),
|
||||
('Conjugate gradient method', 2, None, '___sec34'),
|
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('Conjugate gradient method', 2, None, '___sec35')]}
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<a name="part0025"></a>
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||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec24" class="anchor">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 \( \theta \) 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 \( \theta \) 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 theta</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; font-weight: bold">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))
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||||
</pre></div>
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|
||||
'algortithm',
|
||||
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|
||||
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|
||||
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|
||||
('Steepest descent', 2, None, '___sec1'),
|
||||
('More on Steepest descent', 2, None, '___sec2'),
|
||||
('The ideal', 2, None, '___sec3'),
|
||||
('The sensitiveness of the gradient descent', 2, None, '___sec4'),
|
||||
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|
||||
('And a corresponding example using _scikit-learn_',
|
||||
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|
||||
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|
||||
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|
||||
('Convex functions', 2, None, '___sec7'),
|
||||
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|
||||
('Conditions on convex functions', 2, None, '___sec9'),
|
||||
('More on convex functions', 2, None, '___sec10'),
|
||||
('Some simple problems', 2, None, '___sec11'),
|
||||
('Revisiting our first homework', 2, None, '___sec12'),
|
||||
('Gradient descent example', 2, None, '___sec13'),
|
||||
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|
||||
('The Hessian matrix', 2, None, '___sec15'),
|
||||
('Simple program', 2, None, '___sec16'),
|
||||
('Gradient descent and Ridge', 2, None, '___sec17'),
|
||||
('Stochastic Gradient Descent', 2, None, '___sec18'),
|
||||
('Computation of gradients', 2, None, '___sec19'),
|
||||
('SGD example', 2, None, '___sec20'),
|
||||
('The gradient step', 2, None, '___sec21'),
|
||||
('Simple example code', 2, None, '___sec22'),
|
||||
('When do we stop?', 2, None, '___sec23'),
|
||||
('Slightly different approach', 2, None, '___sec24'),
|
||||
('Conjugate gradient (CG) method', 2, None, '___sec25'),
|
||||
('Conjugate gradient method', 2, None, '___sec26'),
|
||||
("Conjugate gradient method, Newton's method first",
|
||||
2,
|
||||
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|
||||
'___sec27'),
|
||||
('Conjugate gradient method', 2, None, '___sec28'),
|
||||
('Conjugate gradient method', 2, None, '___sec29'),
|
||||
('Conjugate gradient method', 2, None, '___sec30'),
|
||||
('Conjugate gradient method', 2, None, '___sec31'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec32'),
|
||||
('Conjugate gradient method', 2, None, '___sec33'),
|
||||
('Conjugate gradient method', 2, None, '___sec34'),
|
||||
('Conjugate gradient method', 2, None, '___sec35')]}
|
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end of tocinfo -->
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<h2 id="___sec25" class="anchor">Conjugate gradient (CG) method </h2>
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||||
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||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
The success of the CG method 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
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{A}\hat{x} = \hat{b}.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
In the iterative process we end up with a problem like
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{r}= \hat{b}-\hat{A}\hat{x},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
where \( \hat{r} \) is the so-called residual or error in the iterative process.
|
||||
|
||||
<p>
|
||||
When we have found the exact solution, \( \hat{r}=0 \).
|
||||
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- !split -->
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<h2 id="___sec26" class="anchor">Conjugate gradient method </h2>
|
||||
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|
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|
||||
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|
||||
|
||||
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|
||||
The residual is zero when we reach the minimum of the quadratic equation
|
||||
$$
|
||||
\begin{equation*}
|
||||
P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
with the constraint that the matrix \( \hat{A} \) is positive definite and symmetric.
|
||||
If we search for a minimum of the quantum mechanical variance, then the matrix
|
||||
\( \hat{A} \), which is called the Hessian, is given by the second-derivative of the function we want to minimize. This quantity is always positive definite. In our case this corresponds normally to the second derivative of the energy.
|
||||
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<a name="part0028"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec27" class="anchor">Conjugate gradient method, Newton's method first </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
We seek the minimum of the energy or the variance as function of various variational parameters.
|
||||
In our case we have thus a function \( f \) whose minimum we are seeking.
|
||||
In Newton's method we set \( \nabla f = 0 \) and we can thus compute the next iteration point
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{x}-\hat{x}_i=\hat{A}^{-1}\nabla f(\hat{x}_i).
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
Subtracting this equation from that of \( \hat{x}_{i+1} \) we have
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{x}_{i+1}-\hat{x}_i=\hat{A}^{-1}(\nabla f(\hat{x}_{i+1})-\nabla f(\hat{x}_i)).
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
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{'highest level': 2,
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'sections': [('Optimization, the central part of any Machine Learning '
|
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'algortithm',
|
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2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Steepest descent', 2, None, '___sec1'),
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('More on Steepest descent', 2, None, '___sec2'),
|
||||
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|
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|
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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|
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|
||||
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|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
In the CG method we define so-called conjugate directions and two vectors
|
||||
\( \hat{s} \) and \( \hat{t} \)
|
||||
are said to be
|
||||
conjugate if
|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
The philosophy of the CG method is to perform searches in various conjugate directions
|
||||
of our vectors \( \hat{x}_i \) obeying the above criterion, namely
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{x}_i^T\hat{A}\hat{x}_j= 0.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
Two vectors are conjugate if they are orthogonal with respect to
|
||||
this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
|
||||
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<div class="panel-body">
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||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
An example is given by the eigenvectors of the matrix
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
which is zero unless \( i=j \).
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- !split -->
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<h2 id="___sec30" class="anchor">Conjugate gradient method </h2>
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<div class="panel panel-default">
|
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<div class="panel-body">
|
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
|
||||
\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
|
||||
Then the \( \hat{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
|
||||
$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
|
||||
\end{equation*}
|
||||
$$
|
||||
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
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||||
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||||
|
||||
<a name="part0032"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec31" class="anchor">Conjugate gradient method </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
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 \( \hat{p}_k^T \) from the left gives
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
and we can define the coefficients \( \alpha_k \) as
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
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|
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|
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|
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|
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|
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|
||||
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|
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|
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|
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|
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|
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|
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|
||||
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|
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|
||||
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|
||||
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|
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|
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|
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|
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|
||||
('The Hessian matrix', 2, None, '___sec15'),
|
||||
('Simple program', 2, None, '___sec16'),
|
||||
('Gradient descent and Ridge', 2, None, '___sec17'),
|
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|
||||
('Computation of gradients', 2, None, '___sec19'),
|
||||
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|
||||
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|
||||
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|
||||
('When do we stop?', 2, None, '___sec23'),
|
||||
('Slightly different approach', 2, None, '___sec24'),
|
||||
('Conjugate gradient (CG) method', 2, None, '___sec25'),
|
||||
('Conjugate gradient method', 2, None, '___sec26'),
|
||||
("Conjugate gradient method, Newton's method first",
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
('Conjugate gradient method', 2, None, '___sec35')]}
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
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<a name="part0033"></a>
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||||
<!-- !split -->
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||||
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||||
<h2 id="___sec32" class="anchor">Conjugate gradient method and iterations </h2>
|
||||
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||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
|
||||
<p>
|
||||
If we choose the conjugate vectors \( \hat{p}_k \) carefully,
|
||||
then we may not need all of them to obtain a good approximation to the solution
|
||||
\( \hat{x} \).
|
||||
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 \( \hat{x} \) as \( \hat{x}_0 \).
|
||||
We can assume without loss of generality that
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{x}_0=0,
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
or consider the system
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
instead.
|
||||
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
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<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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||||
<!-- navigation toc: --> <li><a href="#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
|
||||
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|
||||
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|
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|
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||||
<a name="part0034"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec33" class="anchor">Conjugate gradient method </h2>
|
||||
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|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
|
||||
$$
|
||||
\begin{equation*}
|
||||
f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
This suggests taking the first basis vector \( \hat{p}_1 \)
|
||||
to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
|
||||
which equals
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{A}\hat{x}_0-\hat{b},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
and
|
||||
\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
|
||||
The other vectors in the basis will be conjugate to the gradient,
|
||||
hence the name conjugate gradient method.
|
||||
</div>
|
||||
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|
||||
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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||||
</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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||||
<a name="part0035"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec34" class="anchor">Conjugate gradient method </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
Note that \( \hat{r}_k \) is the negative gradient of \( f \) at
|
||||
\( \hat{x}=\hat{x}_k \),
|
||||
so the gradient descent method would be to move in the direction \( \hat{r}_k \).
|
||||
Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other,
|
||||
so we take the direction closest to the gradient \( \hat{r}_k \)
|
||||
under the conjugacy constraint.
|
||||
This gives the following expression
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
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|
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{'highest level': 2,
|
||||
'sections': [('Optimization, the central part of any Machine Learning '
|
||||
'algortithm',
|
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2,
|
||||
None,
|
||||
'___sec0'),
|
||||
('Steepest descent', 2, None, '___sec1'),
|
||||
('More on Steepest descent', 2, None, '___sec2'),
|
||||
('The ideal', 2, None, '___sec3'),
|
||||
('The sensitiveness of the gradient descent', 2, None, '___sec4'),
|
||||
('Gradient Descent Example', 2, None, '___sec5'),
|
||||
('And a corresponding example using _scikit-learn_',
|
||||
2,
|
||||
None,
|
||||
'___sec6'),
|
||||
('Convex functions', 2, None, '___sec7'),
|
||||
('Convex function', 2, None, '___sec8'),
|
||||
('Conditions on convex functions', 2, None, '___sec9'),
|
||||
('More on convex functions', 2, None, '___sec10'),
|
||||
('Some simple problems', 2, None, '___sec11'),
|
||||
('Revisiting our first homework', 2, None, '___sec12'),
|
||||
('Gradient descent example', 2, None, '___sec13'),
|
||||
('The derivative of the cost/loss function', 2, None, '___sec14'),
|
||||
('The Hessian matrix', 2, None, '___sec15'),
|
||||
('Simple program', 2, None, '___sec16'),
|
||||
('Gradient descent and Ridge', 2, None, '___sec17'),
|
||||
('Stochastic Gradient Descent', 2, None, '___sec18'),
|
||||
('Computation of gradients', 2, None, '___sec19'),
|
||||
('SGD example', 2, None, '___sec20'),
|
||||
('The gradient step', 2, None, '___sec21'),
|
||||
('Simple example code', 2, None, '___sec22'),
|
||||
('When do we stop?', 2, None, '___sec23'),
|
||||
('Slightly different approach', 2, None, '___sec24'),
|
||||
('Conjugate gradient (CG) method', 2, None, '___sec25'),
|
||||
('Conjugate gradient method', 2, None, '___sec26'),
|
||||
("Conjugate gradient method, Newton's method first",
|
||||
2,
|
||||
None,
|
||||
'___sec27'),
|
||||
('Conjugate gradient method', 2, None, '___sec28'),
|
||||
('Conjugate gradient method', 2, None, '___sec29'),
|
||||
('Conjugate gradient method', 2, None, '___sec30'),
|
||||
('Conjugate gradient method', 2, None, '___sec31'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec32'),
|
||||
('Conjugate gradient method', 2, None, '___sec33'),
|
||||
('Conjugate gradient method', 2, None, '___sec34'),
|
||||
('Conjugate gradient method', 2, None, '___sec35')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
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<a class="navbar-brand" href="Splines-bs.html">Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._Splines-bs001.html#___sec0" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs002.html#___sec1" style="font-size: 80%;">Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs003.html#___sec2" style="font-size: 80%;">More on Steepest descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs004.html#___sec3" style="font-size: 80%;">The ideal</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs005.html#___sec4" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs006.html#___sec5" style="font-size: 80%;">Gradient Descent Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs007.html#___sec6" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs008.html#___sec7" style="font-size: 80%;">Convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs009.html#___sec8" style="font-size: 80%;">Convex function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs010.html#___sec9" style="font-size: 80%;">Conditions on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs011.html#___sec10" style="font-size: 80%;">More on convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs012.html#___sec11" style="font-size: 80%;">Some simple problems</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs013.html#___sec12" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs014.html#___sec13" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs015.html#___sec14" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs016.html#___sec15" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs017.html#___sec16" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs018.html#___sec17" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs019.html#___sec18" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs020.html#___sec19" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs021.html#___sec20" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs022.html#___sec21" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs023.html#___sec22" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient (CG) method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method, Newton's method first</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs035.html#___sec34" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
|
||||
</ul>
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</li>
|
||||
</ul>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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||||
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||||
<a name="part0036"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec35" class="anchor">Conjugate gradient method </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
We can also compute the residual iteratively as
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
which equals
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
or
|
||||
$$
|
||||
\begin{equation*}
|
||||
(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
which gives
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
||||
</div>
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||||
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||||
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<li><a href="._Splines-bs035.html">«</a></li>
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<li><a href="._Splines-bs000.html">1</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Splines-bs028.html">29</a></li>
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<li><a href="._Splines-bs029.html">30</a></li>
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<li><a href="._Splines-bs032.html">33</a></li>
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<li><a href="._Splines-bs033.html">34</a></li>
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<li><a href="._Splines-bs034.html">35</a></li>
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<li><a href="._Splines-bs035.html">36</a></li>
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<li class="active"><a href="._Splines-bs036.html">37</a></li>
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||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -938,68 +938,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec28">Simple example and demonstration </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
The function \( f \) can be either the energy or the variance. If we choose the energy then we have
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{\alpha}_{i+1}-\hat{\alpha}_i=\hat{A}^{-1}(\nabla E(\hat{\alpha}_{i+1})-\nabla E(\hat{\alpha}_i)).
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
In the simple harmonic oscillator model, the gradient and the Hessian \( \hat{A} \) are
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
\frac{d\langle E_L[\alpha]\rangle}{d\alpha} = \alpha-\frac{1}{4\alpha^3}
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and a second derivative which is always positive (meaning that we find a minimum)
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{A}= \frac{d^2\langle E_L[\alpha]\rangle}{d\alpha^2} = 1+\frac{3}{4\alpha^4}
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec29">Simple example and demonstration </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
We get then
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}=\frac{4}{3}\alpha_i-\frac{\alpha_i^4}{3\alpha_{i+1}^3},
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
which can be rewritten as
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}^4-\frac{4}{3}\alpha_i\alpha_{i+1}^4+\frac{1}{3}\alpha_i^4.
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec30">Conjugate gradient method </h2>
|
||||
<h2 id="___sec28">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1032,7 +971,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec31">Conjugate gradient method </h2>
|
||||
<h2 id="___sec29">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1051,7 +990,7 @@ which is zero unless \( i=j \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec32">Conjugate gradient method </h2>
|
||||
<h2 id="___sec30">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1081,7 +1020,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec33">Conjugate gradient method </h2>
|
||||
<h2 id="___sec31">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1118,7 +1057,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec34">Conjugate gradient method and iterations </h2>
|
||||
<h2 id="___sec32">Conjugate gradient method and iterations </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1155,7 +1094,7 @@ instead.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec35">Conjugate gradient method </h2>
|
||||
<h2 id="___sec33">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1188,7 +1127,7 @@ hence the name conjugate gradient method.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec36">Conjugate gradient method </h2>
|
||||
<h2 id="___sec34">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1220,7 +1159,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec37">Conjugate gradient method </h2>
|
||||
<h2 id="___sec35">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
@@ -98,16 +98,14 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
2,
|
||||
None,
|
||||
'___sec27'),
|
||||
('Simple example and demonstration', 2, None, '___sec28'),
|
||||
('Simple example and demonstration', 2, None, '___sec29'),
|
||||
('Conjugate gradient method', 2, None, '___sec28'),
|
||||
('Conjugate gradient method', 2, None, '___sec29'),
|
||||
('Conjugate gradient method', 2, None, '___sec30'),
|
||||
('Conjugate gradient method', 2, None, '___sec31'),
|
||||
('Conjugate gradient method', 2, None, '___sec32'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec32'),
|
||||
('Conjugate gradient method', 2, None, '___sec33'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec34'),
|
||||
('Conjugate gradient method', 2, None, '___sec35'),
|
||||
('Conjugate gradient method', 2, None, '___sec36'),
|
||||
('Conjugate gradient method', 2, None, '___sec37')]}
|
||||
('Conjugate gradient method', 2, None, '___sec34'),
|
||||
('Conjugate gradient method', 2, None, '___sec35')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -879,60 +877,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec28">Simple example and demonstration </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
The function \( f \) can be either the energy or the variance. If we choose the energy then we have
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{\alpha}_{i+1}-\hat{\alpha}_i=\hat{A}^{-1}(\nabla E(\hat{\alpha}_{i+1})-\nabla E(\hat{\alpha}_i)).
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
In the simple harmonic oscillator model, the gradient and the Hessian \( \hat{A} \) are
|
||||
$$
|
||||
\begin{equation*}
|
||||
\frac{d\langle E_L[\alpha]\rangle}{d\alpha} = \alpha-\frac{1}{4\alpha^3}
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
and a second derivative which is always positive (meaning that we find a minimum)
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{A}= \frac{d^2\langle E_L[\alpha]\rangle}{d\alpha^2} = 1+\frac{3}{4\alpha^4}
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec29">Simple example and demonstration </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
We get then
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}=\frac{4}{3}\alpha_i-\frac{\alpha_i^4}{3\alpha_{i+1}^3},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
which can be rewritten as
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}^4-\frac{4}{3}\alpha_i\alpha_{i+1}^4+\frac{1}{3}\alpha_i^4.
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec30">Conjugate gradient method </h2>
|
||||
<h2 id="___sec28">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -962,7 +907,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec31">Conjugate gradient method </h2>
|
||||
<h2 id="___sec29">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -980,7 +925,7 @@ which is zero unless \( i=j \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec32">Conjugate gradient method </h2>
|
||||
<h2 id="___sec30">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1007,7 +952,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec33">Conjugate gradient method </h2>
|
||||
<h2 id="___sec31">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1039,7 +984,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec34">Conjugate gradient method and iterations </h2>
|
||||
<h2 id="___sec32">Conjugate gradient method and iterations </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1075,7 +1020,7 @@ instead.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec35">Conjugate gradient method </h2>
|
||||
<h2 id="___sec33">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1105,7 +1050,7 @@ hence the name conjugate gradient method.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec36">Conjugate gradient method </h2>
|
||||
<h2 id="___sec34">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1134,7 +1079,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec37">Conjugate gradient method </h2>
|
||||
<h2 id="___sec35">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
@@ -103,16 +103,14 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
2,
|
||||
None,
|
||||
'___sec27'),
|
||||
('Simple example and demonstration', 2, None, '___sec28'),
|
||||
('Simple example and demonstration', 2, None, '___sec29'),
|
||||
('Conjugate gradient method', 2, None, '___sec28'),
|
||||
('Conjugate gradient method', 2, None, '___sec29'),
|
||||
('Conjugate gradient method', 2, None, '___sec30'),
|
||||
('Conjugate gradient method', 2, None, '___sec31'),
|
||||
('Conjugate gradient method', 2, None, '___sec32'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec32'),
|
||||
('Conjugate gradient method', 2, None, '___sec33'),
|
||||
('Conjugate gradient method and iterations', 2, None, '___sec34'),
|
||||
('Conjugate gradient method', 2, None, '___sec35'),
|
||||
('Conjugate gradient method', 2, None, '___sec36'),
|
||||
('Conjugate gradient method', 2, None, '___sec37')]}
|
||||
('Conjugate gradient method', 2, None, '___sec34'),
|
||||
('Conjugate gradient method', 2, None, '___sec35')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -884,60 +882,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec28">Simple example and demonstration </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
The function \( f \) can be either the energy or the variance. If we choose the energy then we have
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{\alpha}_{i+1}-\hat{\alpha}_i=\hat{A}^{-1}(\nabla E(\hat{\alpha}_{i+1})-\nabla E(\hat{\alpha}_i)).
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
In the simple harmonic oscillator model, the gradient and the Hessian \( \hat{A} \) are
|
||||
$$
|
||||
\begin{equation*}
|
||||
\frac{d\langle E_L[\alpha]\rangle}{d\alpha} = \alpha-\frac{1}{4\alpha^3}
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
and a second derivative which is always positive (meaning that we find a minimum)
|
||||
$$
|
||||
\begin{equation*}
|
||||
\hat{A}= \frac{d^2\langle E_L[\alpha]\rangle}{d\alpha^2} = 1+\frac{3}{4\alpha^4}
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec29">Simple example and demonstration </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
We get then
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}=\frac{4}{3}\alpha_i-\frac{\alpha_i^4}{3\alpha_{i+1}^3},
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
which can be rewritten as
|
||||
$$
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}^4-\frac{4}{3}\alpha_i\alpha_{i+1}^4+\frac{1}{3}\alpha_i^4.
|
||||
\end{equation*}
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec30">Conjugate gradient method </h2>
|
||||
<h2 id="___sec28">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -967,7 +912,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec31">Conjugate gradient method </h2>
|
||||
<h2 id="___sec29">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -985,7 +930,7 @@ which is zero unless \( i=j \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec32">Conjugate gradient method </h2>
|
||||
<h2 id="___sec30">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1012,7 +957,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec33">Conjugate gradient method </h2>
|
||||
<h2 id="___sec31">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1044,7 +989,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec34">Conjugate gradient method and iterations </h2>
|
||||
<h2 id="___sec32">Conjugate gradient method and iterations </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1080,7 +1025,7 @@ instead.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec35">Conjugate gradient method </h2>
|
||||
<h2 id="___sec33">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1110,7 +1055,7 @@ hence the name conjugate gradient method.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec36">Conjugate gradient method </h2>
|
||||
<h2 id="___sec34">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1139,7 +1084,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec37">Conjugate gradient method </h2>
|
||||
<h2 id="___sec35">Conjugate gradient method </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
@@ -906,88 +906,6 @@
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Simple example and demonstration\n",
|
||||
"The function $f$ can be either the energy or the variance. If we choose the energy then we have"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\hat{\\alpha}_{i+1}-\\hat{\\alpha}_i=\\hat{A}^{-1}(\\nabla E(\\hat{\\alpha}_{i+1})-\\nabla E(\\hat{\\alpha}_i)).\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"In the simple harmonic oscillator model, the gradient and the Hessian $\\hat{A}$ are"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\frac{d\\langle E_L[\\alpha]\\rangle}{d\\alpha} = \\alpha-\\frac{1}{4\\alpha^3}\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"and a second derivative which is always positive (meaning that we find a minimum)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\hat{A}= \\frac{d^2\\langle E_L[\\alpha]\\rangle}{d\\alpha^2} = 1+\\frac{3}{4\\alpha^4}\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Simple example and demonstration\n",
|
||||
"We get then"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\alpha_{i+1}=\\frac{4}{3}\\alpha_i-\\frac{\\alpha_i^4}{3\\alpha_{i+1}^3},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"which can be rewritten as"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\alpha_{i+1}^4-\\frac{4}{3}\\alpha_i\\alpha_{i+1}^4+\\frac{1}{3}\\alpha_i^4.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@@ -630,45 +630,6 @@ Subtracting this equation from that of $\hat{x}_{i+1}$ we have
|
||||
!et
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== Simple example and demonstration =====
|
||||
!bblock
|
||||
The function $f$ can be either the energy or the variance. If we choose the energy then we have
|
||||
!bt
|
||||
\begin{equation*}
|
||||
\hat{\alpha}_{i+1}-\hat{\alpha}_i=\hat{A}^{-1}(\nabla E(\hat{\alpha}_{i+1})-\nabla E(\hat{\alpha}_i)).
|
||||
\end{equation*}
|
||||
!et
|
||||
In the simple harmonic oscillator model, the gradient and the Hessian $\hat{A}$ are
|
||||
!bt
|
||||
\begin{equation*}
|
||||
\frac{d\langle E_L[\alpha]\rangle}{d\alpha} = \alpha-\frac{1}{4\alpha^3}
|
||||
\end{equation*}
|
||||
!et
|
||||
and a second derivative which is always positive (meaning that we find a minimum)
|
||||
!bt
|
||||
\begin{equation*}
|
||||
\hat{A}= \frac{d^2\langle E_L[\alpha]\rangle}{d\alpha^2} = 1+\frac{3}{4\alpha^4}
|
||||
\end{equation*}
|
||||
!et
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== Simple example and demonstration =====
|
||||
!bblock
|
||||
We get then
|
||||
!bt
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}=\frac{4}{3}\alpha_i-\frac{\alpha_i^4}{3\alpha_{i+1}^3},
|
||||
\end{equation*}
|
||||
!et
|
||||
which can be rewritten as
|
||||
!bt
|
||||
\begin{equation*}
|
||||
\alpha_{i+1}^4-\frac{4}{3}\alpha_i\alpha_{i+1}^4+\frac{1}{3}\alpha_i^4.
|
||||
\end{equation*}
|
||||
!et
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== Conjugate gradient method =====
|
||||
|
||||
@@ -44,11 +44,12 @@ system doconce split_html $html.html --method=space10
|
||||
# Bootstrap style
|
||||
html=${name}-bs
|
||||
system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
|
||||
#system doconce split_html $html.html --method=split --pagination --nav_button=bottom
|
||||
system doconce split_html $html.html --method=split --pagination --nav_button=bottom
|
||||
|
||||
# IPython notebook
|
||||
system doconce format ipynb $name $opt
|
||||
|
||||
|
||||
# Ordinary plain LaTeX document
|
||||
rm -f *.aux # important after beamer
|
||||
system doconce format pdflatex $name --minted_latex_style=trac --latex_admon=paragraph $opt
|
||||
|
||||
Reference in New Issue
Block a user