mhjensen
396 lines
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HTML
396 lines
24 KiB
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'algortithm',
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('Revisiting our Logistic Regression case', 2, None, '___sec3'),
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('The equations to solve', 2, None, '___sec4'),
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('Extending to more than one variable', 2, None, '___sec9'),
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('More on Steepest descent', 2, None, '___sec11'),
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('Conjugate gradient method', 2, None, '___sec31'),
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('Conjugate gradient method', 2, None, '___sec32'),
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('Revisiting our first homework', 2, None, '___sec33'),
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('Gradient descent example', 2, None, '___sec34'),
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('The derivative of the cost/loss function', 2, None, '___sec35'),
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('The Hessian matrix', 2, None, '___sec36'),
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('Simple program', 2, None, '___sec37'),
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('Gradient Descent Example', 2, None, '___sec38'),
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('And a corresponding example using _scikit-learn_',
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2,
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None,
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'___sec39'),
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('Gradient descent and Ridge', 2, None, '___sec40'),
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('Program example for gradient descent with Ridge Regression',
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2,
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None,
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'___sec41'),
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('Using gradient descent methods, limitations',
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2,
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None,
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'___sec42'),
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('Friday September 25', 2, None, '___sec43'),
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('Stochastic Gradient Descent', 2, None, '___sec44'),
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('Computation of gradients', 2, None, '___sec45'),
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('SGD example', 2, None, '___sec46'),
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('The gradient step', 2, None, '___sec47'),
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('When do we stop?', 2, None, '___sec49'),
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('Slightly different approach', 2, None, '___sec50'),
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('Program for stochastic gradient', 2, None, '___sec51'),
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('Momentum based GD', 2, None, '___sec52'),
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('More on momentum based approaches', 2, None, '___sec53'),
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('Momentum parameter', 2, None, '___sec54'),
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('Second moment of the gradient', 2, None, '___sec55'),
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('RMS prop', 2, None, '___sec56'),
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('ADAM optimizer', 2, None, '___sec57'),
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('Practical tips', 2, None, '___sec58'),
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('Automatic differentiation', 2, None, '___sec59'),
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('Using autograd', 2, None, '___sec60'),
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('Autograd with more complicated functions', 2, None, '___sec61'),
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('More complicated functions using the elements of their '
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2,
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None,
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('More autograd', 2, None, '___sec64'),
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('And with loops', 2, None, '___sec65'),
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('Using recursion', 2, None, '___sec66'),
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<!-- navigation toc: --> <li><a href="._week39-bs001.html#___sec0" style="font-size: 80%;">Plan for week 39</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs002.html#___sec1" style="font-size: 80%;">Thursday September 24</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs003.html#___sec2" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs004.html#___sec3" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs005.html#___sec4" style="font-size: 80%;">The equations to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs006.html#___sec5" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs007.html#___sec6" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs008.html#___sec7" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs009.html#___sec8" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs010.html#___sec9" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs011.html#___sec10" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs012.html#___sec11" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs013.html#___sec12" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs014.html#___sec13" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs015.html#___sec14" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs016.html#___sec15" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs017.html#___sec16" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs018.html#___sec17" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs019.html#___sec18" style="font-size: 80%;">Some simple problems</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs020.html#___sec19" style="font-size: 80%;">Standard steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs021.html#___sec20" style="font-size: 80%;">Gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs022.html#___sec21" style="font-size: 80%;">Steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs023.html#___sec22" style="font-size: 80%;">Steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs024.html#___sec23" style="font-size: 80%;">Final expressions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs025.html#___sec24" style="font-size: 80%;">Steepest descent example</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs026.html#___sec25" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs028.html#___sec27" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs029.html#___sec28" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs030.html#___sec29" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs032.html#___sec31" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs033.html#___sec32" style="font-size: 80%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs034.html#___sec33" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs035.html#___sec34" style="font-size: 80%;">Gradient descent example</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs036.html#___sec35" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs037.html#___sec36" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs038.html#___sec37" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs039.html#___sec38" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs040.html#___sec39" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs041.html#___sec40" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs042.html#___sec41" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs043.html#___sec42" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs044.html#___sec43" style="font-size: 80%;">Friday September 25</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs045.html#___sec44" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs046.html#___sec45" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs047.html#___sec46" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs048.html#___sec47" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs049.html#___sec48" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs050.html#___sec49" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs051.html#___sec50" style="font-size: 80%;">Slightly different approach</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs052.html#___sec51" style="font-size: 80%;">Program for stochastic gradient</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs053.html#___sec52" style="font-size: 80%;">Momentum based GD</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs054.html#___sec53" style="font-size: 80%;">More on momentum based approaches</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs055.html#___sec54" style="font-size: 80%;">Momentum parameter</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs056.html#___sec55" style="font-size: 80%;">Second moment of the gradient</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs057.html#___sec56" style="font-size: 80%;">RMS prop</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs058.html#___sec57" style="font-size: 80%;">ADAM optimizer</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs059.html#___sec58" style="font-size: 80%;">Practical tips</a></li>
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<!-- navigation toc: --> <li><a href="#___sec59" style="font-size: 80%;">Automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs061.html#___sec60" style="font-size: 80%;">Using autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs062.html#___sec61" style="font-size: 80%;">Autograd with more complicated functions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs063.html#___sec62" style="font-size: 80%;">More complicated functions using the elements of their arguments directly</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs064.html#___sec63" style="font-size: 80%;">Functions using mathematical functions from Numpy</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs065.html#___sec64" style="font-size: 80%;">More autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs066.html#___sec65" style="font-size: 80%;">And with loops</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs067.html#___sec66" style="font-size: 80%;">Using recursion</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs068.html#___sec67" style="font-size: 80%;">Unsupported functions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs069.html#___sec68" style="font-size: 80%;">The syntax a.dot(b) when finding the dot product</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs070.html#___sec69" style="font-size: 80%;">Recommended to avoid</a></li>
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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="part0060"></a>
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<!-- !split -->
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<h2 id="___sec59" class="anchor">Automatic differentiation </h2>
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<p>
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<a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_self">Automatic differentiation (AD)</a>,
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also called algorithmic
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differentiation or computational differentiation,is a set of
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techniques to numerically evaluate the derivative of a function
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specified by a computer program. AD exploits the fact that every
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computer program, no matter how complicated, executes a sequence of
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elementary arithmetic operations (addition, subtraction,
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multiplication, division, etc.) and elementary functions (exp, log,
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sin, cos, etc.). By applying the chain rule repeatedly to these
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operations, derivatives of arbitrary order can be computed
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automatically, accurately to working precision, and using at most a
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small constant factor more arithmetic operations than the original
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program.
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<p>
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Automatic differentiation is neither:
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<ul>
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<li> Symbolic differentiation, nor</li>
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<li> Numerical differentiation (the method of finite differences).</li>
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</ul>
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Symbolic differentiation can lead to inefficient code and faces the
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difficulty of converting a computer program into a single expression,
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while numerical differentiation can introduce round-off errors in the
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discretization process and cancellation
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<p>
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Python has tools for so-called <b>automatic differentiation</b>.
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Consider the following example
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$$
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f(x) = \sin\left(2\pi x + x^2\right)
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$$
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which has the following derivative
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$$
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f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
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$$
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Using <b>autograd</b> we have
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #408080; font-style: italic"># To do elementwise differentiation:</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> elementwise_grad <span style="color: #008000; font-weight: bold">as</span> egrad
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<span style="color: #408080; font-style: italic"># To plot:</span>
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<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>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(x):
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sin(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f_grad_analytic</span>(x):
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>(<span style="color: #666666">2*</span>np<span style="color: #666666">.</span>pi <span style="color: #666666">+</span> <span style="color: #666666">2*</span>x)
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<span style="color: #408080; font-style: italic"># Do the comparison:</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">1</span>,<span style="color: #666666">1000</span>)
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f_grad <span style="color: #666666">=</span> egrad(f)
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computed <span style="color: #666666">=</span> f_grad(x)
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analytic <span style="color: #666666">=</span> f_grad_analytic(x)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'Derivative computed from Autograd compared with the analytical derivative'</span>)
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plt<span style="color: #666666">.</span>plot(x,computed,label<span style="color: #666666">=</span><span style="color: #BA2121">'autograd'</span>)
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plt<span style="color: #666666">.</span>plot(x,analytic,label<span style="color: #666666">=</span><span style="color: #BA2121">'analytic'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'x'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'y'</span>)
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plt<span style="color: #666666">.</span>legend()
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plt<span style="color: #666666">.</span>show()
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<span style="color: #008000">print</span>(<span style="color: #BA2121">"The max absolute difference is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>(np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(computed <span style="color: #666666">-</span> analytic))))
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</pre></div>
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