cleaning up splines
This commit is contained in:
mhjensen
@@ -80,75 +80,73 @@ Automatically generated HTML file from DocOnce source
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('Gradient descent example', 2, None, '___sec28'),
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('The derivative of the cost/loss function', 2, None, '___sec29'),
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('The Hessian matrix', 2, None, '___sec30'),
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('Simple program', 2, None, '___sec31'),
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('Conjugate gradient method', 2, None, '___sec26'),
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('Conjugate gradient method', 2, None, '___sec27'),
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('Gradient descent and Ridge', 2, None, '___sec34'),
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('Automatic differentiation', 2, None, '___sec35'),
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('Using autograd', 2, None, '___sec36'),
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('Gradient descent and Ridge', 2, None, '___sec43'),
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('Automatic differentiation', 2, None, '___sec44'),
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('Using autograd', 2, None, '___sec45'),
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('More complicated functions using the elements of their '
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'___sec38'),
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'___sec47'),
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('Functions using mathematical functions from Numpy',
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'___sec39'),
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('More autograd', 2, None, '___sec40'),
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('And with loops', 2, None, '___sec41'),
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('Using recursion', 2, None, '___sec42'),
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('Unsupported functions', 2, None, '___sec43'),
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'___sec48'),
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('More autograd', 2, None, '___sec49'),
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('And with loops', 2, None, '___sec50'),
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('Using recursion', 2, None, '___sec51'),
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('Unsupported functions', 2, None, '___sec52'),
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('The syntax a.dot(b) when finding the dot product',
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None,
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'___sec44'),
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('Recommended to avoid', 2, None, '___sec45'),
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('Stochastic Gradient Descent', 2, None, '___sec46'),
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('Computation of gradients', 2, None, '___sec47'),
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('SGD example', 2, None, '___sec48'),
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('The gradient step', 2, None, '___sec49'),
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('Simple example code', 2, None, '___sec50'),
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('When do we stop?', 2, None, '___sec51'),
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('Slightly different approach', 2, None, '___sec52'),
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('Program for stochastic gradient', 2, None, '___sec53'),
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('Momentum based methods', 2, None, '___sec54'),
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('Conjugate gradient method', 2, None, '___sec55'),
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('Conjugate gradient method', 2, None, '___sec56'),
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('Conjugate gradient method', 2, None, '___sec57'),
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('Conjugate gradient method', 2, None, '___sec58'),
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('Conjugate gradient method and iterations', 2, None, '___sec59'),
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('Conjugate gradient method', 2, None, '___sec60'),
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('Conjugate gradient method', 2, None, '___sec61'),
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('Conjugate gradient method', 2, None, '___sec62'),
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('Simple implementation of the Conjugate gradient algorithm',
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2,
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None,
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'___sec63'),
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('Broyden–Fletcher–Goldfarb–Shanno algorithm',
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2,
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None,
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'___sec64'),
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'___sec53'),
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('Recommended to avoid', 2, None, '___sec54'),
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('Stochastic Gradient Descent', 2, None, '___sec55'),
|
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('Computation of gradients', 2, None, '___sec56'),
|
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('SGD example', 2, None, '___sec57'),
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('The gradient step', 2, None, '___sec58'),
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('Simple example code', 2, None, '___sec59'),
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('When do we stop?', 2, None, '___sec60'),
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('Slightly different approach', 2, None, '___sec61'),
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('Program for stochastic gradient', 2, None, '___sec62'),
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('Using gradient descent methods, limitations',
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2,
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None,
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'___sec65'),
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('Momentum based GD', 2, None, '___sec66'),
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('More on momentum based approaches', 2, None, '___sec67'),
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('Momentum parameter', 2, None, '___sec68'),
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('Second moment of the gradient', 2, None, '___sec69'),
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('RMS prop', 2, None, '___sec70'),
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('ADAM optimizer', 2, None, '___sec71'),
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('Practical tips', 2, None, '___sec72')]}
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'___sec63'),
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('Momentum based GD', 2, None, '___sec64'),
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('More on momentum based approaches', 2, None, '___sec65'),
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('Momentum parameter', 2, None, '___sec66'),
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('Second moment of the gradient', 2, None, '___sec67'),
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('RMS prop', 2, None, '___sec68'),
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('ADAM optimizer', 2, None, '___sec69'),
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('Practical tips', 2, None, '___sec70')]}
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end of tocinfo -->
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<body>
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@@ -212,53 +210,51 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._Splines-bs024.html#___sec23" style="font-size: 80%;">Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs025.html#___sec24" style="font-size: 80%;">The routine for the steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs026.html#___sec25" style="font-size: 80%;">Steepest descent example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs027.html#___sec26" style="font-size: 80%;">Conjugate gradient</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs028.html#___sec27" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs029.html#___sec28" style="font-size: 80%;">Gradient descent example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs030.html#___sec29" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs032.html#___sec31" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs033.html#___sec32" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs034.html#___sec33" 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-bs035.html#___sec34" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Automatic differentiation</a></li>
|
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<!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;">Using autograd</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs038.html#___sec37" style="font-size: 80%;">Autograd with more complicated functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs039.html#___sec38" 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="._Splines-bs040.html#___sec39" style="font-size: 80%;">Functions using mathematical functions from Numpy</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs041.html#___sec40" style="font-size: 80%;">More autograd</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs042.html#___sec41" style="font-size: 80%;">And with loops</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs043.html#___sec42" style="font-size: 80%;">Using recursion</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs044.html#___sec43" style="font-size: 80%;">Unsupported functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs045.html#___sec44" 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="._Splines-bs046.html#___sec45" style="font-size: 80%;">Recommended to avoid</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs047.html#___sec46" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs048.html#___sec47" style="font-size: 80%;">Computation of gradients</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs049.html#___sec48" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs050.html#___sec49" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs051.html#___sec50" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._Splines-bs052.html#___sec51" style="font-size: 80%;">When do we stop?</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs053.html#___sec52" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs054.html#___sec53" style="font-size: 80%;">Program for stochastic gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs055.html#___sec54" style="font-size: 80%;">Momentum based methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs056.html#___sec55" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs057.html#___sec56" style="font-size: 80%;">Conjugate gradient method</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs058.html#___sec57" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs059.html#___sec58" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs060.html#___sec59" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs061.html#___sec60" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs062.html#___sec61" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs063.html#___sec62" style="font-size: 80%;">Conjugate gradient method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs064.html#___sec63" style="font-size: 80%;">Simple implementation of the Conjugate gradient algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs065.html#___sec64" style="font-size: 80%;">Broyden–Fletcher–Goldfarb–Shanno algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs066.html#___sec65" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs067.html#___sec66" style="font-size: 80%;">Momentum based GD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs068.html#___sec67" style="font-size: 80%;">More on momentum based approaches</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs069.html#___sec68" style="font-size: 80%;">Momentum parameter</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs070.html#___sec69" style="font-size: 80%;">Second moment of the gradient</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs071.html#___sec70" style="font-size: 80%;">RMS prop</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs072.html#___sec71" style="font-size: 80%;">ADAM optimizer</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs073.html#___sec72" style="font-size: 80%;">Practical tips</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</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>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method and iterations</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</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%;">Simple implementation of the Conjugate gradient algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs036.html#___sec35" style="font-size: 80%;">Broyden–Fletcher–Goldfarb–Shanno algorithm</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;">Revisiting our first homework</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs038.html#___sec37" style="font-size: 80%;">Gradient descent example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs039.html#___sec38" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs040.html#___sec39" style="font-size: 80%;">The Hessian matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs041.html#___sec40" style="font-size: 80%;">Simple program</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs042.html#___sec41" style="font-size: 80%;">Gradient Descent Example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Splines-bs043.html#___sec42" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs044.html#___sec43" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs045.html#___sec44" style="font-size: 80%;">Automatic differentiation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs046.html#___sec45" style="font-size: 80%;">Using autograd</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs047.html#___sec46" style="font-size: 80%;">Autograd with more complicated functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs048.html#___sec47" style="font-size: 80%;">More complicated functions using the elements of their arguments directly</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs049.html#___sec48" style="font-size: 80%;">Functions using mathematical functions from Numpy</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs050.html#___sec49" style="font-size: 80%;">More autograd</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs051.html#___sec50" style="font-size: 80%;">And with loops</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs052.html#___sec51" style="font-size: 80%;">Using recursion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs053.html#___sec52" style="font-size: 80%;">Unsupported functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs054.html#___sec53" style="font-size: 80%;">The syntax a.dot(b) when finding the dot product</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs055.html#___sec54" style="font-size: 80%;">Recommended to avoid</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs056.html#___sec55" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs057.html#___sec56" style="font-size: 80%;">Computation of gradients</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs058.html#___sec57" style="font-size: 80%;">SGD example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs059.html#___sec58" style="font-size: 80%;">The gradient step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs060.html#___sec59" style="font-size: 80%;">Simple example code</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs061.html#___sec60" style="font-size: 80%;">When do we stop?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs062.html#___sec61" style="font-size: 80%;">Slightly different approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs063.html#___sec62" style="font-size: 80%;">Program for stochastic gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs064.html#___sec63" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs065.html#___sec64" style="font-size: 80%;">Momentum based GD</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs066.html#___sec65" style="font-size: 80%;">More on momentum based approaches</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs067.html#___sec66" style="font-size: 80%;">Momentum parameter</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs068.html#___sec67" style="font-size: 80%;">Second moment of the gradient</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs069.html#___sec68" style="font-size: 80%;">RMS prop</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs070.html#___sec69" style="font-size: 80%;">ADAM optimizer</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Splines-bs071.html#___sec70" style="font-size: 80%;">Practical tips</a></li>
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||||
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</ul>
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</li>
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@@ -274,36 +270,36 @@ MathJax.Hub.Config({
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<a name="part0037"></a>
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<!-- !split -->
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<h2 id="___sec36" class="anchor">Using autograd </h2>
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<h2 id="___sec36" class="anchor">Revisiting our first homework </h2>
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<p>
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Here we
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experiment with what kind of functions Autograd is capable
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of finding the gradient of. The following Python functions are just
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meant to illustrate what Autograd can do, but please feel free to
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experiment with other, possibly more complicated, functions as well.
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We will use linear regression as a case study for the gradient descent
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methods. Linear regression is a great test case for the gradient
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descent methods discussed in the lectures since it has several
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desirable properties such as:
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<p>
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<ol>
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<li> An analytical solution (recall homework set 1).</li>
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<li> The gradient can be computed analytically.</li>
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<li> The cost function is convex which guarantees that gradient descent converges for small enough learning rates</li>
|
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</ol>
|
||||
|
||||
<!-- 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>
|
||||
<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> grad
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||||
We revisit the example from homework set 1 where we had
|
||||
$$
|
||||
y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100
|
||||
$$
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f1</span>(x):
|
||||
<span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">**3</span> <span style="color: #666666">+</span> <span style="color: #666666">1</span>
|
||||
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
|
||||
$$
|
||||
h_\beta(x) = \hat{y} = \beta_0 + \beta_1 x,
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||||
$$
|
||||
|
||||
f1_grad <span style="color: #666666">=</span> grad(f1)
|
||||
such that
|
||||
$$
|
||||
\hat{y}_i = \beta_0 + \beta_1 x_i.
|
||||
$$
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Remember to send in float as argument to the computed gradient from Autograd!</span>
|
||||
a <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># See the evaluated gradient at a using autograd:</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The gradient of f1 evaluated at a = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> using autograd is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>(a,f1_grad(a)))
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Compare with the analytical derivative, that is f1'(x) = 3*x**2 </span>
|
||||
grad_analytical <span style="color: #666666">=</span> <span style="color: #666666">3*</span>a<span style="color: #666666">**2</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The gradient of f1 evaluated at a = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> by finding the analytic expression is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>(a,grad_analytical))
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -330,7 +326,7 @@ grad_analytical <span style="color: #666666">=</span> <span style="color: #66666
|
||||
<li><a href="._Splines-bs045.html">46</a></li>
|
||||
<li><a href="._Splines-bs046.html">47</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Splines-bs073.html">74</a></li>
|
||||
<li><a href="._Splines-bs071.html">72</a></li>
|
||||
<li><a href="._Splines-bs038.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
Reference in New Issue
Block a user