mhjensen
291 lines
16 KiB
HTML
291 lines
16 KiB
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('The Hessian matrix', 2, None, '___sec37'),
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<a class="navbar-brand" href="week39-bs.html">Week 39: 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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<!-- 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="#___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%;">Friday September 25</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs021.html#___sec20" style="font-size: 80%;">Standard steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs022.html#___sec21" style="font-size: 80%;">Gradient 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%;">Steepest descent method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs025.html#___sec24" style="font-size: 80%;">Final expressions</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs026.html#___sec25" style="font-size: 80%;">Steepest descent example</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</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs031.html#___sec30" style="font-size: 80%;">Conjugate gradient method and iterations</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%;">Conjugate gradient method</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs035.html#___sec34" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs036.html#___sec35" style="font-size: 80%;">Gradient descent example</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs037.html#___sec36" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs038.html#___sec37" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs039.html#___sec38" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs040.html#___sec39" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs041.html#___sec40" 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-bs042.html#___sec41" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs043.html#___sec42" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week39-bs044.html#___sec43" style="font-size: 80%;">Using gradient descent methods, limitations</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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</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0013"></a>
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<!-- !split -->
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<h2 id="___sec12" class="anchor">The ideal </h2>
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<p>
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Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
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minimum of the function \( F \). In general we do not know if we are in a
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global or local minimum. In the special case when \( F \) is a convex
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function, all local minima are also global minima, so in this case
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gradient descent can converge to the global solution. The advantage of
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this scheme is that it is conceptually simple and straightforward to
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implement. However the method in this form has some severe
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limitations:
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<p>
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In machine learing we are often faced with non-convex high dimensional
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cost functions with many local minima. Since GD is deterministic we
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will get stuck in a local minimum, if the method converges, unless we
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have a very good intial guess. This also implies that the scheme is
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sensitive to the chosen initial condition.
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<p>
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Note that the gradient is a function of \( \mathbf{x} =
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(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.
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<p>
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<p>
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