Morten Hjorth-Jensen
386 lines
22 KiB
HTML
386 lines
22 KiB
HTML
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('Minimizing the cross entropy',
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('A more compact expression',
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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="._week38-bs001.html#plans-for-week-38" style="font-size: 80%;">Plans for week 38</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs002.html#ridge-and-lasso-regression-reminder" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs003.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs004.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs005.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs006.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs007.html#logistic-regression" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs008.html#classification-problems" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#optimization-and-deep-learning" style="font-size: 80%;">Optimization and Deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#basics" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs011.html#linear-classifier" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs012.html#some-selected-properties" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#simple-example" style="font-size: 80%;">Simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#plotting-the-mean-value-for-each-group" style="font-size: 80%;">Plotting the mean value for each group</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs015.html#the-logistic-function" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs016.html#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#two-parameters" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs018.html#maximum-likelihood" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs019.html#the-cost-function-rewritten" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#minimizing-the-cross-entropy" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs021.html#a-more-compact-expression" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#extending-to-more-predictors" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs023.html#including-more-classes" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="#more-classes" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#friday-september-23" style="font-size: 80%;">Friday September 23</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#searching-for-optimal-regularization-parameters-lambda" style="font-size: 80%;">Searching for Optimal Regularization Parameters \( \lambda \)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#grid-search" style="font-size: 80%;">Grid Search</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs028.html#randomized-grid-search" style="font-size: 80%;">Randomized Grid Search</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#wisconsin-cancer-data" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#using-the-correlation-matrix" style="font-size: 80%;">Using the correlation matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs031.html#discussing-the-correlation-data" style="font-size: 80%;">Discussing the correlation data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs032.html#other-measures-in-classification-studies-cancer-data-again" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs033.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#revisiting-our-logistic-regression-case" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#the-equations-to-solve" style="font-size: 80%;">The equations to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#solving-using-newton-raphson-s-method" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#brief-reminder-on-newton-raphson-s-method" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs043.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs044.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs045.html#convex-functions" style="font-size: 80%;">Convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs046.html#convex-function" style="font-size: 80%;">Convex function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs047.html#conditions-on-convex-functions" style="font-size: 80%;">Conditions on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs048.html#more-on-convex-functions" style="font-size: 80%;">More on convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs049.html#some-simple-problems" style="font-size: 80%;">Some simple problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs050.html#revisiting-our-first-homework" style="font-size: 80%;">Revisiting our first homework</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs055.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs052.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs053.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs054.html#simple-program" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs055.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs056.html#and-a-corresponding-example-using-scikit-learn" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs057.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs058.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs059.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs060.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs061.html#challenge-yourself-this-weekend" style="font-size: 80%;">Challenge yourself this weekend</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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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="part0024"></a>
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<!-- !split -->
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<h2 id="more-classes" class="anchor">More classes </h2>
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<p>In our discussion of neural networks we will encounter the above again
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in terms of a slightly modified function, the so-called <b>Softmax</b> function.
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</p>
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<p>The softmax function is used in various multiclass classification
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methods, such as multinomial logistic regression (also known as
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softmax regression), multiclass linear discriminant analysis, naive
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Bayes classifiers, and artificial neural networks. Specifically, in
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multinomial logistic regression and linear discriminant analysis, the
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input to the function is the result of \( K \) distinct linear functions,
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and the predicted probability for the \( k \)-th class given a sample
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vector \( \boldsymbol{x} \) and a weighting vector \( \boldsymbol{\beta} \) is (with two
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predictors):
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</p>
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$$
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p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
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$$
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<p>It is easy to extend to more predictors. The final class is </p>
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$$
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p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
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|
$$
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|
|
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<p>and they sum to one. Our earlier discussions were all specialized to
|
|
the case with two classes only. It is easy to see from the above that
|
|
what we derived earlier is compatible with these equations.
|
|
</p>
|
|
|
|
<p>To find the optimal parameters we would typically use a gradient
|
|
descent method. Newton's method and gradient descent methods are
|
|
discussed in the material on <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_self">optimization
|
|
methods</a>.
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</p>
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<p>
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