Morten Hjorth-Jensen
2217 lines
187 KiB
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2217 lines
187 KiB
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Applied Data Analysis and Machine Learning
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From Regression to Support Vector Machines
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1. Linear Regression, basic Elements
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2. Resampling Methods
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3. Ridge and Lasso Regression
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4. Logistic Regression
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Contents
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#id1">
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4.1. Logistic Regression
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#basics">
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4.2. Basics
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-logistic-function">
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4.3. The logistic function
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks">
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4.4. Examples of likelihood functions used in logistic regression and nueral networks
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#wisconsin-cancer-data">
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4.5. Wisconsin Cancer Data
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#optimization-the-central-part-of-any-machine-learning-algortithm">
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4.6. Optimization, the central part of any Machine Learning algortithm
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#revisiting-our-logistic-regression-case">
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4.7. Revisiting our Logistic Regression case
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-equations-to-solve">
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4.8. The equations to solve
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#solving-using-newton-raphson-s-method">
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4.9. Solving using Newton-Raphson’s method
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#brief-reminder-on-newton-raphson-s-method">
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4.10. Brief reminder on Newton-Raphson’s method
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-equations">
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4.11. The equations
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#simple-geometric-interpretation">
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4.12. Simple geometric interpretation
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#extending-to-more-than-one-variable">
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4.13. Extending to more than one variable
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#steepest-descent">
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4.14. Steepest descent
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#more-on-steepest-descent">
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4.15. More on Steepest descent
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-ideal">
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4.16. The ideal
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-sensitiveness-of-the-gradient-descent">
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4.17. The sensitiveness of the gradient descent
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#convex-functions">
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4.18. Convex functions
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#convex-function">
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4.19. Convex function
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#conditions-on-convex-functions">
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4.20. Conditions on convex functions
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#more-on-convex-functions">
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4.21. More on convex functions
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#some-simple-problems">
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4.22. Some simple problems
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||
</a>
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</li>
|
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#friday-september-25">
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||
4.23. Friday September 25
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||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
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<a class="reference internal nav-link" href="#standard-steepest-descent">
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4.24. Standard steepest descent
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||
</a>
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||
</li>
|
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<li class="toc-h2 nav-item toc-entry">
|
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<a class="reference internal nav-link" href="#gradient-method">
|
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4.25. Gradient method
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||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#steepest-descent-method">
|
||
4.26. Steepest descent method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id2">
|
||
4.27. Steepest descent method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-expressions">
|
||
4.28. Final expressions
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#steepest-descent-example">
|
||
4.29. Steepest descent example
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#conjugate-gradient-method">
|
||
4.30. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id3">
|
||
4.31. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id4">
|
||
4.32. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id5">
|
||
4.33. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#conjugate-gradient-method-and-iterations">
|
||
4.34. Conjugate gradient method and iterations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id6">
|
||
4.35. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id7">
|
||
4.36. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id8">
|
||
4.37. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#revisiting-our-first-homework">
|
||
4.38. Revisiting our first homework
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#gradient-descent-example">
|
||
4.39. Gradient descent example
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-derivative-of-the-cost-loss-function">
|
||
4.40. The derivative of the cost/loss function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-hessian-matrix">
|
||
4.41. The Hessian matrix
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#simple-program">
|
||
4.42. Simple program
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id9">
|
||
4.43. Gradient Descent Example
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#and-a-corresponding-example-using-scikit-learn">
|
||
4.44. And a corresponding example using
|
||
<strong>
|
||
scikit-learn
|
||
</strong>
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#gradient-descent-and-ridge">
|
||
4.45. Gradient descent and Ridge
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#program-example-for-gradient-descent-with-ridge-regression">
|
||
4.46. Program example for gradient descent with Ridge Regression
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-gradient-descent-methods-limitations">
|
||
4.47. Using gradient descent methods, limitations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#stochastic-gradient-descent">
|
||
4.48. Stochastic Gradient Descent
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#computation-of-gradients">
|
||
4.49. Computation of gradients
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#sgd-example">
|
||
4.50. SGD example
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-gradient-step">
|
||
4.51. The gradient step
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#simple-example-code">
|
||
4.52. Simple example code
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#when-do-we-stop">
|
||
4.53. When do we stop?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#slightly-different-approach">
|
||
4.54. Slightly different approach
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#program-for-stochastic-gradient">
|
||
4.55. Program for stochastic gradient
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
|
||
</nav>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="main-content" class="row">
|
||
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
|
||
|
||
<div>
|
||
|
||
<div class="section" id="logistic-regression">
|
||
<h1><span class="section-number">4. </span>Logistic Regression<a class="headerlink" href="#logistic-regression" title="Permalink to this headline">¶</a></h1>
|
||
<p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureSeptember18.mp4?vrtx=view-as-webpage">Video of Lecture</a></p>
|
||
<div class="section" id="id1">
|
||
<h2><span class="section-number">4.1. </span>Logistic Regression<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
|
||
<p>In linear regression our main interest was centered on learning the
|
||
coefficients of a functional fit (say a polynomial) in order to be
|
||
able to predict the response of a continuous variable on some unseen
|
||
data. The fit to the continuous variable <span class="math notranslate nohighlight">\(y_i\)</span> is based on some
|
||
independent variables <span class="math notranslate nohighlight">\(\hat{x}_i\)</span>. Linear regression resulted in
|
||
analytical expressions for standard ordinary Least Squares or Ridge
|
||
regression (in terms of matrices to invert) for several quantities,
|
||
ranging from the variance and thereby the confidence intervals of the
|
||
parameters <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> to the mean squared error. If we can invert
|
||
the product of the design matrices, linear regression gives then a
|
||
simple recipe for fitting our data.</p>
|
||
<p>Classification problems, however, are concerned with outcomes taking
|
||
the form of discrete variables (i.e. categories). We may for example,
|
||
on the basis of DNA sequencing for a number of patients, like to find
|
||
out which mutations are important for a certain disease; or based on
|
||
scans of various patients’ brains, figure out if there is a tumor or
|
||
not; or given a specific physical system, we’d like to identify its
|
||
state, say whether it is an ordered or disordered system (typical
|
||
situation in solid state physics); or classify the status of a
|
||
patient, whether she/he has a stroke or not and many other similar
|
||
situations.</p>
|
||
<p>The most common situation we encounter when we apply logistic
|
||
regression is that of two possible outcomes, normally denoted as a
|
||
binary outcome, true or false, positive or negative, success or
|
||
failure etc.</p>
|
||
<p>Logistic regression will also serve as our stepping stone towards
|
||
neural network algorithms and supervised deep learning. For logistic
|
||
learning, the minimization of the cost function leads to a non-linear
|
||
equation in the parameters <span class="math notranslate nohighlight">\(\hat{\beta}\)</span>. The optimization of the
|
||
problem calls therefore for minimization algorithms. This forms the
|
||
bottle neck of all machine learning algorithms, namely how to find
|
||
reliable minima of a multi-variable function. This leads us to the
|
||
family of gradient descent methods. The latter are the working horses
|
||
of basically all modern machine learning algorithms.</p>
|
||
<p>We note also that many of the topics discussed here on logistic
|
||
regression are also commonly used in modern supervised Deep Learning
|
||
models, as we will see later.</p>
|
||
</div>
|
||
<div class="section" id="basics">
|
||
<h2><span class="section-number">4.2. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this headline">¶</a></h2>
|
||
<p>We consider the case where the dependent variables, also called the
|
||
responses or the outcomes, <span class="math notranslate nohighlight">\(y_i\)</span> are discrete and only take values
|
||
from <span class="math notranslate nohighlight">\(k=0,\dots,K-1\)</span> (i.e. <span class="math notranslate nohighlight">\(K\)</span> classes).</p>
|
||
<p>The goal is to predict the
|
||
output classes from the design matrix <span class="math notranslate nohighlight">\(\hat{X}\in\mathbb{R}^{n\times p}\)</span>
|
||
made of <span class="math notranslate nohighlight">\(n\)</span> samples, each of which carries <span class="math notranslate nohighlight">\(p\)</span> features or predictors. The
|
||
primary goal is to identify the classes to which new unseen samples
|
||
belong.</p>
|
||
<p>Let us specialize to the case of two classes only, with outputs
|
||
<span class="math notranslate nohighlight">\(y_i=0\)</span> and <span class="math notranslate nohighlight">\(y_i=1\)</span>. Our outcomes could represent the status of a
|
||
credit card user that could default or not on her/his credit card
|
||
debt. That is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
|
||
\end{split}\]</div>
|
||
<p>Before moving to the logistic model, let us try to use our linear
|
||
regression model to classify these two outcomes. We could for example
|
||
fit a linear model to the default case if <span class="math notranslate nohighlight">\(y_i > 0.5\)</span> and the no
|
||
default case <span class="math notranslate nohighlight">\(y_i \leq 0.5\)</span>.</p>
|
||
<p>We would then have our
|
||
weighted linear combination, namely</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto1"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
|
||
\label{_auto1} \tag{1}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\hat{y}\)</span> is a vector representing the possible outcomes, <span class="math notranslate nohighlight">\(\hat{X}\)</span> is our
|
||
<span class="math notranslate nohighlight">\(n\times p\)</span> design matrix and <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> represents our estimators/predictors.</p>
|
||
<p>The main problem with our function is that it takes values on the
|
||
entire real axis. In the case of logistic regression, however, the
|
||
labels <span class="math notranslate nohighlight">\(y_i\)</span> are discrete variables. A typical example is the credit
|
||
card data discussed below here, where we can set the state of
|
||
defaulting the debt to <span class="math notranslate nohighlight">\(y_i=1\)</span> and not to <span class="math notranslate nohighlight">\(y_i=0\)</span> for one the persons
|
||
in the data set (see the full example below).</p>
|
||
<p>One simple way to get a discrete output is to have sign
|
||
functions that map the output of a linear regressor to values <span class="math notranslate nohighlight">\(\{0,1\}\)</span>,
|
||
<span class="math notranslate nohighlight">\(f(s_i)=sign(s_i)=1\)</span> if <span class="math notranslate nohighlight">\(s_i\ge 0\)</span> and 0 if otherwise.
|
||
We will encounter this model in our first demonstration of neural networks. Historically it is called the <code class="docutils literal notranslate"><span class="pre">perceptron"</span> <span class="pre">model</span> <span class="pre">in</span> <span class="pre">the</span> <span class="pre">machine</span> <span class="pre">learning</span> <span class="pre">literature.</span> <span class="pre">This</span> <span class="pre">model</span> <span class="pre">is</span> <span class="pre">extremely</span> <span class="pre">simple.</span> <span class="pre">However,</span> <span class="pre">in</span> <span class="pre">many</span> <span class="pre">cases</span> <span class="pre">it</span> <span class="pre">is</span> <span class="pre">more</span> <span class="pre">favorable</span> <span class="pre">to</span> <span class="pre">use</span> <span class="pre">a</span> </code>soft” classifier that outputs
|
||
the probability of a given category. This leads us to the logistic function.</p>
|
||
<p>The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person’s against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
|
||
|
||
<span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span><span class="p">,</span> <span class="n">Ridge</span><span class="p">,</span> <span class="n">Lasso</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||
<span class="kn">from</span> <span class="nn">pylab</span> <span class="kn">import</span> <span class="n">plt</span><span class="p">,</span> <span class="n">mpl</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">'seaborn'</span><span class="p">)</span>
|
||
<span class="n">mpl</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'font.family'</span><span class="p">]</span> <span class="o">=</span> <span class="s1">'serif'</span>
|
||
|
||
<span class="c1"># Where to save the figures and data files</span>
|
||
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">"Results"</span>
|
||
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">"Results/FigureFiles"</span>
|
||
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">"DataFiles/"</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
|
||
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"chddata.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Read the chd data as csv file and organize the data into arrays with age group, age, and chd</span>
|
||
<span class="n">chd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">'ID'</span><span class="p">,</span> <span class="s1">'Age'</span><span class="p">,</span> <span class="s1">'Agegroup'</span><span class="p">,</span> <span class="s1">'CHD'</span><span class="p">))</span>
|
||
<span class="n">chd</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'ID'</span><span class="p">,</span> <span class="s1">'Age'</span><span class="p">,</span> <span class="s1">'Agegroup'</span><span class="p">,</span> <span class="s1">'CHD'</span><span class="p">]</span>
|
||
<span class="n">output</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'CHD'</span><span class="p">]</span>
|
||
<span class="n">age</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'Age'</span><span class="p">]</span>
|
||
<span class="n">agegroup</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'Agegroup'</span><span class="p">]</span>
|
||
<span class="n">numberID</span> <span class="o">=</span> <span class="n">chd</span><span class="p">[</span><span class="s1">'ID'</span><span class="p">]</span>
|
||
<span class="n">display</span><span class="p">(</span><span class="n">chd</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">age</span><span class="p">,</span> <span class="n">output</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">'o'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">18</span><span class="p">,</span><span class="mf">70.0</span><span class="p">,</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Age'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'CHD'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Age distribution and Coronary heart disease'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||
<span class="ne">FileNotFoundError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">a77d5ac269b2</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">38</span> <span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">39</span>
|
||
<span class="ne">---> </span><span class="mi">40</span> <span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"chddata.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">41</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">42</span> <span class="c1"># Read the chd data as csv file and organize the data into arrays with age group, age, and chd</span>
|
||
|
||
<span class="ne">FileNotFoundError</span>: [Errno 2] No such file or directory: 'DataFiles/chddata.csv'
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>What we could attempt however is to plot the mean value for each group.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">agegroupmean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.133</span><span class="p">,</span> <span class="mf">0.250</span><span class="p">,</span> <span class="mf">0.333</span><span class="p">,</span> <span class="mf">0.462</span><span class="p">,</span> <span class="mf">0.625</span><span class="p">,</span> <span class="mf">0.765</span><span class="p">,</span> <span class="mf">0.800</span><span class="p">])</span>
|
||
<span class="n">group</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">agegroupmean</span><span class="p">,</span> <span class="s2">"r-"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Age group'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'CHD mean values'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Mean values for each age group'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We are now trying to find a function <span class="math notranslate nohighlight">\(f(y\vert x)\)</span>, that is a function which gives us an expected value for the output <span class="math notranslate nohighlight">\(y\)</span> with a given input <span class="math notranslate nohighlight">\(x\)</span>.
|
||
In standard linear regression with a linear dependence on <span class="math notranslate nohighlight">\(x\)</span>, we would write this in terms of our model</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
|
||
\]</div>
|
||
<p>This expression implies however that <span class="math notranslate nohighlight">\(f(y_i\vert x_i)\)</span> could take any
|
||
value from minus infinity to plus infinity. If we however let
|
||
<span class="math notranslate nohighlight">\(f(y\vert y)\)</span> be represented by the mean value, the above example
|
||
shows us that we can constrain the function to take values between
|
||
zero and one, that is we have <span class="math notranslate nohighlight">\(0 \le f(y_i\vert x_i) \le 1\)</span>. Looking
|
||
at our last curve we see also that it has an S-shaped form. This leads
|
||
us to a very popular model for the function <span class="math notranslate nohighlight">\(f\)</span>, namely the so-called
|
||
Sigmoid function or logistic model. We will consider this function as
|
||
representing the probability for finding a value of <span class="math notranslate nohighlight">\(y_i\)</span> with a given
|
||
<span class="math notranslate nohighlight">\(x_i\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="the-logistic-function">
|
||
<h2><span class="section-number">4.3. </span>The logistic function<a class="headerlink" href="#the-logistic-function" title="Permalink to this headline">¶</a></h2>
|
||
<p>Another widely studied model, is the so-called
|
||
perceptron model, which is an example of a “hard classification” model. We
|
||
will encounter this model when we discuss neural networks as
|
||
well. Each datapoint is deterministically assigned to a category (i.e
|
||
<span class="math notranslate nohighlight">\(y_i=0\)</span> or <span class="math notranslate nohighlight">\(y_i=1\)</span>). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a “soft”
|
||
classifier that outputs the probability of a given category rather
|
||
than a single value. For example, given <span class="math notranslate nohighlight">\(x_i\)</span>, the classifier
|
||
outputs the probability of being in a category <span class="math notranslate nohighlight">\(k\)</span>. Logistic regression
|
||
is the most common example of a so-called soft classifier. In logistic
|
||
regression, the probability that a data point <span class="math notranslate nohighlight">\(x_i\)</span>
|
||
belongs to a category <span class="math notranslate nohighlight">\(y_i=\{0,1\}\)</span> is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
|
||
\]</div>
|
||
<p>Note that <span class="math notranslate nohighlight">\(1-p(t)= p(-t)\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks">
|
||
<h2><span class="section-number">4.4. </span>Examples of likelihood functions used in logistic regression and nueral networks<a class="headerlink" href="#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" title="Permalink to this headline">¶</a></h2>
|
||
<p>The following code plots the logistic function, the step function and other functions we will encounter from here and on.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">"""The sigmoid function (or the logistic curve) is a</span>
|
||
<span class="sd">function that takes any real number, z, and outputs a number (0,1).</span>
|
||
<span class="sd">It is useful in neural networks for assigning weights on a relative scale.</span>
|
||
<span class="sd">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
|
||
|
||
<span class="kn">import</span> <span class="nn">numpy</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">math</span> <span class="k">as</span> <span class="nn">mt</span>
|
||
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">sigma_fn</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">numpy</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)))</span>
|
||
<span class="n">sigma</span> <span class="o">=</span> <span class="n">sigma_fn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'sigmoid function'</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="sd">"""Step Function"""</span>
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="o">.</span><span class="mi">02</span><span class="p">)</span>
|
||
<span class="n">step_fn</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mf">1.0</span> <span class="k">if</span> <span class="n">z</span> <span class="o">>=</span> <span class="mf">0.0</span> <span class="k">else</span> <span class="mf">0.0</span><span class="p">)</span>
|
||
<span class="n">step</span> <span class="o">=</span> <span class="n">step_fn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'step function'</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="sd">"""tanh Function"""</span>
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
|
||
<span class="n">t</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">mt</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'z'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'tanh function'</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We assume now that we have two classes with <span class="math notranslate nohighlight">\(y_i\)</span> either <span class="math notranslate nohighlight">\(0\)</span> or <span class="math notranslate nohighlight">\(1\)</span>. Furthermore we assume also that we have only two parameters <span class="math notranslate nohighlight">\(\beta\)</span> in our fitting of the Sigmoid function, that is we define probabilities</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
|
||
p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> are the weights we wish to extract from data, in our case <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.</p>
|
||
<p>Note that we used</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
|
||
\]</div>
|
||
<p>In order to define the total likelihood for all possible outcomes from a<br />
|
||
dataset <span class="math notranslate nohighlight">\(\mathcal{D}=\{(y_i,x_i)\}\)</span>, with the binary labels
|
||
<span class="math notranslate nohighlight">\(y_i\in\{0,1\}\)</span> and where the data points are drawn independently, we use the so-called <a class="reference external" href="https://en.wikipedia.org/wiki/Maximum_likelihood_estimation">Maximum Likelihood Estimation</a> (MLE) principle.
|
||
We aim thus at maximizing
|
||
the probability of seeing the observed data. We can then approximate the
|
||
likelihood in terms of the product of the individual probabilities of a specific outcome <span class="math notranslate nohighlight">\(y_i\)</span>, that is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>from which we obtain the log-likelihood and our <strong>cost/loss</strong> function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
|
||
\]</div>
|
||
<p>Reordering the logarithms, we can rewrite the <strong>cost/loss</strong> function as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
|
||
\]</div>
|
||
<p>The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to <span class="math notranslate nohighlight">\(\beta\)</span>.
|
||
Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
|
||
\]</div>
|
||
<p>This equation is known in statistics as the <strong>cross entropy</strong>. Finally, we note that just as in linear regression,
|
||
in practice we often supplement the cross-entropy with additional regularization terms, usually <span class="math notranslate nohighlight">\(L_1\)</span> and <span class="math notranslate nohighlight">\(L_2\)</span> regularization as we did for Ridge and Lasso regression.</p>
|
||
<p>The cross entropy is a convex function of the weights <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> and,
|
||
therefore, any local minimizer is a global minimizer.</p>
|
||
<p>Minimizing this
|
||
cost function with respect to the two parameters <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span> we obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
|
||
\]</div>
|
||
<p>Let us now define a vector <span class="math notranslate nohighlight">\(\hat{y}\)</span> with <span class="math notranslate nohighlight">\(n\)</span> elements <span class="math notranslate nohighlight">\(y_i\)</span>, an
|
||
<span class="math notranslate nohighlight">\(n\times p\)</span> matrix <span class="math notranslate nohighlight">\(\hat{X}\)</span> which contains the <span class="math notranslate nohighlight">\(x_i\)</span> values and a
|
||
vector <span class="math notranslate nohighlight">\(\hat{p}\)</span> of fitted probabilities <span class="math notranslate nohighlight">\(p(y_i\vert x_i,\hat{\beta})\)</span>. We can rewrite in a more compact form the first
|
||
derivative of cost function as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).
|
||
\]</div>
|
||
<p>If we in addition define a diagonal matrix <span class="math notranslate nohighlight">\(\hat{W}\)</span> with elements
|
||
<span class="math notranslate nohighlight">\(p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})\)</span>, we can obtain a compact expression of the second derivative as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.
|
||
\]</div>
|
||
<p>Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with <span class="math notranslate nohighlight">\(p\)</span> predictors</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
|
||
\]</div>
|
||
<p>Here we defined <span class="math notranslate nohighlight">\(\hat{x}=[1,x_1,x_2,\dots,x_p]\)</span> and <span class="math notranslate nohighlight">\(\hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p]\)</span> leading to</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
|
||
\]</div>
|
||
<p>Till now we have mainly focused on two classes, the so-called binary
|
||
system. Suppose we wish to extend to <span class="math notranslate nohighlight">\(K\)</span> classes. Let us for the sake
|
||
of simplicity assume we have only two predictors. We have then following model</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
|
||
\]</div>
|
||
<p>and so on till the class <span class="math notranslate nohighlight">\(C=K-1\)</span> class</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
|
||
\]</div>
|
||
<p>and the model is specified in term of <span class="math notranslate nohighlight">\(K-1\)</span> so-called log-odds or
|
||
<strong>logit</strong> transformations.</p>
|
||
<p>In our discussion of neural networks we will encounter the above again
|
||
in terms of a slightly modified function, the so-called <strong>Softmax</strong> function.</p>
|
||
<p>The softmax function is used in various multiclass classification
|
||
methods, such as multinomial logistic regression (also known as
|
||
softmax regression), multiclass linear discriminant analysis, naive
|
||
Bayes classifiers, and artificial neural networks. Specifically, in
|
||
multinomial logistic regression and linear discriminant analysis, the
|
||
input to the function is the result of <span class="math notranslate nohighlight">\(K\)</span> distinct linear functions,
|
||
and the predicted probability for the <span class="math notranslate nohighlight">\(k\)</span>-th class given a sample
|
||
vector <span class="math notranslate nohighlight">\(\hat{x}\)</span> and a weighting vector <span class="math notranslate nohighlight">\(\hat{\beta}\)</span> is (with two
|
||
predictors):</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
|
||
\]</div>
|
||
<p>It is easy to extend to more predictors. The final class is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
|
||
\]</div>
|
||
<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 class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html">optimization
|
||
methods</a>.</p>
|
||
</div>
|
||
<div class="section" id="wisconsin-cancer-data">
|
||
<h2><span class="section-number">4.5. </span>Wisconsin Cancer Data<a class="headerlink" href="#wisconsin-cancer-data" title="Permalink to this headline">¶</a></h2>
|
||
<p>We show here how we can use a simple regression case on the breast
|
||
cancer data using Logistic regression as our algorithm for
|
||
classification.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
|
||
<span class="c1"># Load the data</span>
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">)</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Logistic Regression: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1">#now scale the data</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy Logistic Regression with scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>In addition to the above scores, we could also study the covariance (and the correlation matrix).
|
||
We use <strong>Pandas</strong> to compute the correlation matrix.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="c1"># Making a data frame</span>
|
||
<span class="n">cancerpd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">20</span><span class="p">))</span>
|
||
<span class="n">malignant</span> <span class="o">=</span> <span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
|
||
<span class="n">benign</span> <span class="o">=</span> <span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">30</span><span class="p">):</span>
|
||
<span class="n">_</span><span class="p">,</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">histogram</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span><span class="mi">50</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">malignant</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">bins</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">benign</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">bins</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(())</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Feature magnitude"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Frequency"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s2">"Malignant"</span><span class="p">,</span> <span class="s2">"Benign"</span><span class="p">],</span> <span class="n">loc</span> <span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
|
||
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
|
||
<span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">cancerpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="c1"># use the heatmap function from seaborn to plot the correlation matrix</span>
|
||
<span class="c1"># annot = True to print the values inside the square</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">correlation_matrix</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>In the above example we note two things. In the first plot we display
|
||
the overlap of benign and malignant tumors as functions of the various
|
||
features in the Wisconsing breast cancer data set. We see that for
|
||
some of the features we can distinguish clearly the benign and
|
||
malignant cases while for other features we cannot. This can point to
|
||
us which features may be of greater interest when we wish to classify
|
||
a benign or not benign tumour.</p>
|
||
<p>In the second figure we have computed the so-called correlation
|
||
matrix, which in our case with thirty features becomes a <span class="math notranslate nohighlight">\(30\times 30\)</span>
|
||
matrix.</p>
|
||
<p>We constructed this matrix using <strong>pandas</strong> via the statements</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">cancerpd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>and then</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">cancerpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Diagonalizing this matrix we can in turn say something about which
|
||
features are of relevance and which are not. This leads us to
|
||
the classical Principal Component Analysis (PCA) theorem with
|
||
applications. This will be discussed later this semester (<a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html">week 43</a>).</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
|
||
<span class="c1"># Load the data</span>
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">)</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Logistic Regression: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1">#now scale the data</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy Logistic Regression with scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">LabelEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
|
||
<span class="c1">#Cross validation</span>
|
||
<span class="n">accuracy</span> <span class="o">=</span> <span class="n">cross_validate</span><span class="p">(</span><span class="n">logreg</span><span class="p">,</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">,</span><span class="n">cv</span><span class="o">=</span><span class="mi">10</span><span class="p">)[</span><span class="s1">'test_score'</span><span class="p">]</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Logistic Regression and scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
|
||
|
||
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">logreg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">logreg</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="optimization-the-central-part-of-any-machine-learning-algortithm">
|
||
<h2><span class="section-number">4.6. </span>Optimization, the central part of any Machine Learning algortithm<a class="headerlink" href="#optimization-the-central-part-of-any-machine-learning-algortithm" title="Permalink to this headline">¶</a></h2>
|
||
<p>Almost every problem in machine learning and data science starts with
|
||
a dataset <span class="math notranslate nohighlight">\(X\)</span>, a model <span class="math notranslate nohighlight">\(g(\beta)\)</span>, which is a function of the
|
||
parameters <span class="math notranslate nohighlight">\(\beta\)</span> and a cost function <span class="math notranslate nohighlight">\(C(X, g(\beta))\)</span> that allows
|
||
us to judge how well the model <span class="math notranslate nohighlight">\(g(\beta)\)</span> explains the observations
|
||
<span class="math notranslate nohighlight">\(X\)</span>. The model is fit by finding the values of <span class="math notranslate nohighlight">\(\beta\)</span> that minimize
|
||
the cost function. Ideally we would be able to solve for <span class="math notranslate nohighlight">\(\beta\)</span>
|
||
analytically, however this is not possible in general and we must use
|
||
some approximative/numerical method to compute the minimum.</p>
|
||
</div>
|
||
<div class="section" id="revisiting-our-logistic-regression-case">
|
||
<h2><span class="section-number">4.7. </span>Revisiting our Logistic Regression case<a class="headerlink" href="#revisiting-our-logistic-regression-case" title="Permalink to this headline">¶</a></h2>
|
||
<p>In our discussion on Logistic Regression we studied the
|
||
case of
|
||
two classes, with <span class="math notranslate nohighlight">\(y_i\)</span> either
|
||
<span class="math notranslate nohighlight">\(0\)</span> or <span class="math notranslate nohighlight">\(1\)</span>. Furthermore we assumed also that we have only two
|
||
parameters <span class="math notranslate nohighlight">\(\beta\)</span> in our fitting, that is we
|
||
defined probabilities</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
|
||
p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> are the weights we wish to extract from data, in our case <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="the-equations-to-solve">
|
||
<h2><span class="section-number">4.8. </span>The equations to solve<a class="headerlink" href="#the-equations-to-solve" title="Permalink to this headline">¶</a></h2>
|
||
<p>Our compact equations used a definition of a vector <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> with <span class="math notranslate nohighlight">\(n\)</span>
|
||
elements <span class="math notranslate nohighlight">\(y_i\)</span>, an <span class="math notranslate nohighlight">\(n\times p\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which contains the
|
||
<span class="math notranslate nohighlight">\(x_i\)</span> values and a vector <span class="math notranslate nohighlight">\(\boldsymbol{p}\)</span> of fitted probabilities
|
||
<span class="math notranslate nohighlight">\(p(y_i\vert x_i,\boldsymbol{\beta})\)</span>. We rewrote in a more compact form
|
||
the first derivative of the cost function as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
|
||
\]</div>
|
||
<p>If we in addition define a diagonal matrix <span class="math notranslate nohighlight">\(\boldsymbol{W}\)</span> with elements
|
||
<span class="math notranslate nohighlight">\(p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})\)</span>, we can obtain a compact expression of the second derivative as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
|
||
\]</div>
|
||
<p>This defines what is called the Hessian matrix.</p>
|
||
</div>
|
||
<div class="section" id="solving-using-newton-raphson-s-method">
|
||
<h2><span class="section-number">4.9. </span>Solving using Newton-Raphson’s method<a class="headerlink" href="#solving-using-newton-raphson-s-method" title="Permalink to this headline">¶</a></h2>
|
||
<p>If we can set up these equations, Newton-Raphson’s iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.</p>
|
||
<p>Our iterative scheme is then given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T}\right)^{-1}_{\boldsymbol{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right)_{\boldsymbol{\beta}^{\mathrm{old}}},
|
||
\]</div>
|
||
<p>or in matrix form as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X} \right)^{-1}\times \left(-\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{p}) \right)_{\boldsymbol{\beta}^{\mathrm{old}}}.
|
||
\]</div>
|
||
<p>The right-hand side is computed with the old values of <span class="math notranslate nohighlight">\(\beta\)</span>.</p>
|
||
<p>If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.</p>
|
||
</div>
|
||
<div class="section" id="brief-reminder-on-newton-raphson-s-method">
|
||
<h2><span class="section-number">4.10. </span>Brief reminder on Newton-Raphson’s method<a class="headerlink" href="#brief-reminder-on-newton-raphson-s-method" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let us quickly remind ourselves how we derive the above method.</p>
|
||
<p>Perhaps the most celebrated of all one-dimensional root-finding
|
||
routines is Newton’s method, also called the Newton-Raphson
|
||
method. This method requires the evaluation of both the
|
||
function <span class="math notranslate nohighlight">\(f\)</span> and its derivative <span class="math notranslate nohighlight">\(f'\)</span> at arbitrary points.
|
||
If you can only calculate the derivative
|
||
numerically and/or your function is not of the smooth type, we
|
||
normally discourage the use of this method.</p>
|
||
</div>
|
||
<div class="section" id="the-equations">
|
||
<h2><span class="section-number">4.11. </span>The equations<a class="headerlink" href="#the-equations" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Newton-Raphson formula consists geometrically of extending the
|
||
tangent line at a current point until it crosses zero, then setting
|
||
the next guess to the abscissa of that zero-crossing. The mathematics
|
||
behind this method is rather simple. Employing a Taylor expansion for
|
||
<span class="math notranslate nohighlight">\(x\)</span> sufficiently close to the solution <span class="math notranslate nohighlight">\(s\)</span>, we have</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="eq:taylornr"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
|
||
\label{eq:taylornr} \tag{2}
|
||
\]</div>
|
||
<p>For small enough values of the function and for well-behaved
|
||
functions, the terms beyond linear are unimportant, hence we obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(x)+(s-x)f'(x)\approx 0,
|
||
\]</div>
|
||
<p>yielding</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
s\approx x-\frac{f(x)}{f'(x)}.
|
||
\]</div>
|
||
<p>Having in mind an iterative procedure, it is natural to start iterating with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="simple-geometric-interpretation">
|
||
<h2><span class="section-number">4.12. </span>Simple geometric interpretation<a class="headerlink" href="#simple-geometric-interpretation" title="Permalink to this headline">¶</a></h2>
|
||
<p>The above is Newton-Raphson’s method. It has a simple geometric
|
||
interpretation, namely <span class="math notranslate nohighlight">\(x_{n+1}\)</span> is the point where the tangent from
|
||
<span class="math notranslate nohighlight">\((x_n,f(x_n))\)</span> crosses the <span class="math notranslate nohighlight">\(x\)</span>-axis. Close to the solution,
|
||
Newton-Raphson converges fast to the desired result. However, if we
|
||
are far from a root, where the higher-order terms in the series are
|
||
important, the Newton-Raphson formula can give grossly inaccurate
|
||
results. For instance, the initial guess for the root might be so far
|
||
from the true root as to let the search interval include a local
|
||
maximum or minimum of the function. If an iteration places a trial
|
||
guess near such a local extremum, so that the first derivative nearly
|
||
vanishes, then Newton-Raphson may fail totally</p>
|
||
</div>
|
||
<div class="section" id="extending-to-more-than-one-variable">
|
||
<h2><span class="section-number">4.13. </span>Extending to more than one variable<a class="headerlink" href="#extending-to-more-than-one-variable" title="Permalink to this headline">¶</a></h2>
|
||
<p>Newton’s method can be generalized to systems of several non-linear equations
|
||
and variables. Consider the case with two equations</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{array}{cc} f_1(x_1,x_2) &=0\\
|
||
f_2(x_1,x_2) &=0,\end{array}
|
||
\end{split}\]</div>
|
||
<p>which we Taylor expand to obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
|
||
\partial f_1/\partial x_1+h_2
|
||
\partial f_1/\partial x_2+\dots\\
|
||
0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
|
||
\partial f_2/\partial x_1+h_2
|
||
\partial f_2/\partial x_2+\dots
|
||
\end{array}.
|
||
\end{split}\]</div>
|
||
<p>Defining the Jacobian matrix <span class="math notranslate nohighlight">\({\bf \boldsymbol{J}}\)</span> we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
{\bf \boldsymbol{J}}=\left( \begin{array}{cc}
|
||
\partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
|
||
\partial f_2/\partial x_1 &\partial f_2/\partial x_2
|
||
\end{array} \right),
|
||
\end{split}\]</div>
|
||
<p>we can rephrase Newton’s method as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
|
||
\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
|
||
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
|
||
\end{split}\]</div>
|
||
<p>where we have defined</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
|
||
-{\bf \boldsymbol{J}}^{-1}
|
||
\left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
|
||
\end{split}\]</div>
|
||
<p>We need thus to compute the inverse of the Jacobian matrix and it
|
||
is to understand that difficulties may
|
||
arise in case <span class="math notranslate nohighlight">\({\bf \boldsymbol{J}}\)</span> is nearly singular.</p>
|
||
<p>It is rather straightforward to extend the above scheme to systems of
|
||
more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.</p>
|
||
</div>
|
||
<div class="section" id="steepest-descent">
|
||
<h2><span class="section-number">4.14. </span>Steepest descent<a class="headerlink" href="#steepest-descent" title="Permalink to this headline">¶</a></h2>
|
||
<p>The basic idea of gradient descent is
|
||
that a function <span class="math notranslate nohighlight">\(F(\mathbf{x})\)</span>,
|
||
<span class="math notranslate nohighlight">\(\mathbf{x} \equiv (x_1,\cdots,x_n)\)</span>, decreases fastest if one goes from <span class="math notranslate nohighlight">\(\bf {x}\)</span> in the
|
||
direction of the negative gradient <span class="math notranslate nohighlight">\(-\nabla F(\mathbf{x})\)</span>.</p>
|
||
<p>It can be shown that if</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(\gamma_k > 0\)</span>.</p>
|
||
<p>For <span class="math notranslate nohighlight">\(\gamma_k\)</span> small enough, then <span class="math notranslate nohighlight">\(F(\mathbf{x}_{k+1}) \leq
|
||
F(\mathbf{x}_k)\)</span>. This means that for a sufficiently small <span class="math notranslate nohighlight">\(\gamma_k\)</span>
|
||
we are always moving towards smaller function values, i.e a minimum.</p>
|
||
</div>
|
||
<div class="section" id="more-on-steepest-descent">
|
||
<h2><span class="section-number">4.15. </span>More on Steepest descent<a class="headerlink" href="#more-on-steepest-descent" title="Permalink to this headline">¶</a></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 <span class="math notranslate nohighlight">\(\mathbf{x}_0\)</span> for a minimum of <span class="math notranslate nohighlight">\(F\)</span> and
|
||
computes new approximations according to</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
|
||
\]</div>
|
||
<p>The parameter <span class="math notranslate nohighlight">\(\gamma_k\)</span> is often referred to as the step length or
|
||
the learning rate within the context of Machine Learning.</p>
|
||
</div>
|
||
<div class="section" id="the-ideal">
|
||
<h2><span class="section-number">4.16. </span>The ideal<a class="headerlink" href="#the-ideal" title="Permalink to this headline">¶</a></h2>
|
||
<p>Ideally the sequence <span class="math notranslate nohighlight">\(\{\mathbf{x}_k \}_{k=0}\)</span> converges to a global
|
||
minimum of the function <span class="math notranslate nohighlight">\(F\)</span>. In general we do not know if we are in a
|
||
global or local minimum. In the special case when <span class="math notranslate nohighlight">\(F\)</span> 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>
|
||
<p>In machine learing we are often faced with non-convex high dimensional
|
||
cost functions with many local minima. 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>
|
||
<p>Note that the gradient is a function of <span class="math notranslate nohighlight">\(\mathbf{x} =
|
||
(x_1,\cdots,x_n)\)</span> which makes it expensive to compute numerically.</p>
|
||
</div>
|
||
<div class="section" id="the-sensitiveness-of-the-gradient-descent">
|
||
<h2><span class="section-number">4.17. </span>The sensitiveness of the gradient descent<a class="headerlink" href="#the-sensitiveness-of-the-gradient-descent" title="Permalink to this headline">¶</a></h2>
|
||
<p>The gradient descent method
|
||
is sensitive to the choice of learning rate <span class="math notranslate nohighlight">\(\gamma_k\)</span>. This is due
|
||
to the fact that we are only guaranteed that <span class="math notranslate nohighlight">\(F(\mathbf{x}_{k+1}) \leq
|
||
F(\mathbf{x}_k)\)</span> for sufficiently small <span class="math notranslate nohighlight">\(\gamma_k\)</span>. The problem is to
|
||
determine an optimal learning rate. If the learning rate is chosen too
|
||
small the method will take a long time to converge and if it is too
|
||
large we can experience erratic behavior.</p>
|
||
<p>Many of these shortcomings can be alleviated by introducing
|
||
randomness. One such method is that of Stochastic Gradient Descent
|
||
(SGD), see below.</p>
|
||
</div>
|
||
<div class="section" id="convex-functions">
|
||
<h2><span class="section-number">4.18. </span>Convex functions<a class="headerlink" href="#convex-functions" title="Permalink to this headline">¶</a></h2>
|
||
<p>Ideally we want our cost/loss function to be convex(concave).</p>
|
||
<p>First we give the definition of a convex set: A set <span class="math notranslate nohighlight">\(C\)</span> in
|
||
<span class="math notranslate nohighlight">\(\mathbb{R}^n\)</span> is said to be convex if, for all <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> in <span class="math notranslate nohighlight">\(C\)</span> and
|
||
all <span class="math notranslate nohighlight">\(t \in (0,1)\)</span> , the point <span class="math notranslate nohighlight">\((1 − t)x + ty\)</span> also belongs to
|
||
C. Geometrically this means that every point on the line segment
|
||
connecting <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> is in <span class="math notranslate nohighlight">\(C\)</span> as discussed below.</p>
|
||
<p>The convex subsets of <span class="math notranslate nohighlight">\(\mathbb{R}\)</span> are the intervals of
|
||
<span class="math notranslate nohighlight">\(\mathbb{R}\)</span>. Examples of convex sets of <span class="math notranslate nohighlight">\(\mathbb{R}^2\)</span> are the
|
||
regular polygons (triangles, rectangles, pentagons, etc…).</p>
|
||
</div>
|
||
<div class="section" id="convex-function">
|
||
<h2><span class="section-number">4.19. </span>Convex function<a class="headerlink" href="#convex-function" title="Permalink to this headline">¶</a></h2>
|
||
<p><strong>Convex function</strong>: Let <span class="math notranslate nohighlight">\(X \subset \mathbb{R}^n\)</span> be a convex set. Assume that the function <span class="math notranslate nohighlight">\(f: X \rightarrow \mathbb{R}\)</span> is continuous, then <span class="math notranslate nohighlight">\(f\)</span> is said to be convex if $<span class="math notranslate nohighlight">\(f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) \)</span><span class="math notranslate nohighlight">\( for all \)</span>x_1, x_2 \in X<span class="math notranslate nohighlight">\( and for all \)</span>t \in [0,1]<span class="math notranslate nohighlight">\(. If \)</span>\leq<span class="math notranslate nohighlight">\( is replaced with a strict inequaltiy in the definition, we demand \)</span>x_1 \neq x_2<span class="math notranslate nohighlight">\( and \)</span>t\in(0,1)<span class="math notranslate nohighlight">\( then \)</span>f<span class="math notranslate nohighlight">\( is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \)</span>f(x_1)<span class="math notranslate nohighlight">\( and \)</span>f(x_2)<span class="math notranslate nohighlight">\(, the value of the function on the interval \)</span>[x_1,x_2]$ is always below the line as illustrated below.</p>
|
||
</div>
|
||
<div class="section" id="conditions-on-convex-functions">
|
||
<h2><span class="section-number">4.20. </span>Conditions on convex functions<a class="headerlink" href="#conditions-on-convex-functions" title="Permalink to this headline">¶</a></h2>
|
||
<p>In the following we state first and second-order conditions which
|
||
ensures convexity of a function <span class="math notranslate nohighlight">\(f\)</span>. We write <span class="math notranslate nohighlight">\(D_f\)</span> to denote the
|
||
domain of <span class="math notranslate nohighlight">\(f\)</span>, i.e the subset of <span class="math notranslate nohighlight">\(R^n\)</span> where <span class="math notranslate nohighlight">\(f\)</span> is defined. For more
|
||
details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](<a class="reference external" href="http://stanford.edu/boyd/cvxbook/">http://stanford.edu/boyd/cvxbook/</a>, 2004).</p>
|
||
<p><strong>First order condition.</strong></p>
|
||
<p>Suppose <span class="math notranslate nohighlight">\(f\)</span> is differentiable (i.e <span class="math notranslate nohighlight">\(\nabla f(x)\)</span> is well defined for
|
||
all <span class="math notranslate nohighlight">\(x\)</span> in the domain of <span class="math notranslate nohighlight">\(f\)</span>). Then <span class="math notranslate nohighlight">\(f\)</span> is convex if and only if <span class="math notranslate nohighlight">\(D_f\)</span>
|
||
is a convex set and $<span class="math notranslate nohighlight">\(f(y) \geq f(x) + \nabla f(x)^T (y-x) \)</span><span class="math notranslate nohighlight">\( holds
|
||
for all \)</span>x,y \in D_f<span class="math notranslate nohighlight">\(. This condition means that for a convex function
|
||
the first order Taylor expansion (right hand side above) at any point
|
||
a global under estimator of the function. To convince yourself you can
|
||
make a drawing of \)</span>f(x) = x^2+1<span class="math notranslate nohighlight">\( and draw the tangent line to \)</span>f(x)$ and
|
||
note that it is always below the graph.</p>
|
||
<p><strong>Second order condition.</strong></p>
|
||
<p>Assume that <span class="math notranslate nohighlight">\(f\)</span> is twice
|
||
differentiable, i.e the Hessian matrix exists at each point in
|
||
<span class="math notranslate nohighlight">\(D_f\)</span>. Then <span class="math notranslate nohighlight">\(f\)</span> is convex if and only if <span class="math notranslate nohighlight">\(D_f\)</span> is a convex set and its
|
||
Hessian is positive semi-definite for all <span class="math notranslate nohighlight">\(x\in D_f\)</span>. For a
|
||
single-variable function this reduces to <span class="math notranslate nohighlight">\(f''(x) \geq 0\)</span>. Geometrically this means that <span class="math notranslate nohighlight">\(f\)</span> has nonnegative curvature
|
||
everywhere.</p>
|
||
<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.</p>
|
||
</div>
|
||
<div class="section" id="more-on-convex-functions">
|
||
<h2><span class="section-number">4.21. </span>More on convex functions<a class="headerlink" href="#more-on-convex-functions" title="Permalink to this headline">¶</a></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.</p>
|
||
<p>Ideally we want the
|
||
global minimum (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>
|
||
<p><strong>Any minimum is global for convex functions.</strong></p>
|
||
<p>Consider the problem of finding <span class="math notranslate nohighlight">\(x \in \mathbb{R}^n\)</span> such that <span class="math notranslate nohighlight">\(f(x)\)</span>
|
||
is minimal, where <span class="math notranslate nohighlight">\(f\)</span> is convex and differentiable. Then, any point
|
||
<span class="math notranslate nohighlight">\(x^*\)</span> that satisfies <span class="math notranslate nohighlight">\(\nabla f(x^*) = 0\)</span> is a global minimum.</p>
|
||
<p>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.</p>
|
||
</div>
|
||
<div class="section" id="some-simple-problems">
|
||
<h2><span class="section-number">4.22. </span>Some simple problems<a class="headerlink" href="#some-simple-problems" title="Permalink to this headline">¶</a></h2>
|
||
<ol class="simple">
|
||
<li><p>Show that <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> is convex for <span class="math notranslate nohighlight">\(x \in \mathbb{R}\)</span> using the definition of convexity. Hint: If you re-write the definition, <span class="math notranslate nohighlight">\(f\)</span> is convex if the following holds for all <span class="math notranslate nohighlight">\(x,y \in D_f\)</span> and any <span class="math notranslate nohighlight">\(\lambda \in [0,1]\)</span> <span class="math notranslate nohighlight">\(\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0\)</span>.</p></li>
|
||
<li><p>Using the second order condition show that the following functions are convex on the specified domain.</p></li>
|
||
</ol>
|
||
<ul class="simple">
|
||
<li><p><span class="math notranslate nohighlight">\(f(x) = e^x\)</span> is convex for <span class="math notranslate nohighlight">\(x \in \mathbb{R}\)</span>.</p></li>
|
||
<li><p><span class="math notranslate nohighlight">\(g(x) = -\ln(x)\)</span> is convex for <span class="math notranslate nohighlight">\(x \in (0,\infty)\)</span>.</p></li>
|
||
</ul>
|
||
<ol class="simple">
|
||
<li><p>Let <span class="math notranslate nohighlight">\(f(x) = x^2\)</span> and <span class="math notranslate nohighlight">\(g(x) = e^x\)</span>. Show that <span class="math notranslate nohighlight">\(f(g(x))\)</span> and <span class="math notranslate nohighlight">\(g(f(x))\)</span> is convex for <span class="math notranslate nohighlight">\(x \in \mathbb{R}\)</span>. Also show that if <span class="math notranslate nohighlight">\(f(x)\)</span> is any convex function than <span class="math notranslate nohighlight">\(h(x) = e^{f(x)}\)</span> is convex.</p></li>
|
||
<li><p>A norm is any function that satisfy the following properties</p></li>
|
||
</ol>
|
||
<ul class="simple">
|
||
<li><p><span class="math notranslate nohighlight">\(f(\alpha x) = |\alpha| f(x)\)</span> for all <span class="math notranslate nohighlight">\(\alpha \in \mathbb{R}\)</span>.</p></li>
|
||
<li><p><span class="math notranslate nohighlight">\(f(x+y) \leq f(x) + f(y)\)</span></p></li>
|
||
<li><p><span class="math notranslate nohighlight">\(f(x) \leq 0\)</span> for all <span class="math notranslate nohighlight">\(x \in \mathbb{R}^n\)</span> with equality if and only if <span class="math notranslate nohighlight">\(x = 0\)</span></p></li>
|
||
</ul>
|
||
<p>Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).</p>
|
||
</div>
|
||
<div class="section" id="friday-september-25">
|
||
<h2><span class="section-number">4.23. </span>Friday September 25<a class="headerlink" href="#friday-september-25" title="Permalink to this headline">¶</a></h2>
|
||
<p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember25.mp4?vrtx=view-as-webpage">Video of Lecture</a> and <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember25.pdf">link to handwritten notes</a>.</p>
|
||
</div>
|
||
<div class="section" id="standard-steepest-descent">
|
||
<h2><span class="section-number">4.24. </span>Standard steepest descent<a class="headerlink" href="#standard-steepest-descent" title="Permalink to this headline">¶</a></h2>
|
||
<p>Before we proceed, we would like to discuss the approach called the
|
||
<strong>standard Steepest descent</strong> (different from the above steepest descent discussion), which again leads to us having to be able
|
||
to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).</p>
|
||
<p><a class="reference external" href="https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf">The success of the CG method</a>
|
||
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</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}.
|
||
\]</div>
|
||
<p>In the iterative process we end up with a problem like</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{r}= \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x},
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{r}\)</span> is the so-called residual or error in the iterative process.</p>
|
||
<p>When we have found the exact solution, <span class="math notranslate nohighlight">\(\boldsymbol{r}=0\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="gradient-method">
|
||
<h2><span class="section-number">4.25. </span>Gradient method<a class="headerlink" href="#gradient-method" title="Permalink to this headline">¶</a></h2>
|
||
<p>The residual is zero when we reach the minimum of the quadratic equation</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T\boldsymbol{b},
|
||
\]</div>
|
||
<p>with the constraint that the matrix <span class="math notranslate nohighlight">\(\boldsymbol{A}\)</span> is positive definite and
|
||
symmetric. This defines also the Hessian and we want it to be positive definite.</p>
|
||
</div>
|
||
<div class="section" id="steepest-descent-method">
|
||
<h2><span class="section-number">4.26. </span>Steepest descent method<a class="headerlink" href="#steepest-descent-method" title="Permalink to this headline">¶</a></h2>
|
||
<p>We denote the initial guess for <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> as <span class="math notranslate nohighlight">\(\boldsymbol{x}_0\)</span>.
|
||
We can assume without loss of generality that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x}_0=0,
|
||
\]</div>
|
||
<p>or consider the system</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
|
||
\]</div>
|
||
<p>instead.</p>
|
||
</div>
|
||
<div class="section" id="id2">
|
||
<h2><span class="section-number">4.27. </span>Steepest descent method<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
|
||
<p>One can show that the solution <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> is also the unique minimizer of the quadratic form</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
|
||
\]</div>
|
||
<p>This suggests taking the first basis vector <span class="math notranslate nohighlight">\(\boldsymbol{r}_1\)</span> (see below for definition)
|
||
to be the gradient of <span class="math notranslate nohighlight">\(f\)</span> at <span class="math notranslate nohighlight">\(\boldsymbol{x}=\boldsymbol{x}_0\)</span>,
|
||
which equals</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
|
||
\]</div>
|
||
<p>and
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{x}_0=0\)</span> it is equal <span class="math notranslate nohighlight">\(-\boldsymbol{b}\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="final-expressions">
|
||
<h2><span class="section-number">4.28. </span>Final expressions<a class="headerlink" href="#final-expressions" title="Permalink to this headline">¶</a></h2>
|
||
<p>We can compute the residual iteratively as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1},
|
||
\]</div>
|
||
<p>which equals</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{r}_k),
|
||
\]</div>
|
||
<p>or</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{r}_k,
|
||
\]</div>
|
||
<p>which gives</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\alpha_k = \frac{\boldsymbol{r}_k^T\boldsymbol{r}_k}{\boldsymbol{r}_k^T\boldsymbol{A}\boldsymbol{r}_k}
|
||
\]</div>
|
||
<p>leading to the iterative scheme</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x}_{k+1}=\boldsymbol{x}_k-\alpha_k\boldsymbol{r}_{k},
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="steepest-descent-example">
|
||
<h2><span class="section-number">4.29. </span>Steepest descent example<a class="headerlink" href="#steepest-descent-example" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">numpy.linalg</span> <span class="k">as</span> <span class="nn">la</span>
|
||
|
||
<span class="kn">import</span> <span class="nn">scipy.optimize</span> <span class="k">as</span> <span class="nn">sopt</span>
|
||
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">pt</span>
|
||
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">axes3d</span>
|
||
|
||
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mf">2.5</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span>
|
||
|
||
<span class="k">def</span> <span class="nf">df</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">pt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s2">"3d"</span><span class="p">)</span>
|
||
|
||
<span class="n">xmesh</span><span class="p">,</span> <span class="n">ymesh</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mgrid</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="mi">50</span><span class="n">j</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="mi">50</span><span class="n">j</span><span class="p">]</span>
|
||
<span class="n">fmesh</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">xmesh</span><span class="p">,</span> <span class="n">ymesh</span><span class="p">]))</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">xmesh</span><span class="p">,</span> <span class="n">ymesh</span><span class="p">,</span> <span class="n">fmesh</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>And then as countor plot</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">"equal"</span><span class="p">)</span>
|
||
<span class="n">pt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xmesh</span><span class="p">,</span> <span class="n">ymesh</span><span class="p">,</span> <span class="n">fmesh</span><span class="p">)</span>
|
||
<span class="n">guesses</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mf">2.</span><span class="o">/</span><span class="mi">5</span><span class="p">])]</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Find guesses</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">guesses</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">s</span> <span class="o">=</span> <span class="o">-</span><span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Run it!</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">f1d</span><span class="p">(</span><span class="n">alpha</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">alpha</span><span class="o">*</span><span class="n">s</span><span class="p">)</span>
|
||
|
||
<span class="n">alpha_opt</span> <span class="o">=</span> <span class="n">sopt</span><span class="o">.</span><span class="n">golden</span><span class="p">(</span><span class="n">f1d</span><span class="p">)</span>
|
||
<span class="n">next_guess</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">alpha_opt</span> <span class="o">*</span> <span class="n">s</span>
|
||
<span class="n">guesses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">next_guess</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">next_guess</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>What happened?</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">"equal"</span><span class="p">)</span>
|
||
<span class="n">pt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xmesh</span><span class="p">,</span> <span class="n">ymesh</span><span class="p">,</span> <span class="n">fmesh</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
|
||
<span class="n">it_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">guesses</span><span class="p">)</span>
|
||
<span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">it_array</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">it_array</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s2">"x-"</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="conjugate-gradient-method">
|
||
<h2><span class="section-number">4.30. </span>Conjugate gradient method<a class="headerlink" href="#conjugate-gradient-method" title="Permalink to this headline">¶</a></h2>
|
||
<p>In the CG method we define so-called conjugate directions and two vectors
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{s}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{t}\)</span>
|
||
are said to be
|
||
conjugate if</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{s}^T\boldsymbol{A}\boldsymbol{t}= 0.
|
||
\]</div>
|
||
<p>The philosophy of the CG method is to perform searches in various conjugate directions
|
||
of our vectors <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> obeying the above criterion, namely</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x}_i^T\boldsymbol{A}\boldsymbol{x}_j= 0.
|
||
\]</div>
|
||
<p>Two vectors are conjugate if they are orthogonal with respect to
|
||
this inner product. Being conjugate is a symmetric relation: if <span class="math notranslate nohighlight">\(\boldsymbol{s}\)</span> is conjugate to <span class="math notranslate nohighlight">\(\boldsymbol{t}\)</span>, then <span class="math notranslate nohighlight">\(\boldsymbol{t}\)</span> is conjugate to <span class="math notranslate nohighlight">\(\boldsymbol{s}\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="id3">
|
||
<h2><span class="section-number">4.31. </span>Conjugate gradient method<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h2>
|
||
<p>An example is given by the eigenvectors of the matrix</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{v}_i^T\boldsymbol{A}\boldsymbol{v}_j= \lambda\boldsymbol{v}_i^T\boldsymbol{v}_j,
|
||
\]</div>
|
||
<p>which is zero unless <span class="math notranslate nohighlight">\(i=j\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="id4">
|
||
<h2><span class="section-number">4.32. </span>Conjugate gradient method<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
|
||
<p>Assume now that we have a symmetric positive-definite matrix <span class="math notranslate nohighlight">\(\boldsymbol{A}\)</span> of size
|
||
<span class="math notranslate nohighlight">\(n\times n\)</span>. At each iteration <span class="math notranslate nohighlight">\(i+1\)</span> we obtain the conjugate direction of a vector</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+\alpha_i\boldsymbol{p}_{i}.
|
||
\]</div>
|
||
<p>We assume that <span class="math notranslate nohighlight">\(\boldsymbol{p}_{i}\)</span> is a sequence of <span class="math notranslate nohighlight">\(n\)</span> mutually conjugate directions.
|
||
Then the <span class="math notranslate nohighlight">\(\boldsymbol{p}_{i}\)</span> form a basis of <span class="math notranslate nohighlight">\(R^n\)</span> and we can expand the solution
|
||
<span class="math notranslate nohighlight">\( \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}\)</span> in this basis, namely</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i \boldsymbol{p}_i.
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="id5">
|
||
<h2><span class="section-number">4.33. </span>Conjugate gradient method<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h2>
|
||
<p>The coefficients are given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
|
||
\]</div>
|
||
<p>Multiplying with <span class="math notranslate nohighlight">\(\boldsymbol{p}_k^T\)</span> from the left gives</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{p}_i= \boldsymbol{p}_k^T \boldsymbol{b},
|
||
\]</div>
|
||
<p>and we can define the coefficients <span class="math notranslate nohighlight">\(\alpha_k\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\alpha_k = \frac{\boldsymbol{p}_k^T \boldsymbol{b}}{\boldsymbol{p}_k^T \boldsymbol{A} \boldsymbol{p}_k}
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="conjugate-gradient-method-and-iterations">
|
||
<h2><span class="section-number">4.34. </span>Conjugate gradient method and iterations<a class="headerlink" href="#conjugate-gradient-method-and-iterations" title="Permalink to this headline">¶</a></h2>
|
||
<p>If we choose the conjugate vectors <span class="math notranslate nohighlight">\(\boldsymbol{p}_k\)</span> carefully,
|
||
then we may not need all of them to obtain a good approximation to the solution
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>.
|
||
We want to regard the conjugate gradient method as an iterative method.
|
||
This will us to solve systems where <span class="math notranslate nohighlight">\(n\)</span> is so large that the direct
|
||
method would take too much time.</p>
|
||
<p>We denote the initial guess for <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> as <span class="math notranslate nohighlight">\(\boldsymbol{x}_0\)</span>.
|
||
We can assume without loss of generality that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x}_0=0,
|
||
\]</div>
|
||
<p>or consider the system</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
|
||
\]</div>
|
||
<p>instead.</p>
|
||
</div>
|
||
<div class="section" id="id6">
|
||
<h2><span class="section-number">4.35. </span>Conjugate gradient method<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h2>
|
||
<p>One can show that the solution <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> is also the unique minimizer of the quadratic form</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
|
||
\]</div>
|
||
<p>This suggests taking the first basis vector <span class="math notranslate nohighlight">\(\boldsymbol{p}_1\)</span>
|
||
to be the gradient of <span class="math notranslate nohighlight">\(f\)</span> at <span class="math notranslate nohighlight">\(\boldsymbol{x}=\boldsymbol{x}_0\)</span>,
|
||
which equals</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
|
||
\]</div>
|
||
<p>and
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{x}_0=0\)</span> it is equal <span class="math notranslate nohighlight">\(-\boldsymbol{b}\)</span>.
|
||
The other vectors in the basis will be conjugate to the gradient,
|
||
hence the name conjugate gradient method.</p>
|
||
</div>
|
||
<div class="section" id="id7">
|
||
<h2><span class="section-number">4.36. </span>Conjugate gradient method<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let <span class="math notranslate nohighlight">\(\boldsymbol{r}_k\)</span> be the residual at the <span class="math notranslate nohighlight">\(k\)</span>-th step:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{r}_k=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k.
|
||
\]</div>
|
||
<p>Note that <span class="math notranslate nohighlight">\(\boldsymbol{r}_k\)</span> is the negative gradient of <span class="math notranslate nohighlight">\(f\)</span> at
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{x}=\boldsymbol{x}_k\)</span>,
|
||
so the gradient descent method would be to move in the direction <span class="math notranslate nohighlight">\(\boldsymbol{r}_k\)</span>.
|
||
Here, we insist that the directions <span class="math notranslate nohighlight">\(\boldsymbol{p}_k\)</span> are conjugate to each other,
|
||
so we take the direction closest to the gradient <span class="math notranslate nohighlight">\(\boldsymbol{r}_k\)</span><br />
|
||
under the conjugacy constraint.
|
||
This gives the following expression</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{p}_{k+1}=\boldsymbol{r}_k-\frac{\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{r}_k}{\boldsymbol{p}_k^T\boldsymbol{A}\boldsymbol{p}_k} \boldsymbol{p}_k.
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="id8">
|
||
<h2><span class="section-number">4.37. </span>Conjugate gradient method<a class="headerlink" href="#id8" title="Permalink to this headline">¶</a></h2>
|
||
<p>We can also compute the residual iteratively as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1},
|
||
\]</div>
|
||
<p>which equals</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{p}_k),
|
||
\]</div>
|
||
<p>or</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{p}_k,
|
||
\]</div>
|
||
<p>which gives</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{r}_{k+1}=\boldsymbol{r}_k-\boldsymbol{A}\boldsymbol{p}_{k},
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="revisiting-our-first-homework">
|
||
<h2><span class="section-number">4.38. </span>Revisiting our first homework<a class="headerlink" href="#revisiting-our-first-homework" title="Permalink to this headline">¶</a></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:</p>
|
||
<ol class="simple">
|
||
<li><p>An analytical solution (recall homework set 1).</p></li>
|
||
<li><p>The gradient can be computed analytically.</p></li>
|
||
<li><p>The cost function is convex which guarantees that gradient descent converges for small enough learning rates</p></li>
|
||
</ol>
|
||
<p>We revisit an example similar to what we had in the first homework set. We had a function of the type</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>with <span class="math notranslate nohighlight">\(x_i \in [0,1] \)</span> is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution <span class="math notranslate nohighlight">\(\cal {N}(0,1)\)</span>.
|
||
The linear regression model is given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
h_\beta(x) = \boldsymbol{y} = \beta_0 + \beta_1 x,
|
||
\]</div>
|
||
<p>such that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{y}_i = \beta_0 + \beta_1 x_i.
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="gradient-descent-example">
|
||
<h2><span class="section-number">4.39. </span>Gradient descent example<a class="headerlink" href="#gradient-descent-example" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let <span class="math notranslate nohighlight">\(\mathbf{y} = (y_1,\cdots,y_n)^T\)</span>, <span class="math notranslate nohighlight">\(\mathbf{\boldsymbol{y}} = (\boldsymbol{y}_1,\cdots,\boldsymbol{y}_n)^T\)</span> and <span class="math notranslate nohighlight">\(\beta = (\beta_0, \beta_1)^T\)</span></p>
|
||
<p>It is convenient to write <span class="math notranslate nohighlight">\(\mathbf{\boldsymbol{y}} = X\beta\)</span> where <span class="math notranslate nohighlight">\(X \in \mathbb{R}^{100 \times 2} \)</span> is the design matrix given by (we keep the intercept here)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
X \equiv \begin{bmatrix}
|
||
1 & x_1 \\
|
||
\vdots & \vdots \\
|
||
1 & x_{100} & \\
|
||
\end{bmatrix}.
|
||
\end{split}\]</div>
|
||
<p>The cost/loss/risk function is given by (</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right]
|
||
\]</div>
|
||
<p>and we want to find <span class="math notranslate nohighlight">\(\beta\)</span> such that <span class="math notranslate nohighlight">\(C(\beta)\)</span> is minimized.</p>
|
||
</div>
|
||
<div class="section" id="the-derivative-of-the-cost-loss-function">
|
||
<h2><span class="section-number">4.40. </span>The derivative of the cost/loss function<a class="headerlink" href="#the-derivative-of-the-cost-loss-function" title="Permalink to this headline">¶</a></h2>
|
||
<p>Computing <span class="math notranslate nohighlight">\(\partial C(\beta) / \partial \beta_0\)</span> and <span class="math notranslate nohighlight">\(\partial C(\beta) / \partial \beta_1\)</span> we can show that the gradient can be written as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
|
||
\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
|
||
\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}),
|
||
\end{split}\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(X\)</span> is the design matrix defined above.</p>
|
||
</div>
|
||
<div class="section" id="the-hessian-matrix">
|
||
<h2><span class="section-number">4.41. </span>The Hessian matrix<a class="headerlink" href="#the-hessian-matrix" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Hessian matrix of <span class="math notranslate nohighlight">\(C(\beta)\)</span> is given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\boldsymbol{H} \equiv \begin{bmatrix}
|
||
\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
|
||
\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
|
||
\end{bmatrix} = \frac{2}{n}X^T X.
|
||
\end{split}\]</div>
|
||
<p>This result implies that <span class="math notranslate nohighlight">\(C(\beta)\)</span> is a convex function since the matrix <span class="math notranslate nohighlight">\(X^T X\)</span> always is positive semi-definite.</p>
|
||
</div>
|
||
<div class="section" id="simple-program">
|
||
<h2><span class="section-number">4.42. </span>Simple program<a class="headerlink" href="#simple-program" title="Permalink to this headline">¶</a></h2>
|
||
<p>We can now write a program that minimizes <span class="math notranslate nohighlight">\(C(\beta)\)</span> using the gradient descent method with a constant learning rate <span class="math notranslate nohighlight">\(\gamma\)</span> according to</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
|
||
\]</div>
|
||
<p>We can use the expression we computed for the gradient and let use a
|
||
<span class="math notranslate nohighlight">\(\beta_0\)</span> be chosen randomly and let <span class="math notranslate nohighlight">\(\gamma = 0.001\)</span>. Stop iterating
|
||
when <span class="math notranslate nohighlight">\(||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8}\)</span>. <strong>Note that the code below does not include the latter stop criterion</strong>.</p>
|
||
<p>And finally we can compare our solution for <span class="math notranslate nohighlight">\(\beta\)</span> with the analytic result given by
|
||
<span class="math notranslate nohighlight">\(\beta= (X^TX)^{-1} X^T \mathbf{y}\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="id9">
|
||
<h2><span class="section-number">4.43. </span>Gradient Descent Example<a class="headerlink" href="#id9" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here our simple example</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Importing various packages</span>
|
||
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib.ticker</span> <span class="kn">import</span> <span class="n">LinearLocator</span><span class="p">,</span> <span class="n">FormatStrFormatter</span>
|
||
<span class="kn">import</span> <span class="nn">sys</span>
|
||
|
||
<span class="c1"># the number of datapoints</span>
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
|
||
<span class="c1"># Hessian matrix</span>
|
||
<span class="n">H</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span>
|
||
<span class="c1"># Get the eigenvalues</span>
|
||
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">EigValues</span><span class="p">)</span>
|
||
|
||
<span class="n">beta_linreg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">beta_linreg</span><span class="p">)</span>
|
||
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">eta</span> <span class="o">=</span> <span class="mf">1.0</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">EigValues</span><span class="p">)</span>
|
||
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
|
||
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
|
||
<span class="n">gradient</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">(</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="n">beta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradient</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
|
||
<span class="n">xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">],[</span><span class="mi">2</span><span class="p">]])</span>
|
||
<span class="n">xbnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">xnew</span><span class="p">]</span>
|
||
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">xbnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
|
||
<span class="n">ypredict2</span> <span class="o">=</span> <span class="n">xbnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">beta_linreg</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">"r-"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict2</span><span class="p">,</span> <span class="s2">"b-"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="p">,</span><span class="s1">'ro'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x$'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$y$'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Gradient descent example'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="and-a-corresponding-example-using-scikit-learn">
|
||
<h2><span class="section-number">4.44. </span>And a corresponding example using <strong>scikit-learn</strong><a class="headerlink" href="#and-a-corresponding-example-using-scikit-learn" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Importing various packages</span>
|
||
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">SGDRegressor</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
|
||
<span class="n">beta_linreg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">beta_linreg</span><span class="p">)</span>
|
||
<span class="n">sgdreg</span> <span class="o">=</span> <span class="n">SGDRegressor</span><span class="p">(</span><span class="n">max_iter</span> <span class="o">=</span> <span class="mi">50</span><span class="p">,</span> <span class="n">penalty</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">eta0</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
|
||
<span class="n">sgdreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">sgdreg</span><span class="o">.</span><span class="n">intercept_</span><span class="p">,</span> <span class="n">sgdreg</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="gradient-descent-and-ridge">
|
||
<h2><span class="section-number">4.45. </span>Gradient descent and Ridge<a class="headerlink" href="#gradient-descent-and-ridge" title="Permalink to this headline">¶</a></h2>
|
||
<p>We have also discussed Ridge regression where the loss function contains a regularized term given by the <span class="math notranslate nohighlight">\(L_2\)</span> norm of <span class="math notranslate nohighlight">\(\beta\)</span>,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0.
|
||
\]</div>
|
||
<p>In order to minimize <span class="math notranslate nohighlight">\(C_{\text{ridge}}(\beta)\)</span> using GD we only have adjust the gradient as follows</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
|
||
\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
|
||
\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
|
||
\end{split}\]</div>
|
||
<p>We can easily extend our program to minimize <span class="math notranslate nohighlight">\(C_{\text{ridge}}(\beta)\)</span> using gradient descent and compare with the analytical solution given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="program-example-for-gradient-descent-with-ridge-regression">
|
||
<h2><span class="section-number">4.46. </span>Program example for gradient descent with Ridge Regression<a class="headerlink" href="#program-example-for-gradient-descent-with-ridge-regression" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib.ticker</span> <span class="kn">import</span> <span class="n">LinearLocator</span><span class="p">,</span> <span class="n">FormatStrFormatter</span>
|
||
<span class="kn">import</span> <span class="nn">sys</span>
|
||
|
||
<span class="c1"># the number of datapoints</span>
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
|
||
<span class="n">XT_X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span>
|
||
|
||
<span class="c1">#Ridge parameter lambda</span>
|
||
<span class="n">lmbda</span> <span class="o">=</span> <span class="mf">0.001</span>
|
||
<span class="n">Id</span> <span class="o">=</span> <span class="n">lmbda</span><span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">XT_X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
|
||
<span class="n">beta_linreg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">XT_X</span><span class="o">+</span><span class="n">Id</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">beta_linreg</span><span class="p">)</span>
|
||
<span class="c1"># Start plain gradient descent</span>
|
||
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.1</span>
|
||
<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">100</span>
|
||
|
||
<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">(</span><span class="n">X</span> <span class="o">@</span> <span class="p">(</span><span class="n">beta</span><span class="p">)</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">lmbda</span><span class="o">*</span><span class="n">beta</span>
|
||
<span class="n">beta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
|
||
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span>
|
||
<span class="n">ypredict2</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta_linreg</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">"r-"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ypredict2</span><span class="p">,</span> <span class="s2">"b-"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="p">,</span><span class="s1">'ro'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x$'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$y$'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Gradient descent example for Ridge'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="using-gradient-descent-methods-limitations">
|
||
<h2><span class="section-number">4.47. </span>Using gradient descent methods, limitations<a class="headerlink" href="#using-gradient-descent-methods-limitations" title="Permalink to this headline">¶</a></h2>
|
||
<ul class="simple">
|
||
<li><p><strong>Gradient descent (GD) finds local minima of our function</strong>. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.</p></li>
|
||
<li><p><strong>GD is sensitive to initial conditions</strong>. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.</p></li>
|
||
<li><p><strong>Gradients are computationally expensive to calculate for large datasets</strong>. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, <span class="math notranslate nohighlight">\(E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2\)</span>; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over <em>all</em> <span class="math notranslate nohighlight">\(n\)</span> data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called “mini batches”. This has the added benefit of introducing stochasticity into our algorithm.</p></li>
|
||
<li><p><strong>GD is very sensitive to choices of learning rates</strong>. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would <em>adaptively</em> choose the learning rates to match the landscape.</p></li>
|
||
<li><p><strong>GD treats all directions in parameter space uniformly.</strong> Another major drawback of GD is that unlike Newton’s method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.</p></li>
|
||
<li><p>GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.</p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="stochastic-gradient-descent">
|
||
<h2><span class="section-number">4.48. </span>Stochastic Gradient Descent<a class="headerlink" href="#stochastic-gradient-descent" title="Permalink to this headline">¶</a></h2>
|
||
<p>Stochastic gradient descent (SGD) and variants thereof address some of
|
||
the shortcomings of the Gradient descent method discussed above.</p>
|
||
<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 <span class="math notranslate nohighlight">\(n\)</span> data points <span class="math notranslate nohighlight">\(\{\mathbf{x}_i\}_{i=1}^n\)</span>,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
|
||
\mathbf{\beta}).
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="computation-of-gradients">
|
||
<h2><span class="section-number">4.49. </span>Computation of gradients<a class="headerlink" href="#computation-of-gradients" title="Permalink to this headline">¶</a></h2>
|
||
<p>This in turn means that the gradient can be
|
||
computed as a sum over <span class="math notranslate nohighlight">\(i\)</span>-gradients</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
|
||
\mathbf{\beta}).
|
||
\]</div>
|
||
<p>Stochasticity/randomness is introduced by only taking the
|
||
gradient on a subset of the data called minibatches. If there are <span class="math notranslate nohighlight">\(n\)</span>
|
||
data points and the size of each minibatch is <span class="math notranslate nohighlight">\(M\)</span>, there will be <span class="math notranslate nohighlight">\(n/M\)</span>
|
||
minibatches. We denote these minibatches by <span class="math notranslate nohighlight">\(B_k\)</span> where
|
||
<span class="math notranslate nohighlight">\(k=1,\cdots,n/M\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="sgd-example">
|
||
<h2><span class="section-number">4.50. </span>SGD example<a class="headerlink" href="#sgd-example" title="Permalink to this headline">¶</a></h2>
|
||
<p>As an example, suppose we have <span class="math notranslate nohighlight">\(10\)</span> data points <span class="math notranslate nohighlight">\((\mathbf{x}_1,\cdots, \mathbf{x}_{10})\)</span>
|
||
and we choose to have <span class="math notranslate nohighlight">\(M=5\)</span> minibathces,
|
||
then each minibatch contains two data points. In particular we have
|
||
<span class="math notranslate nohighlight">\(B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
|
||
(\mathbf{x}_9,\mathbf{x}_{10})\)</span>. Note that if you choose <span class="math notranslate nohighlight">\(M=1\)</span> you
|
||
have only a single batch with all data points and on the other extreme,
|
||
you may choose <span class="math notranslate nohighlight">\(M=n\)</span> resulting in a minibatch for each datapoint, i.e
|
||
<span class="math notranslate nohighlight">\(B_k = \mathbf{x}_k\)</span>.</p>
|
||
<p>The idea is now to approximate the gradient by replacing the sum over
|
||
all data points with a sum over the data points in one the minibatches
|
||
picked at random in each gradient descent step</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\nabla_{\beta}
|
||
C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
|
||
\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
|
||
c_i(\mathbf{x}_i, \mathbf{\beta}).
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="the-gradient-step">
|
||
<h2><span class="section-number">4.51. </span>The gradient step<a class="headerlink" href="#the-gradient-step" title="Permalink to this headline">¶</a></h2>
|
||
<p>Thus a gradient descent step now looks like</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
|
||
\mathbf{\beta})
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(k\)</span> is picked at random with equal
|
||
probability from <span class="math notranslate nohighlight">\([1,n/M]\)</span>. 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.</p>
|
||
</div>
|
||
<div class="section" id="simple-example-code">
|
||
<h2><span class="section-number">4.52. </span>Simple example code<a class="headerlink" href="#simple-example-code" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1">#100 datapoints </span>
|
||
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
|
||
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
|
||
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">10</span> <span class="c1">#number of epochs</span>
|
||
|
||
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n_epochs</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
|
||
<span class="n">k</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="c1">#Pick the k-th minibatch at random</span>
|
||
<span class="c1">#Compute the gradient using the data in minibatch Bk</span>
|
||
<span class="c1">#Compute new suggestion for </span>
|
||
<span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</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 (<span class="math notranslate nohighlight">\(M < n\)</span>), the computation of the gradient is much
|
||
cheaper since we sum over the datapoints in the <span class="math notranslate nohighlight">\(k-th\)</span> minibatch and not
|
||
all <span class="math notranslate nohighlight">\(n\)</span> datapoints.</p>
|
||
</div>
|
||
<div class="section" id="when-do-we-stop">
|
||
<h2><span class="section-number">4.53. </span>When do we stop?<a class="headerlink" href="#when-do-we-stop" title="Permalink to this headline">¶</a></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
|
||
is zero is valid also for local minima, so this would only tell us
|
||
that we are close to a local/global minimum. However, we could also
|
||
evaluate the cost function at this point, store the result and
|
||
continue the search. If the test kicks in at a later stage we can
|
||
compare the values of the cost function and keep the <span class="math notranslate nohighlight">\(\beta\)</span> that
|
||
gave the lowest value.</p>
|
||
</div>
|
||
<div class="section" id="slightly-different-approach">
|
||
<h2><span class="section-number">4.54. </span>Slightly different approach<a class="headerlink" href="#slightly-different-approach" title="Permalink to this headline">¶</a></h2>
|
||
<p>Another approach is to let the step length <span class="math notranslate nohighlight">\(\gamma_j\)</span> 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>
|
||
<p>As an example, let <span class="math notranslate nohighlight">\(e = 0,1,2,3,\cdots\)</span> denote the current epoch and let <span class="math notranslate nohighlight">\(t_0, t_1 > 0\)</span> be two fixed numbers. Furthermore, let <span class="math notranslate nohighlight">\(t = e \cdot m + i\)</span> where <span class="math notranslate nohighlight">\(m\)</span> is the number of minibatches and <span class="math notranslate nohighlight">\(i=0,\cdots,m-1\)</span>. Then the function $<span class="math notranslate nohighlight">\(\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} \)</span><span class="math notranslate nohighlight">\( goes to zero as the number of epochs gets large. I.e. we start with a step length \)</span>\gamma_j (0; t_0, t_1) = t_0/t_1<span class="math notranslate nohighlight">\( which decays in *time* \)</span>t$.</p>
|
||
<p>In this way we can fix the number of epochs, compute <span class="math notranslate nohighlight">\(\beta\)</span> 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 <span class="math notranslate nohighlight">\(\beta\)</span> that gives the lowest value of the cost
|
||
function.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="k">def</span> <span class="nf">step_length</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">t0</span><span class="p">,</span><span class="n">t1</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">t0</span><span class="o">/</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">t1</span><span class="p">)</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1">#100 datapoints </span>
|
||
<span class="n">M</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1">#size of each minibatch</span>
|
||
<span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">M</span><span class="p">)</span> <span class="c1">#number of minibatches</span>
|
||
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">500</span> <span class="c1">#number of epochs</span>
|
||
<span class="n">t0</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<span class="n">t1</span> <span class="o">=</span> <span class="mi">10</span>
|
||
|
||
<span class="n">gamma_j</span> <span class="o">=</span> <span class="n">t0</span><span class="o">/</span><span class="n">t1</span>
|
||
<span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n_epochs</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
|
||
<span class="n">k</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="c1">#Pick the k-th minibatch at random</span>
|
||
<span class="c1">#Compute the gradient using the data in minibatch Bk</span>
|
||
<span class="c1">#Compute new suggestion for beta</span>
|
||
<span class="n">t</span> <span class="o">=</span> <span class="n">epoch</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">i</span>
|
||
<span class="n">gamma_j</span> <span class="o">=</span> <span class="n">step_length</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">t0</span><span class="p">,</span><span class="n">t1</span><span class="p">)</span>
|
||
<span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"gamma_j after </span><span class="si">%d</span><span class="s2"> epochs: </span><span class="si">%g</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">n_epochs</span><span class="p">,</span><span class="n">gamma_j</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
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<div class="section" id="program-for-stochastic-gradient">
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<h2><span class="section-number">4.55. </span>Program for stochastic gradient<a class="headerlink" href="#program-for-stochastic-gradient" title="Permalink to this headline">¶</a></h2>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Importing various packages</span>
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<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span>
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<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
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<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
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<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">SGDRegressor</span>
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<span class="n">m</span> <span class="o">=</span> <span class="mi">100</span>
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<span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
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<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
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<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">x</span><span class="p">]</span>
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<span class="n">theta_linreg</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"Own inversion"</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
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<span class="n">sgdreg</span> <span class="o">=</span> <span class="n">SGDRegressor</span><span class="p">(</span><span class="n">max_iter</span> <span class="o">=</span> <span class="mi">50</span><span class="p">,</span> <span class="n">penalty</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">eta0</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
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<span class="n">sgdreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"sgdreg from scikit"</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">sgdreg</span><span class="o">.</span><span class="n">intercept_</span><span class="p">,</span> <span class="n">sgdreg</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
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<span class="n">theta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
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<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.1</span>
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<span class="n">Niterations</span> <span class="o">=</span> <span class="mi">1000</span>
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<span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Niterations</span><span class="p">):</span>
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<span class="n">gradients</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">/</span><span class="n">m</span><span class="o">*</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">((</span><span class="n">X</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
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<span class="n">theta</span> <span class="o">-=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own gd"</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
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<span class="n">xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">],[</span><span class="mi">2</span><span class="p">]])</span>
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<span class="n">Xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">xnew</span><span class="p">]</span>
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<span class="n">ypredict</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
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<span class="n">ypredict2</span> <span class="o">=</span> <span class="n">Xnew</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
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<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</span>
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<span class="n">t0</span><span class="p">,</span> <span class="n">t1</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">50</span>
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<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
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<span class="k">return</span> <span class="n">t0</span><span class="o">/</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">t1</span><span class="p">)</span>
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<span class="n">theta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
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<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_epochs</span><span class="p">):</span>
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<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
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<span class="n">random_index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
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<span class="n">xi</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
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<span class="n">yi</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">random_index</span><span class="p">:</span><span class="n">random_index</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
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<span class="n">gradients</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">xi</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">((</span><span class="n">xi</span> <span class="o">@</span> <span class="n">theta</span><span class="p">)</span><span class="o">-</span><span class="n">yi</span><span class="p">)</span>
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<span class="n">eta</span> <span class="o">=</span> <span class="n">learning_schedule</span><span class="p">(</span><span class="n">epoch</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">i</span><span class="p">)</span>
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<span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span> <span class="o">-</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own sdg"</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">"r-"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict2</span><span class="p">,</span> <span class="s2">"b-"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="p">,</span><span class="s1">'ro'</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">])</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$x$'</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$y$'</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Random numbers '</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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</pre></div>
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</div>
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</div>
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</div>
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<p><strong>Challenge</strong>: try to write a similar code for a Logistic Regression case.</p>
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