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Morten Hjorth-Jensen
@@ -7,8 +7,8 @@
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Applied Data Analysis and Machine Learning, FYS-STK3155/4155 at the University of Oslo, Norway
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Applied Data Analysis and Machine Learning
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About the course
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@@ -117,7 +119,7 @@
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<span class="caption-text">
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Review of Statistics with Resampling Techniques and Linear Algebra
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@@ -134,7 +136,7 @@
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<span class="caption-text">
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From Regression to Support Vector Machines
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@@ -171,7 +173,7 @@
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Decision Trees, Ensemble Methods and Boosting
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@@ -188,7 +190,7 @@
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Dimensionality Reduction
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@@ -205,7 +207,7 @@
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Deep Learning Methods
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@@ -282,7 +284,7 @@
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@@ -300,7 +302,7 @@
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@@ -408,7 +410,118 @@
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<h1>Optimization, the central part of any Machine Learning algortithm</h1>
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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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7.1. 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="#convex-functions">
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7.2. Convex functions
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</a>
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<ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#some-simple-problems">
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7.2.1. Some simple problems
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</a>
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</li>
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</ul>
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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="#standard-steepest-descent">
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7.3. 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="#conjugate-gradient-method">
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7.4. Conjugate gradient 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="#revisiting-our-linear-regression-solvers">
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7.5. Revisiting our Linear Regression Solvers
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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="#using-gradient-descent-methods-limitations">
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7.6. Using gradient descent methods, limitations
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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="#stochastic-gradient-descent-sgd">
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7.7. Stochastic Gradient Descent (SGD)
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</a>
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<ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#program-for-stochastic-gradient">
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7.7.1. Program for stochastic gradient
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</a>
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</li>
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</ul>
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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="#momentum-based-gd">
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7.8. Momentum based GD
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</a>
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<ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#rms-prop">
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7.8.1. RMS prop
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</a>
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</li>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#adam-optimizer">
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7.8.2. ADAM optimizer
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</a>
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</li>
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</ul>
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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="#practical-tips">
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7.9. Practical tips
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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="#automatic-differentiation">
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7.10. Automatic differentiation
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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="#using-autograd-with-ols">
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7.11. Using Autograd with OLS
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</a>
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<ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#including-stochastic-gradient-descent-with-autograd">
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7.11.1. Including Stochastic Gradient Descent with Autograd
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</a>
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</li>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#and-logistic-regression">
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7.11.2. And Logistic Regression
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</a>
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</li>
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</li>
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</ul>
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<div>
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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@@ -783,11 +896,11 @@ which equals</p>
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</div>
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<div class="cell_output docutils container">
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/483257001.py:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-1-7f2b3a6174c2>:18: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot().
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ax = fig.gca(projection="3d")
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</pre></div>
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</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fd098df3280>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x11da42d90>
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</pre></div>
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</div>
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<img alt="_images/chapteroptimization_61_2.png" src="_images/chapteroptimization_61_2.png" />
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@@ -845,7 +958,7 @@ which equals</p>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[<matplotlib.lines.Line2D at 0x7fd0a9063be0>]
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[<matplotlib.lines.Line2D at 0x11e031e50>]
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</pre></div>
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</div>
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<img alt="_images/chapteroptimization_69_1.png" src="_images/chapteroptimization_69_1.png" />
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@@ -1102,11 +1215,11 @@ when <span class="math notranslate nohighlight">\(||\nabla_\beta C(\beta_k) || \
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.28001319 4.21265216]
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[[3.96987657]
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[3.02493054]]
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[[3.96987657]
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[3.02493054]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.29588901 4.59955801]
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[[4.21985165]
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[2.77627886]]
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[[4.21985165]
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[2.77627886]]
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</pre></div>
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</div>
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<img alt="_images/chapteroptimization_123_1.png" src="_images/chapteroptimization_123_1.png" />
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@@ -1135,9 +1248,9 @@ when <span class="math notranslate nohighlight">\(||\nabla_\beta C(\beta_k) || \
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[4.00275135]
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[2.99724883]]
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[3.97065296] [3.07656896]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[3.54163136]
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[3.31866499]]
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[3.59833875] [3.3594821]
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</pre></div>
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</div>
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</div>
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@@ -1208,10 +1321,10 @@ C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta|
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[4.1533795 ]
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[2.92819235]]
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[[4.06858699]
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[2.99829953]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[4.00673393]
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[2.84569271]]
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[[3.98017611]
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[2.86623151]]
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</pre></div>
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</div>
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<img alt="_images/chapteroptimization_132_1.png" src="_images/chapteroptimization_132_1.png" />
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@@ -1461,15 +1574,15 @@ function.</p>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Own inversion
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[[3.99775949]
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[2.94659383]]
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Eigenvalues of Hessian Matrix:[0.36102113 4.18276924]
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[[3.99519225]
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[3.01509543]]
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Eigenvalues of Hessian Matrix:[0.29229741 4.7643536 ]
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theta from own gd
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[[3.99775949]
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[2.94659383]]
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[[3.99519225]
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[3.01509543]]
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theta from own sdg
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[[3.96489434]
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[2.98399675]]
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[[3.98756882]
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[3.02625928]]
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</pre></div>
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</div>
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<img alt="_images/chapteroptimization_148_1.png" src="_images/chapteroptimization_148_1.png" />
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@@ -2081,47 +2194,47 @@ The analytical derivative of f7 at n = 2 is: 1
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<div class="cell_output docutils container">
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<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span>---------------------------------------------------------------------------
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TypeError Traceback (most recent call last)
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/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/1122558214.py in <module>
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<ipython-input-22-cabc613b8702> in <module>
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9 x = 8.4
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10
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---> 11 print("The derivative of f8 is:",f8_grad(x))
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~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
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~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py in nary_f(*args, **kwargs)
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18 else:
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19 x = tuple(args[i] for i in argnum)
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---> 20 return unary_operator(unary_f, x, *nary_op_args, **nary_op_kwargs)
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21 return nary_f
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22 return nary_operator
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~/anaconda3/lib/python3.8/site-packages/autograd/differential_operators.py in grad(fun, x)
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~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/differential_operators.py in grad(fun, x)
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23 arguments as `fun`, but returns the gradient instead. The function `fun`
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24 should be scalar-valued. The gradient has the same type as the argument."""
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---> 25 vjp, ans = _make_vjp(fun, x)
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26 if not vspace(ans).size == 1:
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27 raise TypeError("Grad only applies to real scalar-output functions. "
|
||||
|
||||
~/anaconda3/lib/python3.8/site-packages/autograd/core.py in make_vjp(fun, x)
|
||||
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/core.py in make_vjp(fun, x)
|
||||
8 def make_vjp(fun, x):
|
||||
9 start_node = VJPNode.new_root()
|
||||
---> 10 end_value, end_node = trace(start_node, fun, x)
|
||||
11 if end_node is None:
|
||||
12 def vjp(g): return vspace(x).zeros()
|
||||
|
||||
~/anaconda3/lib/python3.8/site-packages/autograd/tracer.py in trace(start_node, fun, x)
|
||||
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/tracer.py in trace(start_node, fun, x)
|
||||
8 with trace_stack.new_trace() as t:
|
||||
9 start_box = new_box(x, t, start_node)
|
||||
---> 10 end_box = fun(start_box)
|
||||
11 if isbox(end_box) and end_box._trace == start_box._trace:
|
||||
12 return end_box._value, end_box._node
|
||||
|
||||
~/anaconda3/lib/python3.8/site-packages/autograd/wrap_util.py in unary_f(x)
|
||||
~/miniforge3/envs/myenv/lib/python3.9/site-packages/autograd/wrap_util.py in unary_f(x)
|
||||
13 else:
|
||||
14 subargs = subvals(args, zip(argnum, x))
|
||||
---> 15 return fun(*subargs, **kwargs)
|
||||
16 if isinstance(argnum, int):
|
||||
17 x = args[argnum]
|
||||
|
||||
/var/folders/jy/g42mrgv128v34gnnhxwk9nrc0000gp/T/ipykernel_47735/1122558214.py in f8(x)
|
||||
<ipython-input-22-cabc613b8702> in f8(x)
|
||||
2 from autograd import grad
|
||||
3 def f8(x): # Assume x is an array
|
||||
----> 4 x[2] = 3
|
||||
@@ -2396,54 +2509,42 @@ first example shows results with ordinary leats squares.</p>
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
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<footer class="footer">
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