Morten Hjorth-Jensen
2765 lines
243 KiB
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Applied Data Analysis and Machine Learning
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About the course
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Teachers and Grading
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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<a class="current reference internal" href="#">
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7. Optimization, the central part of any Machine Learning algortithm
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<a class="reference internal" href="chapter5.html">
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8. Support Vector Machines, overarching aims
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</a>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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13. Neural networks
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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17. Recurrent neural networks: Overarching view
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Weekly material, notes and exercises
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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Exercises week 35
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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Exercises week 36
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Week 36: Linear Rgeression and Statistical interpretations
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7.1. Steepest descent
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7.2. Convex functions
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7.2.1. Some simple problems
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7.3. Standard steepest descent
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7.4. Conjugate gradient method
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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 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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<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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<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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<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 class="reference internal nav-link" href="#adam-optimizer">
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7.8.2. ADAM optimizer
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7.9. Practical tips
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7.10. Automatic differentiation
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<a class="reference internal nav-link" href="#replace-or-not">
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7.11. Replace or not
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7.12. Using Autograd
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#same-code-but-now-with-momentum-gradient-descent">
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7.13. Same code but now with momentum gradient descent
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</a>
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<li class="toc-h2 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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7.14. Including Stochastic Gradient Descent with Autograd
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<a class="reference internal nav-link" href="#similar-second-order-function-now-problem-but-now-with-adagrad">
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7.14.1. Similar (second order function now) problem but now with AdaGrad
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</a>
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</li>
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</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#introducing-jax">
|
||
7.15. Introducing JAX
|
||
</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">
|
||
<!-- Table of contents that is only displayed when printing the page -->
|
||
<div id="jb-print-docs-body" class="onlyprint">
|
||
<h1>Optimization, the central part of any Machine Learning algortithm</h1>
|
||
<!-- Table of contents -->
|
||
<div id="print-main-content">
|
||
<div id="jb-print-toc">
|
||
|
||
<div>
|
||
<h2> Contents </h2>
|
||
</div>
|
||
<nav aria-label="Page">
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#steepest-descent">
|
||
7.1. Steepest descent
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#convex-functions">
|
||
7.2. Convex functions
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#some-simple-problems">
|
||
7.2.1. Some simple problems
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#standard-steepest-descent">
|
||
7.3. Standard steepest descent
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#conjugate-gradient-method">
|
||
7.4. Conjugate gradient method
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#revisiting-our-linear-regression-solvers">
|
||
7.5. Revisiting our Linear Regression Solvers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-gradient-descent-methods-limitations">
|
||
7.6. 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-sgd">
|
||
7.7. Stochastic Gradient Descent (SGD)
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#program-for-stochastic-gradient">
|
||
7.7.1. Program for stochastic gradient
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#momentum-based-gd">
|
||
7.8. Momentum based GD
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#rms-prop">
|
||
7.8.1. RMS prop
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#adam-optimizer">
|
||
7.8.2. ADAM optimizer
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#practical-tips">
|
||
7.9. Practical tips
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#automatic-differentiation">
|
||
7.10. Automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#replace-or-not">
|
||
7.11. Replace or not
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-autograd">
|
||
7.12. Using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#same-code-but-now-with-momentum-gradient-descent">
|
||
7.13. Same code but now with momentum gradient descent
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#including-stochastic-gradient-descent-with-autograd">
|
||
7.14. Including Stochastic Gradient Descent with Autograd
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#similar-second-order-function-now-problem-but-now-with-adagrad">
|
||
7.14.1. Similar (second order function now) problem but now with AdaGrad
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#introducing-jax">
|
||
7.15. Introducing JAX
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div>
|
||
|
||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html chapteroptimization.do.txt --><div class="tex2jax_ignore mathjax_ignore section" id="optimization-the-central-part-of-any-machine-learning-algortithm">
|
||
<h1><span class="section-number">7. </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></h1>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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">
|
||
\[
|
||
f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
|
||
\label{eq:taylornr} \tag{1}
|
||
\]</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>
|
||
<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>
|
||
<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">\(\boldsymbol{J}\)</span> we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\boldsymbol{J}=\left( \begin{array}{cc}
|
||
\partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
|
||
\partial f_2/\partial x_1 &\partial f_2/\partial x_2
|
||
\end{array} \right),
|
||
\end{split}\]</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">
|
||
\[\begin{split}
|
||
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
|
||
-\boldsymbol{J}^{-1}
|
||
\left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
|
||
\end{split}\]</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">\(\boldsymbol{J}\)</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 class="section" id="steepest-descent">
|
||
<h2><span class="section-number">7.1. </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>
|
||
<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>
|
||
<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>
|
||
<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">7.2. </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>
|
||
<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>
|
||
for all
|
||
<span class="math notranslate nohighlight">\(x_1, x_2 \in X\)</span> and for all <span class="math notranslate nohighlight">\(t \in [0,1]\)</span>.</p>
|
||
<p>If <span class="math notranslate nohighlight">\(\leq\)</span> is replaced with a strict inequality in the
|
||
definition, we demand <span class="math notranslate nohighlight">\(x_1 \neq x_2\)</span> and <span class="math notranslate nohighlight">\(t\in(0,1)\)</span> then <span class="math notranslate nohighlight">\(f\)</span> is said
|
||
to be strictly convex. For a single variable function, convexity means
|
||
that if you draw a straight line connecting <span class="math notranslate nohighlight">\(f(x_1)\)</span> and <span class="math notranslate nohighlight">\(f(x_2)\)</span>, the
|
||
value of the function on the interval <span class="math notranslate nohighlight">\([x_1,x_2]\)</span> is always below the
|
||
line as discussed below.</p>
|
||
<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>: 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> holds
|
||
for all <span class="math notranslate nohighlight">\(x,y \in D_f\)</span>. This condition means that for a convex function
|
||
the first order Taylor expansion (right hand side above) at any point
|
||
is a global under estimator of the function. To convince yourself you can
|
||
make a drawing of <span class="math notranslate nohighlight">\(f(x) = x^2+1\)</span> and draw the tangent line to <span class="math notranslate nohighlight">\(f(x)\)</span> and
|
||
note that it is always below the graph.</p>
|
||
<p><strong>Second order condition</strong>: 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>
|
||
<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 class="section" id="some-simple-problems">
|
||
<h3><span class="section-number">7.2.1. </span>Some simple problems<a class="headerlink" href="#some-simple-problems" title="Permalink to this headline">¶</a></h3>
|
||
<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>
|
||
<div class="section" id="standard-steepest-descent">
|
||
<h2><span class="section-number">7.3. </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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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 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="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 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">TypeError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">18</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">15</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="g g-Whitespace"> </span><span class="mi">17</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="ne">---> </span><span class="mi">18</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="g g-Whitespace"> </span><span class="mi">20</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="g g-Whitespace"> </span><span class="mi">21</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="ne">TypeError</span>: gca() got an unexpected keyword argument 'projection'
|
||
</pre></div>
|
||
</div>
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><Figure size 640x480 with 0 Axes>
|
||
</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">7.4. </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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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-linear-regression-solvers">
|
||
<h2><span class="section-number">7.5. </span>Revisiting our Linear Regression Solvers<a class="headerlink" href="#revisiting-our-linear-regression-solvers" 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.</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">m</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">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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<p>Here is 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>
|
||
<p>Alternatively, we can use <strong>Scikit-Learn</strong> as done here</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">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>
|
||
<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 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">7.6. </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-sgd">
|
||
<h2><span class="section-number">7.7. </span>Stochastic Gradient Descent (SGD)<a class="headerlink" href="#stochastic-gradient-descent-sgd" title="Permalink to this headline">¶</a></h2>
|
||
<p>In stochastic gradient descent, the extreme case is the case where we
|
||
have only one batch, that is we include the whole data set.</p>
|
||
<p>This process is called Stochastic Gradient
|
||
Descent (SGD) (or also sometimes on-line gradient descent). This is
|
||
relatively less common to see because in practice due to vectorized
|
||
code optimizations it can be computationally much more efficient to
|
||
evaluate the gradient for 100 examples, than the gradient for one
|
||
example 100 times. Even though SGD technically refers to using a
|
||
single example at a time to evaluate the gradient, you will hear
|
||
people use the term SGD even when referring to mini-batch gradient
|
||
descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD
|
||
for “Batch gradient descent” are rare to see), where it is usually
|
||
assumed that mini-batches are used. The size of the mini-batch is a
|
||
hyperparameter but it is not very common to cross-validate or bootstrap it. It is
|
||
usually based on memory constraints (if any), or set to some value,
|
||
e.g. 32, 64 or 128. We use powers of 2 in practice because many
|
||
vectorized operation implementations work faster when their inputs are
|
||
sized in powers of 2.</p>
|
||
<p>In our notes with SGD we mean stochastic gradient descent with mini-batches.</p>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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 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 mini-batche</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>
|
||
<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>
|
||
<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>
|
||
<p>We note that we have defined several hyperparameters. These are now the number of epochs, the number of mini-batches and the parameters <span class="math notranslate nohighlight">\(t_0\)</span> and <span class="math notranslate nohighlight">\(t_1\)</span>.</p>
|
||
<div class="section" id="program-for-stochastic-gradient">
|
||
<h3><span class="section-number">7.7.1. </span>Program for stochastic gradient<a class="headerlink" href="#program-for-stochastic-gradient" title="Permalink to this headline">¶</a></h3>
|
||
<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="c1"># Importing various packages</span>
|
||
<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>
|
||
<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="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="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>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</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">XT_X</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="sa">f</span><span class="s2">"Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<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>
|
||
<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">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="n">theta</span><span class="p">)</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="n">theta</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="s2">"theta from own gd"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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">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>
|
||
<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>
|
||
<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>
|
||
|
||
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</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">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>
|
||
<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</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">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>
|
||
|
||
<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>
|
||
<span class="c1"># Can you figure out a better way of setting up the contributions to each batch?</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">random_index</span> <span class="o">=</span> <span class="n">M</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="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="n">M</span><span class="p">]</span>
|
||
<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="n">M</span><span class="p">]</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</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>
|
||
<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>
|
||
<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>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own sdg"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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">'Random numbers '</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 code, we have use replacement in setting up the
|
||
mini-batches. The discussion
|
||
<a class="reference external" href="https://sebastianraschka.com/faq/docs/sgd-methods.html">here</a> may be
|
||
useful. More material will be added later.</p>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="momentum-based-gd">
|
||
<h2><span class="section-number">7.8. </span>Momentum based GD<a class="headerlink" href="#momentum-based-gd" title="Permalink to this headline">¶</a></h2>
|
||
<p>The stochastic gradient descent (SGD) is almost always used with a
|
||
<em>momentum</em> or inertia term that serves as a memory of the direction we
|
||
are moving in parameter space. This is typically implemented as
|
||
follows</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{v}_{t}=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber
|
||
\]</div>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto1"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\boldsymbol{\theta}_{t+1}= \boldsymbol{\theta}_t -\mathbf{v}_{t},
|
||
\label{_auto1} \tag{2}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where we have introduced a momentum parameter <span class="math notranslate nohighlight">\(\gamma\)</span>, with
|
||
<span class="math notranslate nohighlight">\(0\le\gamma\le 1\)</span>, and for brevity we dropped the explicit notation to
|
||
indicate the gradient is to be taken over a different mini-batch at
|
||
each step. We call this algorithm gradient descent with momentum
|
||
(GDM). From these equations, it is clear that <span class="math notranslate nohighlight">\(\mathbf{v}_t\)</span> is a
|
||
running average of recently encountered gradients and
|
||
<span class="math notranslate nohighlight">\((1-\gamma)^{-1}\)</span> sets the characteristic time scale for the memory
|
||
used in the averaging procedure. Consistent with this, when
|
||
<span class="math notranslate nohighlight">\(\gamma=0\)</span>, this just reduces down to ordinary SGD as discussed
|
||
earlier. An equivalent way of writing the updates is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
|
||
\]</div>
|
||
<p>where we have defined <span class="math notranslate nohighlight">\(\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}\)</span>.</p>
|
||
<p>Let us try to get more intuition from these equations. It is helpful
|
||
to consider a simple physical analogy with a particle of mass <span class="math notranslate nohighlight">\(m\)</span>
|
||
moving in a viscous medium with drag coefficient <span class="math notranslate nohighlight">\(\mu\)</span> and potential
|
||
<span class="math notranslate nohighlight">\(E(\mathbf{w})\)</span>. If we denote the particle’s position by <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>,
|
||
then its motion is described by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
|
||
\]</div>
|
||
<p>We can discretize this equation in the usual way to get</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
|
||
\]</div>
|
||
<p>Rearranging this equation, we can rewrite this as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
|
||
\]</div>
|
||
<p>Notice that this equation is identical to previous one if we identify
|
||
the position of the particle, <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>, with the parameters
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span>. This allows us to identify the momentum
|
||
parameter and learning rate with the mass of the particle and the
|
||
viscous drag as:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
|
||
\]</div>
|
||
<p>Thus, as the name suggests, the momentum parameter is proportional to
|
||
the mass of the particle and effectively provides inertia.
|
||
Furthermore, in the large viscosity/small learning rate limit, our
|
||
memory time scales as <span class="math notranslate nohighlight">\((1-\gamma)^{-1} \approx m/(\mu \Delta t)\)</span>.</p>
|
||
<p>Why is momentum useful? SGD momentum helps the gradient descent
|
||
algorithm gain speed in directions with persistent but small gradients
|
||
even in the presence of stochasticity, while suppressing oscillations
|
||
in high-curvature directions. This becomes especially important in
|
||
situations where the landscape is shallow and flat in some directions
|
||
and narrow and steep in others. It has been argued that first-order
|
||
methods (with appropriate initial conditions) can perform comparable
|
||
to more expensive second order methods, especially in the context of
|
||
complex deep learning models.</p>
|
||
<p>These beneficial properties of momentum can sometimes become even more
|
||
pronounced by using a slight modification of the classical momentum
|
||
algorithm called Nesterov Accelerated Gradient (NAG).</p>
|
||
<p>In the NAG algorithm, rather than calculating the gradient at the
|
||
current parameters, <span class="math notranslate nohighlight">\(\nabla_\theta E(\boldsymbol{\theta}_t)\)</span>, one
|
||
calculates the gradient at the expected value of the parameters given
|
||
our current momentum, <span class="math notranslate nohighlight">\(\nabla_\theta E(\boldsymbol{\theta}_t +\gamma
|
||
\mathbf{v}_{t-1})\)</span>. This yields the NAG update rule</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{v}_{t}=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber
|
||
\]</div>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto2"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\boldsymbol{\theta}_{t+1}= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
|
||
\label{_auto2} \tag{3}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of <span class="math notranslate nohighlight">\(\gamma\)</span>.</p>
|
||
<p>In stochastic gradient descent, with and without momentum, we still
|
||
have to specify a schedule for tuning the learning rates <span class="math notranslate nohighlight">\(\eta_t\)</span>
|
||
as a function of time. As discussed in the context of Newton’s
|
||
method, this presents a number of dilemmas. The learning rate is
|
||
limited by the steepest direction which can change depending on the
|
||
current position in the landscape. To circumvent this problem, ideally
|
||
our algorithm would keep track of curvature and take large steps in
|
||
shallow, flat directions and small steps in steep, narrow directions.
|
||
Second-order methods accomplish this by calculating or approximating
|
||
the Hessian and normalizing the learning rate by the
|
||
curvature. However, this is very computationally expensive for
|
||
extremely large models. Ideally, we would like to be able to
|
||
adaptively change the step size to match the landscape without paying
|
||
the steep computational price of calculating or approximating
|
||
Hessians.</p>
|
||
<p>Recently, a number of methods have been introduced that accomplish
|
||
this by tracking not only the gradient, but also the second moment of
|
||
the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and
|
||
ADAM.</p>
|
||
<div class="section" id="rms-prop">
|
||
<h3><span class="section-number">7.8.1. </span>RMS prop<a class="headerlink" href="#rms-prop" title="Permalink to this headline">¶</a></h3>
|
||
<p>In RMS prop, in addition to keeping a running average of the first
|
||
moment of the gradient, we also keep track of the second moment
|
||
denoted by <span class="math notranslate nohighlight">\(\mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2]\)</span>. The update rule
|
||
for RMS prop is given by</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto3"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\mathbf{g}_t = \nabla_\theta E(\boldsymbol{\theta})
|
||
\label{_auto3} \tag{4}
|
||
\end{equation}
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{s}_t =\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\theta}_{t+1}=\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\beta\)</span> controls the averaging time of the second moment and is
|
||
typically taken to be about <span class="math notranslate nohighlight">\(\beta=0.9\)</span>, <span class="math notranslate nohighlight">\(\eta_t\)</span> is a learning rate
|
||
typically chosen to be <span class="math notranslate nohighlight">\(10^{-3}\)</span>, and <span class="math notranslate nohighlight">\(\epsilon\sim 10^{-8} \)</span> is a
|
||
small regularization constant to prevent divergences. Multiplication
|
||
and division by vectors is understood as an element-wise operation. It
|
||
is clear from this formula that the learning rate is reduced in
|
||
directions where the norm of the gradient is consistently large. This
|
||
greatly speeds up the convergence by allowing us to use a larger
|
||
learning rate for flat directions.</p>
|
||
</div>
|
||
<div class="section" id="adam-optimizer">
|
||
<h3><span class="section-number">7.8.2. </span>ADAM optimizer<a class="headerlink" href="#adam-optimizer" title="Permalink to this headline">¶</a></h3>
|
||
<p>A related algorithm is the ADAM optimizer. In ADAM, we keep a running
|
||
average of both the first and second moment of the gradient and use
|
||
this information to adaptively change the learning rate for different
|
||
parameters. In addition to keeping a running average of the first and
|
||
second moments of the gradient
|
||
(i.e. <span class="math notranslate nohighlight">\(\mathbf{m}_t=\mathbb{E}[\mathbf{g}_t]\)</span> and
|
||
<span class="math notranslate nohighlight">\(\mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t]\)</span>, respectively), ADAM
|
||
performs an additional bias correction to account for the fact that we
|
||
are estimating the first two moments of the gradient using a running
|
||
average (denoted by the hats in the update rule below). The update
|
||
rule for ADAM is given by (where multiplication and division are once
|
||
again understood to be element-wise operations below)</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto4"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\mathbf{g}_t = \nabla_\theta E(\boldsymbol{\theta})
|
||
\label{_auto4} \tag{5}
|
||
\end{equation}
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{m}_t = \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathbf{s}_t =\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\mathbf{m}}_t={\mathbf{m}_t \over 1-\beta_1^t} \nonumber
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\mathbf{s}}_t ={\mathbf{s}_t \over1-\beta_2^t} \nonumber
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\theta}_{t+1}=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber
|
||
\]</div>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto5"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\label{_auto5} \tag{6}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\beta_1\)</span> and <span class="math notranslate nohighlight">\(\beta_2\)</span> set the memory lifetime of the first and
|
||
second moment and are typically taken to be <span class="math notranslate nohighlight">\(0.9\)</span> and <span class="math notranslate nohighlight">\(0.99\)</span>
|
||
respectively, and <span class="math notranslate nohighlight">\(\eta\)</span> and <span class="math notranslate nohighlight">\(\epsilon\)</span> are identical to RMSprop.</p>
|
||
<p>Like in RMSprop, the effective step size of a parameter depends on the
|
||
magnitude of its gradient squared. To understand this better, let us
|
||
rewrite this expression in terms of the variance
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t -
|
||
(\boldsymbol{\mathbf{m}}_t)^2\)</span>. Consider a single parameter <span class="math notranslate nohighlight">\(\theta_t\)</span>. The
|
||
update rule for this parameter is given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
|
||
\]</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="practical-tips">
|
||
<h2><span class="section-number">7.9. </span>Practical tips<a class="headerlink" href="#practical-tips" title="Permalink to this headline">¶</a></h2>
|
||
<ul class="simple">
|
||
<li><p><strong>Randomize the data when making mini-batches</strong>. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.</p></li>
|
||
<li><p><strong>Transform your inputs</strong>. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.</p></li>
|
||
<li><p><strong>Monitor the out-of-sample performance.</strong> Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This <em>early stopping</em> significantly improves performance in many settings.</p></li>
|
||
<li><p><strong>Adaptive optimization methods don’t always have good generalization.</strong> Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.</p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="automatic-differentiation">
|
||
<h2><span class="section-number">7.10. </span>Automatic differentiation<a class="headerlink" href="#automatic-differentiation" title="Permalink to this headline">¶</a></h2>
|
||
<p><a class="reference external" href="https://en.wikipedia.org/wiki/Automatic_differentiation">Automatic differentiation (AD)</a>,
|
||
also called algorithmic
|
||
differentiation or computational differentiation,is a set of
|
||
techniques to numerically evaluate the derivative of a function
|
||
specified by a computer program. AD exploits the fact that every
|
||
computer program, no matter how complicated, executes a sequence of
|
||
elementary arithmetic operations (addition, subtraction,
|
||
multiplication, division, etc.) and elementary functions (exp, log,
|
||
sin, cos, etc.). By applying the chain rule repeatedly to these
|
||
operations, derivatives of arbitrary order can be computed
|
||
automatically, accurately to working precision, and using at most a
|
||
small constant factor more arithmetic operations than the original
|
||
program.</p>
|
||
<p>Automatic differentiation is neither:</p>
|
||
<ul class="simple">
|
||
<li><p>Symbolic differentiation, nor</p></li>
|
||
<li><p>Numerical differentiation (the method of finite differences).</p></li>
|
||
</ul>
|
||
<p>Symbolic differentiation can lead to inefficient code and faces the
|
||
difficulty of converting a computer program into a single expression,
|
||
while numerical differentiation can introduce round-off errors in the
|
||
discretization process and cancellation</p>
|
||
<p>Python has tools for so-called <strong>automatic differentiation</strong>.
|
||
Consider the following example</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(x) = \sin\left(2\pi x + x^2\right)
|
||
\]</div>
|
||
<p>which has the following derivative</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
|
||
\]</div>
|
||
<p>Using <strong>autograd</strong> we have</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="c1"># To do elementwise differentiation:</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">elementwise_grad</span> <span class="k">as</span> <span class="n">egrad</span>
|
||
|
||
<span class="c1"># To plot:</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</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="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">f_grad_analytic</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">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Do the comparison:</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1000</span><span class="p">)</span>
|
||
|
||
<span class="n">f_grad</span> <span class="o">=</span> <span class="n">egrad</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
|
||
|
||
<span class="n">computed</span> <span class="o">=</span> <span class="n">f_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">analytic</span> <span class="o">=</span> <span class="n">f_grad_analytic</span><span class="p">(</span><span class="n">x</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="s1">'Derivative computed from Autograd compared with the analytical derivative'</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">computed</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">'autograd'</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">analytic</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">'analytic'</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="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="s1">'y'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</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="nb">print</span><span class="p">(</span><span class="s2">"The max absolute difference is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">computed</span> <span class="o">-</span> <span class="n">analytic</span><span class="p">))))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here we
|
||
experiment with what kind of functions Autograd is capable
|
||
of finding the gradient of. The following Python functions are just
|
||
meant to illustrate what Autograd can do, but please feel free to
|
||
experiment with other, possibly more complicated, functions as well.</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="k">def</span> <span class="nf">f1</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span>
|
||
|
||
<span class="n">f1_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Remember to send in float as argument to the computed gradient from Autograd!</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
|
||
<span class="c1"># See the evaluated gradient at a using autograd:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The gradient of f1 evaluated at a = </span><span class="si">%g</span><span class="s2"> using autograd is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">f1_grad</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
|
||
|
||
<span class="c1"># Compare with the analytical derivative, that is f1'(x) = 3*x**2 </span>
|
||
<span class="n">grad_analytical</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The gradient of f1 evaluated at a = </span><span class="si">%g</span><span class="s2"> by finding the analytic expression is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">grad_analytical</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>To differentiate with respect to two (or more) arguments of a Python
|
||
function, Autograd need to know at which variable the function if
|
||
being differentiated with respect to.</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f2</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">3</span><span class="o">*</span><span class="n">x1</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x2</span><span class="o">*</span><span class="p">(</span><span class="n">x1</span> <span class="o">-</span> <span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
|
||
|
||
<span class="c1"># By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1</span>
|
||
<span class="n">f2_grad_x1</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad</span>
|
||
<span class="n">f2_grad_x2</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">x1</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<span class="n">x2</span> <span class="o">=</span> <span class="mf">3.0</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Evaluating at x1 = </span><span class="si">%g</span><span class="s2">, x2 = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"-"</span><span class="o">*</span><span class="mi">30</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Compare with the analytical derivatives:</span>
|
||
|
||
<span class="c1"># Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:</span>
|
||
<span class="n">f2_grad_x1_analytical</span> <span class="o">=</span> <span class="mi">9</span><span class="o">*</span><span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x2</span>
|
||
|
||
<span class="c1"># Derivative of f2 w.r.t x2 is: x1 - 5:</span>
|
||
<span class="n">f2_grad_x2_analytical</span> <span class="o">=</span> <span class="n">x1</span> <span class="o">-</span> <span class="mi">5</span>
|
||
|
||
<span class="c1"># See the evaluated derivations:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The derivative of f2 w.r.t x1: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x1</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The analytical derivative of f2 w.r.t x1: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x1</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
|
||
|
||
<span class="nb">print</span><span class="p">()</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The derivative of f2 w.r.t x2: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x2</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The analytical derivative of f2 w.r.t x2: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span> <span class="n">f2_grad_x2</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">)</span> <span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f3</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> <span class="c1"># Assumes x is an array of length 5 or higher</span>
|
||
<span class="k">return</span> <span class="mi">2</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">3</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">5</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">+</span> <span class="mi">11</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span>
|
||
|
||
<span class="n">f3_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f3</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">linspace</span><span class="p">(</span><span class="mi">0</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="c1"># Print the computed gradient:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The computed gradient of f3 is: "</span><span class="p">,</span> <span class="n">f3_grad</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
|
||
|
||
<span class="c1"># The analytical gradient is: (2, 3, 5, 7, 22*x[4])</span>
|
||
<span class="n">f3_grad_analytical</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">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">22</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">4</span><span class="p">]])</span>
|
||
|
||
<span class="c1"># Print the analytical gradient:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The analytical gradient of f3 is: "</span><span class="p">,</span> <span class="n">f3_grad_analytical</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Note that in this case, when sending an array as input argument, the
|
||
output from Autograd is another array. This is the true gradient of
|
||
the function, as opposed to the function in the previous example. By
|
||
using arrays to represent the variables, the output from Autograd
|
||
might be easier to work with, as the output is closer to what one
|
||
could expect form a gradient-evaluting 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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f4</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">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">f4_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f4</span><span class="p">)</span>
|
||
|
||
<span class="n">x</span> <span class="o">=</span> <span class="mf">2.7</span>
|
||
|
||
<span class="c1"># Print the computed derivative:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The computed derivative of f4 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f4_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
|
||
|
||
<span class="c1"># The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi</span>
|
||
<span class="n">f4_grad_analytical</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">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</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">pi</span>
|
||
|
||
<span class="c1"># Print the analytical gradient:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The analytical gradient of f4 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f4_grad_analytical</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f5</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">if</span> <span class="n">x</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="k">return</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="mi">1</span>
|
||
|
||
<span class="n">f5_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f5</span><span class="p">)</span>
|
||
|
||
<span class="n">x</span> <span class="o">=</span> <span class="mf">2.7</span>
|
||
|
||
<span class="c1"># Print the computed derivative:</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The computed derivative of f5 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f5_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f6_for</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="n">val</span> <span class="o">=</span> <span class="mi">0</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">10</span><span class="p">):</span>
|
||
<span class="n">val</span> <span class="o">=</span> <span class="n">val</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span>
|
||
<span class="k">return</span> <span class="n">val</span>
|
||
|
||
<span class="k">def</span> <span class="nf">f6_while</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="n">val</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="k">while</span> <span class="n">i</span> <span class="o"><</span> <span class="mi">10</span><span class="p">:</span>
|
||
<span class="n">val</span> <span class="o">=</span> <span class="n">val</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span>
|
||
<span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
|
||
<span class="k">return</span> <span class="n">val</span>
|
||
|
||
<span class="n">f6_for_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f6_for</span><span class="p">)</span>
|
||
<span class="n">f6_while_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f6_while</span><span class="p">)</span>
|
||
|
||
<span class="n">x</span> <span class="o">=</span> <span class="mf">0.5</span>
|
||
|
||
<span class="c1"># Print the computed derivaties of f6_for and f6_while</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The computed derivative of f6_for at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f6_for_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The computed derivative of f6_while at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f6_while_grad</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="c1"># Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9</span>
|
||
<span class="c1"># The analytical derivative is: sum(i*x**(i-1)) </span>
|
||
<span class="n">f6_grad_analytical</span> <span class="o">=</span> <span class="mi">0</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">10</span><span class="p">):</span>
|
||
<span class="n">f6_grad_analytical</span> <span class="o">+=</span> <span class="n">i</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The analytical derivative of f6 at x = </span><span class="si">%g</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">f6_grad_analytical</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="k">def</span> <span class="nf">f7</span><span class="p">(</span><span class="n">n</span><span class="p">):</span> <span class="c1"># Assume that n is an integer</span>
|
||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="mi">1</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="n">n</span><span class="o">*</span><span class="n">f7</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">f7_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f7</span><span class="p">)</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mf">2.0</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The computed derivative of f7 at n = </span><span class="si">%d</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">f7_grad</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
|
||
|
||
<span class="c1"># The function f7 is an implementation of the factorial of n.</span>
|
||
<span class="c1"># By using the product rule, one can find that the derivative is:</span>
|
||
|
||
<span class="n">f7_grad_analytical</span> <span class="o">=</span> <span class="mi">0</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="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">tmp</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="k">if</span> <span class="n">k</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
|
||
<span class="n">tmp</span> <span class="o">*=</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
|
||
<span class="n">f7_grad_analytical</span> <span class="o">+=</span> <span class="n">tmp</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The analytical derivative of f7 at n = </span><span class="si">%d</span><span class="s2"> is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">f7_grad_analytical</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.</p>
|
||
<p>Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.</p>
|
||
<p>Assigning a value to the variable being differentiated with respect to</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">"""</span>
|
||
<span class="sd">import autograd.numpy as np</span>
|
||
<span class="sd">from autograd import grad</span>
|
||
<span class="sd">def f8(x): # Assume x is an array</span>
|
||
<span class="sd"> x[2] = 3</span>
|
||
<span class="sd"> return x*2</span>
|
||
|
||
<span class="sd">f8_grad = grad(f8)</span>
|
||
|
||
<span class="sd">x = 8.4</span>
|
||
|
||
<span class="sd">print("The derivative of f8 is:",f8_grad(x))</span>
|
||
<span class="sd">"""</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here, Autograd tells us that an ‘ArrayBox’ does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f9</span><span class="p">(</span><span class="n">a</span><span class="p">):</span> <span class="c1"># Assume a is an array with 2 elements</span>
|
||
<span class="n">b</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">1.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">])</span>
|
||
<span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
|
||
|
||
<span class="n">f9_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f9</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">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">])</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The derivative of f9 is:"</span><span class="p">,</span><span class="n">f9_grad</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here we are told that the ‘dot’ function does not belong to Autograd’s
|
||
version of a Numpy array. To overcome this, an alternative syntax
|
||
which also computed the dot product can be used:</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">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
<span class="k">def</span> <span class="nf">f9_alternative</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> <span class="c1"># Assume a is an array with 2 elements</span>
|
||
<span class="n">b</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">1.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">])</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">b</span><span class="p">)</span> <span class="c1"># The same as x_1*b_1 + x_2*b_2</span>
|
||
|
||
<span class="n">f9_alternative_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">f9_alternative</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">array</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">])</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The gradient of f9 is:"</span><span class="p">,</span><span class="n">f9_alternative_grad</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
|
||
|
||
<span class="c1"># The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively</span>
|
||
<span class="c1"># w.r.t x is (b_1, b_2).</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>The documentation recommends to avoid inplace operations such as</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">a</span> <span class="o">+=</span> <span class="n">b</span>
|
||
<span class="n">a</span> <span class="o">-=</span> <span class="n">b</span>
|
||
<span class="n">a</span><span class="o">*=</span> <span class="n">b</span>
|
||
<span class="n">a</span> <span class="o">/=</span><span class="n">b</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="replace-or-not">
|
||
<h2><span class="section-number">7.11. </span>Replace or not<a class="headerlink" href="#replace-or-not" title="Permalink to this headline">¶</a></h2>
|
||
<p>In the above code, we have use replacement in setting up the
|
||
mini-batches. The discussion
|
||
<a class="reference external" href="https://sebastianraschka.com/faq/docs/sgd-methods.html">here</a> may be
|
||
useful.</p>
|
||
</div>
|
||
<div class="section" id="using-autograd">
|
||
<h2><span class="section-number">7.12. </span>Using Autograd<a class="headerlink" href="#using-autograd" title="Permalink to this headline">¶</a></h2>
|
||
<p>We conclude the part on optmization by showing how we can make codes
|
||
for linear regression and logistic regression using <strong>autograd</strong>. The
|
||
first example shows results with ordinary leats squares.</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"># Using Autograd to calculate gradients for OLS</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">beta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</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">XT_X</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="sa">f</span><span class="s2">"Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<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>
|
||
<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="c1"># define the gradient</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">)</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="n">training_gradient</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
<span class="n">theta</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="s2">"theta from own gd"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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">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>
|
||
<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>
|
||
<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>
|
||
|
||
<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">'Random numbers '</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="same-code-but-now-with-momentum-gradient-descent">
|
||
<h2><span class="section-number">7.13. </span>Same code but now with momentum gradient descent<a class="headerlink" href="#same-code-but-now-with-momentum-gradient-descent" 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"># Using Autograd to calculate gradients for OLS</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">beta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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="c1">#+np.random.randn(n,1)</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</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">XT_X</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="sa">f</span><span class="s2">"Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<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>
|
||
<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">30</span>
|
||
|
||
<span class="c1"># define the gradient</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">)</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="n">training_gradient</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
<span class="n">theta</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="nb">iter</span><span class="p">,</span><span class="n">gradients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">gradients</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own gd"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Now improve with momentum gradient descent</span>
|
||
<span class="n">change</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<span class="n">delta_momentum</span> <span class="o">=</span> <span class="mf">0.3</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="c1"># calculate gradient</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="n">training_gradient</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
<span class="c1"># calculate update</span>
|
||
<span class="n">new_change</span> <span class="o">=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span><span class="o">+</span><span class="n">delta_momentum</span><span class="o">*</span><span class="n">change</span>
|
||
<span class="c1"># take a step</span>
|
||
<span class="n">theta</span> <span class="o">-=</span> <span class="n">new_change</span>
|
||
<span class="c1"># save the change</span>
|
||
<span class="n">change</span> <span class="o">=</span> <span class="n">new_change</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span><span class="n">gradients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">gradients</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own gd wth momentum"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We note indeed a considerable increase in efficiency here, we less iterations needed.
|
||
However, if we can invert the Hessian matrix, this is the preferred approach, as shown in the example here.</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"># Using Newton's method</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">beta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</span><span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</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="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">XT_X</span>
|
||
<span class="c1"># Note that here the Hessian does not depend on the parameters beta</span>
|
||
<span class="n">invH</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">pinv</span><span class="p">(</span><span class="n">H</span><span class="p">)</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="sa">f</span><span class="s2">"Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">"</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">Niterations</span> <span class="o">=</span> <span class="mi">5</span>
|
||
|
||
<span class="c1"># define the gradient</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">)</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="n">training_gradient</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
|
||
<span class="n">beta</span> <span class="o">-=</span> <span class="n">invH</span> <span class="o">@</span> <span class="n">gradients</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span><span class="n">gradients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">gradients</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"beta from own Newton code"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="including-stochastic-gradient-descent-with-autograd">
|
||
<h2><span class="section-number">7.14. </span>Including Stochastic Gradient Descent with Autograd<a class="headerlink" href="#including-stochastic-gradient-descent-with-autograd" title="Permalink to this headline">¶</a></h2>
|
||
<p>In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using <strong>autograd</strong>.</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"># Using Autograd to calculate gradients using SGD</span>
|
||
<span class="c1"># OLS example</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="c1"># Note change from previous example</span>
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</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="mi">2</span><span class="p">)</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</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">XT_X</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="sa">f</span><span class="s2">"Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<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>
|
||
<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="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</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="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
|
||
<span class="n">theta</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="s2">"theta from own gd"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</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">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>
|
||
<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>
|
||
<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>
|
||
|
||
<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">'Random numbers '</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">n_epochs</span> <span class="o">=</span> <span class="mi">50</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">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>
|
||
<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</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">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>
|
||
|
||
<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>
|
||
<span class="c1"># Can you figure out a better way of setting up the contributions to each batch?</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">random_index</span> <span class="o">=</span> <span class="n">M</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="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="n">M</span><span class="p">]</span>
|
||
<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="n">M</span><span class="p">]</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
|
||
<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>
|
||
<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>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own sdg"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here we include momentum in the standard gradient descent approach.</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"># Using Autograd to calculate gradients using SGD</span>
|
||
<span class="c1"># OLS example</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="c1"># Note change from previous example</span>
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</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="mi">2</span><span class="p">)</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</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">XT_X</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="sa">f</span><span class="s2">"Eigenvalues of Hessian Matrix:</span><span class="si">{</span><span class="n">EigValues</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<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>
|
||
<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">100</span>
|
||
|
||
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</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="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
|
||
<span class="n">theta</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="s2">"theta from own gd"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
|
||
|
||
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</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">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>
|
||
<span class="k">def</span> <span class="nf">learning_schedule</span><span class="p">(</span><span class="n">t</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">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>
|
||
|
||
<span class="n">change</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<span class="n">delta_momentum</span> <span class="o">=</span> <span class="mf">0.3</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="n">n_epochs</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">random_index</span> <span class="o">=</span> <span class="n">M</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="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="n">M</span><span class="p">]</span>
|
||
<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="n">M</span><span class="p">]</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
|
||
<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>
|
||
<span class="c1"># calculate update</span>
|
||
<span class="n">new_change</span> <span class="o">=</span> <span class="n">eta</span><span class="o">*</span><span class="n">gradients</span><span class="o">+</span><span class="n">delta_momentum</span><span class="o">*</span><span class="n">change</span>
|
||
<span class="c1"># take a step</span>
|
||
<span class="n">theta</span> <span class="o">-=</span> <span class="n">new_change</span>
|
||
<span class="c1"># save the change</span>
|
||
<span class="n">change</span> <span class="o">=</span> <span class="n">new_change</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own sdg with momentum"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="similar-second-order-function-now-problem-but-now-with-adagrad">
|
||
<h3><span class="section-number">7.14.1. </span>Similar (second order function now) problem but now with AdaGrad<a class="headerlink" href="#similar-second-order-function-now-problem-but-now-with-adagrad" title="Permalink to this headline">¶</a></h3>
|
||
<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"># Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent</span>
|
||
<span class="c1"># OLS example</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="c1"># Note change from previous example</span>
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</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="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</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">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="mf">2.0</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="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="c1"># +np.random.randn(n,1)</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">x</span><span class="o">*</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
|
||
|
||
|
||
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="c1"># Define parameters for Stochastic Gradient Descent</span>
|
||
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</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="c1"># Guess for unknown parameters theta</span>
|
||
<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">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Value for learning rate</span>
|
||
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
<span class="c1"># Including AdaGrad parameter to avoid possible division by zero</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</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="n">n_epochs</span><span class="p">):</span>
|
||
<span class="c1"># The outer product is calculated from scratch for each epoch</span>
|
||
<span class="n">Giter</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</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">random_index</span> <span class="o">=</span> <span class="n">M</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="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="n">M</span><span class="p">]</span>
|
||
<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="n">M</span><span class="p">]</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
|
||
<span class="c1"># Calculate the outer product of the gradients</span>
|
||
<span class="n">Giter</span> <span class="o">+=</span><span class="n">gradients</span> <span class="o">@</span> <span class="n">gradients</span><span class="o">.</span><span class="n">T</span>
|
||
<span class="c1"># Simpler algorithm with only diagonal elements</span>
|
||
<span class="n">Ginverse</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">eta</span><span class="o">/</span><span class="p">(</span><span class="n">delta</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="n">Giter</span><span class="p">)))]</span>
|
||
<span class="c1"># compute update</span>
|
||
<span class="n">update</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">Ginverse</span><span class="p">,</span><span class="n">gradients</span><span class="p">)</span>
|
||
<span class="n">theta</span> <span class="o">-=</span> <span class="n">update</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own AdaGrad"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Running this code we note an almost perfect agreement with the results from matrix inversion.</p>
|
||
<p>Similarly, here is our implementation of RMSprop.</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"># Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent</span>
|
||
<span class="c1"># OLS example</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">autograd.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">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="c1"># Note change from previous example</span>
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">theta</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y</span><span class="o">-</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="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</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">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="mf">2.0</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="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="c1"># +np.random.randn(n,1)</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">x</span><span class="o">*</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="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">pinv</span><span class="p">(</span><span class="n">XT_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="s2">"Own inversion"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta_linreg</span><span class="p">)</span>
|
||
|
||
|
||
<span class="c1"># Note that we request the derivative wrt third argument (theta, 2 here)</span>
|
||
<span class="n">training_gradient</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">CostOLS</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="c1"># Define parameters for Stochastic Gradient Descent</span>
|
||
<span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">50</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="c1"># Guess for unknown parameters theta</span>
|
||
<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">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Value for learning rate</span>
|
||
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
<span class="c1"># Value for parameter rho</span>
|
||
<span class="n">rho</span> <span class="o">=</span> <span class="mf">0.99</span>
|
||
<span class="c1"># Including AdaGrad parameter to avoid possible division by zero</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</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="n">n_epochs</span><span class="p">):</span>
|
||
<span class="n">Giter</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</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">random_index</span> <span class="o">=</span> <span class="n">M</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="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="n">M</span><span class="p">]</span>
|
||
<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="n">M</span><span class="p">]</span>
|
||
<span class="n">gradients</span> <span class="o">=</span> <span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">M</span><span class="p">)</span><span class="o">*</span><span class="n">training_gradient</span><span class="p">(</span><span class="n">yi</span><span class="p">,</span> <span class="n">xi</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
|
||
<span class="c1"># Previous value for the outer product of gradients</span>
|
||
<span class="n">Previous</span> <span class="o">=</span> <span class="n">Giter</span>
|
||
<span class="c1"># Accumulated gradient</span>
|
||
<span class="n">Giter</span> <span class="o">+=</span><span class="n">gradients</span> <span class="o">@</span> <span class="n">gradients</span><span class="o">.</span><span class="n">T</span>
|
||
<span class="c1"># Scaling with rho the new and the previous results</span>
|
||
<span class="n">Gnew</span> <span class="o">=</span> <span class="p">(</span><span class="n">rho</span><span class="o">*</span><span class="n">Previous</span><span class="o">+</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">rho</span><span class="p">)</span><span class="o">*</span><span class="n">Giter</span><span class="p">)</span>
|
||
<span class="c1"># Taking the diagonal only and inverting</span>
|
||
<span class="n">Ginverse</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">eta</span><span class="o">/</span><span class="p">(</span><span class="n">delta</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="n">Gnew</span><span class="p">)))]</span>
|
||
<span class="c1"># Hadamard product</span>
|
||
<span class="n">update</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">Ginverse</span><span class="p">,</span><span class="n">gradients</span><span class="p">)</span>
|
||
<span class="n">theta</span> <span class="o">-=</span> <span class="n">update</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"theta from own RMSprop"</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="introducing-jax">
|
||
<h2><span class="section-number">7.15. </span>Introducing <a class="reference external" href="https://jax.readthedocs.io/en/latest/">JAX</a><a class="headerlink" href="#introducing-jax" title="Permalink to this headline">¶</a></h2>
|
||
<p>Presently, instead of using <strong>autograd</strong>, we recommend using <a class="reference external" href="https://jax.readthedocs.io/en/latest/">JAX</a></p>
|
||
<p><strong>JAX</strong> is Autograd and <a class="reference external" href="https://www.tensorflow.org/xla">XLA (Accelerated Linear Algebra))</a>,
|
||
brought together for high-performance numerical computing and machine learning research.
|
||
It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.</p>
|
||
<p>Here’s a simple example on how you can use <strong>JAX</strong> to compute the derivate of the logistic 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">jax.numpy</span> <span class="k">as</span> <span class="nn">jnp</span>
|
||
<span class="kn">from</span> <span class="nn">jax</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">jit</span><span class="p">,</span> <span class="n">vmap</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sum_logistic</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">+</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)))</span>
|
||
|
||
<span class="n">x_small</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">3.</span><span class="p">)</span>
|
||
<span class="n">derivative_fn</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">sum_logistic</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">derivative_fn</span><span class="p">(</span><span class="n">x_small</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<script type="text/x-thebe-config">
|
||
{
|
||
requestKernel: true,
|
||
binderOptions: {
|
||
repo: "binder-examples/jupyter-stacks-datascience",
|
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ref: "master",
|
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},
|
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codeMirrorConfig: {
|
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theme: "abcdef",
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mode: "python"
|
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},
|
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kernelOptions: {
|
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kernelName: "python3",
|
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path: "./."
|
||
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|
||
predefinedOutput: true
|
||
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|
||
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|
||
<script>kernelName = 'python3'</script>
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