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‎docs/.doctrees/Module3_IntroducingNumpy/AutoDiff.doctree‎

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‎docs/.doctrees/nbsphinx/Module3_IntroducingNumpy/AutoDiff.ipynb‎

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},
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"# Automatic Differentiation"
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]
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"source": [
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"## Introduction to MyGrad"
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"\n",
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},
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"source": [
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"It is important to reiterate that MyGrad *never gives us the actual function* $\\frac{\\mathrm{d}f}{\\mathrm{d}x}$; it only computes the derivative evaluated at a specific input $x=10$."
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"source": [
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"### MyGrad Adds \"Drop-In\" AutoDiff to NumPy\n"
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"\n",
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{
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"source": [
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"## Vectorized Auto-Differentiation"
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"Like NumPy's array, MyGrad's tensor supports [vectorized operations](https://www.pythonlikeyoumeanit.com/Module3_IntroducingNumpy/VectorizedOperations.html), allowing us to evaluate the derivative of a function at multiple points simultaneously.\n",
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},
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"<div class=\"alert alert-info\">"
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"## Visualizing the Derivative\n",
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"execution": {
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"source": [
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"## Seek and Derive"
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"Computers equipped with automatic differentiation libraries can make short work of derivatives that are well-beyond the reach of mere mortals.\n",
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},
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"cell_type": "markdown",
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"cell_type": "markdown",
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"\n",
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"Let's take a simple example.\n",
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"We'll choose the function $f(x) = (x-8)^2$ and the starting point $x=-1.5$.\n",
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"As we search for $x_\\mathrm{min}$ we don't want to make our updates to $x_o$\n",
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"too big, so we will scale our updates by a factor of $3/10$ (which is somewhat haphazardly here).\n",
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"too big, so we will scale our updates by a factor of $3/10$ (the value of which is chosen somewhat haphazardly here).\n",
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"\n",
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"```python\n",
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"# Performing gradient descent on f(x) = (x - 8) ** 2\n",
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"## Reading Comprehension Exercise Solutions"
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]
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"**Auto-differentiation: Solution**"
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‎docs/Module3_IntroducingNumpy/AutoDiff.html‎

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@@ -594,7 +594,7 @@ <h2>Applying Automatic Differentiation: Solving Optimization Problems<a class="h
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<p>How does automatic differentiation help us to solve such a problem? The derivative of a function evaluated at some <span class="math notranslate nohighlight">\(x_o\)</span> tells us the slope of the function – whether it is decreasing or increasing – at <span class="math notranslate nohighlight">\(x_o\)</span>. This is certainly useful information for helping us search for <span class="math notranslate nohighlight">\(x_\mathrm{min}\)</span>: always look in the direction of decreasing slope, until the slope goes to <span class="math notranslate nohighlight">\(0\)</span>.</p>
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<p>We start our search for <span class="math notranslate nohighlight">\(x_{\mathrm{min}}\)</span> by picking a random starting for value for <span class="math notranslate nohighlight">\(x_o\)</span>, use the autodiff library to compute <span class="math notranslate nohighlight">\(\frac{\mathrm{d}f}{\mathrm{d}x}\big|_{x=x_{o}}\)</span> and then use that information to “step” <span class="math notranslate nohighlight">\(x_o\)</span> in the direction that “descends” <span class="math notranslate nohighlight">\(f(x)\)</span>. We repeat this process until we see that <span class="math notranslate nohighlight">\(\frac{\mathrm{d}f}{\mathrm{d}x}\big|_{x=x_{o}} \approx 0\)</span>. It must be noted that this approach towards finding <span class="math notranslate nohighlight">\(x_\mathrm{min}\)</span> is highly limited;
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saddle-points can stop us in our tracks, and we will only be able to find <em>local</em> minima with this strategy. Nonetheless, it is still very useful!</p>
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<p>Let’s take a simple example. We’ll choose the function <span class="math notranslate nohighlight">\(f(x) = (x-8)^2\)</span> and the starting point <span class="math notranslate nohighlight">\(x=-1.5\)</span>. As we search for <span class="math notranslate nohighlight">\(x_\mathrm{min}\)</span> we don’t want to make our updates to <span class="math notranslate nohighlight">\(x_o\)</span> too big, so we will scale our updates by a factor of <span class="math notranslate nohighlight">\(3/10\)</span> (which is somewhat haphazardly here).</p>
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<p>Let’s take a simple example. We’ll choose the function <span class="math notranslate nohighlight">\(f(x) = (x-8)^2\)</span> and the starting point <span class="math notranslate nohighlight">\(x=-1.5\)</span>. As we search for <span class="math notranslate nohighlight">\(x_\mathrm{min}\)</span> we don’t want to make our updates to <span class="math notranslate nohighlight">\(x_o\)</span> too big, so we will scale our updates by a factor of <span class="math notranslate nohighlight">\(3/10\)</span> (the value of which is chosen somewhat haphazardly here).</p>
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<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Performing gradient descent on f(x) = (x - 8) ** 2</span>
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<span class="n">x</span> <span class="o">=</span> <span class="n">mg</span><span class="o">.</span><span class="n">Tensor</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">)</span>
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<span class="n">step_scale</span> <span class="o">=</span> <span class="mf">0.3</span>

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