Morten Hjorth-Jensen
416 lines
25 KiB
HTML
416 lines
25 KiB
HTML
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('Same code but now with momentum gradient descent',
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<a class="navbar-brand" href="week37-bs.html">Week 37: Gradient descent methods</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week37-bs001.html#plans-for-week-37-lecture-monday" style="font-size: 80%;">Plans for week 37, lecture Monday</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#readings-and-videos" style="font-size: 80%;">Readings and Videos:</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.html#material-for-lecture-monday-september-8" style="font-size: 80%;">Material for lecture Monday September 8</a></li>
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|
<!-- navigation toc: --> <li><a href="._week37-bs004.html#gradient-descent-and-revisiting-ordinary-least-squares-from-last-week" style="font-size: 80%;">Gradient descent and revisiting Ordinary Least Squares from last week</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs006.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs007.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.html#simple-program" style="font-size: 80%;">Simple program</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs010.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs012.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs014.html#momentum-based-gd" style="font-size: 80%;">Momentum based GD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs015.html#improving-gradient-descent-with-momentum" style="font-size: 80%;">Improving gradient descent with momentum</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs053.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;">Same code but now with momentum gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#overview-video-on-stochastic-gradient-descent-sgd" style="font-size: 80%;">Overview video on Stochastic Gradient Descent (SGD)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#batches-and-mini-batches" style="font-size: 80%;">Batches and mini-batches</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#pros-and-cons" style="font-size: 80%;">Pros and cons</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#convergence-rates" style="font-size: 80%;">Convergence rates</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#accuracy" style="font-size: 80%;">Accuracy</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs022.html#stochastic-gradient-descent-sgd" style="font-size: 80%;">Stochastic Gradient Descent (SGD)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#stochastic-gradient-descent" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs024.html#computation-of-gradients" style="font-size: 80%;">Computation of gradients</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#sgd-example" style="font-size: 80%;">SGD example</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.html#the-gradient-step" style="font-size: 80%;">The gradient step</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs027.html#simple-example-code" style="font-size: 80%;">Simple example code</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs028.html#when-do-we-stop" style="font-size: 80%;">When do we stop?</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#slightly-different-approach" style="font-size: 80%;">Slightly different approach</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs030.html#time-decay-rate" style="font-size: 80%;">Time decay rate</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs031.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;">Code with a Number of Minibatches which varies</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs032.html#replace-or-not" style="font-size: 80%;">Replace or not</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#second-moment-of-the-gradient" style="font-size: 80%;">Second moment of the gradient</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs034.html#challenge-choosing-a-fixed-learning-rate" style="font-size: 80%;">Challenge: Choosing a Fixed Learning Rate</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#motivation-for-adaptive-step-sizes" style="font-size: 80%;">Motivation for Adaptive Step Sizes</a></li>
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|
<!-- navigation toc: --> <li><a href="._week37-bs035.html#adagrad-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;">AdaGrad algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</a></li>
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<!-- navigation toc: --> <li><a href="#derivation-of-the-adagrad-algorithm" style="font-size: 80%;">Derivation of the AdaGrad Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs037.html#adagrad-update-rule-derivation" style="font-size: 80%;">AdaGrad Update Rule Derivation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs038.html#adagrad-properties" style="font-size: 80%;">AdaGrad Properties</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs039.html#rmsprop-adaptive-learning-rates" style="font-size: 80%;">RMSProp: Adaptive Learning Rates</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs039.html#rmsprop-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;">RMSProp algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs040.html#adam-optimizer" style="font-size: 80%;">Adam Optimizer</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs041.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;">"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.html#why-combine-momentum-and-rmsprop" style="font-size: 80%;">Why Combine Momentum and RMSProp?</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs043.html#adam-exponential-moving-averages-moments" style="font-size: 80%;">Adam: Exponential Moving Averages (Moments)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs044.html#adam-bias-correction" style="font-size: 80%;">Adam: Bias Correction</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs045.html#adam-update-rule-derivation" style="font-size: 80%;">Adam: Update Rule Derivation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs046.html#adam-vs-adagrad-and-rmsprop" style="font-size: 80%;">Adam vs. AdaGrad and RMSProp</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs047.html#adaptivity-across-dimensions" style="font-size: 80%;">Adaptivity Across Dimensions</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs047.html#adam-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;">ADAM algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs048.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;">Algorithms and codes for Adagrad, RMSprop and Adam</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs049.html#practical-tips" style="font-size: 80%;">Practical tips</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs050.html#sneaking-in-automatic-differentiation-using-autograd" style="font-size: 80%;">Sneaking in automatic differentiation using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs053.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;">Same code but now with momentum gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs052.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;">Including Stochastic Gradient Descent with Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs053.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;">Same code but now with momentum gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs054.html#but-none-of-these-can-compete-with-newton-s-method" style="font-size: 80%;">But none of these can compete with Newton's method</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs055.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;">Similar (second order function now) problem but now with AdaGrad</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs056.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;">RMSprop for adaptive learning rate with Stochastic Gradient Descent</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs057.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;">And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs058.html#material-for-the-lab-sessions" style="font-size: 80%;">Material for the lab sessions</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs059.html#reminder-on-different-scaling-methods" style="font-size: 80%;">Reminder on different scaling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs060.html#functionality-in-scikit-learn" style="font-size: 80%;">Functionality in Scikit-Learn</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs061.html#more-preprocessing" style="font-size: 80%;">More preprocessing</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs062.html#frequently-used-scaling-functions" style="font-size: 80%;">Frequently used scaling functions</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0036"></a>
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<!-- !split -->
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<h2 id="derivation-of-the-adagrad-algorithm" class="anchor">Derivation of the AdaGrad Algorithm </h2>
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<ol>
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<li> AdaGrad maintains a running sum of squared gradients for each parameter (coordinate)</li>
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<li> Let \( g_t = \nabla C_{i_t}(x_t) \) be the gradient at step \( t \) (or a subgradient for nondifferentiable cases).</li>
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<li> Initialize \( r_0 = 0 \) (an all-zero vector in \( \mathbb{R}^d \)).</li>
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<li> At each iteration \( t \), update the accumulation:</li>
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</ol>
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$$
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r_t = r_{t-1} + g_t \circ g_t,
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$$
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<ol>
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<li> Here \( g_t \circ g_t \) denotes element-wise square of the gradient vector. \( g_t^{(j)} = g_{t-1}^{(j)} + (g_{t,j})^2 \) for each parameter \( j \).</li>
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<li> We can view \( H_t = \mathrm{diag}(r_t) \) as a diagonal matrix of past squared gradients. Initially \( H_0 = 0 \).</li>
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