Morten Hjorth-Jensen
432 lines
27 KiB
HTML
432 lines
27 KiB
HTML
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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%;"><b>Plans for week 37, lecture Monday</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#readings-and-videos" style="font-size: 80%;"><b>Readings and Videos:</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.html#material-for-lecture-monday-september-8" style="font-size: 80%;"><b>Material for lecture Monday September 8</b></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%;"><b>Gradient descent and revisiting Ordinary Least Squares from last week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#gradient-descent-example" style="font-size: 80%;"><b>Gradient descent example</b></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%;"><b>The derivative of the cost/loss function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs007.html#the-hessian-matrix" style="font-size: 80%;"><b>The Hessian matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.html#simple-program" style="font-size: 80%;"><b>Simple program</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#gradient-descent-example" style="font-size: 80%;"><b>Gradient Descent Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs010.html#gradient-descent-and-ridge" style="font-size: 80%;"><b>Gradient descent and Ridge</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;"><b>The Hessian matrix for Ridge Regression</b></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%;"><b>Program example for gradient descent with Ridge Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#using-gradient-descent-methods-limitations" style="font-size: 80%;"><b>Using gradient descent methods, limitations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs014.html#momentum-based-gd" style="font-size: 80%;"><b>Momentum based GD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs015.html#improving-gradient-descent-with-momentum" style="font-size: 80%;"><b>Improving gradient descent with momentum</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs056.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></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%;"><b>Overview video on Stochastic Gradient Descent (SGD)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#batches-and-mini-batches" style="font-size: 80%;"><b>Batches and mini-batches</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#pros-and-cons" style="font-size: 80%;"><b>Pros and cons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#convergence-rates" style="font-size: 80%;"><b>Convergence rates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#accuracy" style="font-size: 80%;"><b>Accuracy</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs022.html#stochastic-gradient-descent-sgd" style="font-size: 80%;"><b>Stochastic Gradient Descent (SGD)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#stochastic-gradient-descent" style="font-size: 80%;"><b>Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs024.html#computation-of-gradients" style="font-size: 80%;"><b>Computation of gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#sgd-example" style="font-size: 80%;"><b>SGD example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.html#the-gradient-step" style="font-size: 80%;"><b>The gradient step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs027.html#simple-example-code" style="font-size: 80%;"><b>Simple example code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs028.html#when-do-we-stop" style="font-size: 80%;"><b>When do we stop?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#slightly-different-approach" style="font-size: 80%;"><b>Slightly different approach</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs030.html#time-decay-rate" style="font-size: 80%;"><b>Time decay rate</b></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%;"><b>Code with a Number of Minibatches which varies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs032.html#replace-or-not" style="font-size: 80%;"><b>Replace or not</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week37-bs033.html#sgd-vs-full-batch-gd-convergence-speed-and-memory-comparison" style="font-size: 80%;"><b>SGD vs Full-Batch GD: Convergence Speed and Memory Comparison</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#theoretical-convergence-speed-and-convex-optimization" style="font-size: 80%;"> Theoretical Convergence Speed and convex optimization</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#strongly-convex-case" style="font-size: 80%;"> Strongly Convex Case</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#non-convex-problems" style="font-size: 80%;"> Non-Convex Problems</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs034.html#memory-usage-and-scalability" style="font-size: 80%;"><b>Memory Usage and Scalability</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#empirical-evidence-convergence-time-and-memory-in-practice" style="font-size: 80%;"><b>Empirical Evidence: Convergence Time and Memory in Practice</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#deep-neural-networks" style="font-size: 80%;"> Deep Neural Networks</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#memory-constraints" style="font-size: 80%;"> Memory constraints</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs036.html#second-moment-of-the-gradient" style="font-size: 80%;"><b>Second moment of the gradient</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs037.html#challenge-choosing-a-fixed-learning-rate" style="font-size: 80%;"><b>Challenge: Choosing a Fixed Learning Rate</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs038.html#motivation-for-adaptive-step-sizes" style="font-size: 80%;"><b>Motivation for Adaptive Step Sizes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs038.html#adagrad-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;"><b>AdaGrad algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs039.html#derivation-of-the-adagrad-algorithm" style="font-size: 80%;"><b>Derivation of the AdaGrad Algorithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs040.html#adagrad-update-rule-derivation" style="font-size: 80%;"><b>AdaGrad Update Rule Derivation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs041.html#adagrad-properties" style="font-size: 80%;"><b>AdaGrad Properties</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.html#rmsprop-adaptive-learning-rates" style="font-size: 80%;"><b>RMSProp: Adaptive Learning Rates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.html#rmsprop-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;"><b>RMSProp algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs043.html#adam-optimizer" style="font-size: 80%;"><b>Adam Optimizer</b></a></li>
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<!-- navigation toc: --> <li><a href="#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;"><b>"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs045.html#why-combine-momentum-and-rmsprop" style="font-size: 80%;"><b>Why Combine Momentum and RMSProp?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs046.html#adam-exponential-moving-averages-moments" style="font-size: 80%;"><b>Adam: Exponential Moving Averages (Moments)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs047.html#adam-bias-correction" style="font-size: 80%;"><b>Adam: Bias Correction</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs048.html#adam-update-rule-derivation" style="font-size: 80%;"><b>Adam: Update Rule Derivation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs049.html#adam-vs-adagrad-and-rmsprop" style="font-size: 80%;"><b>Adam vs. AdaGrad and RMSProp</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs050.html#adaptivity-across-dimensions" style="font-size: 80%;"><b>Adaptivity Across Dimensions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs050.html#adam-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;"><b>ADAM algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs051.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;"><b>Algorithms and codes for Adagrad, RMSprop and Adam</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs052.html#practical-tips" style="font-size: 80%;"><b>Practical tips</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs053.html#sneaking-in-automatic-differentiation-using-autograd" style="font-size: 80%;"><b>Sneaking in automatic differentiation using Autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs056.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs055.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;"><b>Including Stochastic Gradient Descent with Autograd</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs056.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;"><b>Same code but now with momentum gradient descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs057.html#but-none-of-these-can-compete-with-newton-s-method" style="font-size: 80%;"><b>But none of these can compete with Newton's method</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs058.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;"><b>Similar (second order function now) problem but now with AdaGrad</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs059.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;"><b>RMSprop for adaptive learning rate with Stochastic Gradient Descent</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs060.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;"><b>And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs061.html#material-for-the-lab-sessions" style="font-size: 80%;"><b>Material for the lab sessions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs062.html#reminder-on-different-scaling-methods" style="font-size: 80%;"><b>Reminder on different scaling methods</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs063.html#functionality-in-scikit-learn" style="font-size: 80%;"><b>Functionality in Scikit-Learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs064.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs065.html#frequently-used-scaling-functions" style="font-size: 80%;"><b>Frequently used scaling functions</b></a></li>
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<a name="part0044"></a>
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<!-- !split -->
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<h2 id="adam-optimizer-https-arxiv-org-abs-1412-6980" class="anchor"><a href="https://arxiv.org/abs/1412.6980" target="_self">ADAM optimizer</a> </h2>
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<p>In <a href="https://arxiv.org/abs/1412.6980" target="_self">ADAM</a>, we keep a running average of
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both the first and second moment of the gradient and use this
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information to adaptively change the learning rate for different
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parameters. The method is efficient when working with large
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problems involving lots data and/or parameters. It is a combination of the
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gradient descent with momentum algorithm and the RMSprop algorithm
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discussed above.
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