Jekyll2018-10-11T22:22:13-07:00https://raghavsomani.github.io/Raghav SomaniResearch Fellow @ Microsoft Research Lab - IndiaRaghav Somaniraghavsomani1995@gmail.comNon-asymptotic rate for Random Shuffling for Quadratic functions2018-07-12T00:00:00-07:002018-07-12T00:00:00-07:00https://raghavsomani.github.io/posts/2018/07/blog-post-6<p>This article is in continuation of my <a href="https://raghavsomani.github.io/posts/2018/04/blog-post-4/" target="_blank">previous blog</a>, and discusses about a section of the work by <a href="https://arxiv.org/pdf/1806.10077.pdf" target="_blank">Jeffery Z. HaoChen and Suvrit Sra 2018</a>, in which the authors come up with a non-asymptotic rate of <script type="math/tex">\mathcal{O}\left(\frac{1}{T^2} + \frac{n^3}{T^3} \right)</script> for Random Shuffling Stochastic algorithm which is strictly better than that of SGD. The article talks about the simple case when the objective function is a sum of quadratic functions where with a fixed step-size and after a reasonable number of epochs, we can guarentee a faster rate for Random Shuffling.</p>
<p>The complete PDF post can be viewed <a href="\files\RRQuadratic.pdf" target="_blank">here</a>.</p>Raghav Somaniraghavsomani1995@gmail.comThis article is in continuation of my [previous blog](https://raghavsomani.github.io/posts/2018/04/blog-post-4/), and discusses about a section of the work by [Jeffery Z. HaoChen and Suvrit Sra 2018](https://arxiv.org/pdf/1806.10077.pdf){:target="_blank"}, in which the authors come up with a non-asymptotic rate of $$\mathcal{O}\left(\frac{1}{T^2} + \frac{n^3}{T^3} \right)$$ for Random Shuffling Stochastic algorithm which is strictly better than that of SGD.Bias-Variance Trade-offs for Averaged SGD in Least Mean Squares2018-07-04T00:00:00-07:002018-07-04T00:00:00-07:00https://raghavsomani.github.io/posts/2018/07/blog-post-5<p>This article is on the work by <a href="https://arxiv.org/pdf/1412.0156.pdf" target="_blank">Défossez and Bach 2014</a>, in which the authors develop an operator view point for analyzing Averaged SGD updates to show the Bias-Variance Trade-off and provide tight convergence rates of Least Mean Squared problem.</p>
<p>The complete PDF post can be viewed <a href="\files\BiasVariance.pdf" target="_blank">here</a>.</p>Raghav Somaniraghavsomani1995@gmail.comThis article is on the work by [Défossez and Bach 2014](https://arxiv.org/pdf/1412.0156.pdf){:target="_blank"}, in which the authors develop an operator view point for analyzing Averaged SGD updates to show the Bias-Variance Trade-off and provide tight convergence rates of Least Mean Squared problem.Random Reshuffling converges to a smaller neighborhood than SGD2018-04-01T00:00:00-07:002018-04-01T00:00:00-07:00https://raghavsomani.github.io/posts/2018/04/blog-post-4<p>This article is on the recent work by <a href="https://arxiv.org/pdf/1803.07964.pdf" target="_blank">Ying et. al. 2018</a>, in which the authors show that SGD with Random Reshuffling outperforms independent sampling with replacement by showing that the MSE of the iterates at the end of each epoch is of the order <script type="math/tex">O(\eta^2)</script> for constant step-size <script type="math/tex">\eta</script>. This is a significant improvement compared to the traditional SGD with i.i.d. sampling where the same quantity is of the order <script type="math/tex">O(\eta)</script>.</p>
<p>The complete PDF post can be viewed <a href="\files\SGDvsRR.pdf" target="_blank">here</a>.</p>Raghav Somaniraghavsomani1995@gmail.comThis article is on the recent work by [Ying et. al. 2018](https://arxiv.org/pdf/1803.07964.pdf){:target="_blank"}, in which the authors show that SGD with Random Reshuffling outperforms independent sampling with replacement.Nesterov’s Acceleration2018-03-30T00:00:00-07:002018-03-30T00:00:00-07:00https://raghavsomani.github.io/posts/2018/03/blog-post-3<p>This post contains a summary and survey of the Nesterov’s accelerated gradient descent method and some insightful implications that can be derived from it. We analyze the simple convex quadratic case and have a close look at the dynamics of the error vectors.</p>
<p>Refer the document <a href="\files\nesterov.pdf" target="_blank">here</a>.</p>Raghav Somaniraghavsomani1995@gmail.comThis post contains a summary and survey of the Nesterov’s accelerated gradient descent method and some insightful implications that can be derived from it. We analyze the simple convex quadratic case and have a close look at the dynamics of the error vectors.Some resources to start with Fundamentals of Machine Learning2018-01-06T00:00:00-08:002018-01-06T00:00:00-08:00https://raghavsomani.github.io/posts/2018/01/blog-post-2<p>With a number of courses, books and reading material out there here is a list of some which I personally find useful for building a fundamental understanding in Machine Learning.</p>
<p>Machine Learning at a higher level requires some mathematical prerequisites which are at the heart of it.</p>
<ol>
<li>Learning Theory</li>
<li>Optimization</li>
<li>Statistical learning and high dimensional probability theory</li>
</ol>
<p>Some really nice resources might be the ones below</p>
<ol>
<li>Learning Theory
<ol>
<li><a href="https://www.youtube.com/watch?v=mbyG85GZ0PI&list=PLD63A284B7615313A">Learning from Data - Caltech</a>.</li>
<li>The initial chapters from <a href="https://libgen.pw/download/book/5a1f05453a044650f50e3ec5">Foundations of Machine Learning - Mohri</a>, or Part I from <a href="http://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf">Understanding Machine Learning From Theory to Algorithms - Shai Shalev-Shwartz and Shai Ben-David</a>.</li>
</ol>
</li>
<li>Optimization for Machine Learning
<ol>
<li>Large scale optimization for Machine Learning - Talks by Suvrit Sra - <a href="https://www.youtube.com/watch?v=AYcfpq5hH5g">Part 1</a>, <a href="https://www.youtube.com/watch?v=rNwkvvm2Hes">Part 2</a>, and <a href="https://www.youtube.com/watch?v=YwaWFto9KoQ">Part 3</a> - <a href="http://suvrit.de/teach/msr2015/index.html">Slides</a>.</li>
<li>Convex Optimization literature - <a href="https://www.youtube.com/watch?v=McLq1hEq3UY&list=PL3940DD956CDF0622">Convex Optimization course by Stephen Boyd</a> <a href="http://mlss11.bordeaux.inria.fr/docs/mlss11Bordeaux_Vandenberghe.pdf">Slides</a>, and the classical book on <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.693.855&rep=rep1&type=pdf">Introductory Lectures on Convex Programming - Yuri Nesterov</a>.</li>
<li><a href="https://arxiv.org/pdf/1712.07897.pdf">Non-convex Optimization for Machine Learning - Jain and Kar</a>.</li>
<li><a href="http://suvrit.de/mit/optml++/index.html">OPTML++ page by Suvrit Sra</a>.</li>
</ol>
</li>
<li>Statistical Learning and Probabilistic Machine Lerning
<ol>
<li><a href="https://www.youtube.com/user/dataschool/playlists?sort=dd&view=50&shelf_id=4">Introduction to Statistical Learning - Trevor Hastie and Robert Tibshirani</a> - <a href="http://www-bcf.usc.edu/~gareth/ISL/ISLR\%20First\%20Printing.pdf">Introductory Book</a>, <a href="https://web.stanford.edu/~hastie/Papers/ESLII.pdf">Advanced Book</a>.</li>
<li><a href="http://dsd.future-lab.cn/members/2015nlp/Machine_Learning.pdf">Machine Learning: A Probabilistic Perspective - Kevin P Murphy</a>.</li>
</ol>
</li>
</ol>Raghav Somaniraghavsomani1995@gmail.comWith a number of courses, books and reading material out there here is a list of some which I personally find useful for building a fundamental understanding in Machine Learning.A survey on Large Scale Optimization2017-12-11T00:00:00-08:002017-12-11T00:00:00-08:00https://raghavsomani.github.io/posts/2017/12/blog-post-1<p>A very important aspect of Machine Learning is Optimization, therefore to have the best results one requires fast and scalable methods before one can appreciate a learning model. Such algorithms involve minimization of a class of functions <script type="math/tex">f(\mathbf{x})</script>, that usually do not have a closed form solution, or even if they have, computing them is expensive in both memory and computation time. Here is where iterative methods turn up to be easy and handy. Analyzing such algorithms involve mathematical analysis of both the function to optimize and the algorithm. This post contains a summary and survey of the theoretical understandings of Large Scale Optimization by referring some talks, papers, and lectures that I have come across in the recent. I hope that the insights of the working of these optimization algorithms will allow the reader to appreciate the rich literature of large scale optimization methods.</p>
<p>The complete PDF post can be viewed <a href="\files\largescaleopt.pdf" target="_blank">here</a>.</p>Raghav Somaniraghavsomani1995@gmail.comThis post contains a summary and survey of the theoretical understandings of Large Scale Optimization by referring some talks, papers, and lectures that I have come across in the recent.