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Rim and Sai
TITLE Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods
ABSTRACT
In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent. We prove global non-assymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates (in L2 sense), and dual function values. We also show that the primal iterates converge at an accelerated linear rate in the L1 sense. This is the first time a linear rate is shown for the stochastic heavy ball method (i.e., stochastic gradient descent method with momentum). Under somewhat weaker conditions, we establish a sublinear convergence rate for Cesaro averages of primal iterates. Moreover, we propose a novel concept, which we call stochastic momentum, aimed at decreasing the cost of performing the momentum step. We prove linear convergence of several stochastic methods with stochastic momentum, and show that in some sparse data regimes and for sufficiently small momentum parameters, these methods enjoy better overall complexity than methods with deterministic momentum. Finally, we perform extensive numerical testing on artificial and real datasets, including data coming from average consensus problems.
BIO
Nicolas is a final year PhD student at The University of Edinburgh in School of Mathematics. More specifically he is a member of the Operational Research and Optimization Group (ERGO) under the supervision of Dr. Peter Richtarik.
Before he moved to Edinburgh he spent 4 years in Athens as undergraduate student in department of Mathematics at National and Kapodistrian University of Athens and 1 year as postgraduate student at Imperial College London where he obtained an MSc in Computing (Computational Management Science). His research interests include (but are not limited to): Large Scale Optimization, Machine Learning, Deep Learning, Randomized numerical linear algebra, Randomized and Distributed Algorithms .