Stochastic Mirror Descent in Variationally Coherent Optimization Problems
Zhengyuan Zhou · Panayotis Mertikopoulos · Nicholas Bambos · Stephen Boyd · Peter W Glynn

Mon Dec 4th 06:30 -- 10:30 PM @ Pacific Ballroom #165 #None
In this paper, we examine a class of non-convex stochastic optimization problems which we call \emph{variationally coherent}, and which properly includes pseudo-/quasi-convex and star-convex optimization problems. To solve such problems, we focus on the widely used \ac{SMD} family of algorithms (which contains stochastic gradient descent as a special case), and we show that the last iterate of \ac{SMD} converges to the problem's solution set with probability $1$. This result contributes to the landscape of non-convex stochastic optimization by clarifying that neither pseudo-/quasi-convexity nor star-convexity is essential for (almost sure) global convergence; rather, variational coherence, a much weaker requirement, suffices. Characterization of convergence rates for the subclass of strongly variationally coherent optimization problems as well as simulation results are also presented.

Author Information

Zhengyuan Zhou (Stanford University)
Panayotis Mertikopoulos (CNRS (French National Center for Scientific Research))
Nicholas Bambos
Stephen Boyd (Stanford University)
Peter W Glynn (Stanford University)

Peter W. Glynn is the Thomas Ford Professor in the Department of Management Science and Engineering (MS&E) at Stanford University, and also holds a courtesy appointment in the Department of Electrical Engineering. He received his Ph.D in Operations Research from Stanford University in 1982. He then joined the faculty of the University of Wisconsin at Madison, where he held a joint appointment between the Industrial Engineering Department and Mathematics Research Center, and courtesy appointments in Computer Science and Mathematics. In 1987, he returned to Stanford, where he joined the Department of Operations Research. He was Director of Stanford's Institute for Computational and Mathematical Engineering from 2006 until 2010 and served as Chair of MS&E from 2011 through 2015. He is a Fellow of INFORMS and a Fellow of the Institute of Mathematical Statistics, and was an IMS Medallion Lecturer in 1995 and INFORMS Markov Lecturer in 2014. He was co-winner of the Outstanding Publication Awards from the INFORMS Simulation Society in 1993, 2008, and 2016, was a co-winner of the Best (Biannual) Publication Award from the INFORMS Applied Probability Society in 2009, and was the co-winner of the John von Neumann Theory Prize from INFORMS in 2010. In 2012, he was elected to the National Academy of Engineering. He was Founding Editor-in-Chief of Stochastic Systems and is currently Editor-in-Chief of Journal of Applied Probability and Advances in Applied Probability. His research interests lie in simulation, computational probability, queueing theory, statistical inference for stochastic processes, and stochastic modeling.

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