SILO: Variational Principles for Mirror Descent and Mirror Langevin Dynamics

Mirror descent, introduced by Nemirovsky and Yudin in the 1970s, is a primal-dual convex optimization method that can be tailored to the geometry of the optimization problem at hand through the choice of a strongly convex distance-generating potential function. It arises as a basic primitive in a variety of applications, including large-scale optimization, machine learning, and control. In this talk, based on joint work with Belinda Tzen, Anant Raj, and Francis Bach, I will discuss a variational formulation of mirror descent and of its stochastic variant, mirror Langevin dynamics. The main idea, inspired by classic work of Brezis and Ekeland, is to show that mirror descent emerges as a closed-loop solution for a certain optimal control problem, and the Bellman value function is given by the dual-space Bregman divergence between the initial condition and the global minimizer of the objective function. This formulation has several interesting corollaries and implications, including a form of implicit regularization, which I will discuss.

Bio: 

Maxim Raginsky received the B.S. and M.S. degrees in 2000 and the Ph.D. degree in 2002 from Northwestern University, all in Electrical Engineering. He has held research positions with Northwestern, the University of Illinois at Urbana-Champaign (where he was a Beckman Foundation Postdoctoral Fellow from 2004 to 2007), and Duke University. In 2012, he has returned to the UIUC, where he is currently a Professor and William L. Everett Fellow with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory. He also holds a courtesy appointment with the Department of Computer Science. Prof. Raginsky’s interests cover probability and stochastic processes, deterministic and stochastic control, machine learning, optimization, and information theory. Much of his recent research is motivated by fundamental questions in modeling, learning, and simulation of nonlinear dynamical systems, with applications to advanced electronics, autonomy, and artificial intelligence.