Events

Variational Principles for Mirror Descent and Mirror Langevin Dynamics

Speaker: Maxim Raginsky

Location: 60 Fifth Avenue, Room 150

Date: Thursday, March 16, 2023

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.