Stochastic mirror descent dynamics and their convergence in monotone variational inequalities

TL;DR

This paper studies stochastic mirror descent dynamics for monotone variational inequalities, demonstrating global convergence, convergence rates, and exponential concentration of trajectories through stochastic differential equations.

Contribution

It introduces a tunable stochastic differential equation framework for mirror descent, achieving global convergence and detailed probabilistic trajectory behavior analysis.

Findings

Global convergence in the ergodic sense

Estimated average rate of convergence

Exponential concentration of trajectories

Abstract

We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences that respectively pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around this average.