The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities

TL;DR

This paper investigates the local convergence rate of optimistic mirror descent in stochastic variational inequalities, linking the rate to the local geometry via a new measure called the Legendre exponent.

Contribution

It introduces the Legendre exponent to quantify the local geometry's effect on convergence and characterizes how it influences optimal step-size and convergence rates.

Findings

Legendre exponent measures Bregman divergence growth near solutions.

Optimal step-size policies depend on the Legendre exponent.

Different Bregman functions exhibit distinct convergence behaviors.

Abstract

In this paper, we analyze the local convergence rate of optimistic mirror descent methods in stochastic variational inequalities, a class of optimization problems with important applications to learning theory and machine learning. Our analysis reveals an intricate relation between the algorithm's rate of convergence and the local geometry induced by the method's underlying Bregman function. We quantify this relation by means of the Legendre exponent, a notion that we introduce to measure the growth rate of the Bregman divergence relative to the ambient norm near a solution. We show that this exponent determines both the optimal step-size policy of the algorithm and the optimal rates attained, explaining in this way the differences observed for some popular Bregman functions (Euclidean projection, negative entropy, fractional power, etc.).