Generalized Smooth Stochastic Variational Inequalities: Almost Sure Convergence and Convergence Rates

TL;DR

This paper develops convergence guarantees for stochastic variational inequality algorithms under a relaxed smoothness condition, broadening applicability to non-monotone operators in machine learning.

Contribution

It introduces the first almost-sure convergence results and unbiased convergence rates for clipped projection and extragradient methods under generalized smoothness assumptions.

Findings

Proves almost-sure convergence without boundedness assumptions.

Establishes unbiased convergence rates for $rac{1}{2}$-smooth operators.

Extends analysis to non-monotone operators in stochastic variational inequalities.

Abstract

This paper focuses on solving a stochastic variational inequality (SVI) problem under relaxed smoothness assumption for a class of structured non-monotone operators. The SVI problem has attracted significant interest in the machine learning community due to its immediate application to adversarial training and multi-agent reinforcement learning. In many such applications, the resulting operators do not satisfy the smoothness assumption. To address this issue, we focus on a weaker generalized smoothness assumption called $\alpha$-symmetric. Under $p$-quasi sharpness and $\alpha$-symmetric assumptions on the operator, we study clipped projection (gradient descent-ascent) and clipped Korpelevich (extragradient) methods. For these clipped methods, we provide the first almost-sure convergence results without making any assumptions on the boundedness of either the stochastic operator or the stochastic samples. We also provide the first in-expectation unbiased convergence rate results for these methods under a relaxed smoothness assumption for $\alpha \leq \frac{1}{2}$.