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

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 the generalized smoothness assumption and consider two well-known stochastic methods with clipping, namely, projection and Korpelevich. 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. Furthermore, we provide the first in-expectation convergence rate results for these methods under a relaxed smoothness assumption.