On the convergence of single-call stochastic extra-gradient methods

TL;DR

This paper analyzes single-call stochastic extra-gradient methods, demonstrating they achieve an optimal convergence rate in smooth deterministic problems and a local rate in certain non-monotone stochastic variational inequalities.

Contribution

It provides a unified analysis showing single-call algorithms retain optimal convergence rates in both deterministic and non-monotone stochastic settings.

Findings

Single-call methods achieve $oldsymbol{ ext{O}(1/t)}$ ergodic convergence in smooth deterministic problems.

Last iterate of stochastic single-call methods converges at $oldsymbol{ ext{O}(1/t)}$ rate in non-monotone variational inequalities.

The analysis extends the understanding of efficiency of single-call algorithms in variational inequality problems.

Abstract

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal $\mathcal{O}(1/t)$ convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning applications), several algorithms have been proposed as surrogates to Extra-Gradient with a \emph{single} oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a $\mathcal{O}(1/t)$ ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, \emph{stochastic} extra-gradient methods still enjoys a $\mathcal{O}(1/t)$ local convergence rate to solutions of \emph{non-monotone} variational inequalities that satisfy a second-order sufficient condition.