Simple and optimal methods for stochastic variational inequalities, I: operator extrapolation

by   Georgios Kotsalis, et al.

In this paper we first present a novel operator extrapolation (OE) method for solving deterministic variational inequality (VI) problems. Similar to the gradient (operator) projection method, OE updates one single search sequence by solving a single projection subproblem in each iteration. We show that OE can achieve the optimal rate of convergence for solving a variety of VI problems in a much simpler way than existing approaches. We then introduce the stochastic operator extrapolation (SOE) method and establish its optimal convergence behavior for solving different stochastic VI problems. In particular, SOE achieves the optimal complexity for solving a fundamental problem, i.e., stochastic smooth and strongly monotone VI, for the first time in the literature. We also present a stochastic block operator extrapolations (SBOE) method to further reduce the iteration cost for the OE method applied to large-scale deterministic VIs with a certain block structure. Numerical experiments have been conducted to demonstrate the potential advantages of the proposed algorithms. In fact, all these algorithms are applied to solve generalized monotone variational inequality (GMVI) problems whose operator is not necessarily monotone. We will also discuss optimal OE-based policy evaluation methods for reinforcement learning in a companion paper.


page 1

page 2

page 3

page 4


Optimistic Dual Extrapolation for Coherent Non-monotone Variational Inequalities

The optimization problems associated with training generative adversaria...

A Single-Timescale Analysis For Stochastic Approximation With Multiple Coupled Sequences

Stochastic approximation (SA) with multiple coupled sequences has found ...

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

Variational inequalities have recently attracted considerable interest i...

A Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusion Problems

We study stochastic monotone inclusion problems, which widely appear in ...

Forward-backward-forward methods with variance reduction for stochastic variational inequalities

We develop a new stochastic algorithm with variance reduction for solvin...

Geometry-Aware Universal Mirror-Prox

Mirror-prox (MP) is a well-known algorithm to solve variational inequali...