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

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.