Stochastic Variance Reduction for Variational Inequality Methods

Abstract

We propose stochastic variance reduced algorithms for solving convex-concave saddle point problems, monotone variational inequalities, and monotone inclusions. Our framework applies to extragradient, forward-backward-forward, and forward-reflected-backward methods both in Euclidean and Bregman setups. All proposed methods converge in exactly the same setting as their deterministic counterparts and they either match or improve the best-known complexities for solving structured min-max problems. Our results reinforce the correspondence between variance reduction in variational inequalities and minimization. We also illustrate the improvements of our approach with numerical evaluations on matrix games.

Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming

“…We show that the proposed algorithm exhibit linear convergence rate with high probability. To the best of our knowledge, this is the first stochastic algorithm with linear rate for the general standard-form LP problems (2). For unconstrained bilinear problems, our restarted scheme improves the complexity of existing linear convergence of stochastic algorithms [2] by a factor of the condition number.…”

“…However, beyond matrix games, their method requires extra assumptions such as bounded domain and involves a three-loop algorithm. More recently, [2] proposes a stochastic extragradient method with variance reduction for solving variational inequalities. Under Euclidean setting, their method is based on a loopless variant of variance-reduced method [27,31].…”

“…We here assume there is an unbiased stochastic oracle F ξ (z) such that E[F ξ (z)] = F (z) (see Section 4 for examples on how to construct the stochastic oracles). To efficiently solve (2), our key ideas are variance reduction and restarts.…”

“…Variance reduction can usually improve the convergence property of stochastic minimization algorithms [28,17,49]. Recently, [2] extends the variance reduction scheme to EGM for solving monotonic variational inequality (with some ambiguity, we call this algorithm sEGM), which was shown to have O(1/ǫ) convergence rate.…”

“…On the other hand, most of the previous works on stochastic algorithms have sublinear rate. The only exception is [2], where the authors show the linear convergence of SPDHG for solving problems satisfying global metric sub-regularity. Indeed, unconstrained bilinear problems satisfy the global metric sub-regularity, while LP does not satisfy it globally.…”