Extragradient Method with Variance Reduction for Stochastic Variational Inequalities

Abstract

Abstract. We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudo-monotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, non-uniform variance of the oracle and, also, we do not require any regularization. Alongside the stochastic approximation procedure, we iteratively reduce the variance of the stochastic error. Our method attains the optimal oracle complexity O(1/ǫ 2 ) (up to a logarithmic term) and a faster rate O(1/K) in terms of the mean (quadratic) natural residual and the D-gap function, where K is the number of iterations required for a given tolerance ǫ > 0. Such convergence rate represents an acceleration with respect to the stochastic error. The generated sequence also enjoys a new feature: the sequence is bounded in L p if the stochastic error has finite p-moment. Explicit estimates for the convergence rate, the oracle complexity and the p-moments are given depending on problem parameters and distance of the initial iterate to the solution set. Moreover, sharper constants are possible if the variance is uniform over the solution set or the feasible set. Our results provide new classes of stochastic variational inequalities for which a convergence rate of O(1/K) holds in terms of the mean-squared distance to the solution set. Our analysis includes the distributed solution of pseudo-monotone Cartesian variational inequalities under partial coordination of parameters between users of a network.

A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization

“…The reader is referred to the recent publications [13] and [17] for a discussion of the challenges associated with this problem and its applications (our use of the "≤" relation instead of the common "≥" is only motivated by the easiness to show the conversion to our formulation). We propose to convert problem (1.3) to the nested form (1.1) by defining the lifted gap function f : Ê n × Ê n → Ê as 4) and the function g : Ê n → Ê n × Ê n as g(x) = x, [H(x)] .…”

On variance reduction for stochastic smooth convex optimization with multiplicative noise

“…Their complexity bounds typically grow with the oracle variance σ 2 in (8). See [8,15,58,20,25,26] and references therein. An essential point related to (Q) is if such increased effort in computation per iteration used is worth.…”

“…We review some works besides [20,8]. A variation of Assumption 2 was proposed by Iusem, Jofré, Oliveira and Thompson in [25], but their method is tailored at solving monotone variational inequalities with the extragradient method. Hence, on the class of smooth convex functions, the suboptimal iteration complexity of O(ǫ −1 ) is achieved with the use of an additional proximal step (not required for optimization).…”

“…See e.g. [29,32,24,25,26,5] and references therein. VI is a framework which generalizes unconstrained system of equations and the first order necessary condition of constrained minimization (including a broader class of problems such as equilibrium, saddle-point and complementarity problems).…”

On Stochastic Mirror-prox Algorithms for Stochastic Cartesian Variational Inequalities: Randomized Block Coordinate and Optimal Averaging Schemes

“…Kannan and Shanbhag (see [14,15]) studied almost sure convergence of extragradient algorithms in solving stochastic VIs with pseudo-monotone mappings and derived optimal rate statements under a strong pseudo-monotone condition. Recently, Iusem et al [10] developed an extragradient method with variance reduction for solving stochastic variational inequalities requiring only pseudo-monotonicity. Motivated by the recent developments in extragradient methods and their generalizations, in this paper, we consider SMP methods.…”