An Accelerated Variance Reduced Extra-Point Approach to Finite-Sum VI and Optimization

Abstract: In this paper, we develop stochastic variance reduced algorithms for solving a class of finite-sum monotone VI, where the operator consists of the sum of finitely many monotone VI mappings and the sum of finitely many monotone gradient mappings. We study the gradient complexities of the proposed algorithms under the settings when the sum of VI mappings is either strongly monotone or merely monotone. Furthermore, we consider the case when each of the VI mapping and gradient mapping is only accessible via noisy stochastic estimators and establish the sample gradient complexity. We demonstrate the application of the proposed algorithms for solving finite-sum convex optimization with finite-sum inequality constraints and develop a zeroth-order approach when only noisy and biased samples of objective/constraint function values are available.