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

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In this paper, we develop stochastic variance reduced algorithms for solving a class of finite-sum hemivariational inequality (HVI) problem.

• In this HVI problem, the associated function is assumed to be differentiable, and both the vector mapping and the function are of finite-sum structure.
• We propose two algorithms to solve the cases when the vector mapping is either merely monotone or strongly monotone, while the function is assumed to be convex.
• We show how to apply variance reduction in the proposed algorithms when such an HVI problem has a finite-sum structure, and the resulting accelerated gradient complexities can match the best bound established for finite-sum VI problem, as well as the bound given by the direct Katyusha for finite-sum optimization respectively, in terms of the corresponding parameters such as (gradient) Lipschitz constants and the sizes of the finite-sums.