A distributed stochastic forward-backward-forward self-adaptive algorithm for Cartesian stochastic variational inequalities

In this paper, we consider a Cartesian stochastic variational inequality with a high dimensional solution space. This mathematical formulation captures a wide range of optimization problems including stochastic Nash games and stochastic minimization problems. By combining the advantages of the forward-backward-forward method and the stochastic approximated method, a novel distributed algorithm is developed for addressing this large-scale problem without any kind of monotonicity. A salient feature of the proposed algorithm is to compute two independent queries of a stochastic oracle at each iteration. The main contributions include: (i) The necessary condition imposed on the involved operator is related merely to the Lipschitz continuity, which are quite general. (ii) At each iteration, the suggested algorithm only requires one computation of the projection onto each feasible set, which can be easily evaluated. (iii) The distributed implementation of the stochastic approximation based Armijo-type line search strategy is adopted to weaken the line search condition and define variable adaptive non-monotonic stepsizes, when the Lipschitz constant is unknown. Some theoretical results of the almost sure convergence, the optimal rate statement, and the oracle complexity bound are established with conditions weaker than the conditions of other methods studied in the literature. Finally, preliminary numerical results are presented to show the efficiency and the competitiveness of our algorithm.