On the analysis of distributed splitting methods with variance reduction for stochastic generalized Nash equilibrium problems

We focus on problems of stochastic generalized Nash equilibrium seeking with joint constraints and expectation-valued operators. In this work, the stochastic variance-reduced gradient (SVRG) technique is modified to contend with infinite sample space and then, a stochastic forward-backward-forward splitting scheme with variance reduction (DVRSFBF) is proposed for resolving structured monotone inclusion problems. In DVRSFBF, the average gradient is computed periodically in the outer loop, while only cheap sampling is required in the frequently activated inner loop, thus achieving significant speedups when sampling costs cannot be overlooked. The algorithm is fully distributed and it guarantees almost sure convergence under appropriate batch size and strong monotonicity assumptions. Moreover, it exhibits a linear rate with possible biased estimators, which is relatively mild and adopted by many optimization schemes especially for those based on simulations. A numerical study on a class of networked Cournot games reflects the performance of DVRSFBF.