Complexity guarantees for risk-neutral generalized Nash equilibrium problems

TL;DR

This paper introduces a stochastic variance-reduced gradient method within a distributed scheme to efficiently solve risk-neutral generalized Nash equilibrium problems, providing convergence guarantees and analyzing sample complexity.

Contribution

It develops a novel stochastic algorithm with variance reduction for structured monotone inclusion problems in GNEPs, ensuring convergence and efficiency.

Findings

Algorithm guarantees almost sure convergence.

Sample complexity of psilon-approximate solutions is (psilon^{-3}).

Numerical results demonstrate improved performance on networked Cournot games.

Abstract

In this paper, we address \ac{SGNEP} seeking with risk-neutral agents. Our main contribution lies the development of a stochastic variance-reduced gradient (SVRG) technique, modified to contend with general sample spaces, within a stochastic forward-backward-forward splitting scheme for resolving structured monotone inclusion problems. This stochastic scheme is a double-loop method, in which the mini-batch gradient estimator is computed periodically in the outer loop, while only cheap sampling is required in a frequently activated inner loop, thus achieving significant speed-ups 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 rather mild and imposed in many simulation-based optimization schemes. Under monotone regimes, the expectation of the gap function of an averaged iterate diminishes at a suitable sublinear rate while the sample-complexity of computing an $\epsilon$-solution is provably $\mathcal{O}(\epsilon^{-3})$. A numerical study on a class of networked Cournot games reflects the performance of our proposed algorithm.