Smoothing-Enabled Randomized Stochastic Gradient Schemes for Solving Nonconvex Nonsmooth Potential Games under Uncertainty

Abstract: The state of the art in solving nonconvex nonsmooth games under uncertainty remains in its infancy. Existing studies primarily rely on stringent growth conditions or local convexity-like properties, making the development of alternative algorithms desirable. In this work, we study a class of stochastic $N$-player noncooperative games characterized by a potential function. We first consider the nonconvex smooth setting and develop a randomized stochastic gradient (RSG) scheme. The RSG scheme achieves the optimal sample complexity of $\mathcal{O}(N^{2}ε^{-4})$ for reaching a point whose expected residual has norm at most $ε$. Building on this result, we introduce a randomized smoothed RSG (RS-RSG) scheme for solving stochastic potential games afflicted by nonconvexity and nonsmoothness. We show that RS-RSG asymptotically converges to an equilibrium of the smoothed game with sample complexity $\mathcal{O}(L^{4}{\max}n^{3/2}{\max}N^{3}η^{-1}ε^{-4})$, where $η>0$ is the smoothing parameter. Under Lipschitz continuity of the Clarke subdifferentials, we show that the expected residual evaluated at the smoothed equilibrium is $\mathcal{O}(η^{2})$. In addition, we discuss the biased RSG and RS-RSG variants and demonstrate the effectiveness of the biased RS-RSG scheme on a class of stochastic potential hierarchical games where the exact lower-level solution is unavailable in finite time. Collectively, our results provide a new pathway that goes beyond classical conditions for solving stochastic nonconvex nonsmooth games. Some preliminary numerics are also provided.