Smoothing-Enabled Randomized Stochastic Gradient Schemes for Solving Nonconvex Nonsmooth Potential Games under Uncertainty
Zhuoyu Xiao
TL;DR
A randomized smoothed RSG (RS-RSG) scheme for solving stochastic potential games afflicted by nonconvexity and nonsmoothness is introduced and it is shown that RS-RSG asymptotically converges to an equilibrium of the smoothed game with sample complexity.
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.