Distributed Generalized Distributionally Robust Equilibrium Seeking for Dynamical Games Under Unknown Time-Varying Interference

This paper investigates a distributed generalized Nash equilibrium-seeking problem in stochastic dynamical systems, focusing on two key challenges: 1) nonlinear coupled constraints and nonlinear dynamics, and 2) nonconvex objectives influenced by disturbances with unknown time-varying distributions. To address these challenges, a distributionally robust game framework with an exact penalty is proposed. We introduce a first-order equilibrium concept suitable for nonconvex-nons-mooth settings and ensure finite-sample guarantees. Furthermore, a distributed zeroth-order feedback algorithm is proposed to solve the problem. This algorithm utilizes gradient estimators for the objective functions and subgradient estimators for the exact penalty terms. We provide a detailed analysis of the relationship between communication errors and the dynamic energy of the system, along with an expected upper bound for the zeroth-order gradient estimation. Our findings indicate that the expectation of the time-accumulated regret grows at a sublinear rate. Furthermore, as the distribution stabilizes, we show that the empirical distribution converges with $O(1)$ sampling complexity.