While traditional distributionally robust optimization (DRO) aims to minimize the maximal risk over a set of distributions, Agarwal & Zhang (2022) recently proposed a variant that replaces risk with *excess risk*. Compared to DRO, the new formulation—minimax excess risk optimization (MERO) has the advantage of suppressing the effect of heterogeneous noise in different distributions. However, the choice of excess risk leads to a very challenging minimax optimization problem, and currently there exists only an inefficient algorithm for empirical MERO. In this paper, we develop efficient stochastic approximation approaches which directly target MERO. Specifically, we leverage techniques from stochastic convex optimization to estimate the minimal risk of every distribution, and solve MERO as a stochastic convex-concave optimization (SCCO) problem with biased gradients. The presence of bias makes existing theoretical guarantees of SCCO inapplicable, and fortunately, we demonstrate that the bias, caused by the estimation error of the minimal risk, is under-control. Thus, MERO can still be optimized with a nearly optimal convergence rate. Moreover, we investigate a practical scenario where the quantity of samples drawn from each distribution may differ, and propose a stochastic approach that delivers *distribution-dependent* convergence rates.

This paper is concerned with the generalized Nash equilibrium (GNE) seeking problem of noncooperative games in dynamic environments, where the cost function and coupled constraint of each agent are time-varying. In this case, each agent is required to make a decision before obtaining its cost function and a local inequality constraint. The purpose of the addressed problem is to establish a new distributed primary-dual and mirror descent online algorithm with inertia that is capable of seeking the GNE via time-varying communication graphs and has the potential of achieving a low average regret. Then, two information transmission modes and two estimate update strategies are discussed, respectively. Finally, a simulation example is presented to illustrate the effectiveness of the algorithms, and to further compare their performances.