The minimax excess risk optimization (MERO) problem is a new variation of the traditional distributionally robust optimization (DRO) problem, which achieves uniformly low regret across all test distributions under suitable conditions. In this paper, we propose a zeroth-order stochastic mirror descent (ZO-SMD) algorithm available for both smooth and non-smooth MERO to estimate the minimal risk of each distrbution, and finally solve MERO as (non-)smooth stochastic convex-concave (linear) minimax optimization problems. The proposed algorithm is proved to converge at optimal convergence rates of $\mathcal{O}\left(1/\sqrt{t}\right)$ on the estimate of $R_i^*$ and $\mathcal{O}\left(1/\sqrt{t}\right)$ on the optimization error of both smooth and non-smooth MERO. Numerical results show the efficiency of the proposed algorithm.

翻译：极小化极大超额风险优化（MERO）问题是传统分布鲁棒优化（DRO）问题的一种新变体，在适当条件下，它能在所有测试分布上实现一致的低遗憾。本文提出了一种适用于光滑与非光滑MERO的零阶随机镜像下降（ZO-SMD）算法，用于估计每个分布的最小风险，并最终将MERO作为（非）光滑随机凸凹（线性）极小化极大优化问题进行求解。理论证明，所提算法在$R_i^*$的估计上以$\mathcal{O}\left(1/\sqrt{t}\right)$的最优收敛速率收敛，在光滑与非光滑MERO的优化误差上也均以$\mathcal{O}\left(1/\sqrt{t}\right)$的速率收敛。数值结果验证了所提算法的有效性。