Zeroth-Order Stochastic Mirror Descent Algorithms for Minimax Excess Risk Optimization

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.