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

• URL: http://arxiv.org/abs/2408.12209v1
• Date: Thu, 22 Aug 2024 08:35:41 GMT
• Title: Zeroth-Order Stochastic Mirror Descent Algorithms for Minimax Excess Risk Optimization
• Authors: Zhihao Gu, Zi Xu,
• Abstract summary: We propose a zeroth-order mirror descent (ZO-SMD) algorithm available for both smooth and non-smooth MERO. The proposed algorithm is proved to converge at optimal convergence rates of $mathcalOleft (1/sqrttright)$ on the estimate of $R_i*$ and $mathcalOleft (1/sqrttright)$ on the optimization error of both smooth and non-smooth MERO.
• Score: 3.802030070947573
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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