Mean-semideviation-based distributionally robust learning with weakly convex losses: convergence rates and finite-sample guarantees

• 成果类型: Article; Early Access
• 署名作者: Zhu, Landi; Gurbuzbalaban, Mert; Ruszczynski, Andrzej
• 署名单位: Rutgers University System; Rutgers University New Brunswick
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02218-z
• 发表日期: 2025
• 关键词: VARIABLE SELECTION; RISK; algorithms; models; optimization

摘要

We consider a distributionally robust stochastic optimization problem where the ambiguity sets are implicitly defined by the dual representation of the mean-semideviation risk measure. Utilizing the specific form of this risk measure, we reformulate the problem as a stochastic two-level composition optimization problem. In this setting, we consider a single time-scale algorithm, involving two versions of the inner function value tracking: linearized tracking of a continuously differentiable loss function with Lipschitz gradients, and SPIDER tracking of a weakly convex loss function. We adopt the squared norm of the gradient of the Moreau envelope as our measure of stationarity and show that the sample complexity of O(epsilon-3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}(\varepsilon <^>{-3})$$\end{document} is possible in both cases, with only the constant larger in the second case. Finally, we demonstrate the performance of our algorithm with a deep learning example and a weakly convex, non-smooth regression example.