Stochastic subgradient methods with guaranteed global stability in nonsmooth nonconvex optimization

• 成果类型: Article; Early Access
• 署名作者: Xiao, Nachuan; Hu, Xiaoyin; Toh, Kim-Chuan
• 署名单位: The Chinese University of Hong Kong, Shenzhen; Shenzhen University; National University of Singapore; National University of Singapore
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02409-2
• 发表日期: 2026-08-04
• 关键词: nonsmooth optimization; Stochastic subgradient methods; Nonconvex Optimization; Global stability; Differential inclusion; dynamical-system; approximations; CONVERGENCE; composite

摘要

In this paper, we focus on providing convergence guarantees for stochastic subgradient methods in minimizing nonsmooth nonconvex functions. We first investigate the global stability of a general framework for stochastic subgradient methods, where the corresponding differential inclusion admits a coercive Lyapunov function. We prove that, for any sequence of sufficiently small stepsizes and approximation parameters, coupled with sufficiently controlled noises, the iterates are uniformly bounded and asymptotically stabilize around the stable set of its corresponding differential inclusion. Moreover, we develop an improved analysis to apply our proposed framework to establish the global stability of a wide range of stochastic subgradient methods, where the corresponding Lyapunov functions are possibly non-coercive. These theoretical results illustrate the promising potential of our proposed framework for establishing the global stability of various stochastic subgradient methods.