Some primal-dual theory for subgradient methods for strongly convex optimization

• 成果类型: Article
• 署名作者: Grimmer, Benjamin; Li, Danlin
• 署名单位: Johns Hopkins University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-025-02201-8
• 发表日期: 2025-11
• 页码: 759-788
• 关键词: Subgradient method; strong convexity; convergence rates; Proximal; switching; Non-Lipschitz; convergence-rates

摘要

We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable O(1/T) convergence guarantees in terms of both their classic primal gap and a not previously analyzed dual gap for strongly convex optimization. Consequently, our theory provides these classic methods with simple, optimal stopping criteria and optimality certificates at no added computational cost. Our results apply to a wide range of stepsize selections and to non-Lipschitz ill-conditioned problems where the early iterations of the subgradient method may diverge exponentially quickly (a phenomenon which, to the best of our knowledge, no prior works address). Even in the presence of such undesirable behaviors, our theory still ensures and bounds eventual convergence.