ASYMPTOTIC OPTIMALITY IN STOCHASTIC OPTIMIZATION

• 成果类型: Article
• 署名作者: Duchi, John C.; Ruan, Feng
• 署名单位: Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1831
• 发表日期: 2021
• 页码: 21-48
• 关键词: tilt stability; approximation

摘要

We study local complexity measures for stochastic convex optimization problems, providing a local minimax theory analogous to that of Hajek and Le Cam for classical statistical problems. We give complementary optimality results, developing fully online methods that adaptively achieve optimal convergence guarantees. Our results provide function-specific lower bounds and convergence results that make precise a correspondence between statistical difficulty and the geometric notion of tilt-stability from optimization. As part of this development, we show how variants of Nesterov's dual averaging-a stochastic gradient-based procedure-guarantee finite time identification of constraints in optimization problems, while stochastic gradient procedures fail. Additionally, we highlight a gap between problems with linear and nonlinear constraints: standard stochastic-gradient-based procedures are suboptimal even for the simplest nonlinear constraints, necessitating the development of asymptotically optimal Riemannian stochastic gradient methods.