Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints

• 成果类型: Article
• 署名作者: Oliveira, Roberto, I; Thompson, Philip
• 署名单位: Purdue University System; Purdue University; Purdue University System; Purdue University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01810-x
• 发表日期: 2023
• 页码: 1-48
• 关键词: statistical estimators; sharp minima; error-bounds; complexities; inequalities

摘要

We derive new and improved non-asymptotic deviation inequalities for the sample average approximation (SAA) of an optimization problem. Our results give strong error probability bounds that are sub-Gaussian even when the randomness of the problem is fairly heavy tailed. Additionally, we obtain good (often optimal) dependence on the sample size and geometrical parameters of the problem. Finally, we allow for random constraints on the SAA and unbounded feasible sets, which also do not seem to have been considered before in the non-asymptotic literature. Our proofs combine different ideas of potential independent interest: an adaptation of Talagrand's generic chaining bound for sub-Gaussian processes; localization ideas from the Statistical Learning literature; and the use of standard conditions in Optimization (metric regularity, Slater-type conditions) to control fluctuations of the feasible set.