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

TL;DR

This paper develops new non-asymptotic deviation bounds for the sample average approximation in optimization, accommodating heavy-tailed randomness, random constraints, and unbounded feasible sets, with strong error guarantees.

Contribution

It introduces improved deviation inequalities for SAA under weak assumptions, including heavy tails and random constraints, with near-optimal dependence on sample size and problem geometry.

Findings

Provides sub-Gaussian deviation bounds for heavy-tailed data

Handles random constraints and unbounded feasible sets

Achieves near-optimal dependence on sample size

Abstract

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.