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

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 randomness problem is fairly heavy tailed. Additionally, we obtain good (often optimal) dependence on size geometrical parameters Finally, allow random constraints SAA unbounded feasible sets, which also do not seem to have been considered before in literature. proofs combine different ideas potential independent interest: adaptation Talagrand’s “generic chaining” bound sub-Gaussian processes; “localization” from Statistical Learning literature; use standard conditions Optimization (metric regularity, Slater-type conditions) control fluctuations set.