Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints
arXiv:1705.00822
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.
Accepted at MPA. Added comments in intro and Sec 5.2. Added props. 3-4 in Sec.5.3: concrete case of (Thm 3) when "locally metric regular solution set". This is a analog to Stoch. Opt. of the "local Bernstein condition" from Stat. Learning. E.g.: locally strongly or locally weakly sharp convex problems. (3) Further correction of typos