Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

TL;DR

This paper provides non-asymptotic exponential tail bounds for solutions to sample average approximations of strongly convex stochastic programs in Hilbert spaces, applicable to infinite-dimensional PDE-constrained problems.

Contribution

It establishes tail bounds without requiring compactness of the feasible set, extending analysis to infinite-dimensional stochastic optimization problems.

Findings

Non-asymptotic exponential tail bounds derived

Applicable to infinite-dimensional PDE problems with random inputs

Numerical results confirm theoretical predictions

Abstract

We analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program's solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs. We present numerical results illustrating our theoretical findings.