General Feasibility Bounds for Sample Average Approximation via Vapnik-Chervonenkis Dimension

TL;DR

This paper establishes explicit feasibility bounds for sample average approximation in stochastic optimization using VC dimension, showing infeasibility decreases exponentially with computable rates under minimal assumptions.

Contribution

It introduces a general framework leveraging VC dimension and PAC learning to derive feasibility bounds for SAA without relying on specific problem structures.

Findings

Feasibility bounds decrease exponentially with sample size.

Bounds apply broadly to stochastic optimization problems.

Explicit constants and rates are provided for practical use.

Abstract

We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse assumption. Instead of analyzing problems with specific structures, we utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework that offers explicit feasibility bounds for SAA solutions under minimal structural or distributional assumption. We show that, as long as the hypothesis class formed by the feasbible region has a finite VC dimension, the infeasibility of SAA solutions decreases exponentially with computable rates and explicitly identifiable accompanying constants. We demonstrate how our bounds apply more generally and competitively compared to existing results.