Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces

TL;DR

This paper establishes the asymptotic consistency of empirical solutions in nonconvex risk-averse stochastic optimization problems within infinite dimensional spaces, a setting common in control and machine learning.

Contribution

It provides the first consistency analysis for infinite dimensional stochastic optimization problems, leveraging variational convergence techniques.

Findings

Proves consistency of empirical estimators in infinite dimensional spaces.

Demonstrates results for key problem classes in the literature.

Highlights the role of hidden norm compactness in solution sets.

Abstract

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the literature.