Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces
arXiv:2207.14756 · doi:10.1287/moor.2022.0200
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.
arXiv comment24 pages
References in corpus (7)
- Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
- The Random Feature Model for Input-Output Maps between Banach Spaces
- Single-stage gradient-based stellarator coil design: stochastic optimization
- Adaptive sampling strategies for risk-averse stochastic optimization with constraints
- Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces
- Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces
- Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization