Stability and Sample-based Approximations of Composite Stochastic Optimization Problems

TL;DR

This paper investigates the stability of composite risk functionals in stochastic optimization under measure perturbations, analyzing the asymptotic behavior of data-driven estimators like kernels and wavelets, with implications for risk-averse optimization.

Contribution

It introduces new theoretical results on the stability and asymptotic properties of measure-perturbed composite risk functionals using empirical and smoothing estimators.

Findings

Established strong law of large numbers for estimators

Proved consistency and bias reduction potential

Analyzed properties of kernel and wavelet-based estimators

Abstract

Optimization under uncertainty and risk is indispensable in many practical situations. Our paper addresses stability of optimization problems using composite risk functionals which are subjected to measure perturbations. Our main focus is the asymptotic behavior of data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions. We analyze the properties of the new estimators and we establish strong law of large numbers, consistency, and bias reduction potential under fairly general assumptions. Our results are germane to risk-averse optimization and to data science in general.