Distributionally Robust Variance Minimization: Tight Variance Bounds over $f$-Divergence Neighborhoods

TL;DR

This paper develops a finite-dimensional convex reformulation for distributionally robust optimization problems that include variance penalties, providing tight bounds on variance within $f$-divergence neighborhoods, enhancing robustness analysis.

Contribution

It introduces a novel convex reformulation for variance-penalized DRO problems over $f$-divergence neighborhoods, extending prior work limited to expected value objectives.

Findings

Reformulation as finite-dimensional convex optimization.

Provides tight bounds on variance under model uncertainty.

Extends DRO framework to variance-penalized objectives.

Abstract

Distributionally robust optimization (DRO) is a widely used framework for optimizing objective functionals in the presence of both randomness and model-form uncertainty. A key step in the practical solution of many DRO problems is a tractable reformulation of the optimization over the chosen model ambiguity set, which is generally infinite dimensional. Previous works have solved this problem in the case where the objective functional is an expected value. In this paper we study objective functionals that are the sum of an expected value and a variance penalty term. We prove that the corresponding variance-penalized DRO problem over an $f$-divergence neighborhood can be reformulated as a finite-dimensional convex optimization problem. This result also provides tight uncertainty quantification bounds on the variance.