Robustness Measures in Distributionally Robust Optimization

TL;DR

This paper interprets distributionally robust optimization as a tradeoff between performance and robustness, introducing a systematic way to measure and select uncertainty sets based on worst-case sensitivity.

Contribution

It reveals that DRO's regularizer is a robustness measure, providing a new perspective and systematic approach for choosing uncertainty sets in robust decision-making.

Findings

WCS characterizes the sensitivity of expected costs to model deviations.

Varying the uncertainty set size traces a Pareto frontier of performance and robustness.

The approach helps identify instances with high robustness costs and guides system redesign.

Abstract

Distributionally Robust Optimization (DRO) is a worst-case approach to decision making when there is model uncertainty. It is also well known that for certain uncertainty sets, DRO is approximated by a regularized nominal problem. We show that the regularizer is not just a penalty function but the worst-case sensitivity (WCS) of the expected cost with respect to deviations from the nominal model, giving it the interpretation of a robustness measure. This has substantial consequences for robust modeling. It shows that DRO is fundamentally a tradeoff between performance and robustness, where the robustness measure is determined by the uncertainty set. The robustness measure reveals properties of a cost distribution that affect sensitivity to misspecification. This leads to a systematic approach to selecting uncertainty sets. The family of DRO solutions obtained by varying the size of the uncertainty set traces a near Pareto-optimal performance--robustness frontier that can be used to select its size. The frontier identifies problem instances where the price of robustness is high and provides insight into effective ways of redesigning the system to reduce this cost. We derive WCS for a collection of commonly used uncertainty sets, and illustrate these ideas in a number of applications.