Worst-case sensitivity

TL;DR

This paper introduces Worst-Case Sensitivity as a measure of how the expected cost in Distributionally Robust Optimization models reacts to small changes in the uncertainty set, unifying various models and providing practical tools for set selection.

Contribution

It formalizes Worst-Case Sensitivity, connects DRO to regularized optimization, and derives explicit formulas for common uncertainty sets, aiding in model design and analysis.

Findings

Worst-Case Sensitivity acts as a generalized deviation measure.

DRO models approximate mean-worst-case sensitivity problems with small uncertainty sets.

Explicit formulas for sensitivity are provided for key uncertainty sets.

Abstract

We introduce the notion of Worst-Case Sensitivity, defined as the worst-case rate of increase in the expected cost of a Distributionally Robust Optimization (DRO) model when the size of the uncertainty set vanishes. We show that worst-case sensitivity is a Generalized Measure of Deviation and that a large class of DRO models are essentially mean-(worst-case) sensitivity problems when uncertainty sets are small, unifying recent results on the relationship between DRO and regularized empirical optimization with worst-case sensitivity playing the role of the regularizer. More generally, DRO solutions can be sensitive to the family and size of the uncertainty set, and reflect the properties of its worst-case sensitivity. We derive closed-form expressions of worst-case sensitivity for well known uncertainty sets including smooth $\phi$-divergence, total variation, "budgeted" uncertainty sets, uncertainty sets corresponding to a convex combination of expected value and CVaR, and the Wasserstein metric. These can be used to select the uncertainty set and its size for a given application.