  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.
