  We develop two notions of instance optimality in differential privacy,
inspired by classical statistical theory: one by defining a local minimax risk
and the other by considering unbiased mechanisms and analogizing the Cramer-Rao
bound, and we show that the local modulus of continuity of the estimand of
interest completely determines these quantities. We also develop a
complementary collection mechanisms, which we term the inverse sensitivity
mechanisms, which are instance optimal (or nearly instance optimal) for a large
class of estimands. Moreover, these mechanisms uniformly outperform the smooth
sensitivity framework on each instance for several function classes of
interest, including real-valued continuous functions. We carefully present two
instantiations of the mechanisms for median and robust regression estimation
with corresponding experiments.
