Universally Instance-Optimal Mechanisms for Private Statistical Estimation

We consider the problem of instance-optimal statistical estimation under the constraint of differential privacy where mechanisms must adapt to the difficulty of the input dataset. We prove a
new instance specific lower bound using a new divergence and show it characterizes the local minimax optimal rates for private statistical estimation. We propose two new mechanisms that are
universally instance-optimal for general estimation problems up to logarithmic factors. Our first
mechanism, the total variation mechanism, builds on the exponential mechanism with stable approximations of the total variation distance, and is universally instance-optimal in the high privacy regime $\varepsilon \le 1/\sqrt{n}$

† Northwestern University
‡ Stanford University