Geometrizing rates of convergence under local differential privacy constraints

We study the problem of estimating a functional $\theta(\mathbb P)$ of an unknown probability distribution $\mathbb P \in\mathcal P$ in which the original iid sample $X_1,\dots, X_n$ is kept private even from the statistician via an $\alpha$-local differential privacy constraint. Let $\omega_{TV}$ denote the modulus of continuity of the functional $\theta$ over $\mathcal P$, with respect to total variation distance. For a large class of loss functions $l$ and a fixed privacy level $\alpha$, we prove that the privatized minimax risk is equivalent to $l(\omega_{TV}(n^{-1/2}))$ to within constants, under regularity conditions that are satisfied, in particular, if $\theta$ is linear and $\mathcal P$ is convex. Our results complement the theory developed by Donoho and Liu (1991) with the nowadays highly relevant case of privatized data. Somewhat surprisingly, the difficulty of the estimation problem in the private case is characterized by $\omega_{TV}$, whereas, it is characterized by the Hellinger modulus of continuity if the original data $X_1,\dots, X_n$ are available. We also find that for locally private estimation of linear functionals over a convex model a simple sample mean estimator, based on independently and binary privatized observations, always achieves the minimax rate. We further provide a general recipe for choosing the functional parameter in the optimal binary privatization mechanisms and illustrate the general theory in numerous examples. Our theory allows to quantify the price to be paid for local differential privacy in a large class of estimation problems. This price appears to be highly problem specific.