Geometrizing rates of convergence under differential privacy constraints
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We study estimation of a functional θ( P) of an unknown probability distribution P ∈ P in which the original iid sample X_1,..., X_n is kept private even from the statistician via an α-local differential privacy constraint. Let ω_1 denote the modulus of continuity of the functional θ over P, with respect to total variation distance. For a large class of loss functions l, we prove that the privatized minimax risk is equivalent to l(ω_1(n^-1/2)) to within constants, under regularity conditions that are satisfied, in particular, if θ is linear and 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 ω_1, whereas, it is characterized by the Hellinger modulus of continuity if the original data X_1,..., X_n are available. We also provide a general recipe for constructing rate optimal 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.
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