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# Angelika Rohde (University of Freiburg) – “Geometrizing rates of convergence under privacy constraints “

## April 5, 2018, 2:00 pm - 3:15 pm

The Statistical Seminar: Every Monday at 2:00 pm.

Time: 2:00 pm – 3:15 pm
Date: 5th of April 2018 exceptionally on thursday
Place: Room 2002.

Angelika Rohde (University of Freiburg)

Geometrizing rates of convergence under privacy constraints

Abstract:

We study estimation of a functional $\theta(\Pr)$ of an unknown probability distribution $\Pr \in\P$ 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 $\omega_1$ denote the modulus of continuity of the functional $\theta$ 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 $l(\omega_1((n\alpha^2)^{-1/2}))$ to within constants, under regularity conditions that are satisfied, in particular, if $\theta$ is linear and $\P$ is convex. Our results extend the theory developed by Donoho and Liu (1991) to the nowadays highly relevant case of privatized data. Somewhat surprisingly, the difficulty of the estimation problem in the private case is characterized by $\omega_1$, whereas, it is characterized by the Hellinger modulus of continuity if the original data $X_1,\dots, X_n$ 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.

Organizers:
Cristina BUTUCEA, Alexandre TSYBAKOV, Eric MOULINES, Mathieu ROSENBAUM