Density estimation under local differential privacy and Hellinger loss

Abstract

In this paper, we carry out a piecewise constant estimator of the density for privatised data. We establish a non-asymptotic oracle inequality for the Hellinger loss and deduce that our estimator is adaptive and (almost) rate optimal over a wide range of Besov classes. In particular, we show that a lower bound condition improves the convergence rates. This is in contrast to what happens with the L 2 loss where the rates can differ depending on whether the density is bounded or not.