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 rate optimal over a wide range of Besov classes (up to possible logarithmic factors). We also get better estimation rates when the density is not only in a Besov class but also bounded away from 0. These rates are optimal within possible log factors. This result is in contrast to what happens with the ${\mathbb{L}}^2$ loss where the privatised minimax rates over Besov classes can be improved in some cases by assuming the target bounded from above.