Title: Nonlinear functionals estimation under local differential privacy

Abstract: The concept of differential privacy provides a rigorous formalism to randomize data and quantify the amount of privacy. We consider i.i.d.  individuals with outcomes distributed according to the common probability distribution P. This original data is further randomized using a privacy mechanism into observations that the statistician is allowed to use in order to recover information about the distribution P. We will consider local differential privacy where each sample from the original data is privatized on the user’s local machine before its release.
We build privatized samples and nonparametric estimation methods of nonlinear functionals of the probability density and prove their optimality. We show that for the estimation of a quadratic functional, interactive procedures that use previously released private data are faster than the non interactive ones. In both cases we show how to produce privacy mechanisms and estimators adaptive to (free of) the smoothness of the underlying density. We extend these results to non smooth functionals of the density.
This is based on joint work with A. Rohde, L. Steinberger and Y. Issartel.