Adaptive pointwise density estimation under local differential privacy

TL;DR

This paper develops adaptive density estimators under local differential privacy constraints, quantifies the privacy-induced deterioration in estimation rates, and proposes methods to optimally tune estimators adaptively.

Contribution

It introduces privacy-aware projection and kernel density estimators with minimax optimal rates and adapts classical bandwidth selection methods for privacy settings.

Findings

Minimax rates deteriorate due to privacy constraints

Proposed adaptive estimators are nearly minimax optimal

Lower bounds confirm the unavoidable impact of privacy on estimation accuracy

Abstract

We consider the estimation of a density at a fixed point under a local differential privacy constraint, where the observations are anonymised before being available for statistical inference. We propose both a privatised version of a projection density estimator as well as a kernel density estimator and derive their minimax rates under a privacy constraint. There is a twofold deterioration of the minimax rates due to the anonymisation, which we show to be unavoidable by providing lower bounds. In both estimation procedures a tuning parameter has to be chosen. We suggest a variant of the classical Goldenshluger-Lepski method for choosing the bandwidth and the cut-off dimension, respectively, and analyse its performance. It provides adaptive minimax-optimal (up to log-factors) estimators. We discuss in detail how the lower and upper bound depend on the privacy constraints, which in turn is reflected by a modification of the adaptive method.