Multivariate density estimation from privatised data: universal consistency and minimax rates

TL;DR

This paper develops a new method for multivariate density estimation under local differential privacy constraints, achieving universal consistency and optimal convergence rates with a privacy-preserving histogram-based estimator.

Contribution

It introduces a novel privatized histogram estimator for density estimation that guarantees universal consistency and derives optimal convergence rates under privacy constraints.

Findings

The estimator is universally consistent in pointwise and L1 sense.

Achieves minimax optimal convergence rates over Lipschitz classes.

Demonstrates the privacy-utility trade-off through simulations.

Abstract

We revisit the classical problem of nonparametric density estimation but impose local differential privacy constraints. Under such constraints, the original multivariate data $X_1,\ldots,X_n \in \mathbb{R}^d$ cannot be directly observed, and all estimators are functions of the randomised output of a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and in this work we propose to add Laplace distributed noise to a discretisation of the location of a vector $X_i$. Based on these randomised data, we design a novel estimator of the density function, which can be viewed as a privatised version of the well-studied histogram density estimator. Our theoretical results include universal pointwise consistency and strong universal $L_1$-consistency. In addition, a convergence rate over classes of Lipschitz functions is derived, which is complemented by a matching minimax lower bound. We illustrate the trade-off between data utility and privacy by means of a small simulation study.