Privately Learning Smooth Distributions on the Hypercube by Projections

TL;DR

This paper develops a differentially private method for estimating smooth probability densities on high-dimensional hypercubes, extending previous one-dimensional results and introducing an adaptive, privacy-preserving estimator that balances bias and variance.

Contribution

It generalizes existing results to higher dimensions and non-integer smoothness levels, and proposes a data-driven, privacy-aware adaptive estimation strategy using a modified Lepskii method.

Findings

Extends Sobolev density estimation to high dimensions and non-integer smoothness.

Introduces a privacy-aware adaptive projection estimator.

Demonstrates improved bias-variance trade-off under differential privacy constraints.

Abstract

Fueled by the ever-increasing need for statistics that guarantee the privacy of their training sets, this article studies the centrally-private estimation of Sobolev-smooth densities of probability over the hypercube in dimension d. The contributions of this article are two-fold : Firstly, it generalizes the one dimensional results of (Lalanne et al., 2023) to non-integer levels of smoothness and to a high-dimensional setting, which is important for two reasons : it is more suited for modern learning tasks, and it allows understanding the relations between privacy, dimensionality and smoothness, which is a central question with differential privacy. Secondly, this article presents a private strategy of estimation that is data-driven (usually referred to as adaptive in Statistics) in order to privately choose an estimator that achieves a good bias-variance trade-off among a finite family of private projection estimators without prior knowledge of the ground-truth smoothness $\beta$. This is achieved by adapting the Lepskii method for private selection, by adding a new penalization term that makes the estimation privacy-aware.