About the Cost of Central Privacy in Density Estimation

TL;DR

This paper analyzes the impact of different privacy regimes on non-parametric density estimation, revealing how privacy constraints affect estimator optimality across various smoothness classes and privacy notions.

Contribution

It extends existing results on histogram estimators under privacy, explores the effects of non-constant privacy budgets, and compares classical and concentrated differential privacy in density estimation.

Findings

Histogram estimators are optimal for Lipschitz densities under classical privacy.

Privacy can degrade minimax risk for Sobolev densities when privacy budgets are not constant.

Zero concentrated differential privacy achieves optimal estimation without relaxation.

Abstract

We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of central differential privacy, but also the more recent notion of central concentrated differential privacy. We recover the result of Barber and Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk, and under regular differential privacy, and we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman and Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.