Optimal locally private estimation under $\ell_p$ loss for $1\le p\le 2$

TL;DR

This paper proves the asymptotic optimality of a privatization scheme for estimating discrete distributions under local differential privacy constraints across a range of loss functions.

Contribution

It extends previous work by establishing asymptotic optimality of the privatization scheme under all $\, ext{ell}_p$ losses for $1 \, ext{to}\, 2$ and provides a lower bound on the minimax risk for any loss function $\, ext{ell}_p^p$ with $p \, ext{ge}\, 1.$

Findings

The scheme is asymptotically optimal for all $p$ in [1,2].

The minimax risk lower bound applies to any $\, ext{ell}_p^p$ loss with $p \, ext{ge}\, 1.$

The ratio of the scheme's loss to the optimal approaches 1 as sample size increases.

Abstract

We consider the minimax estimation problem of a discrete distribution with support size $k$ under locally differential privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number $\epsilon$ measures the privacy level of a privatization scheme. In our previous work (IEEE Trans. Inform. Theory, 2018), we proposed a family of new privatization schemes and the corresponding estimator. We also proved that our scheme and estimator are order optimal in the regime $e^{\epsilon} \ll k$ under both $\ell_2^2$ (mean square) and $\ell_1$ loss. In this paper, we sharpen this result by showing asymptotic optimality of the proposed scheme under the $\ell_p^p$ loss for all $1\le p\le 2.$ More precisely, we show that for any $p\in[1,2]$ and any $k$ and $\epsilon,$ the ratio between the worst-case $\ell_p^p$ estimation loss of our scheme and the optimal value approaches $1$ as the number of samples tends to infinity. The lower bound on the minimax risk of private estimation that we establish as a part of the proof is valid for any loss function $\ell_p^p, p\ge 1.$