Optimal locally private estimation under $\ell_p$ loss for $1 \le p \le 2$
Min Ye
Department of Electrical Engineering Princeton University Princeton, NJ, 08544 e-mail: yeemmi@gmail.com
Alexander Barg*
Department of Electrical and Computer Engineering and Institute for Systems Research University of Maryland College Park, MD 20742 e-mail: abarg@umd.edu
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 $\ell_p^p, p \ge 1$.
AMS 2000 subject classifications: 62G05. Keywords and phrases: Minimax estimation, local differential privacy.
Received August 2018.
1. Introduction
This paper continues our work [28]. The context of the problem that we consider is related to a major challenge in the statistical analysis of user data, namely, the conflict between learning accurate statistics and protecting sensitive information about the individuals. As in [28], we rely on a particular formalization
- A. Barg is also with Institute for Problems of Information Transmission (IITP), Russian Academy of Sciences, 127051 Moscow, Russia. His research was partially supported by NSF grants CCF1814487 and CCF1618603.