Optimal locally private estimation under $\ell_{p}$ loss for $1\le p\le 2$

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 $\ell_{p}^{p},p\ge 1.$