Optimal schemes for discrete distribution estimation under local differential privacy

Abstract

We consider the minimax estimation problem of a discrete distribution with support size k under 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 ∊ measures the privacy level of a privatization scheme. For a given ∊, we want to find the optimal privatization scheme which minimizes the expected estimation loss for the worst-case distribution. Two schemes in the literature provide order optimal performance in the high-privacy regime when ∊ is very close to 0, and in the low-privacy regime when e∊ ≈ k, respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when 1 ≪ e∊ ≪ k. More concretely, we prove that when 3.8 2 metric and 30% under l 1 metric over the existing schemes. We also prove a tight lower bound for the whole region e∊ ≪ k, which implies that our schemes are order optimal in this regime.