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 $\epsilon$ measures the privacy level of a privatization scheme. For a given $\epsilon,$ we consider the problem of constructing optimal privatization schemes with $\epsilon$-privacy level, i.e., schemes that minimize the expected estimation loss for the worst-case distribution. Two schemes in the literature provide order optimal performance in the high privacy regime where $\epsilon$ is very close to $0,$ and in the low privacy regime where $e^{\epsilon}\approx k,$ 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 $1\ll e^{\epsilon} \ll k.$ More concretely, we prove that when $3.8 < \epsilon <\ln(k/9) ,$ our schemes reduce the expected estimation loss by $50\%$ under $\ell_2^2$ metric and by $30\%$ under $\ell_1$ metric over the existing schemes. We also prove a lower bound for the region $e^{\epsilon} \ll k,$ which implies that our schemes are order optimal in this regime.

本文研究在隐私限制下，离散分布的Minimax估计问题。通过将保密方案分别应用于每个原始样本，我们需要从保密样本中估计原始样本的分布。对于给定的ε，我们考虑构造具有ε-隐私级别、即能够最小化最坏情况下的预期估计损失的最优保密方案问题。本文提出了一种新的保密方案族，它在中等隐私度量ε的情况下显著提高了现有方案的性能。

局部差分隐私下的离散分布估计最优方案