In this paper we study the problem of estimating the unknown mean $\theta$ of a unit variance Gaussian distribution in a locally differentially private (LDP) way. In the high-privacy regime ($\epsilon\le 0.67$), we identify the exact optimal privacy mechanism that minimizes the variance of the estimator asymptotically. It turns out to be the extraordinarily simple sign mechanism that applies randomized response to the sign of $X_i-\theta$. However, since this optimal mechanism depends on the unknown mean $\theta$, we employ a two-stage LDP parameter estimation procedure which requires splitting agents into two groups. The first $n_1$ observations are used to consistently but not necessarily efficiently estimate the parameter $\theta$ by $\tilde{\theta}_{n_1}$. Then this estimate is updated by applying the sign mechanism with $\tilde{\theta}_{n_1}$ instead of $\theta$ to the remaining $n-n_1$ observations, to obtain an LDP and efficient estimator of the unknown mean.

翻译：本文研究在局部差分隐私（LDP）框架下，对单位方差高斯分布的未知均值 $\theta$ 进行估计的问题。在高隐私保护区域（$\epsilon\le 0.67$），我们确定了渐近意义下方差最小化的最优隐私机制。该机制极为简洁，即对 $X_i-\theta$ 的符号应用随机化应答的符号机制。然而，由于该最优机制依赖于未知均值 $\theta$，我们采用两阶段LDP参数估计流程，需将样本分为两组。第一阶段使用前 $n_1$ 个观测值通过 $\tilde{\theta}_{n_1}$ 对参数 $\theta$ 进行相合估计（无需高效性）；第二阶段用 $\tilde{\theta}_{n_1}$ 替代 $\theta$，对剩余 $n-n_1$ 个观测值应用符号机制，从而获得未知均值的LDP高效估计量。