Efficient Estimation of a Gaussian Mean with Local Differential Privacy

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 ($\epsilon\le 1$), we identify an optimal privacy mechanism that minimizes the variance of the estimator asymptotically. Our main technical contribution is the maximization of the Fisher-Information of the sanitized data with respect to the local privacy mechanism $Q$. We find that the exact solution $Q_{\theta,\epsilon}$ of this maximization is the sign mechanism that applies randomized response to the sign of $X_i-\theta$, where $X_1,\dots, X_n$ are the confidential iid original samples. However, since this optimal local 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.