Efficient estimation of a Gaussian mean with local differential privacy

In this paper, we study the problem of estimating the unknown mean θ of a unit variance Gaussian distribution in a locally differentially private (LDP) way. In the high-privacy regime (ϵ≤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 Qθ,ϵ of this maximization is the sign mechanism that applies randomized response to the sign of Xi−θ, where X1,…,Xn are the confidential iid original samples. However, since this optimal local mechanism depends on the unknown mean θ, we employ a two-stage LDP parameter estimation procedure which requires splitting agents into two groups. The first n1 observations are used to consistently but not necessarily efficiently estimate the parameter θ by θn1~ . Then this estimate is updated by applying the sign mechanism with θ~n1 instead of θ to the remaining n−n1 observations, to obtain an LDP and efficient estimator of the unknown mean.