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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.04840 |
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Table of Contents:
- 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 ($ε\le 1$), 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 $Q_{θ,ε}$ of this maximization is the sign mechanism that applies randomized response to the sign of $X_i-θ$, where $X_1,\dots, X_n$ 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 $n_1$ observations are used to consistently but not necessarily efficiently estimate the parameter $θ$ by $\tildeθ_{n_1}$. Then this estimate is updated by applying the sign mechanism with $\tildeθ_{n_1}$ instead of $θ$ to the remaining $n-n_1$ observations, to obtain an LDP and efficient estimator of the unknown mean.