We develop a theory of asymptotic efficiency in regular parametric models when data confidentiality is ensured by local differential privacy (LDP). Even though efficient parameter estimation is a classical and well-studied problem in mathematical statistics, it leads to several non-trivial obstacles that need to be tackled when dealing with the LDP case. Starting from a standard parametric model $\mathcal P=(P_\theta)_{\theta\in\Theta}$, $\Theta\subseteq\mathbb R^p$, for the iid unobserved sensitive data $X_1,\dots, X_n$, we establish local asymptotic mixed normality (along subsequences) of the model $$Q^{(n)}\mathcal P=(Q^{(n)}P_\theta^n)_{\theta\in\Theta}$$ generating the sanitized observations $Z_1,\dots, Z_n$, where $Q^{(n)}$ is an arbitrary sequence of sequentially interactive privacy mechanisms. This result readily implies convolution and local asymptotic minimax theorems. In case $p=1$, the optimal asymptotic variance is found to be the inverse of the supremal Fisher-Information $\sup_{Q\in\mathcal Q_\alpha} I_\theta(Q\mathcal P)\in\mathbb R$, where the supremum runs over all $\alpha$-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism $\hat{Q}$ and an estimator $\hat{\theta}_n(Z_1,\dots, Z_n)$ based on the corresponding sanitized data that achieves this asymptotically optimal variance.

翻译：本文针对在局部差分隐私（LDP）保障数据机密性的正则参数模型中，建立了渐近效率理论。尽管有效参数估计是数理统计中经典且充分研究的问题，但在处理LDP情形时仍面临若干需克服的非平凡障碍。从独立同分布未观测敏感数据$X_1,\dots, X_n$的标准参数模型$\mathcal P=(P_\theta)_{\theta\in\Theta}$（其中$\Theta\subseteq\mathbb R^p$）出发，我们建立了生成净化观测值$Z_1,\dots, Z_n$的模型$$Q^{(n)}\mathcal P=(Q^{(n)}P_\theta^n)_{\theta\in\Theta}$$的局部渐近混合正态性（沿子序列），其中$Q^{(n)} $为任意序列的序列交互式隐私机制。该结果直接推导出卷积定理和局部渐近极小极大定理。当$p=1$时，最优渐近方差为超最大Fisher信息量$\sup_{Q\in\mathcal Q_\alpha} I_\theta(Q\mathcal P)\in\mathbb R$的逆，其中上确界遍历所有$\alpha$-差分隐私（边际）马尔可夫核。我们提出了一种算法，用以寻找（近似）最优隐私机制$\hat{Q}$，并基于相应净化数据构造估计量$\hat{\theta}_n(Z_1,\dots, Z_n)$，使其达到该渐近最优方差。