Efficiency in local differential privacy
The Annals of Statistics · 2024 · Volume 52 · Issue 5
Abstract
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 nontrivial obstacles that need to be tackled when dealing with the LDP case. Starting from a regular parametric model P=(Pθ)θ∈Θ, Θ⊆Rp, for the i.i.d. unobserved sensitive data X1,…,Xn, we establish local asymptotic mixed normality (along subsequences) of the model Q(n)P=(Q(n)Pθn)θ∈Θ generating the sanitized observations Z1,…,Zn, 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 supQ∈QαIθ(QP)∈R, where the supremum runs over all α-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism Qˆ and an estimator θˆn(Z1,…,Zn) based on the corresponding sanitized data that achieves this asymptotically optimal variance.
Keywords
62B15, 62F12, 62F30, 62L05, efficiency, Fisher information, local asymptotic mixed normality, Local differential privacy
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