Efficiency in local differential privacy

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.