Efficiency in local differential privacy

01/25/2023

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by   Lukas Steinberger, et al.

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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 𝒫=(P_θ)_θ∈Θ, Θ⊆ℝ^p, for the iid unobserved sensitive data X_1,…, X_n, we establish local asymptotic mixed normality (along subsequences) of the model Q^(n)𝒫=(Q^(n)P_θ^n)_θ∈Θ generating the sanitized observations Z_1,…, 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∈𝒬_α I_θ(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 Q̂ and an estimator θ̂_n(Z_1,…, Z_n) based on the corresponding sanitized data that achieves this asymptotically optimal variance.