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 non-trivial obstacles that need to be tackled when dealing with the LDP case. Starting from a standard parametric model π«=(P_ΞΈ)_ΞΈβΞ, Ξββ^p, for the iid unobserved sensitive data X_1,β¦, X_n, we establish local asymptotic mixed normality (along subsequences) of the model Q^(n)π«=(Q^(n)P_ΞΈ^n)_ΞΈβΞ generating the sanitized observations Z_1,β¦, 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_Qβπ¬_Ξ± I_ΞΈ(Qπ«)ββ, where the supremum runs over all Ξ±-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism QΜ and an estimator ΞΈΜ_n(Z_1,β¦, Z_n) based on the corresponding sanitized data that achieves this asymptotically optimal variance.
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