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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2301.10600 |
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| _version_ | 1866917606840074240 |
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| author | Steinberger, Lukas |
| author_facet | Steinberger, Lukas |
| contents | 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 $\mathcal P=(P_θ)_{θ\inΘ}$, $Θ\subseteq\mathbb R^p$, for the iid unobserved sensitive data $X_1,\dots, X_n$, we establish local asymptotic mixed normality (along subsequences) of the model $$Q^{(n)}\mathcal P=(Q^{(n)}P_θ^n)_{θ\inΘ}$$ 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 $\sup_{Q\in\mathcal Q_α} I_θ(Q\mathcal P)\in\mathbb R$, where the supremum runs over all $α$-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism $\hat{Q}$ and an estimator $\hatθ_n(Z_1,\dots, Z_n)$ based on the corresponding sanitized data that achieves this asymptotically optimal variance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10600 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficiency in local differential privacy Steinberger, Lukas Statistics Theory 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 $\mathcal P=(P_θ)_{θ\inΘ}$, $Θ\subseteq\mathbb R^p$, for the iid unobserved sensitive data $X_1,\dots, X_n$, we establish local asymptotic mixed normality (along subsequences) of the model $$Q^{(n)}\mathcal P=(Q^{(n)}P_θ^n)_{θ\inΘ}$$ 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 $\sup_{Q\in\mathcal Q_α} I_θ(Q\mathcal P)\in\mathbb R$, where the supremum runs over all $α$-differentially private (marginal) Markov kernels. We present an algorithm for finding a (nearly) optimal privacy mechanism $\hat{Q}$ and an estimator $\hatθ_n(Z_1,\dots, Z_n)$ based on the corresponding sanitized data that achieves this asymptotically optimal variance. |
| title | Efficiency in local differential privacy |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2301.10600 |