February 2024 The right complexity measure in locally private estimation: It is not the Fisher information
John C. Duchi, Feng Ruan
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Ann. Statist. 52(1): 1-51 (February 2024). DOI: 10.1214/22-AOS2227


We identify fundamental tradeoffs between statistical utility and privacy under local models of privacy in which data is kept private even from the statistician, providing instance-specific bounds for private estimation and learning problems by developing the local minimax risk. In contrast to approaches based on worst-case (minimax) error, which are conservative, this allows us to evaluate the difficulty of individual problem instances and delineate the possibilities for adaptation in private estimation and inference. Our main results show that the local modulus of continuity of the estimand with respect to the variation distance—as opposed to the Hellinger distance central to classical statistics—characterizes rates of convergence under locally private estimation for many notions of privacy, including differential privacy and its relaxations. As consequences of these results, we identify an alternative to the Fisher information for private estimation, giving a more nuanced understanding of the challenges of adaptivity and optimality.


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John C. Duchi. Feng Ruan. "The right complexity measure in locally private estimation: It is not the Fisher information." Ann. Statist. 52 (1) 1 - 51, February 2024. https://doi.org/10.1214/22-AOS2227


Received: 1 September 2020; Revised: 1 July 2022; Published: February 2024
First available in Project Euclid: 7 March 2024

MathSciNet: MR4718406
Digital Object Identifier: 10.1214/22-AOS2227

Primary: 62G05 , 62G10 , 68P27

Keywords: L1 information , local minimax complexity , Locally private estimation , strong data processing inequalities

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 1 • February 2024
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