Abstract
It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners’ privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber’s contamination model and the local differential privacy (LDP) constraints.
In this paper, we start with a general minimax lower bound result, which disentangles the costs of being robust against Huber contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.
Funding Statement
The second author was supported by Engineering and Physical Sciences Reseach Council (EPSRC) New Investigator Award EP/W016117/1.
Acknowledgements
We are thankful to the anonymous referees for detailed comments and suggestions, which greatly improved the paper.
Citation
Mengchu Li. Thomas B. Berrett. Yi Yu. "On robustness and local differential privacy." Ann. Statist. 51 (2) 717 - 737, April 2023. https://doi.org/10.1214/23-AOS2267
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