Privacy and Statistical Risk: Formalisms and Minimax Bounds

TL;DR

This paper compares various formal definitions of privacy in statistical analysis, establishes their equivalences, and analyzes their impact on minimax risk bounds for key estimation problems, offering insights into privacy-utility trade-offs.

Contribution

It provides formal equivalence results between different privacy definitions and derives minimax risk bounds under these privacy constraints for multiple estimation tasks.

Findings

Equivalence established between privacy definitions.

Minimax risk bounds derived for mean, support, and density estimation.

Insights into privacy-utility trade-offs in statistical estimation.

Abstract

We explore and compare a variety of definitions for privacy and disclosure limitation in statistical estimation and data analysis, including (approximate) differential privacy, testing-based definitions of privacy, and posterior guarantees on disclosure risk. We give equivalence results between the definitions, shedding light on the relationships between different formalisms for privacy. We also take an inferential perspective, where---building off of these definitions---we provide minimax risk bounds for several estimation problems, including mean estimation, estimation of the support of a distribution, and nonparametric density estimation. These bounds highlight the statistical consequences of different definitions of privacy and provide a second lens for evaluating the advantages and disadvantages of different techniques for disclosure limitation.