Minimax rate for multivariate data under componentwise local differential privacy constraints

Abstract

Our research analyses the balance between maintaining privacy and preserving statistical accuracy when dealing with multivariate data that is subject to componentwise local differential privacy (CLDP). With CLDP, each component of the private data is made public through a separate privacy channel. This allows for varying levels of privacy protection for different components or for the privatization of each component by different entities, each with their own distinct privacy policies. It also covers the practical situations where it is impossible to privatize jointly all the components of the raw data. We develop general techniques for establishing minimax bounds that shed light on the statistical cost of privacy in this context, as a function of the privacy levels α1, . . . , αd of the d components.

We demonstrate the versatility and efficiency of these techniques by presenting various statistical applications. Specifically, we examine nonparametric density and joint moments estimation under CLDP, providing upper and lower bounds that match up to constant factors, as well as an associated data-driven adaptive procedure. Additionally, we conduct a detailed analysis of the effective privacy level, exploring how information about a private characteristic of an individual may be inferred from the publicly visible characteristics of the same individual.