Heavy Hitters and the Structure of Local Privacy

TL;DR

This paper introduces a new locally differentially private algorithm for heavy hitters that achieves optimal error bounds, and explores the structure of local privacy, including group privacy degradation and transformations between privacy models.

Contribution

The paper presents an optimal error algorithm for heavy hitters under local differential privacy and develops new theoretical insights into local privacy structures and transformations.

Findings

Optimal error bounds for heavy hitters in local privacy

Group privacy degrades proportionally to √k in local models

Transformations from approximate to pure local privacy protocols

Abstract

We present a new locally differentially private algorithm for the heavy hitters problem which achieves optimal worst-case error as a function of all standardly considered parameters. Prior work obtained error rates which depend optimally on the number of users, the size of the domain, and the privacy parameter, but depend sub-optimally on the failure probability. We strengthen existing lower bounds on the error to incorporate the failure probability, and show that our new upper bound is tight with respect to this parameter as well. Our lower bound is based on a new understanding of the structure of locally private protocols. We further develop these ideas to obtain the following general results beyond heavy hitters. $\bullet$ Advanced Grouposition: In the local model, group privacy for $k$ users degrades proportionally to $\approx \sqrt{k}$, instead of linearly in $k$ as in the central model. Stronger group privacy yields improved max-information guarantees, as well as stronger lower bounds (via "packing arguments"), over the central model. $\bullet$ Building on a transformation of Bassily and Smith (STOC 2015), we give a generic transformation from any non-interactive approximate-private local protocol into a pure-private local protocol. Again in contrast with the central model, this shows that we cannot obtain more accurate algorithms by moving from pure to approximate local privacy.