Pan-Private Uniformity Testing

TL;DR

This paper introduces and analyzes the concept of pan-privacy, a model that balances between local and central differential privacy, and applies it to uniformity testing over a domain, revealing new sample complexity bounds.

Contribution

The paper formalizes pan-privacy, establishes its equivalence to sequentially interactive local privacy under multiple intrusions, and determines the sample complexity for uniformity testing in this model.

Findings

Pure pan-privacy against multiple intrusions equals sequentially interactive local privacy.

Sample complexity for pan-private uniformity testing is Θ(k^{2/3}).

Lower bound of Ω(k) for sequentially interactive setting separates it from pan-privacy.

Abstract

A centrally differentially private algorithm maps raw data to differentially private outputs. In contrast, a locally differentially private algorithm may only access data through public interaction with data holders, and this interaction must be a differentially private function of the data. We study the intermediate model of pan-privacy. Unlike a locally private algorithm, a pan-private algorithm receives data in the clear. Unlike a centrally private algorithm, the algorithm receives data one element at a time and must maintain a differentially private internal state while processing this stream. First, we show that pure pan-privacy against multiple intrusions on the internal state is equivalent to sequentially interactive local privacy. Next, we contextualize pan-privacy against a single intrusion by analyzing the sample complexity of uniformity testing over domain $[k]$. Focusing on the dependence on $k$, centrally private uniformity testing has sample complexity $\Theta(\sqrt{k})$, while noninteractive locally private uniformity testing has sample complexity $\Theta(k)$. We show that the sample complexity of pure pan-private uniformity testing is $\Theta(k^{2/3})$. By a new $\Omega(k)$ lower bound for the sequentially interactive setting, we also separate pan-private from sequentially interactive locally private and multi-intrusion pan-private uniformity testing.