Test without Trust: Optimal Locally Private Distribution Testing

TL;DR

This paper investigates distribution testing under local differential privacy constraints, introducing a new mechanism called RAPTOR and analyzing the sample complexity for identity and independence testing.

Contribution

It introduces RAPTOR, a simple, one-bit communication mechanism for private testing, and provides sample complexity bounds for tests using existing and new mechanisms.

Findings

RAPTOR achieves sample optimality for private testing.

Existing mechanisms like RAPPOR require more samples than RAPTOR.

Proposed tests are computationally efficient.

Abstract

We study the problem of distribution testing when the samples can only be accessed using a locally differentially private mechanism and focus on two representative testing questions of identity (goodness-of-fit) and independence testing for discrete distributions. We are concerned with two settings: First, when we insist on using an already deployed, general-purpose locally differentially private mechanism such as the popular RAPPOR or the recently introduced Hadamard Response for collecting data, and must build our tests based on the data collected via this mechanism; and second, when no such restriction is imposed, and we can design a bespoke mechanism specifically for testing. For the latter purpose, we introduce the Randomized Aggregated Private Testing Optimal Response (RAPTOR) mechanism which is remarkably simple and requires only one bit of communication per sample. We propose tests based on these mechanisms and analyze their sample complexities. Each proposed test can be implemented efficiently. In each case (barring one), we complement our performance bounds for algorithms with information-theoretic lower bounds and establish sample optimality of our proposed algorithm. A peculiar feature that emerges is that our sample-optimal algorithm based on RAPTOR uses public-coins, and any test based on RAPPOR or Hadamard Response, which are both private-coin mechanisms, requires significantly more samples.