Test Without Trust: Optimal Locally Private Distribution Testing

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. First, we construct tests that use existing, general-purpose locally differentially private mechanisms such as the popular RAPPOR or the recently introduced Hadamard Response for collecting data and show that our proposed tests are sample optimal, when we insist on using these mechanisms. Next, we allow bespoke mechanisms designed specifically for testing and introduce the Randomized Aggregated Private Testing Optimal Response (RAPTOR) mechanism which is remarkably simple and requires only one bit of communication per sample. We show that our proposed mechanism yields sample-optimal tests, and in particular outperforms any test based on RAPPOR or Hadamard response. A distinguishing feature of our optimal mechanism is that, in contrast to existing mechanisms, it uses public randomness.