Minimax optimal goodness-of-fit testing for densities and multinomials under a local differential privacy constraint

TL;DR

This paper develops a minimax optimal goodness-of-fit testing method under local differential privacy constraints, providing the first such optimal test for continuous densities and discrete distributions, with adaptive and non-adaptive versions.

Contribution

It introduces the first minimax optimal goodness-of-fit test under local differential privacy, applicable to both continuous and discrete data, with adaptive capabilities.

Findings

Established upper bounds on separation distance for the proposed test.

Derived matching lower bounds on minimax separation rates.

Demonstrated the test's optimality in continuous and discrete settings.

Abstract

Finding anonymization mechanisms to protect personal data is at the heart of recent machine learning research. Here, we consider the consequences of local differential privacy constraints on goodness-of-fit testing, i.e. the statistical problem assessing whether sample points are generated from a fixed density $f_0$, or not. The observations are kept hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a testing procedure which is based on an estimation of the quadratic distance between the density $f$ of the unobserved samples and $f_0$. We establish an upper bound on the separation distance associated with this test, and a matching lower bound on the minimax separation rates of testing under non-interactive privacy in the case that $f_0$ is uniform, in discrete and continuous settings. To the best of our knowledge, we provide the first minimax optimal test and associated private transformation under a local differential privacy constraint over Besov balls in the continuous setting, quantifying the price to pay for data privacy. We also present a test that is adaptive to the smoothness parameter of the unknown density and remains minimax optimal up to a logarithmic factor. Finally, we note that our results can be translated to the discrete case, where the treatment of probability vectors is shown to be equivalent to that of piecewise constant densities in our setting. That is why we work with a unified setting for both the continuous and the discrete cases.