Goodness-of-fit testing for H\"older continuous densities under local differential privacy

TL;DR

This paper investigates the limits of goodness-of-fit testing for H"older continuous densities under local differential privacy constraints, proposing optimal privacy mechanisms and testing procedures, and analyzing the impact of interaction on test performance.

Contribution

It introduces privacy mechanisms and testing procedures that are nearly optimal for private goodness-of-fit testing of various distributions, and compares interactive and non-interactive approaches.

Findings

Proposed privacy mechanisms achieve near-optimal minimax rates.

Sequentially interactive mechanisms outperform non-interactive ones.

Results show deterioration in testing performance under privacy constraints.

Abstract

We address the problem of goodness-of-fit testing for H\"older continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of $f_0$ including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.