A Differentially Private Kernel Two-Sample Test

TL;DR

This paper introduces a differentially private kernel two-sample test that maintains high utility and privacy by approximating complex test statistics with finite-dimensional representations and perturbing them for privacy guarantees.

Contribution

It proposes a novel framework for differentially private kernel two-sample testing using finite-dimensional approximations and a simple chi-squared test, addressing privacy concerns in sensitive data analysis.

Findings

Requires only a modest increase in sample size for comparable power to non-private tests.

Effective in two realistic data settings, maintaining utility under privacy constraints.

Provides a practical approach to privacy-preserving statistical testing without heavily compromising accuracy.

Abstract

Kernel two-sample testing is a useful statistical tool in determining whether data samples arise from different distributions without imposing any parametric assumptions on those distributions. However, raw data samples can expose sensitive information about individuals who participate in scientific studies, which makes the current tests vulnerable to privacy breaches. Hence, we design a new framework for kernel two-sample testing conforming to differential privacy constraints, in order to guarantee the privacy of subjects in the data. Unlike existing differentially private parametric tests that simply add noise to data, kernel-based testing imposes a challenge due to a complex dependence of test statistics on the raw data, as these statistics correspond to estimators of distances between representations of probability measures in Hilbert spaces. Our approach considers finite dimensional approximations to those representations. As a result, a simple chi-squared test is obtained, where a test statistic depends on a mean and covariance of empirical differences between the samples, which we perturb for a privacy guarantee. We investigate the utility of our framework in two realistic settings and conclude that our method requires only a relatively modest increase in sample size to achieve a similar level of power to the non-private tests in both settings.