Canonical Noise Distributions and Private Hypothesis Tests
This work addresses the challenge of achieving optimal privacy-preserving statistical inference for data analysts and researchers, offering a novel theoretical foundation and practical tools within the f-DP framework, though it is incremental in building upon existing DP generalizations.
The paper tackles the problem of designing optimal privacy mechanisms within the f-DP framework by introducing canonical noise distributions (CNDs), which enable lossless privacy guarantees and applications to private hypothesis testing, resulting in more powerful and accurate tests compared to existing methods, such as showing empirical improvements in power and type I error accuracy for specific DP cases.
$f$-DP has recently been proposed as a generalization of differential privacy allowing a lossless analysis of composition, post-processing, and privacy amplification via subsampling. In the setting of $f$-DP, we propose the concept of a canonical noise distribution (CND), the first mechanism designed for an arbitrary $f$-DP guarantee. The notion of CND captures whether an additive privacy mechanism perfectly matches the privacy guarantee of a given $f$. We prove that a CND always exists, and give a construction that produces a CND for any $f$. We show that private hypothesis tests are intimately related to CNDs, allowing for the release of private $p$-values at no additional privacy cost as well as the construction of uniformly most powerful (UMP) tests for binary data, within the general $f$-DP framework. We apply our techniques to the problem of difference of proportions testing, and construct a UMP unbiased (UMPU) "semi-private" test which upper bounds the performance of any $f$-DP test. Using this as a benchmark we propose a private test, based on the inversion of characteristic functions, which allows for optimal inference for the two population parameters and is nearly as powerful as the semi-private UMPU. When specialized to the case of $(ε,0)$-DP, we show empirically that our proposed test is more powerful than any $(ε/\sqrt 2)$-DP test and has more accurate type I errors than the classic normal approximation test.