A unified interpretation of the Gaussian mechanism for differential privacy through the sensitivity index

TL;DR

This paper introduces the sensitivity index, a unified parameter that encapsulates the core aspects of the Gaussian mechanism in differential privacy, simplifying analysis and comparison across different DP interpretations.

Contribution

It proposes the sensitivity index as a single parameter to unify various interpretations of the Gaussian mechanism in differential privacy.

Findings

The sensitivity index effectively characterizes the Gaussian mechanism.

It links DP interpretations to ROC curves and hypothesis testing.

Provides a practical tool for interpreting and comparing privacy guarantees.

Abstract

The Gaussian mechanism (GM) represents a universally employed tool for achieving differential privacy (DP), and a large body of work has been devoted to its analysis. We argue that the three prevailing interpretations of the GM, namely $(\varepsilon, \delta)$-DP, f-DP and R\'enyi DP can be expressed by using a single parameter $\psi$, which we term the sensitivity index. $\psi$ uniquely characterises the GM and its properties by encapsulating its two fundamental quantities: the sensitivity of the query and the magnitude of the noise perturbation. With strong links to the ROC curve and the hypothesis-testing interpretation of DP, $\psi$ offers the practitioner a powerful method for interpreting, comparing and communicating the privacy guarantees of Gaussian mechanisms.