Almost Perfect Privacy for Additive Gaussian Privacy Filters

TL;DR

This paper investigates the limits of information leakage in Gaussian privacy filters, deriving bounds and approximations for privacy-utility tradeoffs when privacy leakage is minimal, with implications for data privacy mechanisms.

Contribution

It introduces a second-order approximation for the privacy leakage function in Gaussian settings and connects it to data processing inequalities, advancing understanding of privacy-utility tradeoffs.

Findings

Perfect privacy implies zero leaked information for continuous variables.

Derived a second-order approximation for small privacy leakage levels.

Established bounds for privacy-utility tradeoffs using estimation theory.

Abstract

We study the maximal mutual information about a random variable $Y$ (representing non-private information) displayed through an additive Gaussian channel when guaranteeing that only $\epsilon$ bits of information is leaked about a random variable $X$ (representing private information) that is correlated with $Y$. Denoting this quantity by $g_\epsilon(X,Y)$, we show that for perfect privacy, i.e., $\epsilon=0$, one has $g_0(X,Y)=0$ for any pair of absolutely continuous random variables $(X,Y)$ and then derive a second-order approximation for $g_\epsilon(X,Y)$ for small $\epsilon$. This approximation is shown to be related to the strong data processing inequality for mutual information under suitable conditions on the joint distribution $P_{XY}$. Next, motivated by an operational interpretation of data privacy, we formulate the privacy-utility tradeoff in the same setup using estimation-theoretic quantities and obtain explicit bounds for this tradeoff when $\epsilon$ is sufficiently small using the approximation formula derived for $g_\epsilon(X,Y)$.