$\alpha$-leakage by R\'{e}nyi Divergence and Sibson Mutual Information

TL;DR

This paper introduces a new framework for $oldsymbol{ ext{$oldsymbol{ ilde{f}}}$-mean information gain} and demonstrates how Rényi divergence and Sibson mutual information serve as measures of information leakage, providing insights into adversarial knowledge gain.

Contribution

It proposes a unified $ ilde{f}$-mean information gain measure and links existing $oldsymbol{ ext{$oldsymbol{ ext{α}}$-leakage}}$ concepts to this framework, offering new interpretations.

Findings

Rényi divergence is the maximum $ ilde{f}$-mean information gain per event.

Sibson mutual information is the $ ilde{f}$-mean of information gain over all decisions.

Existing $ ext{α}$-leakage can be expressed as scaled $ ilde{f}$-mean measures.

Abstract

For $\tilde{f}(t) = \exp(\frac{\alpha-1}{\alpha}t)$, this paper proposes a $\tilde{f}$-mean information gain measure. R\'{e}nyi divergence is shown to be the maximum $\tilde{f}$-mean information gain incurred at each elementary event $y$ of channel output $Y$ and Sibson mutual information is the $\tilde{f}$-mean of this $Y$-elementary information gain. Both are proposed as $\alpha$-leakage measures, indicating the most information an adversary can obtain on sensitive data. It is shown that the existing $\alpha$-leakage by Arimoto mutual information can be expressed as $\tilde{f}$-mean measures by a scaled probability. Further, Sibson mutual information is interpreted as the maximum $\tilde{f}$-mean information gain over all estimation decisions applied to channel output.