$α$-leakage by Rényi Divergence and Sibson Mutual Information
Ni Ding, Mohammad Amin Zarrabian, Parastoo Sadeghi
TL;DR
This work reframes $\alpha$-leakage through a $\tilde{f}$-mean information gain with $\tilde{f}(t)=\exp(\frac{\alpha-1}{\alpha}t)$, linking it to Rényi divergence and Sibson mutual information. It proves that the per-event leakage at each channel output $y$ is maximized by the Rényi divergence $D_{\alpha}(\mathbf{P}_{X|Y=y} \| \mathbf{P}_X)$, while the overall leakage is the Sibson mutual information $I_{\alpha}^{\text{S}}(\mathbf{P}_X, \mathbf{P}_{Y|X})$, which is the $\tilde{f}$-mean of per-event leakages over $y$. The paper derives explicit maximizers for the adversary’s soft decision and shows how existing $\alpha$-leakage notions arise via scaling priors and post-processing inequalities, unifying multiple privacy measures under a single $\tilde{f}$-mean framework. These results provide exact leakage characterizations in terms of $D_{\alpha}$ and $I_{\alpha}^{\text{S}}$, and suggest fruitful avenues for exploring the relationship between $f$-mean and $\tilde{f}$-mean information measures in privacy analysis.
Abstract
For $\tilde{f}(t) = \exp(\frac{α-1}αt)$, this paper proposes a $\tilde{f}$-mean information gain measure. Rényi 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 $α$-leakage measures, indicating the most information an adversary can obtain on sensitive data. It is shown that the existing $α$-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.