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$α$-leakage Interpretation of Rényi Capacity

Ni Ding, Farhad Farokhi, Tao Guo, Yinfei Xu, Xiang Zhang

TL;DR

This work reinterpretates $\alpha$-leakage through a $\tilde{f}$-mean framework, identifying Sibson mutual information as an instance-wise and instance-conditioned leakage measure tied to the adversary’s optimal inference. It unifies divergence-based leakage, elementary leakage, and capacity via a max–max Blahut-Arimoto approach, and introduces a $Y$-elementary leakage that extends pointwise maximal leakage to the full Rényi order range $\alpha\in[0,\infty)$. A tail-bound analysis yields a practical $\delta$-approximation condition for both $Y$- and $XY$-elementary leakages, while the capacity viewpoint shows that the Rényi capacity $C_{\alpha}$ corresponds to the maximal $\tilde{f}$-mean leakage over both the adversary’s decision and the channel input, enabling an alternating optimization method to compute it. Overall, the paper provides a cohesive information-theoretic interpretation of privacy leakage across the entire Rényi spectrum and a computable framework for its evaluation via generalized Blahut-Arimoto iterations.

Abstract

For $\tilde{f}(t) = \exp(\frac{α-1}αt)$, this paper shows that the Sibson mutual information is an $α$-leakage averaged over the adversary's $\tilde{f}$-mean relative information gain (on the secret) at elementary event of channel output $Y$ as well as the joint occurrence of elementary channel input $X$ and output $Y$. This interpretation is used to derive a sufficient condition that achieves a $δ$-approximation of $ε$-upper bounded $α$-leakage. A $Y$-elementary $α$-leakage is proposed, extending the existing pointwise maximal leakage to the overall Rényi order range $α\in [0,\infty)$. Maximizing this $Y$-elementary leakage over all attributes $U$ of channel input $X$ gives the Rényi divergence. Further, the Rényi capacity is interpreted as the maximal $\tilde{f}$-mean information leakage over both the adversary's malicious inference decision and the channel input $X$ (represents the adversary's prior belief). This suggests an alternating max-max implementation of the existing generalized Blahut-Arimoto method.

$α$-leakage Interpretation of Rényi Capacity

TL;DR

This work reinterpretates -leakage through a -mean framework, identifying Sibson mutual information as an instance-wise and instance-conditioned leakage measure tied to the adversary’s optimal inference. It unifies divergence-based leakage, elementary leakage, and capacity via a max–max Blahut-Arimoto approach, and introduces a -elementary leakage that extends pointwise maximal leakage to the full Rényi order range . A tail-bound analysis yields a practical -approximation condition for both - and -elementary leakages, while the capacity viewpoint shows that the Rényi capacity corresponds to the maximal -mean leakage over both the adversary’s decision and the channel input, enabling an alternating optimization method to compute it. Overall, the paper provides a cohesive information-theoretic interpretation of privacy leakage across the entire Rényi spectrum and a computable framework for its evaluation via generalized Blahut-Arimoto iterations.

Abstract

For , this paper shows that the Sibson mutual information is an -leakage averaged over the adversary's -mean relative information gain (on the secret) at elementary event of channel output as well as the joint occurrence of elementary channel input and output . This interpretation is used to derive a sufficient condition that achieves a -approximation of -upper bounded -leakage. A -elementary -leakage is proposed, extending the existing pointwise maximal leakage to the overall Rényi order range . Maximizing this -elementary leakage over all attributes of channel input gives the Rényi divergence. Further, the Rényi capacity is interpreted as the maximal -mean information leakage over both the adversary's malicious inference decision and the channel input (represents the adversary's prior belief). This suggests an alternating max-max implementation of the existing generalized Blahut-Arimoto method.

Paper Structure

This paper contains 12 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. $\alpha$-Leakage: Maximum Information Gain
  4. Leakage Measure by Divergence
  5. Maximal Leakage by Sibson Mutual Information
  6. Elementary (Pointwise) Information Leakage
  7. $\delta$-approximation of $\alpha$-Leakage: Tail Bound
  8. Capacity of $\alpha$-leakage
  9. Unconditional $\tilde{f}$-mean Relative Information
  10. Decompositions by $\tilde{D}_{\alpha} ( P_{\hat{X}} \| P_X )$
  11. Generalized Blahurt-Arimoto Method Arimoto1976_CompErrExp
  12. Conclusion

Key Result

Proposition 1

The maximal leakage over all attributes $U$ of $X$ for given $P_{X}$ and $P_{Y|X}$ over Markov chain $U-X-Y-\hat{U}$ is

Theorems & Definitions (5)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • Theorem 1