$α$-leakage Interpretation of Rényi Capacity

2025-10-08

AI Summary

This paper addresses the interpretation and quantification of α-leakage within Rényi capacity. Prior work is limited to pointwise maximal leakage (only at α = 1) and lacks semantic grounding in adversarial inference capability. To bridge this gap, we introduce *Y-element α-leakage*, establish its equivalence to a weighted average of Sibson mutual information, and—crucially—reveal an intrinsic unification: Rényi capacity equals the adversary’s maximum *average* α-leakage over the channel output Y regarding input X. Methodologically, we propose an *f-mean information gain* analytical framework, formulate an interpretive model based on dual maximization—over adversarial decision rules and prior beliefs—and design a generalized Blahut–Arimoto-type alternating max–max algorithm. Theoretical contributions include: (i) generalizing pointwise leakage to the full range α ∈ [0, ∞); (ii) showing that Rényi divergence arises as the maximum Y-element α-leakage; and (iii) deriving a sufficient condition for δ-approximate ε-upper bounds on α-leakage, yielding a novel theoretical tool for privacy quantification.

Technology Category

• Machine Learning: Information Theory
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

For $ ilde{f}(t) = exp(frac{α-1}αt)$, this paper shows that the Sibson mutual information is an $α$-leakage averaged over the adversary's $ ilde{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 $αin [0,infty)$. 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 $ ilde{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.

Problem

• Interprets Sibson mutual information as alpha-leakage
• Derives conditions for approximating bounded alpha-leakage
• Extends pointwise leakage to full Renyi order range

Innovation

• Sibson mutual information interprets alpha-leakage
• Y-elementary alpha-leakage extends pointwise leakage
• Alternating max-max implements generalized Blahut-Arimoto method