A Variational Formula for Infinity-R\'{e}nyi Divergence with Applications to Information Leakage

TL;DR

This paper introduces a variational formula for infinity-Rényi divergence, linking it to guessing strategies and enabling new closed-form expressions for information leakage measures.

Contribution

It provides a novel variational characterization of infinity-Rényi divergence that is gain-function agnostic and applies it to derive explicit formulas for maximal α-leakage.

Findings

Derived a variational formula for infinity-Rényi divergence.

Established closed-form expressions for maximal α-leakage.

Linked divergence characterization to guessing and information leakage.

Abstract

We present a variational characterization for the R\'{e}nyi divergence of order infinity. Our characterization is related to guessing: the objective functional is a ratio of maximal expected values of a gain function applied to the probability of correctly guessing an unknown random variable. An important aspect of our variational characterization is that it remains agnostic to the particular gain function considered, as long as it satisfies some regularity conditions. Also, we define two variants of a tunable measure of information leakage, the maximal $\alpha$-leakage, and obtain closed-form expressions for these information measures by leveraging our variational characterization.