An Operational Approach to Information Leakage via Generalized Gain Functions

We introduce a gain function viewpoint of information leakage by proposing maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage, a rich class of operationally meaningful leakage measures that subsumes recently introduced leakage measures — maximal leakage and maximal <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-leakage. In maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage, the gain of an adversary in guessing an unknown random variable is measured using a gain function applied to the probability of correctly guessing. In particular, maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage captures the multiplicative increase, upon observing <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula>, in the expected gain of an adversary in guessing a randomized function of <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>, maximized over all such randomized functions. We also consider the scenario where an adversary can make multiple attempts to guess the randomized function of interest. We show that maximal leakage is an upper bound on maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage under multiple guesses, for any non-negative gain function <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>. We obtain a closed-form expression for maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage under multiple guesses for a class of concave gain functions. We also study maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage measure for a specific class of gain functions related to the <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-loss, that interpolates log-loss (<inline-formula> <tex-math notation="LaTeX">$\alpha =1$ </tex-math></inline-formula>) and (soft) 0–1 loss (<inline-formula> <tex-math notation="LaTeX">$\alpha =\infty $ </tex-math></inline-formula>). In particular, we first completely characterize the minimal expected <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-loss under multiple guesses and analyze how the corresponding leakage measure is affected with the number of guesses. We show that a new measure of divergence that belongs to the class of Bregman divergences captures the relative performance of an arbitrary adversarial strategy with respect to an optimal strategy in minimizing the expected <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-loss. Finally, we study two variants of maximal <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-leakage depending on the type of adversary and obtain closed-form expressions for them, which do not depend on the particular gain function considered as long as it satisfies some mild regularity conditions. We do this by developing a variational characterization for the Rényi divergence of order infinity which naturally generalizes the definition of pointwise maximal leakage to incorporate arbitrary gain functions.