An Operational Approach to Information Leakage via Generalized Gain Functions

We introduce a gain function viewpoint of information leakage by proposing maximal $g$-leakage, a rich class of operationally meaningful leakage measures that subsumes recently introduced measures maximal leakage and maximal $\alpha$-leakage. In maximal $g$-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 $g$-leakage captures the multiplicative increase, upon observing $Y$, in the expected gain of an adversary in guessing a randomized function of $X$, maximized over all such randomized functions. We show that maximal leakage is an upper bound on maximal $g$-leakage. We obtain a closed-form expression for maximal $g$-leakage for a class of concave gain functions. We also study two variants of maximal $g$-leakage depending on the type of an 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 R\'{e}nyi divergence of order infinity which naturally generalizes the definition of pointwise maximal leakage to incorporate arbitrary gain functions. Finally, we study information leakage in the scenario where an adversary can make multiple guesses by focusing on maximizing a specific gain function related to $\alpha$-loss. In particular, we first completely characterize the minimal expected $\alpha$-loss under multiple guesses and analyze how the corresponding leakage measure is affected with the number of guesses. We also 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 $\alpha$-loss.