We introduce a \emph{gain function} viewpoint of information leakage by proposing \emph{maximal $g$-leakage}, a rich class of operationally meaningful leakage measures that subsumes recently introduced leakage 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 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 $g$-leakage under multiple guesses, for any non-negative gain function $g$. We obtain a closed-form expression for maximal $g$-leakage under multiple guesses for a class of concave gain functions. We also study maximal $g$-leakage measure for a specific class of gain functions related to the $\alpha$-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. Finally, we study two variants of maximal $g$-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 R\'{e}nyi divergence of order infinity which naturally generalizes the definition of pointwise maximal leakage to incorporate arbitrary gain functions.

翻译：我们通过提出最大$g$-泄露（一种丰富且具有操作意义的泄露度量类，涵盖了最近引入的{最大泄露}和{最大$\alpha$-泄露}）引入了信息泄露的{增益函数}视角。在最大$g$-泄露中，对手猜测未知随机变量的收益通过{增益函数}作用于正确猜测概率来度量。具体而言，最大$g$-泄露刻画了在观察到$Y$后，对手猜测$X$的随机化函数时预期增益的乘法增量，并对所有此类随机化函数取最大值。我们还考虑了对手可通过多次尝试来猜测目标随机化函数的场景。研究表明，对于任意非负增益函数$g$，在多次猜测下最大泄露是最大$g$-泄露的上界。针对一类凹增益函数，我们得到了多次猜测下最大$g$-泄露的闭式表达式。我们还研究了与$\alpha$-损失相关的特定增益函数类对应的最大$g$-泄露度量。具体地，我们首先完整刻画了多次猜测下的最小期望$\alpha$-损失，并分析了对应泄露度量随猜测次数变化的影响。最后，我们根据对手类型研究了最大$g$-泄露的两种变体，并得到了它们的闭式表达式——这些表达式不依赖于具体增益函数，只要其满足某些温和的正则条件。为此，我们发展了无穷阶Rényi散度的变分刻画，该刻画自然推广了逐点最大泄露的定义以容纳任意增益函数。