Tunable Measures for Information Leakage and Applications to Privacy-Utility Tradeoffs

We introduce a tunable measure for information leakage called maximal alpha-leakage. This measure quantifies the maximal gain of an adversary in inferring any (potentially random) function of a dataset from a release of the data. The inferential capability of the adversary is, in turn, quantified by a class of adversarial loss functions that we introduce as $α$-loss, $α\in[1,\infty]$. The choice of $α$ determines the specific adversarial action and ranges from refining a belief (about any function of the data) for $α=1$ to guessing the most likely value for $α=\infty$ while refining the $α^{th}$ moment of the belief for $α$ in between. Maximal alpha-leakage then quantifies the adversarial gain under $α$-loss over all possible functions of the data. In particular, for the extremal values of $α=1$ and $α=\infty$, maximal alpha-leakage simplifies to mutual information and maximal leakage, respectively. For $α\in(1,\infty)$ this measure is shown to be the Arimoto channel capacity of order $α$. We show that maximal alpha-leakage satisfies data processing inequalities and a sub-additivity property thereby allowing for a weak composition result. Building upon these properties, we use maximal alpha-leakage as the privacy measure and study the problem of data publishing with privacy guarantees, wherein the utility of the released data is ensured via a hard distortion constraint. Unlike average distortion, hard distortion provides a deterministic guarantee of fidelity. We show that under a hard distortion constraint, for $α>1$ the optimal mechanism is independent of $α$, and therefore, the resulting optimal tradeoff is the same for all values of $α>1$. Finally, the tunability of maximal alpha-leakage as a privacy measure is also illustrated for binary data with average Hamming distortion as the utility measure.