Tunable Measures for Information Leakage and Applications to Privacy-Utility Tradeoffs
Jiachun Liao, Oliver Kosut, Lalitha Sankar, Flávio P. Calmon · IEEE Transactions on Information Theory · 2019
We introduce a tunable measure for information leakage calledmaximal$\alpha $-leakage. This measure quantifies the maximal gain of an adversary in inferring any (potentially random) function of a dataset from a release of the data. The inferential capability of the adversary is, in turn, quantified by a class of adversarial loss functions that we introduce as$\alpha $-loss,$\alpha \in [1,\infty) \cup \{\infty \}$. The choice of$\alpha $determines the specific adversarial action and ranges from refining a belief (about any function of the data) for$\alpha =1$to guessing the most likely value for$\alpha = \infty $while refining the$\alpha ^{\text {th}}$moment of the belief for$\alpha $in between. Maximal$\alpha $-leakage then quantifies the adversarial gain under$\alpha $-loss over all possible functions of the data. In particular, for the extremal values of$\alpha =1$and$\alpha =\infty $, maximal$\alpha $-leakage simplifies to mutual information and maximal leakage, respectively. For$\alpha \in (1,\infty)$this measure is shown to be the Arimoto channel capacity of order$\alpha $. We show that maximal$\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 $-leakage as the privacy measure and study the problem of data publishing with privacy guarantees, wherein the utility of the released data is ensured via ahard distortionconstraint. Unlike average distortion, hard distortion provides a deterministic guarantee of fidelity. We show that under a hard distortion constraint, for$\alpha >1$the optimal mechanism is independent of$\alpha $, and therefore, the resulting optimal tradeoff is the same for all values of$\alpha >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.