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

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Abstract
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 $\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 a hard distortion constraint. 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.
Funding Information
  • National Science Foundation (CCF-1422358, CCF-1350914, CIF-1815361, CIF-1901243)

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