Robust Privacy-Utility Tradeoffs under Differential Privacy and Hamming Distortion

A privacy utility tradeoff is developed for any arbitrary set of finite-alphabet source distributions. Privacy is quantified using differential privacy (DP), and utility is quantified using expected Hamming distortion maximized over the set of distributions. The family of source distribution sets (source sets) is categorized into three classes, based on different levels of prior knowledge they capture: (i) for Class I source sets, defined as sets whose convex hull includes the uniform point, the optimal DP mechanism is shown to be symmetric; (ii) for Class II source sets, comprised of distributions that are not in Class I and are restricted to have an order imposed on the distribution, the optimal mechanism is shown to exploit the source knowledge to provide more privacy to the outliers; and (iii) for all remaining source sets, which are neither Class I nor Class II, henceforth defined as Class III, bounds on optimal leakage are developed; these bounds are shown to be tight for Class III sets satisfying a specific property. To get a better understanding of DP under Hamming distortion, numerical comparisons to information theoretic leakage is provided, and analytical comparisons are developed for specific distortion ranges. Index Terms Utility-privacy tradeoff, differential privacy, information leakage, Hamming distortion