An Information Geometric Approach to Local Information Privacy with Applications to Max-lift and Local Differential Privacy

Abstract: We study an information-theoretic privacy mechanism design, where an agent observes useful data $Y$ and wants to reveal the information to a user. Since the useful data is correlated with the private data $X$, the agent uses a privacy mechanism to produce disclosed data $U$ that can be released. We assume that the agent observes $Y$ and has no direct access to $X$, i.e., the private data is hidden. We study the privacy mechanism design that maximizes the revealed information about $Y$ while satisfying a bounded Local Information Privacy (LIP) criterion. When the leakage is sufficiently small, concepts from information geometry allow us to locally approximate the mutual information. By utilizing this approximation the main privacy-utility trade-off problem can be rewritten as a quadratic optimization problem that has closed-form solution under some constraints. For the cases where the closed-form solution is not obtained we provide lower bounds on it. In contrast to the previous works that have complexity issues, here, we provide simple privacy designs with low complexity which are based on finding the maximum singular value and singular vector of a matrix. To do so, we follow two approaches where in the first one we find a lower bound on the main problem and then approximate it, however, in the second approach we approximate the main problem directly. In this work, we present geometrical interpretations of the proposed methods and in a numerical example we compare our results considering both approaches with the optimal solution and the previous methods. Furthermore, we discuss how our method can be generalized considering larger amounts for the privacy leakage. Finally, we discuss how the proposed methods can be applied to deal with differential privacy.