Data Disclosure Mechanism Design with Non-zero Leakage

We study an information-theoretic privacy problem, where an agent observes useful data Y and wants to reveal the information to a user. Since the useful data is correlated with sensitive data X, the agent employs a privacy mechanism to produce data U that can be disclosed. Thus, we study the privacy mechanism design that maximizes the revealed information about Y while satisfying an ℓ
₁
-privacy criterion under the Markov chain X-Y -U. When a sufficiently small leakage is allowed, we show that the optimizer of the design problem has a specific structure which allows us to use a local approximation of mutual information. More specifically, we show that the optimizer vectors are perturbations of fixed distributions. By using this approximation the original optimization problem can be reduced to a linear programming problem and an approximate solution for privacy mechanism design can be obtained.