Optimal Privacy-Utility Trade-off under a Rate Constraint

We study the privacy-utility trade-off in data release under a rate constraint. An agent observes random variable X and reveals information U to the utility provider over a rate-constrained channel, such that I(X; U) ≤ R, in return for utility I(U; Y ), where Y denotes a latent random variable correlated with X. While the objective is to maximize the utility, the agent also wants to protect a private information S, also correlated with X and Y from the utility provider. The trade-off between rate, utility and private information leakage is studied. This problem can be thought of as a generalization of both the information bottleneck and privacy funnel problems, reducing to either of the two problems in special cases. A necessary and sufficient condition for the existence of positive utility under zero private information leakage (or perfect privacy) is established. Subsequently, the problem of maximizing the utility subject to perfect privacy constraint is shown to be a linear program when the rate constraint is inactive. Also, the maximum value of the ratio of utility to infinitesimal private information leakage for an arbitrary rate constraint is obtained.