Multi-User Privacy Mechanism Design with Non-zero Leakage

A privacy mechanism design problem is studied through the lens of information theory. In this work, an agent observes useful data $Y=(Y1,...,YN)$ that is correlated with private data $X=(X1,...,XN)$ which is assumed to be also accessible by the agent. Here, we consider $K$ users where user $i$ demands a sub-vector of $Y$, denoted by $C{i}$. The agent wishes to disclose $C{i}$ to user $i$. Since $C{i}$ is correlated with $X$ it can not be disclosed directly. A privacy mechanism is designed to generate disclosed data $U$ which maximizes a linear combinations of the users utilities while satisfying a bounded privacy constraint in terms of mutual information. In a similar work it has been assumed that $Xi$ is a deterministic function of $Yi$, however in this work we let $Xi$ and $Yi$ be arbitrarily correlated. First, an upper bound on the privacy-utility trade-off is obtained by using a specific transformation, Functional Representation Lemma and Strong Functional Representaion Lemma, then we show that the upper bound can be decomposed into $N$ parallel problems. Next, lower bounds on privacy-utility trade-off are derived using Functional Representation Lemma and Strong Functional Representaion Lemma. The upper bound is tight within a constant and the lower bounds assert that the disclosed data is independent of all ${Xj}_{i=1}^N$ except one which we allocate the maximum allowed leakage to it. Finally, the obtained bounds are studied in special cases.