Multi-User Privacy Mechanism Design with Non-zero Leakage
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A privacy mechanism design problem is studied through the lens of information theory. In this work, an agent observes useful data Y=(Y_1,...,Y_N) that is correlated with private data X=(X_1,...,X_N) 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 X_i is a deterministic function of Y_i, however in this work we let X_i and Y_i 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 {X_j}_i=1^N except one which we allocate the maximum allowed leakage to it. Finally, the obtained bounds are studied in special cases.
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