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.
翻译:通过信息理论的透镜来研究隐私机制设计问题。 在这项工作中, 代理人观察与私人数据X=( X_ 1 ),..., Y_ N) 美元相关的有用数据 $Y = (Y_ 1,...), Y_ N) 美元 美元, 私人数据 $X = (X_ 1,..., X_ N) 美元, 假定代理也可以访问该数据 。 这里, 我们考虑用户$ 需要 $ 的子矢量为 $Y, 以 $ C ⁇ 美元为单位。 代理人希望向用户披露 美元 。 由于 $C ⁇ i} 与 美元是不可直接披露的 $X 。 一个隐私机制旨在生成美元 披露数据 $U$, 使用户的线性组合最大化, 同时满足共同信息方面的封闭性隐私限制。 在类似工作中, 美元是 $Y_ 美元, 但是在这个工作中, 我们允许 $x_ i 美元 和 $ $ $_ 美元 。