基于随机图着色的二元函数差分隐私
Differential Privacy for Binary Functions via Randomized Graph Colorings
摘要：
我们提出了一个框架，通过数据集的图形表示为二进制函数设计差分私有（DP）机制。数据集是图中的节点，任何两个相邻的数据集都由边连接。我们想要近似的真二进制函数为数据集分配一个值（或真颜色）。随机DP机制则等同于图的随机着色。我们使用的一个关键概念是图的边界。任何两个相邻的数据集分配不同的真彩色属于边界。在这个框架下，我们发现，固定在边界的机制行为诱导一个唯一的最优机制。此外，如果该机构是有一个均匀的行为在边界上，我们提出了一个封闭的表达式的最佳机构，这是通过一个\emph{pullback}操作的最佳机构的线图。对于平衡机制，不偏向于一个二进制值而不是另一个二进制值，最佳$(ε,δ)$ -DP机制采用特别简单的形式，仅取决于到边界的最小距离、$ε$和$δ$。
英文摘要：
We present a framework for designing differentially private (DP) mechanisms for binary functions via a graph representation of datasets. Datasets are nodes in the graph and any two neighboring datasets are connected by an edge. The true binary function we want to approximate assigns a value (or true color) to a dataset. Randomized DP mechanisms are then equivalent to randomized colorings of the graph. A key notion we use is that of the boundary of the graph. Any two neighboring datasets assigned a different true color belong to the boundary. Under this framework, we show that fixing the mechanism behavior at the boundary induces a unique optimal mechanism. Moreover, if the mechanism is to have a homogeneous behavior at the boundary, we present a closed expression for the optimal mechanism, which is obtained by means of a \emph{pullback} operation on the optimal mechanism of a line graph. For balanced mechanisms, not favoring one binary value over another, the optimal $(ε,δ)$-DP mechanism takes a particularly simple form, depending only on the minimum distance to the boundary, on $ε$, and on $δ$.
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