We provide a unified framework for characterizing pure and approximate differentially private (DP) learnabiliity. The framework uses the language of graph theory: for a concept class $\mathcal{H}$, we define the contradiction graph $G$ of $\mathcal{H}$. It vertices are realizable datasets, and two datasets $S,S'$ are connected by an edge if they contradict each other (i.e., there is a point $x$ that is labeled differently in $S$ and $S'$). Our main finding is that the combinatorial structure of $G$ is deeply related to learning $\mathcal{H}$ under DP. Learning $\mathcal{H}$ under pure DP is captured by the fractional clique number of $G$. Learning $\mathcal{H}$ under approximate DP is captured by the clique number of $G$. Consequently, we identify graph-theoretic dimensions that characterize DP learnability: the clique dimension and fractional clique dimension. Along the way, we reveal properties of the contradiction graph which may be of independent interest. We also suggest several open questions and directions for future research.

翻译：我们提供了一个统一框架，用于刻画纯差分隐私和近似差分隐私下的可学习性。该框架使用图论语言：对于概念类 $\mathcal{H}$，我们定义其矛盾图 $G$，顶点为可实现数据集，若两个数据集 $S,S'$ 相互矛盾（即存在一个点 $x$ 在 $S$ 和 $S'$ 中被赋予不同标签），则它们之间连边。我们的主要发现是，$G$ 的组合结构与在差分隐私下学习 $\mathcal{H}$ 密切相关：纯差分隐私下的可学习性由 $G$ 的分数团数刻画，而近似差分隐私下的可学习性由 $G$ 的团数刻画。由此，我们识别出刻画差分隐私可学习性的图论维度：团维数和分数团维数。在此过程中，我们揭示了矛盾图可能具有独立意义的性质，并提出了若干开放问题及未来研究方向。