Hierarchical Clustering is a popular unsupervised machine learning method with decades of history and numerous applications. We initiate the study of differentially private approximation algorithms for hierarchical clustering under the rigorous framework introduced by (Dasgupta, 2016). We show strong lower bounds for the problem: that any $\epsilon$-DP algorithm must exhibit $O(|V|^2/ \epsilon)$-additive error for an input dataset $V$. Then, we exhibit a polynomial-time approximation algorithm with $O(|V|^{2.5}/ \epsilon)$-additive error, and an exponential-time algorithm that meets the lower bound. To overcome the lower bound, we focus on the stochastic block model, a popular model of graphs, and, with a separation assumption on the blocks, propose a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly. Finally, we perform an empirical study of our algorithms and validate their performance.

翻译：层次聚类是一种流行的无监督机器学习方法，具有数十年的历史和众多应用。我们基于(Dasgupta, 2016)提出的严格框架，首次开展了层次聚类差分隐私近似算法的研究。我们证明了该问题的强下界：对于输入数据集$V$，任何$\epsilon$-差分隐私算法必须具有$O(|V|^2/ \epsilon)$的加性误差。然后，我们提出了一种具有$O(|V|^{2.5}/ \epsilon)$加性误差的多项式时间近似算法，以及一种达到下界的指数时间算法。为了突破下界，我们关注随机块模型这一流行的图模型，并在块间分离假设下，提出了一种私有的$1+o(1)$近似算法，该算法还能精确恢复块结构。最后，我们对算法进行了实证研究并验证了其性能。