  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.
