  In this work, we explore the unique challenges -- and opportunities -- of
unsupervised federated learning (FL). We develop and analyze a one-shot
federated clustering scheme, $k$-FED, based on the widely-used Lloyd's method
for $k$-means clustering. In contrast to many supervised problems, we show that
the issue of statistical heterogeneity in federated networks can in fact
benefit our analysis. We analyse $k$-FED under a center separation assumption
and compare it to the best known requirements of its centralized counterpart.
Our analysis shows that in heterogeneous regimes where the number of clusters
per device $(k')$ is smaller than the total number of clusters over the network
$k$, $(k'\le \sqrt{k})$, we can use heterogeneity to our advantage --
significantly weakening the cluster separation requirements for $k$-FED. From a
practical viewpoint, $k$-FED also has many desirable properties: it requires
only round of communication, can run asynchronously, and can handle partial
participation or node/network failures. We motivate our analysis with
experiments on common FL benchmarks, and highlight the practical utility of
one-shot clustering through use-cases in personalized FL and device sampling.
