  We propose a parametric family of algorithms for personalized federated
learning with locally convex user costs. The proposed framework is based on a
generalization of convex clustering in which the differences between different
users' models are penalized via a sum-of-norms penalty, weighted by a penalty
parameter $\lambda$. The proposed approach enables "automatic" model
clustering, without prior knowledge of the hidden cluster structure, nor the
number of clusters. Analytical bounds on the weight parameter, that lead to
simultaneous personalization, generalization and automatic model clustering are
provided. The solution to the formulated problem enables personalization, by
providing different models across different clusters, and generalization, by
providing models different than the per-user models computed in isolation. We
then provide an efficient algorithm based on the Parallel Direction Method of
Multipliers (PDMM) to solve the proposed formulation in a federated
server-users setting. Numerical experiments corroborate our findings. As an
interesting byproduct, our results provide several generalizations to convex
clustering.
