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The paper proposes a family of communication efficient methods fordistributed learning in heterogeneous environments in which users obtain datafrom one of $K$ different distributions. In the proposed setup, the grouping ofusers (based on the data distributions they sample), as well as the underlyingstatistical properties of the distributions, are apriori unknown. A family ofOne-shot Distributed Clustered Learning methods (ODCL-$\mathcal{C}$) isproposed, parametrized by the set of admissible clustering algorithms$\mathcal{C}$, with the objective of learning the true model at each user. Theadmissible clustering methods include $K$-means (KM) and convex clustering(CC), giving rise to various one-shot methods within the proposed family, suchas ODCL-KM and ODCL-CC. The proposed one-shot approach, based on localcomputations at the users and a clustering based aggregation step at the serveris shown to provide strong learning guarantees. In particular, for stronglyconvex problems it is shown that, as long as the number of data points per useris above a threshold, the proposed approach achieves order-optimal mean-squarederror (MSE) rates in terms of the sample size. An explicit characterization ofthe threshold is provided in terms of problem parameters. The trade-offs withrespect to selecting various clustering methods (ODCL-CC, ODCL-KM) arediscussed and significant improvements over state-of-the-art are demonstrated.Numerical experiments illustrate the findings and corroborate the performanceof the proposed methods.

A One-shot Framework for Distributed Clustered Learning in Heterogeneous Environments