Federated multi-task learning (FMTL) has emerged as a natural choice to capture the statistical diversity among the clients in federated learning. To unleash the potential of FMTL beyond statistical diversity, we formulate a new FMTL problem FedU using Laplacian regularization, which can explicitly leverage relationships among the clients for multi-task learning. We first show that FedU provides a unified framework covering a wide range of problems such as conventional federated learning, personalized federated learning, few-shot learning, and stratified model learning. We then propose algorithms including both communication-centralized and decentralized schemes to learn optimal models of FedU. Theoretically, we show that the convergence rates of both FedUs algorithms achieve linear speedup for strongly convex and sublinear speedup of order $1/2$ for nonconvex objectives. While the analysis of FedU is applicable to both strongly convex and nonconvex loss functions, the conventional FMTL algorithm MOCHA, which is based on CoCoA framework, is only applicable to convex case. Experimentally, we verify that FedU outperforms the vanilla FedAvg, MOCHA, as well as pFedMe and Per-FedAvg in personalized federated learning.
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