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In this work, we consider the optimization formulation of personalizedfederated learning recently introduced by Hanzely and Richt\'arik (2020) whichwas shown to give an alternative explanation to the workings of local {\tt SGD}methods. Our first contribution is establishing the first lower bounds for thisformulation, for both the communication complexity and the local oraclecomplexity. Our second contribution is the design of several optimal methodsmatching these lower bounds in almost all regimes. These are the first provablyoptimal methods for personalized federated learning. Our optimal methodsinclude an accelerated variant of {\tt FedProx}, and an acceleratedvariance-reduced version of {\tt FedAvg}/Local {\tt SGD}. We demonstrate thepractical superiority of our methods through extensive numerical experiments.

Lower Bounds and Optimal Algorithms for Personalized Federated Learning