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In this work, we consider the optimization formulation of personalized federated learning recently introduced by Hanzely and Richtárik (2020) which was shown to give an alternative explanation to the workings of local SGD methods. Our first contribution is establishing the first lower bounds for this formulation, for both the communication complexity and the local oracle complexity. Our second contribution is the design of several optimal methods matching these lower bounds in almost allarXiv:2010.02372v1 fatcat:botvsoawe5fobf64dj4frjoc2e