A widely recognized difficulty in federated learning arises from the statistical heterogeneity among clients: local datasets often come from different but not entirely unrelated distributions, and personalization is, therefore, necessary to achieve optimal results from each individual's perspective. In this paper, we show how the excess risks of personalized federated learning with a smooth, strongly convex loss depend on data heterogeneity from a minimax point of view. Our analysis reveals a

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... rprising theorem of the alternative for personalized federated learning: there exists a threshold such that (a) if a certain measure of data heterogeneity is below this threshold, the FedAvg algorithm [McMahan et al., 2017] is minimax optimal; (b) when the measure of heterogeneity is above this threshold, then doing pure local training (i.e., clients solve empirical risk minimization problems on their local datasets without any communication) is minimax optimal. As an implication, our results show that the presumably difficult (infinite-dimensional) problem of adapting to client-wise heterogeneity can be reduced to a simple binary decision problem of choosing between the two baseline algorithms. Our analysis relies on a new notion of algorithmic stability that takes into account the nature of federated learning.