We propose an asymptotic framework to analyze the performance of (personalized) federated learning algorithms. In this new framework, we formulate federated learning as a multi-criterion objective, where the goal is to minimize each client's loss using information from all of the clients. We analyze a linear regression model where, for a given client, we may theoretically compare the performance of various algorithms in the high-dimensional asymptotic limit. This asymptotic multi-criterion approach naturally models the high-dimensional, many-device nature of federated learning. These tools make fairly precise predictions about the benefits of personalization and information sharing in federated scenarios -- at least in our (stylized) model -- including that Federated Averaging with simple client fine-tuning achieves the same asymptotic risk as the more intricate meta-learning and proximal-regularized approaches and outperforming Federated Averaging without personalization. We evaluate these predictions on federated versions of the EMNIST, CIFAR-100, Shakespeare, and Stack Overflow datasets, where the experiments corroborate the theoretical predictions, suggesting such frameworks may provide a useful guide to practical algorithmic development.