Curvature Transfer: When Does Collaboration Help Personalized Learning?

Personalized Federated Learning (PFL) improves client-specific models by transferring information across heterogeneous clients, yet existing methods provide limited insight into what recipient-specific information is actually needed for transfer and when collaboration adds value beyond local learning. We characterize this information through local transfer statistics that determine how population information should be adapted to a recipient. For least-squares heads over a shared representation, the characterization is exact and reduces to a curvature matrix and a feature–objective cross-moment, without requiring the full client distribution or client-to-population density ratios. This characterization shows that collaboration is useful when other clients improve estimation of the recipient's transfer statistics through similarity or exploitable structure. We develop pooled and structured estimators for these regimes and provide finite-time guarantees that characterize when collaboration improves personalized transfer. Experiments recover the predicted collaboration regimes and examine whether the transfer principle extends to nonlinear personalized heads. Under scarce homogeneous CIFAR features, pooled estimation reduces error by $10$-$19$% relative to the strongest non-oracle comparator, while structured estimation reduces error by $4$-$21$% under structured CIFAR-10/100 heterogeneity. The results also identify settings in which strong local estimation remains preferable, particularly under higher heterogeneity and on FEMNIST features. Overall, our results characterize when cross-client information improves personalized learning through better estimation of recipient-specific transfer information.