FedAffine: Personalized Federated Learning with Shared Affine Predictor Families

Personalized federated learning must decide which cross-client differences should survive collaboration, yet trained client models conflate persistent population variation with finite-sample estimation error. We call this the population–estimation ambiguity and impose a structural restriction: client population predictors lie near a low-dimensional affine family. A sample-weighted sequence model decomposes risk into unrepresented population variation, client-specific estimation, and pooled estimation, separating personalization capacity from coefficient regularization. This motivates FedAffine, which collaboratively learns a high-dimensional base and shared basis while keeping a short coefficient local. Coefficients can be reconstructed from local data, enabling deployment to held-out clients without persistent training-time state. Under the pFL-Bench FEMNIST protocol, reconstructive FedAffine re-estimates three coefficients from zero and attains weighted accuracy of $90.18\%$ on participating clients and $90.22\%$ on held-out clients, versus $89.48\%$ and $89.41\%$ for Ditto. We additionally report the accuracy–adaptation-compute frontier. On multi-institutional radiotherapy dose prediction, FedAffine achieves site-macro Metric 1 of $3.52$ Gy, compared with $4.03$ for FedAvg-FT. Controlled experiments validate the fixed-family risk calculation and find predictive but incomplete alignment with a planted span; the analysis does not guarantee neural recovery of that span.