Relative-Error Composite Federated Optimization with One Server-Side Proximal Step per Round

We consider deterministic composite federated optimization, in which distributed smooth client objectives are combined with a shared, possibly nonsmooth, regularizer. We propose Relative-Error Composite FedDyn (REC-FedDyn), an inexact method that supports partial participation, with one server-side proximal evaluation per communication round. REC-FedDyn employs a directly testable relative-error criterion that allows each client to independently control the accuracy of its local subproblem, without a server-side acceptance test. We establish almost-sure iterate convergence for strongly convex and convex problems, almost-sure asymptotic stationarity for nonconvex problems, and communication and parallel gradient-evaluation complexity bounds, all under partial participation and without assuming bounded data heterogeneity. Experiments on image inpainting, fMRI prediction, and neural-network classification with structured regularization demonstrate efficiency and robustness across levels of data heterogeneity and client participation.