A Decentralized Lens on Centralized Federated Learning: Unifying FL via Inexact Douglas–Rachford Splitting

Federated learning enables collaborative training without pooling client data, but data heterogeneity remains a major challenge, especially when combined with partial participation and inexact local computation, whose joint effects are difficult to capture in a unified analysis of existing methods. To analyze these interactions, we develop an operator-splitting framework on the full client product space with a consensus constraint. Within this framework, a target field defines the residual and energy, while a conditional mean-square error criterion connects the target dynamics to actual client computations. This connection enables convex convergence guarantees under an additional summable conditional-mean condition, as well as nonconvex stationarity guarantees through expected forward–backward-envelope descent. Under local Kurdyka-Łojasiewicz (KL) geometry, almost-sure boundedness, and explicit process and error-tail conditions, the nonconvex analysis further yields finite length and convergence of the full sequence. Applying the abstract algorithm to federated optimization yields the FedExSplit (FES) family, with variants defined by the target potential. We then use an exact heterogeneous quadratic construction to explain how its mean-anchored variant (FES-MA) can suppress the effect of client disagreement on the mean update in a specified parameter regime. Controlled one-step experiments support this mechanism, while training experiments show that FES-MA improves validation accuracy over FedAvg by approximately 1.7 percentage points on FEMNIST and 2.6 percentage points on CIFAR-100 under high heterogeneity (${\alpha }_{\mathrm{LDA}}=0.1$).