Compressed Proximal Federated Learning for Non-Convex Composite Optimization on Heterogeneous Data

Abstract: Federated Composite Optimization (FCO) has emerged as a promising framework for training models with structural constraints (e.g., sparsity) in distributed edge networks. However, simultaneously achieving communication efficiency and convergence robustness remains a significant challenge, particularly when dealing with non-smooth regularizers, statistical heterogeneity, and the restrictions of biased compression. To address these issues, we propose FedCEF (Federated Composite Error Feedback), a novel algorithm tailored for non-convex FCO. FedCEF introduces a decoupled proximal update scheme that separates the proximal operator from communication, enabling clients to handle non-smooth terms locally while transmitting compressed information. To mitigate the noise from aggressive quantization and the bias from non-IID data, FedCEF integrates a rigorous error feedback mechanism with control variates. Furthermore, we design a communication-efficient pre-proximal downlink strategy that allows clients to exactly reconstruct global control variables without explicit transmission. We theoretically establish that FedCEF achieves sublinear convergence to a bounded residual error under general non-convexity, which is controllable via the step size and batch size. Extensive experiments on real datasets validate FedCEF maintains competitive model accuracy even under extreme compression ratios (e.g., 1%), significantly reducing the total communication volume compared to uncompressed baselines.