Compressed Composite Federated Algorithm with Error Feedback

Federated composite learning—where a smooth empirical loss is augmented with a non-smooth regularizer—is a key paradigm for learning sparse, interpretable models over bandwidth-limited edge networks. Existing approaches, however, impose heavy client-side computation because they require repeated proximal updates on each device, and they incur prohibitive communication costs when transmitting dense, high-dimensional parameters. In this paper we propose FedCEF, a server-centric, communication-efficient algorithm that jointly tackles gradient compression and non-smooth regularization. FedCEF delegates the proximal mapping entirely to the server, eliminating all proximal evaluations from clients and drastically reducing on-device computation. Our algorithm accommodates weakly convex regularizers, significantly broadening the applicability beyond convex regularizers. The algorithm further integrates an error-feedback mechanism that compensates for gradient compression, simultaneously cutting communication overhead and suppressing compression-induced bias. Under standard assumptions we prove that FedCEF converges to a neighborhood of a stationary point for non-convex composite objectives. Numerical results also validate the efficiency of our algorithm.