Achieving Linear Speedup for Composite Federated Learning

Abstract: This paper proposes FedNMap, a normal map-based method for composite federated learning, where the objective consists of a smooth loss and a possibly nonsmooth regularizer. FedNMap leverages a normal map-based update scheme to handle the nonsmooth term and incorporates a local correction strategy to mitigate the impact of data heterogeneity across clients. Under standard assumptions, including smooth local losses, weak convexity of the regularizer, and bounded stochastic gradient variance, FedNMap achieves linear speedup with respect to both the number of clients $n$ and the number of local updates $Q$ for nonconvex losses, both with and without the Polyak-Łojasiewicz (PL) condition. To our knowledge, this is the first result establishing linear speedup for nonconvex composite federated learning.