Federated learning (FL) is a machine learning paradigm that targets model training without gathering the local data dispersed over various data sources. Standard FL, which employs a single server, can only support a limited number of users, leading to degraded learning capability. In this work, we consider a multi-server FL framework, referred to as \emph{Confederated Learning} (CFL), in order to accommodate a larger number of users. A CFL system is composed of multiple networked edge servers, with each server connected to an individual set of users. Decentralized collaboration among servers is leveraged to harness all users' data for model training. Due to the potentially massive number of users involved, it is crucial to reduce the communication overhead of the CFL system. We propose a stochastic gradient method for distributed learning in the CFL framework. The proposed method incorporates a conditionally-triggered user selection (CTUS) mechanism as the central component to effectively reduce communication overhead. Relying on a delicately designed triggering condition, the CTUS mechanism allows each server to select only a small number of users to upload their gradients, without significantly jeopardizing the convergence performance of the algorithm. Our theoretical analysis reveals that the proposed algorithm enjoys a linear convergence rate. Simulation results show that it achieves substantial improvement over state-of-the-art algorithms in terms of communication efficiency.

We study decentralized non-convex finite-sum minimization problems described over a network of nodes, where each node possesses a local batch of data samples. We propose a single-timescale first-order randomized incremental gradient method, termed as GT-SAGA. GT-SAGA is computationally efficient since it evaluates only one component gradient per node per iteration and achieves provably fast and robust performance by leveraging node-level variance reduction and network-level gradient tracking. For general smooth non-convex problems, we show almost sure and mean-squared convergence to a first-order stationary point and describe regimes of practical significance where GT-SAGA achieves a network-independent convergence rate and outperforms the existing approaches respectively. When the global cost function further satisfies the Polyak-Lojaciewisz condition, we show that GT-SAGA exhibits global linear convergence to an optimal solution in expectation and describe regimes of practical interest where the performance is network-independent and improves upon the existing work. Numerical experiments based on real-world datasets are included to highlight the behavior and convergence aspects of the proposed method.