Federated learning (FL) is a challenging setting for optimization due to the heterogeneity of the data across different clients which can cause a client drift phenomenon.

Federated learning (FL) has emerged as an enabling framework for communication-efficient decentralized training.

Biased gradient compression with error feedback (EF) reduces communication in federated learning (FL), but under non-IID data, the residual error can decay slowly, causing gradient mismatch and stalled progress in the early rounds. We propose step-ahead partial error feedback (SA-PEF), which integrates step-ahead (SA) correction with partial error feedback (PEF). SA-PEF recovers EF when the step-ahead coefficient α = 0 and step-ahead EF (SAEF) when α = 1. For non-convex objectives and δ-contractive compressors, we establish a second-moment bound and a residual recursion that guarantee convergence to stationar-ity under heterogeneous data and partial client participation. To balance SAEF's rapid warm-up with EF's long-term stability, we select α near its theory-predicted optimum. Experiments across diverse architectures and datasets show that SA-PEF consistently reaches target accuracy faster than EF. Modern large-scale machine learning increasingly relies on distributed computation, where both data and compute are spread across many devices. Federated learning (FL) enables model training in this setting without centralizing raw data, enhancing privacy and scalability under heterogeneous client distributions (McMahan et al., 2017; Kairouz et al., 2021). In each synchronous FL round, the server broadcasts the current global model to a subset of clients. These clients perform several steps of stochastic gradient descent (SGD) on their local data and return updates to the server, which aggregates them to form the next global iterate (Huang et al., 2022; Wang & Ji, 2022; Li et al., 2024). Although FL leverages rich distributed data, it faces two key challenges.

Decentralized optimization is emerging as a viable alternative for scalable distributed machine learning, but also introduces new challenges in terms of synchronization costs. To this end, several communication-reduction techniques, such as non-blocking communication, quantization, and local steps, have been explored in the decentralized setting. Due to the complexity of analyzing optimization in such a relaxed setting, this line of work often assumes \emph{global} communication rounds, which require additional synchronization. In this paper, we consider decentralized optimization in the simpler, but harder to analyze, \emph{asynchronous gossip} model, in which communication occurs in discrete, randomly chosen pairings among nodes. Perhaps surprisingly, we show that a variant of SGD called \emph{SwarmSGD} still converges in this setting, even if \emph{non-blocking communication}, \emph{quantization}, and \emph{local steps} are all applied \emph{in conjunction}, and even if the node data distributions and underlying graph topology are both \emph{heterogenous}. Our analysis is based on a new connection with multi-dimensional load-balancing processes. We implement this algorithm and deploy it in a super-computing environment, showing that it can outperform previous decentralized methods in terms of end-to-end training time, and that it can even rival carefully-tuned large-batch SGD for certain tasks.

Recent works have explored the use of momentum in local methods to enhance distributed SGD. This is particularly appealing in Federated Learning (FL), where momentum intuitively appears as a solution to mitigate the effects of statistical heterogeneity. Despite recent progress in this direction, it is still unclear if momentum can guarantee convergence under unbounded heterogeneity in decentralized scenarios, where only some workers participate at each round. In this work we analyze momentum under cyclic client participation, and theoretically prove that it remains inevitably affected by statistical heterogeneity. Similarly to SGD, we prove that decreasing step-sizes do not help either: in fact, any schedule decreasing faster than $Θ\left(1/t\right)$ leads to convergence to a constant value that depends on the initialization and the heterogeneity bound. Numerical results corroborate the theory, and deep learning experiments confirm its relevance for realistic settings.

Split Federated Learning is a system-efficient federated learning paradigm that leverages the rich computing resources at a central server to train model partitions. Data heterogeneity across silos, however, presents a major challenge undermining the convergence speed and accuracy of the global model. This paper introduces Step-wise Momentum Fusion (SMoFi), an effective and lightweight framework that counteracts gradient divergence arising from data heterogeneity by synchronizing the momentum buffers across server-side optimizers. To control gradient divergence over the training process, we design a staleness-aware alignment mechanism that imposes constraints on gradient updates of the server-side submodel at each optimization step. Extensive validations on multiple real-world datasets show that SMoFi consistently improves global model accuracy (up to 7.1%) and convergence speed (up to 10.25$\times$). Furthermore, SMoFi has a greater impact with more clients involved and deeper learning models, making it particularly suitable for model training in resource-constrained contexts.