Communication-Efficient Gluon in Federated Learning

SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget. It outperforms SignSGD in practice, yet it can ascend even on a linear function. Signing the gradient before the Linear Minimization Oracle (LMO), rather than after, does not repair this: we construct a small explicit instance on which sign-before (MuonUSign) and sign-on-both-sides (MuonSign) ascend as well, so no placement of the sign around the oracle descends in general. Error feedback, the standard remedy for a biased compressor, does not rescue SignMuon: when applied to Muon's output, error feedback can fail for every smoothness constant, step size, and momentum. Applied to the gradient, error feedback does work, and EF21-MuonUSign and EF21-MuonSign attain the standard $\mathcal{O}(T^{-1/2})$ rate for the squared gradient norm on smooth nonconvex problems, the latter at one bit in each direction. Experiments then reverse the ordering: across centralized CIFAR-10, federated CIFAR-10, and the nanoGPT speedrun, the strongest compressed method is consistently sign-after-the-LMO, precisely the placement we prove divergent, with the provably convergent variants trailing it. Compressing after the LMO, a heuristic, matters more at these scales than the guarantee does.

Agnostic federated learning (AFL) seeks a model that performs reliably across $m$ heterogeneous workers, but communication remains a bottleneck. We improve communication efficiency by reducing the number of synchronization rounds via faster convergence and the communication cost per round via compression. We first propose AFL-BR, which updates the dual weights over workers using online mirror ascent with KL divergence and blockwise restarts. It achieves an $O((\log m)^{1/4}T^{-1/8})$ stationarity rate after $T$ update rounds, reducing the $m$-dependence of the synchronization rounds required for convergence from polynomial to logarithmic order. Building on AFL-BR, we develop AFL-Com by applying bidirectional compression with error feedback (EF). Instead of compressing local gradients, workers apply EF to their dual-weighted gradients, enabling direct control of the aggregated compression error under time-varying weights. We then establish an $O((δ^{-1}+(\log m)^{1/4})T^{-1/8})$ stationarity rate for AFL-Com under general $δ$-approximate compressors and improve the $δ$-dependence from $δ^{-1}$ to $δ^{-1/2}$ for additive-and-idempotent compressors with shared randomness (SR). With suitable compression levels, AFL-Com retains the same convergence rate as AFL-BR at a lower per-round communication cost, yielding reductions in total communication complexity by factors of $(\log m)^{1/4}$ with Top-$k$ and $(\log m)^{1/2}$ with Rand-$k$ and SR. Experiments validate the improved synchronization and communication efficiency of our methods.

Federated fine-tuning adapts large language models (LLMs) to decentralized client data, but its scalability in cross-device training is often limited by the high communication cost. Muon is an optimizer that improves optimization performance by orthogonalizing momentum for matrix-valued parameters. Existing federated Muon methods demonstrate the benefit of matrix-aware optimization in federated learning, but still require transmitting full layer-size updates and optimizer state. A natural way to reduce communication is to directly apply Muon to LoRA factors, but this changes the optimized object and weakens Muon's matrix-aware update geometry. We propose FedSubMuon, a communication-efficient federated Muon fine-tuning method that optimizes compact coefficient matrices within shared structured subspaces. This design keeps Muon on a single matrix-valued trainable object, while reducing the client upload to compact coefficient matrices. We further introduce FedSubMuon-GT, an accuracy-oriented extension that uses projected gradients to adapt tracked subspace bases toward task-relevant gradient directions. Experiments on instruction tuning and mathematical reasoning show that FedSubMuon-GT achieves the best overall accuracy on four of five dataset-model pairs, while FedSubMuon performs best under all matched communication budgets. On Dolly-15K, the closest communication baseline requires 5.5 times and 1.4 times more total communication on Llama-1B and Qwen-4B, respectively.