Recently, the Muon optimizer has demonstrated strong efficiency in large model training through matrix orthogonalization. At each optimization step, Muon orthogonalizes the raw update matrix via Newton-Schulz (NS) iterations before applying it to model parameters. However, orthogonalization requires repeated matrix multiplications whose computational cost scales with O(n³) for an n× n matrix, making the overhead prohibitive at large model scales. In this paper, we observe that the geometric angle derived from cosine similarity between consecutive orthogonal update matrices remains highly stable during training, a phenomenon that we partially explain theoretically and support empirically. Based on this angular stability, we propose LazyMuon, which skips computationally expensive NS iterations through intermittent linear interpolation and employs a residual correction mechanism to mitigate accumulated approximation errors. This design ensures fast iteration while preserving Muon’s orthogonal manifold constraints. Empirical evaluations on language and vision tasks demonstrate that LazyMuon achieves speedups of up to 1.34× and 1.31× over Muon, respectively, with negligible degradation in convergence.

The emergence of large language models (LLMs) has revolutionized AI development, yet their resource demands beyond a single cluster or even datacenter, limiting accessibility to well-resourced organizations. Decentralized training has emerged as a promising paradigm to leverage dispersed resources across clusters, datacenters and even regions, offering the potential to democratize LLM development for broader communities. As the first comprehensive exploration of this emerging field, we present decentralized LLM training as a resource-driven paradigm and categorize existing efforts into community-driven and organizational approaches. We further clarify this through: (1) a comparison with related paradigms, (2) characterization of decentralized resources, and (3) a taxonomy of recent advancements. We also provide up-to-date case studies and outline future directions to advance research in decentralized LLM training.