LionMuon is a new optimizer that alternates between Lion’s sign-based updates and Muon’s spectral matrix-sign updates on a fixed period P, sharing a single dual-EMA momentum buffer. This design keeps the memory footprint the same as Lion and half that of AdamW while reducing the average iteration cost compared to Muon. Experiments on 124M, 355M, and 720M models show LionMuon Pareto-dominates Muon, Lion, Signum, and AdamW across datasets and architectures, achieving lower validation loss with less compute.
By Arman Bolatov, Artem Riabinin, Nikita Kornilov, Andrey Veprikov, Samuel Horv\'ath, Martin Tak\'a\v{c}, Aleksandr Beznosikov
arXiv:2607. 29674v1 Announce Type: cross Abstract: 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.
By Maria Smirnova, Alexey Kravatskiy
The paper introduces a physical response-and-memory model for the Muon optimizer, explaining its semi‑orthogonalized momentum update as the maximally dissipative direction under an output‑side safety budget. It treats the weight matrix as a responsive medium with internal stress, showing that momentum corresponds to accumulated stress whose relaxation occurs over multiple timescales—fast and slow. Based on this, the authors propose the Bi‑Maxwell optimizer, which uses a two‑timescale memory kernel and achieves target loss in fewer steps on a public large‑language‑model benchmark.
By Yinze Hu, Hongjun Xiang, Xingao Gong, Hongyu Yu
arXiv:2605. 17109v3 Announce Type: replace-cross Abstract: In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly.
By Fangzhou Wu, Rikhav Shah, Sandeep Silwal, Qiuyi Zhang
arXiv:2607. 20512v1 Announce Type: cross Abstract: The Muon optimizer reaches the grokking threshold on modular arithmetic faster than AdamW.
By Yufeng Wang
arXiv:2505. 23725v3 Announce Type: replace Abstract: DiLoCo is a powerful framework for training large language models (LLMs), enabling larger optimal batch sizes and increased accelerator utilization under networking constraints.
By Benjamin Th\'erien, Xiaolong Huang, Aaron Defazio, Irina Rish, Eugene Belilovsky
arXiv:2608. 03941v1 Announce Type: new Abstract: Muon is a recent optimizer that orthogonalizes the update to each weight matrix with a Newton-Schulz iteration, which performs steepest descent under the spectral norm.
By Arslan Battalov, Karim Kramin, Alexander Markotenko, Sofia Sinitsina
arXiv:2607. 19771v1 Announce Type: cross Abstract: Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood.
By Jiachun Li
arXiv:2607. 01678v1 Announce Type: new Abstract: Communication increasingly dominates the cost of Large Language Model (LLM) pre-training, especially under data-parallel and sharded training schemes, where gradient synchronization and parameter reconstruction overhead increase with model size and system scale.
By Mingkai Zheng, Junlin Chen, Haotian Xie, Zhao Zhang
arXiv:2606. 04058v1 Announce Type: cross Abstract: Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon.
By Gagik Magakyan, Pablo Parrilo, Asuman Ozdaglar
arXiv:2608.20818v1 Announce Type: cross
Abstract: The matrix-aware optimizer Muon improves large model training by balancing updates across singular directions, yet its scaling behavior and end-to-en...
By Chenghao Li, Xiao Han, Xinxin Huang, Wei Liu, Boyang Li, Bing Xiao, Heran Zhang, Juanma Perez Rua, Ke Xu, Kangning Liu, Linjun Kuang, Na Li, Tan Wang, Tian Xie, Wei Peng, Yang Pei, Yifan Xu, Yuanhao Zhai, Yuwei Lin, Zhe Wang, Zihao He, Daniel Li, Junbiao Tang, Ziyang Jiang, Dake Chen
The paper investigates why the orthogonal optimiser Muon outperforms Adam in large language model pretraining by analysing the spectral properties of Transformer loss landscapes. It finds that Muon’s momentum buffers exhibit an anisotropic spectral profile with a volatile head and a tolerant bulk, enabling larger effective step sizes. Building on this insight, the authors propose Spectral‑Aware Muon (SAMuon) and a lightweight variant, which adjust the bulk scaling while keeping the head unchanged, achieving 13–24 % fewer training tokens than Muon without extra FLOPs.
By Xiaodong Wu, Wenyi Yu, Chao Zhang, Philip Woodland