The paper introduces two variance‑adaptive variants of the Muon optimizer—Muon‑NSR and Muon‑VS—for language model pretraining. Both methods incorporate gradient‑variance information into Muon’s orthogonalization process without adding extra hyperparameters, preserving its spectral normalization structure. Experiments on Llama‑style and GPT‑2 models ranging from 125 M to 1.2 B parameters show that these variants outperform well‑tuned Muon baselines and achieve up to a 1.33× step‑to‑target speedup on Llama‑1.2B.
By Jingru Li, Yibo Fan, Huan Li
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 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:2507. 01598v5 Announce Type: replace Abstract: Muon, a recently proposed optimizer that leverages the inherent matrix structure of neural network parameters, has demonstrated strong empirical performance, indicating its potential as a successor to standard optimizers such as AdamW.
By Naoki Sato, Hiroki Naganuma, Hideaki Iiduka
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:2606. 27153v1 Announce Type: cross Abstract: Matrix-orthogonalization-based optimizers, exemplified by Muon, have demonstrated strong convergence behavior across a wide range of modern deep learning workloads.
By Vincent Chen, Starrick Liu, Regis Cheng, Dance Yang, Shalfun Li, Ryan Yu, Lucy Liang, Hang Su, Roy Gan, Hao Wang, Qian Wang