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:2610.31728v1 Announce Type: new
Abstract: Muon has become a popular optimizer for large language model training, yet its theoretical foundations remain poorly understood. In this work, we study...
By Xiaoran Chen, Binghui Li, Liwei Wang, Pinyan Lu
arXiv:2604. 00004v2 Announce Type: replace-cross Abstract: The extension of context windows in Large Language Models is typically facilitated by scaling positional encodings followed by lightweight Continual Pre-Training (CPT).
By Ning Yang, Hengyu Zhong, Wentao Wang, Baoliang Tian, Haijun Zhang, Jun Wang
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:2606. 03899v1 Announce Type: new Abstract: Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear.
By Xianliang Li, Zihan Zhang, Weiyang Liu, Han Bao
arXiv:2610.27674v1 Announce Type: new
Abstract: Momentum is central to stochastic optimization in deep learning, including the training of large language models. Classical momentum methods typically...
By Haifeng Xu, Yijin Ren, Qi Deng, Chenyu Zhou, Dongdong Ge, Yinyu Ye