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The Muon optimizer has demonstrated remarkable empirical success in handling matrix-structured parameters for training neural networks. However, a significant gap remains between its practical performance and theoretical understanding.…

Machine Learning · Computer Science 2026-05-12 Da Chang , Yongxiang Liu , Ganzhao Yuan

The recently introduced optimizer, Muon, has gained increasing attention due to its superior performance across a wide range of applications. However, its effectiveness in federated learning remains unexplored. To address this gap, this…

Machine Learning · Computer Science 2025-10-07 Xinwen Zhang , Hongchang Gao

To define a steepest descent method over a neural network, we need to choose a norm for each layer, a way to aggregate these norms across layers, and whether to use normalization. We systematically explore different alternatives for…

Machine Learning · Computer Science 2025-10-14 Michael Crawshaw , Chirag Modi , Mingrui Liu , Robert M. Gower

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral…

Batch normalization (BN) is an important technique commonly incorporated into deep learning models to perform standardization within mini-batches. The merits of BN in improving a model's learning efficiency can be further amplified by…

Machine Learning · Computer Science 2021-04-07 Lei Huang , Yi Zhou , Li Liu , Fan Zhu , Ling Shao

The Muon optimizer has demonstrated strong empirical performance in pre-training large language models by performing matrix-level gradient (or momentum) orthogonalization in each layer independently. In this work, we propose TEON, a…

Machine Learning · Computer Science 2026-02-03 Ruijie Zhang , Yequan Zhao , Ziyue Liu , Zhengyang Wang , Dongyang Li , Yupeng Su , Sijia Liu , Zheng Zhang

The growing adoption of spectrum-aware matrix-valued optimizers such as Muon and Shampoo in deep learning motivates a systematic study of their generalization properties and, in particular, when they might outperform competitive algorithms.…

Machine Learning · Computer Science 2025-12-15 Bhavya Vasudeva , Puneesh Deora , Yize Zhao , Vatsal Sharan , Christos Thrampoulidis

Muon-style optimizers take a matrix-valued momentum or preconditioned update $B = U \operatorname{diag}(\sigma_1,\ldots,\sigma_r) V^\top$ and replace it with its canonical partial polar factor $\operatorname{Pol}(B) = U V^\top$. This maps…

Machine Learning · Computer Science 2026-05-27 Albert Yi

Distributed training of large neural networks is bottlenecked by full-precision gradient communication and by coordinatewise optimizers that ignore the matrix structure of weight tensors. We propose Sign-Muon, a 1-bit, matrix-aware…

Machine Learning · Computer Science 2026-05-19 Neel Mishra , Kushagara Trivedi , Pawan Kumar

Recent studies have revealed some issues of Multi-Head Attention (MHA), e.g., redundancy and over-parameterization. Specifically, the heads of MHA were originally designed to attend to information from different representation subspaces,…

Machine Learning · Computer Science 2023-10-17 Jinjie Ni , Rui Mao , Zonglin Yang , Han Lei , Erik Cambria

Large Language Models (LLMs) achieve competitive performance across diverse natural language processing (NLP) tasks, yet pretraining is computationally demanding, making optimizer efficiency an important practical consideration. Muon…

Machine Learning · Computer Science 2026-01-22 Jingru Li , Yibo Fan , Huan Li

Muon optimizer has demonstrated robust results in pretraining of language models but its performance in finetuning of existing public pretrained models is not yet explored. Currently, Muon is used along with AdamW introducing a scope of…

Computation and Language · Computer Science 2025-11-11 Saurabh Page , Advait Joshi , S. S. Sonawane

Modern machine learning models are prone to over-reliance on spurious correlations, which can often lead to poor performance on minority groups. In this paper, we identify surprising and nuanced behavior of finetuned models on worst-group…

Machine Learning · Computer Science 2024-10-29 Tyler LaBonte , John C. Hill , Xinchen Zhang , Vidya Muthukumar , Abhishek Kumar

Muon has recently emerged as a competitive alternative to AdamW for large-scale pre-training, with orthogonalization via Newton-Schulz (NS) iterations as its core operation. Existing Muon variants apply a uniform NS schedule to all…

Machine Learning · Computer Science 2026-05-19 Xinlin Zhuang , Panyi Ouyang , Yichen Li , Jiangming Shi , Yizhang Chen , Shuman Liu , Ying Qian , Weiyang Liu , Haibo Zhang , Imran Razzak

The pursuit of faster optimization algorithms remains an active and important research direction in deep learning. Recently, the Muon optimizer [JJB+24] has demonstrated promising empirical performance, but its theoretical foundation…

Machine Learning · Computer Science 2025-09-30 Lizhang Chen , Jonathan Li , Qiang Liu

The muon optimizer has picked up much attention as of late as a possible replacement to the seemingly omnipresent Adam optimizer. Recently, care has been taken to document the scaling laws of hyper-parameters under muon such as weight decay…

Machine Learning · Computer Science 2025-05-09 Devan Selvaraj

In this note, we inspect the convergence of a new optimizer for pretraining LLMs, namely the Muon optimizer. Such an optimizer is closely related to a specialized steepest descent method where the update direction is the minimizer of the…

Optimization and Control · Mathematics 2025-06-03 Jiaxiang Li , Mingyi Hong

Muon has emerged as an efficient alternative to Adam for pretraining, yet remains underused for fine-tuning. A key obstacle is that most open models are pretrained with Adam, and naively switching to Muon for fine-tuning leads to degraded…

Machine Learning · Computer Science 2026-05-12 Xingyu Qu , Peigeng Huang , Samuel Horvath

Muon-style optimizers leverage Newton-Schulz (NS) iterations to orthogonalize updates, yielding update geometries that often outperform Adam-series methods. However, this orthogonalization discards magnitude information, rendering training…

Machine Learning · Computer Science 2026-03-10 Peng Cheng , Jiucheng Zang , Qingnan Li , Liheng Ma , Yufei Cui , Yingxue Zhang , Boxing Chen , Ming Jian , Wen Tong

Optimizers play an important role in both pretraining and finetuning stages when training large language models (LLMs). In this paper, we present an observation that full finetuning with the same optimizer as in pretraining achieves a…

Machine Learning · Computer Science 2026-05-08 Yuxing Liu , Jianyu Wang , Tong Zhang