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The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce…

Optimization and Control · Mathematics 2026-04-03 Zhehang Du , Weijie Su

The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become…

Machine Learning · Computer Science 2026-05-27 Jiacheng Li , Jianchao Tan , Hongtao Xu , Jiaqi Zhang , Yifan Lu , Yerui Sun , Yuchen Xie , Xunliang Cai

The Muon optimizer, based on matrix orthogonalization, has recently shown faster convergence and better computational efficiency over AdamW in LLM pre-training. However, the memory overhead of maintaining high-precision optimizer states…

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

Recent empirical research has demonstrated that deep learning optimizers based on the linear minimization oracle (LMO) over specifically chosen Non-Euclidean norm balls, such as Muon and Scion, outperform Adam-type methods in the training…

Optimization and Control · Mathematics 2025-12-19 Xun Qian , Hussein Rammal , Dmitry Kovalev , Peter Richtárik

Gradient orthogonalization is a simple strategy that shows great utility in speeding up gradient descent. The Muon optimizer (Jordan, Jin, et al., 2024) combines gradient orthogonalization with first-order momentum and achieves significant…

Machine Learning · Computer Science 2025-10-21 Ahmed Khaled , Kaan Ozkara , Tao Yu , Mingyi Hong , Youngsuk Park

Muon has emerged as a promising optimizer for large-scale foundation model pre-training by exploiting the matrix structure of neural network updates through iterative orthogonalization. However, its practical efficiency is limited by the…

Machine Learning · Computer Science 2026-04-14 Ziyue Liu , Ruijie Zhang , Zhengyang Wang , Yequan Zhao , Yupeng Su , Zi Yang , Zheng Zhang

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

This paper investigates the impact of different optimizers on the grokking phenomenon, where models exhibit delayed generalization. We conducted experiments across seven numerical tasks (primarily modular arithmetic) using a modern…

Machine Learning · Computer Science 2025-04-23 Amund Tveit , Bjørn Remseth , Arve Skogvold

Muon orthogonalizes matrix updates, but multi-head attention naturally operates at the level of heads. This granularity mismatch raises the question of whether Muon should be applied to the full attention projection, to individual heads, or…

Machine Learning · Computer Science 2026-05-12 Hongtao Zhang , Wenjie Zhou , Wei Chen , Xueqi Cheng

Recommender systems (RecSys) are increasingly emphasizing scaling, leveraging larger architectures and more interaction data to improve personalization. Yet, despite the optimizer's pivotal role in training, modern RecSys pipelines almost…

Information Retrieval · Computer Science 2026-03-03 Rong Shan , Aofan Yu , Bo Chen , Kuo Cai , Qiang Luo , Ruiming Tang , Han Li , Weiwen Liu , Weinan Zhang , Jianghao Lin

Matrix-based preconditioned optimizers, such as Muon, have recently been shown to be more efficient than scalar-based optimizers for training large-scale neural networks, including large language models (LLMs). Recent benchmark studies of…

Machine Learning · Computer Science 2026-02-02 Yifeng Liu , Angela Yuan , Quanquan Gu

Optimizers are crucial for the efficient training of Large Language Models (LLMs). While AdamW is the de facto standard, recent structure-aware optimizers like Muon have emerged, which regularize gradient updates by operating on entire…

Machine Learning · Computer Science 2025-10-07 Zehua Liu , Han Wu , Xiaojin Fu , Shuqi Liu , Xiongwei Han , Tao Zhong , Mingxuan Yuan

Multi-Task Learning (MTL) is a foundational machine learning problem that has seen extensive development over the past decade. Recently, various optimization-based MTL approaches have been proposed to learn multiple tasks simultaneously by…

Machine Learning · Computer Science 2026-04-13 Zhipeng Zhou , Linxiao Cao , Pengcheng Wu , Peilin Zhao , Chunyan Miao

DiLoCo is a powerful framework for training large language models (LLMs), enabling larger optimal batch sizes and increased accelerator utilization under networking constraints. However, DiLoCo's performance has been shown to degrade as the…

Machine Learning · Computer Science 2026-02-26 Benjamin Thérien , Xiaolong Huang , Aaron Defazio , Irina Rish , Eugene Belilovsky

Orthogonal momentum gradient updates have emerged to overcome the limitations of vector-based optimizers like Adam. The vector-based optimizer Adam suffers from high memory costs and ill-conditioned momentum gradient updates. However,…

Machine Learning · Computer Science 2025-12-19 Dipan Maity

Mode connectivity has been widely studied, yet the role of the optimizer remains underexplored. We revisit it through optimizer-induced implicit regularization, asking how connectivity behaves when restricted to solutions constrained by a…

Artificial Intelligence · Computer Science 2026-05-12 Fangzhao Zhang , Sungyoon Kim , Erica Zhang , Yiqi Jiang , Mert Pilanci

Orthogonalized-momentum optimizers such as Muon improve transformer training by approximately whitening/orthogonalizing matrix-valued momentum updates via a short polar-decomposition iteration. However, polar-factor approximations typically…

Machine Learning · Computer Science 2026-03-19 Ben S. Southworth , Stephen Thomas

MLP is a heavily used backbone in modern deep learning (DL) architectures for supervised learning on tabular data, and AdamW is the go-to optimizer used to train tabular DL models. Unlike architecture design, however, the choice of…

Machine Learning · Computer Science 2026-04-20 Yury Gorishniy , Ivan Rubachev , Dmitrii Feoktistov , Artem Babenko

Recently, Muon and related spectral optimizers have demonstrated strong empirical performance as scalable stochastic methods, often outperforming Adam. Yet their behaviour remains poorly understood. We analyze stochastic spectral…

Optimization and Control · Mathematics 2026-05-12 Elliot Paquette , Noah Marshall , Lucas Benigni , Guangyuan Wang , Atish Agarwala , Courtney Paquette