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Muon has emerged as a strong competitor to AdamW for language model pre-training, yet its behavior at scale is sensitive to weight decay. Recent work has observed that, for Muon without decoupled weight decay, the spectral norm of weight…

Machine Learning · Computer Science 2026-05-12 Kai Lion , Florian Hübler , Bingcong Li , Antonio Orvieto , Niao He

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

We present a comprehensive theoretical and empirical study of the Muon optimizer for training transformers only with a small to medium decoder (30M - 200M parameters), with an emphasis on its mathematical foundations, convergence properties…

Machine Learning · Computer Science 2025-09-30 Sushant Mehta , Raj Dandekar , Rajat Dandekar , Sreedath Panat

Matrix-based optimizers have attracted growing interest for improving LLM training efficiency, with significant progress centered on orthogonalization/whitening based methods. While yielding substantial performance gains, a fundamental…

Machine Learning · Computer Science 2026-02-10 Wenbo Gong , Javier Zazo , Qijun Luo , Puqian Wang , James Hensman , Chao Ma

A framework is introduced for solving a sequence of slowly changing optimization problems, including those arising in regression and classification applications, using optimization algorithms such as stochastic gradient descent (SGD). The…

Machine Learning · Computer Science 2015-09-25 Craig Wilson , Venugopal V. Veeravalli

Large models recently are widely applied in artificial intelligence, so efficient training of large models has received widespread attention. More recently, a useful Muon optimizer is specifically designed for matrix-structured parameters…

Machine Learning · Computer Science 2025-09-22 Feihu Huang , Yuning Luo , Songcan Chen

We propose a framework for online meta-optimization of parameters that govern optimization, called Amortized Proximal Optimization (APO). We first interpret various existing neural network optimizers as approximate stochastic proximal point…

Machine Learning · Computer Science 2022-03-02 Juhan Bae , Paul Vicol , Jeff Z. HaoChen , Roger Grosse

Sign-based optimization algorithms, such as SignSGD and Muon, have garnered significant attention for their remarkable performance in training large foundation models. Despite this empirical success, we still lack a theoretical…

Machine Learning · Computer Science 2026-05-08 Hongyi Tao , Dingzhi Yu , Lijun Zhang

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint…

Optimization and Control · Mathematics 2023-11-01 Shiyi Jiang , Jianqiang Cheng , Kai Pan , Zuo-Jun Max Shen

Low-rank gradient-based optimization methods have significantly improved memory efficiency during the training of large language models (LLMs), enabling operations within constrained hardware without sacrificing performance. However, these…

Machine Learning · Computer Science 2025-10-28 Yehonathan Refael , Guy Smorodinsky , Tom Tirer , Ofir Lindenbaum

A new adaptive hybrid optimization strategy, entitled squads, is proposed for complex inverse analysis of computationally intensive physical models. The new strategy is designed to be computationally efficient and robust in identification…

Geophysics · Physics 2015-06-03 Velimir V. Vesselinov , Dylan R. Harp

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

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common…

Machine Learning · Computer Science 2025-09-05 Thomas R. Harvey

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…

The Muon optimizer enjoys strong empirical performance and theoretical grounding. However, the super-linear cost of its orthonormalization step introduces increasing overhead with scale. To alleviate this cost, several works have attempted…

Machine Learning · Computer Science 2025-12-22 Kwangjun Ahn , Noah Amsel , John Langford

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…

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

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

Training large language models (LLMs) relies almost exclusively on dense adaptive optimizers with increasingly sophisticated preconditioners. We challenge this by showing that randomly masking parameter updates can be highly effective, with…

Machine Learning · Computer Science 2026-02-18 Taejong Joo , Wenhan Xia , Cheolmin Kim , Ming Zhang , Eugene Ie

Training large language models (LLMs) relies on adaptive optimizers such as Adam, which introduce extra operations and require significantly more memory to maintain first- and second-order moments than SGD. While recent works such as…

Machine Learning · Computer Science 2026-05-22 Athanasios Glentis , Jiaxiang Li , Andi Han , Mingyi Hong