MONA is a new optimizer that extends the Muon optimizer by adding a Nesterov‑style acceleration term derived from an exponential moving average of gradient differences. The paper provides a convergence analysis showing that this term offers curvature‑aware corrections while maintaining Muon’s spectral‑norm regularization. Empirical results demonstrate that MONA outperforms both Muon and AdamW on Mixture‑of‑Experts pretraining across models ranging from 1 B to 68 B parameters, and achieves state‑of‑the‑art performance on downstream benchmarks after fine‑tuning the largest model.
By Jiacheng Li, Jianchao Tan, Hongtao Xu, Jiaqi Zhang, Yifan Lu, Yerui Sun, Yuchen Xie, Xunliang Cai
Musec introduces MomentUm SpEctral Clipping, an optimizer-level, architecture‑agnostic technique that replaces Muon’s spectral flattening with selective spectral clipping to stabilize training. By clipping singular values above a threshold while preserving the momentum’s spectral structure, Musec addresses loss spikes and unbounded weight growth without requiring architecture‑specific changes. Soft Musec, an efficient implementation using smooth spectral saturation via coupled Newton‑Schulz iterations, offers convergence guarantees in nonconvex nonsmooth stochastic optimization and empirically improves stability across diverse learning rates and model sizes.
By Zhuanghua Liu, Menglian Wang, Luo Luo
arXiv:2605. 22432v2 Announce Type: replace Abstract: Modern deep learning commonly relies on AdamW with prescribed learning rate schedules, but recent works challenge both components: Schedule-Free optimization removes explicit schedules via iterate averaging, and Muon improves the update geometry by orthogonalizing momentum for matrix parameters.
By Jueun Kim, Baekrok Shin, Jihun Yun, Beomhan Baek, Minhak Song, Chulhee Yun
The paper investigates why the orthogonal optimiser Muon outperforms Adam in large language model pretraining by analysing the spectral properties of Transformer loss landscapes. It finds that Muon’s momentum buffers exhibit an anisotropic spectral profile with a volatile head and a tolerant bulk, enabling larger effective step sizes. Building on this insight, the authors propose Spectral‑Aware Muon (SAMuon) and a lightweight variant, which adjust the bulk scaling while keeping the head unchanged, achieving 13–24 % fewer training tokens than Muon without extra FLOPs.
By Xiaodong Wu, Wenyi Yu, Chao Zhang, Philip Woodland
arXiv:2605. 17109v3 Announce Type: replace-cross Abstract: In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly.
By Fangzhou Wu, Rikhav Shah, Sandeep Silwal, Qiuyi Zhang
AF‑Muon is an AdamW‑free extension of the Muon optimizer that retains Muon’s matrix update for hidden weights while applying a support‑aware finite‑cap linear minimization oracle to tied vocabulary tables and an RMS‑normalized update for one‑dimensional auxiliary parameters. This design eliminates second‑moment state, reducing optimizer‑state memory by about 20% compared to Hybrid Muon. Across nine tied‑token settings—including decoder‑only language models, T5‑style encoder‑decoders, and ImageGPT‑style variants—AF‑Muon consistently improves mean validation loss and perplexity over both Hybrid Muon and a SCION‑style Sign endpoint, with robust gains confirmed by long‑horizon runs and hyperparameter studies.
By Arash Lagzian, Paniz Halvachi, Junming Zhang, Zhouhan Lin, Dianbo Liu
arXiv:2606. 00442v1 Announce Type: new Abstract: Many machine learning techniques rely on approximating a loss function's curvature, but this is notoriously hard to do at the scale of modern deep networks.
By Artem Artemev, Rui Xia, Benjamin M. Boyd, Youjing Yu, Felix Dangel, Guillaume Hennequin, Alberto Bernacchia
arXiv:2607. 09967v1 Announce Type: cross Abstract: Many neural networks operations have a multiplicative nature rather than additive: halving or doubling a norm are analogous relatively but require unequal optimization distances when taking linear steps.
By Ethan Smith
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
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. 25971v1 Announce Type: new Abstract: Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object.
By Alexander H\"agele, Alejandro Hern\'andez-Cano, Atli Kosson, Martin Jaggi
arXiv:2607. 13246v1 Announce Type: cross Abstract: Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training.
By Ali Parviz, Gal Mishne, Alex Cloninger