@inproceedings{li-etal-2026-variance,
title = "Variance-Adaptive Muon: Pre-Orthogonalization Variance Modulation for Efficient Language Model Pretraining",
author = "li, Jingru and
Fan, Yibo and
Li, Huan",
editor = "Lee, Hung-Yi and
Dev, Sunipa and
Li, Jessy and
Elliott, Desmond",
booktitle = "Findings of the {A}ssociation for {C}omputational {L}inguistics: {EMNLP} 2026",
month = oct,
year = "2026",
address = "Budapest, Hungary",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2026.findings-emnlp.1780/",
pages = "38060--38081",
ISBN = "979-8-89176-462-0",
abstract = "Optimizer design plays a central role in efficient language model pretraining, directly affecting optimization dynamics, convergence speed, and compute cost under fixed training budgets. Muon has emerged as a strong optimizer by orthogonalizing momentum updates, yielding a matrix-valued analogue of sign-based normalization. However, unlike Adam-style methods, Muon does not explicitly incorporate gradient-variance information into its updates. Motivated by Adam{'}s variance-adaptive interpretation, we propose Muon-NSR and Muon-VS, two variance-adaptive Muon variants for language model pretraining. Muon-NSR applies noise-to-signal ratio (NSR) modulation before Newton{--}Schulz orthogonalization, whereas Muon-VS uses variance scaling (VS) without introducing any additional hyperparameters beyond those of Muon. Both methods preserve Muon{'}s spectral normalization structure while requiring only one additional variance buffer. Experiments on Llama-style and GPT-2 pretraining across model scales from 125M to 1.2B parameters show that our methods improve over well-tuned Muon baselines and remain competitive with representative adaptive Muon-family baselines. On Llama-1.2B, Muon-VS achieves a 1.33{\texttimes} step-to-target speedup over a well-tuned Muon baseline, with Muon{'}s final validation loss as the target. These results indicate that variance-adaptive modulation is a simple and effective mechanism for improving Muon-style optimizers in language model pretraining."
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<abstract>Optimizer design plays a central role in efficient language model pretraining, directly affecting optimization dynamics, convergence speed, and compute cost under fixed training budgets. Muon has emerged as a strong optimizer by orthogonalizing momentum updates, yielding a matrix-valued analogue of sign-based normalization. However, unlike Adam-style methods, Muon does not explicitly incorporate gradient-variance information into its updates. Motivated by Adam’s variance-adaptive interpretation, we propose Muon-NSR and Muon-VS, two variance-adaptive Muon variants for language model pretraining. Muon-NSR applies noise-to-signal ratio (NSR) modulation before Newton–Schulz orthogonalization, whereas Muon-VS uses variance scaling (VS) without introducing any additional hyperparameters beyond those of Muon. Both methods preserve Muon’s spectral normalization structure while requiring only one additional variance buffer. Experiments on Llama-style and GPT-2 pretraining across model scales from 125M to 1.2B parameters show that our methods improve over well-tuned Muon baselines and remain competitive with representative adaptive Muon-family baselines. On Llama-1.2B, Muon-VS achieves a 1.33× step-to-target speedup over a well-tuned Muon baseline, with Muon’s final validation loss as the target. These results indicate that variance-adaptive modulation is a simple and effective mechanism for improving Muon-style optimizers in language model pretraining.</abstract>
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%0 Conference Proceedings
%T Variance-Adaptive Muon: Pre-Orthogonalization Variance Modulation for Efficient Language Model Pretraining
%A li, Jingru
%A Fan, Yibo
%A Li, Huan
%Y Lee, Hung-Yi
%Y Dev, Sunipa
%Y Li, Jessy
%Y Elliott, Desmond
%S Findings of the Association for Computational Linguistics: EMNLP 2026
%D 2026
%8 October
%I Association for Computational Linguistics
%C Budapest, Hungary
%@ 979-8-89176-462-0
%F li-etal-2026-variance
%X Optimizer design plays a central role in efficient language model pretraining, directly affecting optimization dynamics, convergence speed, and compute cost under fixed training budgets. Muon has emerged as a strong optimizer by orthogonalizing momentum updates, yielding a matrix-valued analogue of sign-based normalization. However, unlike Adam-style methods, Muon does not explicitly incorporate gradient-variance information into its updates. Motivated by Adam’s variance-adaptive interpretation, we propose Muon-NSR and Muon-VS, two variance-adaptive Muon variants for language model pretraining. Muon-NSR applies noise-to-signal ratio (NSR) modulation before Newton–Schulz orthogonalization, whereas Muon-VS uses variance scaling (VS) without introducing any additional hyperparameters beyond those of Muon. Both methods preserve Muon’s spectral normalization structure while requiring only one additional variance buffer. Experiments on Llama-style and GPT-2 pretraining across model scales from 125M to 1.2B parameters show that our methods improve over well-tuned Muon baselines and remain competitive with representative adaptive Muon-family baselines. On Llama-1.2B, Muon-VS achieves a 1.33× step-to-target speedup over a well-tuned Muon baseline, with Muon’s final validation loss as the target. These results indicate that variance-adaptive modulation is a simple and effective mechanism for improving Muon-style optimizers in language model pretraining.
%U https://aclanthology.org/2026.findings-emnlp.1780/
%P 38060-38081
Markdown (Informal)
[Variance-Adaptive Muon: Pre-Orthogonalization Variance Modulation for Efficient Language Model Pretraining](https://aclanthology.org/2026.findings-emnlp.1780/) (li et al., Findings 2026)
ACL