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arXiv · 2602.21545

MUON+: Towards More Effective Muon via One Additional Normalization Step for LLM Pre-training

Ruijie Zhang·Yequan Zhao·Ziyue Liu·Zhengyang Wang·Yupeng Su·Liyan Tan·Zheng Zhang
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Abstract

Muon has recently emerged as a strong optimizer for large language model pre-training, orthogonalizing the momentum matrix via Newton--Schulz polar iterations. A natural intuition is that polar iterations, by flattening the singular spectrum to all ones, should also eliminate column- and row-wise norm imbalance in the update. We show that this is not true in practice: practical polar steps can substantially amplify the imbalance. We term this the post-polar imbalanced update problem, and prove that such imbalance tightens the second-order term in a blockwise descent analysis, weakening Muon's per-step descent guarantee. Motivated by this analysis, we propose Muon+, a one-line fix that inserts a single normalization step after polar orthogonalization. Muon+ adds no optimizer state. Across pre-training experiments on GPT and LLaMA models from 60M to 7B parameters, spanning both compute-optimal budgets and extended token-to-parameter ratios up to approximately 200, Muon+ consistently outperforms Muon in terms of training and validation perplexity, leading to significant overall pre-training speedup.

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Ruijie Zhang, Yequan Zhao, Ziyue Liu, Zhengyang Wang, Yupeng Su, Liyan Tan, Zheng Zhang. 2026-02-25. MUON+: Towards More Effective Muon via One Additional Normalization Step for LLM Pre-training. https://arxiv.org/abs/2602.21545

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