@inproceedings{lin-etal-2026-lazymuon,
title = "{L}azy{M}uon: Exploiting Angular Stability to Reduce Newton-Schulz Overhead in Muon for Efficient Large Model Training",
author = "Lin, Kainan and
Lu, Rongwei and
Dong, Haotian and
Chen, Le and
Jiang, Jingyan and
Yang, Huan and
Wang, Zhi",
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.173/",
pages = "3646--3659",
ISBN = "979-8-89176-462-0",
abstract = "Recently, the Muon optimizer has demonstrated strong efficiency in large model training through matrix orthogonalization. At each optimization step, Muon orthogonalizes the raw update matrix via Newton-Schulz (NS) iterations before applying it to model parameters. However, orthogonalization requires repeated matrix multiplications whose computational cost scales with $O(n^3)$ for an $n\times n$ matrix, making the overhead prohibitive at large model scales. In this paper, we observe that the geometric angle derived from cosine similarity between consecutive orthogonal update matrices remains highly stable during training, a phenomenon that we partially explain theoretically and support empirically. Based on this angular stability, we propose LazyMuon, which skips computationally expensive NS iterations through intermittent linear interpolation and employs a residual correction mechanism to mitigate accumulated approximation errors. This design ensures fast iteration while preserving Muon{'}s orthogonal manifold constraints. Empirical evaluations on language and vision tasks demonstrate that LazyMuon achieves speedups of up to 1.34$\times$ and 1.31$\times$ over Muon, respectively, with negligible degradation in convergence."
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<abstract>Recently, the Muon optimizer has demonstrated strong efficiency in large model training through matrix orthogonalization. At each optimization step, Muon orthogonalizes the raw update matrix via Newton-Schulz (NS) iterations before applying it to model parameters. However, orthogonalization requires repeated matrix multiplications whose computational cost scales with O(n³) for an n\times n matrix, making the overhead prohibitive at large model scales. In this paper, we observe that the geometric angle derived from cosine similarity between consecutive orthogonal update matrices remains highly stable during training, a phenomenon that we partially explain theoretically and support empirically. Based on this angular stability, we propose LazyMuon, which skips computationally expensive NS iterations through intermittent linear interpolation and employs a residual correction mechanism to mitigate accumulated approximation errors. This design ensures fast iteration while preserving Muon’s orthogonal manifold constraints. Empirical evaluations on language and vision tasks demonstrate that LazyMuon achieves speedups of up to 1.34\times and 1.31\times over Muon, respectively, with negligible degradation in convergence.</abstract>
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%0 Conference Proceedings
%T LazyMuon: Exploiting Angular Stability to Reduce Newton-Schulz Overhead in Muon for Efficient Large Model Training
%A Lin, Kainan
%A Lu, Rongwei
%A Dong, Haotian
%A Chen, Le
%A Jiang, Jingyan
%A Yang, Huan
%A Wang, Zhi
%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 lin-etal-2026-lazymuon
%X Recently, the Muon optimizer has demonstrated strong efficiency in large model training through matrix orthogonalization. At each optimization step, Muon orthogonalizes the raw update matrix via Newton-Schulz (NS) iterations before applying it to model parameters. However, orthogonalization requires repeated matrix multiplications whose computational cost scales with O(n³) for an n\times n matrix, making the overhead prohibitive at large model scales. In this paper, we observe that the geometric angle derived from cosine similarity between consecutive orthogonal update matrices remains highly stable during training, a phenomenon that we partially explain theoretically and support empirically. Based on this angular stability, we propose LazyMuon, which skips computationally expensive NS iterations through intermittent linear interpolation and employs a residual correction mechanism to mitigate accumulated approximation errors. This design ensures fast iteration while preserving Muon’s orthogonal manifold constraints. Empirical evaluations on language and vision tasks demonstrate that LazyMuon achieves speedups of up to 1.34\times and 1.31\times over Muon, respectively, with negligible degradation in convergence.
%U https://aclanthology.org/2026.findings-emnlp.173/
%P 3646-3659
Markdown (Informal)
[LazyMuon: Exploiting Angular Stability to Reduce Newton-Schulz Overhead in Muon for Efficient Large Model Training](https://aclanthology.org/2026.findings-emnlp.173/) (Lin et al., Findings 2026)
ACL