Beyond the Ideal: Analyzing the Inexact Muon Update

Egor Shulgin, Sultan AlRashed, Peter Richtárik, Francesco Orabona
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:5167-5175, 2026.

Abstract

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon’s efficiency relies on fast, approximate orthogonalization, while most theoretical analyses study idealized exact-SVD updates. This work moves beyond the ideal by providing a general analysis of the \emph{inexact} orthogonalized update at Muon’s core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error, $\delta$. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure, such as the number of Newton-Schulz steps, from an implementation detail to a critical parameter that must be \emph{co-tuned} with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-shulgin26a, title = { Beyond the Ideal: Analyzing the Inexact Muon Update }, author = {Shulgin, Egor and AlRashed, Sultan and Richt{\'a}rik, Peter and Orabona, Francesco}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {5167--5175}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/shulgin26a/shulgin26a.pdf}, url = {https://proceedings.mlr.press/v300/shulgin26a.html}, abstract = { The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon’s efficiency relies on fast, approximate orthogonalization, while most theoretical analyses study idealized exact-SVD updates. This work moves beyond the ideal by providing a general analysis of the \emph{inexact} orthogonalized update at Muon’s core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error, $\delta$. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure, such as the number of Newton-Schulz steps, from an implementation detail to a critical parameter that must be \emph{co-tuned} with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes. } }
Endnote
%0 Conference Paper %T Beyond the Ideal: Analyzing the Inexact Muon Update %A Egor Shulgin %A Sultan AlRashed %A Peter Richtárik %A Francesco Orabona %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-shulgin26a %I PMLR %P 5167--5175 %U https://proceedings.mlr.press/v300/shulgin26a.html %V 300 %X The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon’s efficiency relies on fast, approximate orthogonalization, while most theoretical analyses study idealized exact-SVD updates. This work moves beyond the ideal by providing a general analysis of the \emph{inexact} orthogonalized update at Muon’s core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error, $\delta$. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure, such as the number of Newton-Schulz steps, from an implementation detail to a critical parameter that must be \emph{co-tuned} with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.
APA
Shulgin, E., AlRashed, S., Richtárik, P. & Orabona, F.. (2026). Beyond the Ideal: Analyzing the Inexact Muon Update . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:5167-5175 Available from https://proceedings.mlr.press/v300/shulgin26a.html.

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