NorMuon: Making Muon more efficient and scalable

Zichong Li, Liming Liu, Chen Liang, Weizhu Chen, Tuo Zhao
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:68514-68533, 2026.

Abstract

The choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving optimization geometry through better conditioning. Despite Muon’s emergence as a candidate successor to Adam, the potential for jointly leveraging their strengths—has not been systematically explored. In this work, we bridge this gap by proposing NorMuon (Neuron-wise Normalized Muon), an optimizer that synergistically combines orthogonalization with neuron-level adaptive learning rates. Our analysis reveals that while Muon effectively reduces condition numbers, the resulting updates exhibit highly non-uniform neuron norms, causing certain neurons to dominate the optimization process. NorMuon addresses this imbalance by maintaining second-moment statistics for each neuron and applying row-wise normalization after orthogonalization, ensuring balanced parameter utilization while preserving Muon’s conditioning benefits. To enable practical deployment at scale, we develop an efficient distributed implementation under the FSDP2 framework that distributes orthogonalization computations across devices. Experiments across multiple model scales demonstrate that NorMuon consistently outperforms both AdamW and Muon, achieving a 21.74% reduction in training steps relative to AdamW and an 11.31 percentage-point larger efficiency gain than Muon on 1.1B pretraining. Results suggest that orthogonalization and adaptive learning rates are complementary rather than competing, opening new avenues for optimizer design in large-scale deep learning.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-li26dh, title = {{N}or{M}uon: Making Muon more efficient and scalable}, author = {Li, Zichong and Liu, Liming and Liang, Chen and Chen, Weizhu and Zhao, Tuo}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {68514--68533}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/li26dh/li26dh.pdf}, url = {https://proceedings.mlr.press/v306/li26dh.html}, abstract = {The choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving optimization geometry through better conditioning. Despite Muon’s emergence as a candidate successor to Adam, the potential for jointly leveraging their strengths—has not been systematically explored. In this work, we bridge this gap by proposing NorMuon (Neuron-wise Normalized Muon), an optimizer that synergistically combines orthogonalization with neuron-level adaptive learning rates. Our analysis reveals that while Muon effectively reduces condition numbers, the resulting updates exhibit highly non-uniform neuron norms, causing certain neurons to dominate the optimization process. NorMuon addresses this imbalance by maintaining second-moment statistics for each neuron and applying row-wise normalization after orthogonalization, ensuring balanced parameter utilization while preserving Muon’s conditioning benefits. To enable practical deployment at scale, we develop an efficient distributed implementation under the FSDP2 framework that distributes orthogonalization computations across devices. Experiments across multiple model scales demonstrate that NorMuon consistently outperforms both AdamW and Muon, achieving a 21.74% reduction in training steps relative to AdamW and an 11.31 percentage-point larger efficiency gain than Muon on 1.1B pretraining. Results suggest that orthogonalization and adaptive learning rates are complementary rather than competing, opening new avenues for optimizer design in large-scale deep learning.} }
Endnote
%0 Conference Paper %T NorMuon: Making Muon more efficient and scalable %A Zichong Li %A Liming Liu %A Chen Liang %A Weizhu Chen %A Tuo Zhao %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-li26dh %I PMLR %P 68514--68533 %U https://proceedings.mlr.press/v306/li26dh.html %V 306 %X The choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving optimization geometry through better conditioning. Despite Muon’s emergence as a candidate successor to Adam, the potential for jointly leveraging their strengths—has not been systematically explored. In this work, we bridge this gap by proposing NorMuon (Neuron-wise Normalized Muon), an optimizer that synergistically combines orthogonalization with neuron-level adaptive learning rates. Our analysis reveals that while Muon effectively reduces condition numbers, the resulting updates exhibit highly non-uniform neuron norms, causing certain neurons to dominate the optimization process. NorMuon addresses this imbalance by maintaining second-moment statistics for each neuron and applying row-wise normalization after orthogonalization, ensuring balanced parameter utilization while preserving Muon’s conditioning benefits. To enable practical deployment at scale, we develop an efficient distributed implementation under the FSDP2 framework that distributes orthogonalization computations across devices. Experiments across multiple model scales demonstrate that NorMuon consistently outperforms both AdamW and Muon, achieving a 21.74% reduction in training steps relative to AdamW and an 11.31 percentage-point larger efficiency gain than Muon on 1.1B pretraining. Results suggest that orthogonalization and adaptive learning rates are complementary rather than competing, opening new avenues for optimizer design in large-scale deep learning.
APA
Li, Z., Liu, L., Liang, C., Chen, W. & Zhao, T.. (2026). NorMuon: Making Muon more efficient and scalable. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:68514-68533 Available from https://proceedings.mlr.press/v306/li26dh.html.

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