Delving into Muon and Beyond: Deep Analysis and Extensions

• URL: http://arxiv.org/abs/2602.04669v1
• Date: Wed, 04 Feb 2026 15:40:47 GMT
• Title: Delving into Muon and Beyond: Deep Analysis and Extensions
• Authors: Xianbiao Qi, Marco Chen, Jiaquan Ye, Yelin He, Rong Xiao,
• Abstract summary: We study Muon as the p = 0 endpoint of a family of spectral transformations of the form U boldsymbolp V'.<n>We find that RMS-normalized updates yield more stable optimization than first-moment updates.<n>Our results suggest that Muon is best understood as an effective form of spectral normalization, but not a universally superior optimization method.
• Score: 8.297062899157664
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Muon optimizer has recently attracted considerable attention for its strong empirical performance and use of orthogonalized updates on matrix-shaped parameters, yet its underlying mechanisms and relationship to adaptive optimizers such as Adam remain insufficiently understood. In this work, we aim to address these questions through a unified spectral perspective. Specifically, we view Muon as the p = 0 endpoint of a family of spectral transformations of the form U \boldsymbolΣ^{p} V' , and consider additional variants with p = 1/2 , p = 1/4 , and p = 1 . These transformations are applied to both first-moment updates, as in momentum SGD, and to root-mean-square (RMS) normalized gradient updates as in Adam. To enable efficient computation, we develop a coupled Newton iteration that avoids explicit singular value decomposition. Across controlled experiments, we find that RMS-normalized updates yield more stable optimization than first-moment updates. Moreover, while spectral compression provides strong stabilization benefits under first-moment updates, the Muon update (p = 0) does not consistently outperform Adam. These results suggest that Muon is best understood as an effective form of spectral normalization, but not a universally superior optimization method. Our source code will be released at https://github.com/Ocram7/BeyondMuon.

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