Delving into Muon and Beyond: Deep Analysis and Extensions

2026-02-04

AI Summary

The intrinsic mechanism of the Muon optimizer and its relationship to adaptive methods like Adam remain unclear. This work proposes a unified perspective based on spectral transformation, framing Muon as a special case at $ p = 0 $, and systematically investigates variants with $ p = 1/2, 1/4, 1 $ in gradient updates. The spectral transformation is efficiently implemented via coupled Newton iterations, circumventing explicit singular value decomposition. Experiments demonstrate that RMS-normalized updates are more stable than first-moment-based ones, and spectral compression substantially enhances the stability of first-moment updates. However, Muon ($ p = 0 $) does not consistently outperform Adam. This study reveals that Muon is essentially an effective spectral normalization method rather than a universally superior optimizer.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 \boldsymbol{\Sigma}^{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.

Problem

• Muon optimizer
• adaptive optimizers
• spectral transformations
• optimization stability
• first-moment updates

Innovation

• spectral normalization
• Muon optimizer
• adaptive optimization
• Newton iteration
• singular value decomposition