Preconditioning Benefits of Spectral Orthogonalization in Muon

2026-01-20

AI Summary

This work investigates the mechanism of spectral orthogonalization in the Muon optimizer and its advantages in matrix optimization. Focusing on matrix factorization and in-context learning with linear Transformers, we propose a simplified variant of Muon and provide the first rigorous proof that it achieves linear convergence independent of the condition number. Through spectral-domain analysis, we reveal that its dynamics are equivalent to a set of decoupled scalar sequences and elucidate the preconditioning role played by spectral orthogonalization. Both theoretical analysis and empirical experiments demonstrate that the simplified Muon significantly outperforms gradient descent and Adam in terms of linear convergence rate, highlighting its potential for high-dimensional non-convex optimization.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The Muon optimizer, a matrix-structured algorithm that leverages spectral orthogonalization of gradients, is a milestone in the pretraining of large language models. However, the underlying mechanisms of Muon -- particularly the role of gradient orthogonalization -- remain poorly understood, with very few works providing end-to-end analyses that rigorously explain its advantages in concrete applications. We take a step by studying the effectiveness of a simplified variant of Muon through two case studies: matrix factorization, and in-context learning of linear transformers. For both problems, we prove that simplified Muon converges linearly with iteration complexities independent of the relevant condition number, provably outperforming gradient descent and Adam. Our analysis reveals that the Muon dynamics decouple into a collection of independent scalar sequences in the spectral domain, each exhibiting similar convergence behavior. Our theory formalizes the preconditioning effect induced by spectral orthogonalization, offering insight into Muon's effectiveness in these matrix optimization problems and potentially beyond.

Problem

• spectral orthogonalization
• Muon optimizer
• preconditioning
• gradient optimization
• matrix optimization

Innovation

• spectral orthogonalization
• preconditioning
• linear convergence
• matrix factorization
• in-context learning