Reassessing Muon for Matrix Factorization

2026-07-14

AI Summary

This study systematically evaluates whether the performance advantages of the Muon optimizer are intrinsic or contingent upon the complex factors prevalent in large-scale deep learning, by examining its behavior on low-rank matrix factorization—a task with clear spectral structure and strong theoretical guarantees. Through comparisons with carefully tuned adaptive optimizers such as AdamW, and interpreted via the lens of steepest descent in spectral norm, the analysis reveals that Muon does not consistently outperform baseline methods in this controlled setting. Its previously reported superiority appears highly sensitive to specific hyperparameter choices and task configurations. These findings challenge the assumption of Muon’s universal efficacy and underscore the importance of rigorously assessing modern optimizers on simplified yet theoretically well-understood problems.

Abstract

Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training. Its empirical success has motivated a growing body of theoretical work that interprets Muon as steepest descent under the spectral norm. Yet it remains unclear which of Muon's advantages stem from its update rule itself and which are artifacts of the scale, architecture, and data of modern deep networks. In this work, we isolate the optimizer from these confounding factors by studying Muon on a simple, well-understood, and spectrally structured problem: low-rank matrix factorization. Through a controlled comparison against carefully tuned adaptive baselines, we find that Muon does not consistently outperform AdamW in this setting and that several previously reported advantages are sensitive to hyperparameter choices. Our results provide a more nuanced picture of when spectrum-aware orthogonalization is beneficial and argue for evaluating modern optimizers on controlled problems in addition to end-to-end benchmarks.

Problem

• Muon
• matrix factorization
• optimization
• spectral norm
• adaptive optimizers

Innovation

• Muon optimizer
• matrix factorization
• spectral norm
• adaptive optimization
• controlled evaluation