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arXiv · 2601.19156

Convergence of Muon with Newton-Schulz

Gyu Yeol Kim·Min-hwan Oh
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Abstract

We analyze Muon as originally proposed and used in practice -- using the momentum orthogonalization with a few Newton-Schulz steps. The prior theoretical results replace this key step in Muon with an exact SVD-based polar factor. We prove that Muon with Newton-Schulz converges to a stationary point at the same rate as the SVD-polar idealization, up to a constant factor for a given number $q$ of Newton-Schulz steps. We further analyze this constant factor and prove that it converges to 1 doubly exponentially in $q$ and improves with the degree of the polynomial used in Newton-Schulz for approximating the orthogonalization direction. We also prove that Muon removes the typical square-root-of-rank loss compared to its vector-based counterpart, SGD with momentum. Our results explain why Muon with a few low-degree Newton-Schulz steps matches exact-polar (SVD) behavior at a much faster wall-clock time and explain how much momentum matrix orthogonalization via Newton-Schulz benefits over the vector-based optimizer. Overall, our theory justifies the practical Newton-Schulz design of Muon, narrowing its practice-theory gap.

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Gyu Yeol Kim, Min-hwan Oh. 2026-01-27. Convergence of Muon with Newton-Schulz. https://arxiv.org/abs/2601.19156

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