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Polar Switch: Fast Muon Orthogonalization with Polynomial Circuits

Zhehang Du, Weijie Su · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.26742 · cited by 0 works (OpenAlex) · open-access PDF

Chinese

Muangcase 设计的这些较低的参数比2024年出生的语言模型优化器,主要不同于2015年出生的光谱降幅,主要是在其快速的,GPU友好的Newton--Schulz半多元格式调节格式调节的方面。因此,一系列的研究建议更快的多元格式调节,如Polar Express和Gram Newton--Schulz的估计,这两者都是基于金属多元格式调节的。在这篇论文中,我们介绍了Polar Switch,一种新的方法来调节多元格式调节格式调节。使用简单多元格式调节的路线,Polar Switch 达到给定的高多元格式调节值的程度,低于 quintications的FLOP。我们定义了新电路作为一个短

Machine translation (Meta M2M100); the English original follows.

Abstract

Muon, a 2024-born optimizer for language models, differs from the 2015-born spectral gradient descent mainly in its fast, GPU-friendly Newton--Schulz polynomial for orthogonalizing the gradient matrix. Accordingly, a line of research proposes faster polynomial iterations for gradient orthogonalization, such as Polar Express and Gram Newton--Schulz, both built on quintic polynomials. In this paper, we introduce Polar Switch, a new approach to polynomial gradient orthogonalization. Using simple polynomial circuits, Polar Switch reaches a given high polynomial degree at a lower FLOP cost than quintic iterations. We define a circuit as a short sequence of symmetric matrix products on a few square buffers, and circuits with $k$ products form a restricted family of odd polynomials of degree up to $2^{k+1}+1$. We choose the sequence of products in each circuit, their coefficients, and how the circuits are composed by numerically minimizing a worst-case bound on the polar approximation error. This minimization is subject to a FLOP budget that accounts for symmetric matrix multiplication kernels and a rounding constraint for numerical stability in half precision. Our two recommended Polar Switch methods have 15.5--18.7\% lower runtime than Polar Express on square matrices and 8.6--16.2\% lower runtime than Gram Newton--Schulz on rectangular matrices with aspect ratios 3.25, 3.5, and 8, with comparable approximation accuracy. In pretraining Modded-NanoGPT \#91, Qwen3-0.6B, and Gemma 3-1B, Polar Switch reaches final validation losses similar to those of established orthogonalization methods. Across the more than 14 million matrices that each of the two recommended methods orthogonalized in these runs, the estimated output spectral norms never exceed the designed bounds by more than 3\%, demonstrating their numerical stability.

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Topics: Orthogonalization, Mathematics, Polynomial, Polar decomposition, Quintic function, Polar

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