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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

Korean

Muangcase는 2024 년에 태어난 언어 모델을위한 더 낮은 스펙트럼 최적화를 설계한 Gemimized 측면으로, 2015 년에 태어난 스펙트럼 gradient descent에서 주로 GPU 친화적 인 Acximic Newton-Schulz polynomial을 사용하여 빠른 스펙트럼 차이를 나타냅니다. 따라서 연구 라인은 Polynomial gradient orthogonalization을위한 더 빠른 polynomial iterations를 제안합니다. Polar Switch는 Polar Express와 Gram Newton-Schulz와 같은 GPU 친화적 인 Newton-Schulz polynomials에 모두 구축됩니다. 이 논문에서 우리는 Polar Switch, polynomial gradient orthogonalization에 대한 새로운 접근 방식을 소개합니다. 따라서

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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