Newest papers

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

Russian

Muangcase, более низкий спектральный оптимизатор, созданный для языковых моделей в 2024 году, отличается от спектрального спуска, созданного в 2015 году, в основном в своем быстром, GPU-friendly Newton-Schulz polynomial, созданном для ортогонализации градиентной матрицы. Соответственно, линия исследований предлагает более быстрые полиномические итерации для градиентной ортогонализации, такие как Polar Express и Gram Newton-Schulz, как они построены на квинтических полиномах. В этой статье мы вводим Polar Switch, новый подход к ортогонализации полиномы. Используя простые полиномические схемы, Polar Switch достигает определенной высокой степени полиномической ошибки при более низ

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.

Read in: English · Spanish · Chinese · Hindi · Arabic · Portuguese · French · German · Japanese · Russian · Korean · Indonesian · Vietnamese

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

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange