Polar Switch: Fast Muon Orthogonalization with Polynomial Circuits
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 sq...
| Authors | Zhehang Du, Weijie Su |
|---|---|
| Primary category | math.NA |
| Published | 19 October 2026 |
| Last updated at source | 19 October 2026 |
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