Muon$^p$: Muon with Fractional Spectral Powers

• URL: http://arxiv.org/abs/2606.13867v1
• Date: Thu, 11 Jun 2026 19:48:12 GMT
• Title: Muon$^p$: Muon with Fractional Spectral Powers
• Abstract summary: Muon is an increasingly widely used gradient that replaces a $G=USVtop$ with its polar factor $UVtop$.<n>We introduce Muon$p$ that instead uses fractional spectral-power updates $USpVtop$ for rational $pin(0,1)$.<n>Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.
• Score: 3.223825674658464
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Muon is an increasingly widely used optimizer that replaces a gradient $G=USV^\top$ with its polar factor $UV^\top$, thereby flattening the singular spectrum. However, full flattening discards singular-value information that may matter for adaptation. We introduce Muon$^p$, a Muon-style optimizer that instead uses fractional spectral-power updates $US^pV^\top$ for rational $p\in(0,1)$, interpolating between Muon and gradient descent. To make it practical, we prove that fractional spectral powers cannot be computed by any fixed univariate polynomial iteration, and furthermore derive low-degree odd bivariate recurrences that approximate $US^pV^\top$ using only matrix multiplications, preserving Muon's matrix-multiplication-only structure and compute complexity. We show that Muon$^p$ maximizes the linear improvement in loss under the Schatten $q$-norm for $q=1+\frac{1}{p}$. Empirically, Muon$^p$ is especially effective for finetuning: on billion-scale models, Muon$^p$ improves validation perplexity and downstream task performance. We further analyze when Muon$^p$ is less suitable, through the lens of spectral geometry. Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.

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