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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.26459 |
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| _version_ | 1866918523459076096 |
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| author | Yi, Albert |
| author_facet | Yi, Albert |
| contents | Muon-style optimizers take a matrix-valued momentum or preconditioned update $B = U \operatorname{diag}(σ_1,\ldots,σ_r) V^\top$ and replace it with its canonical partial polar factor $\operatorname{Pol}(B) = U V^\top$. This maps every nonzero singular value to one. MuCon is the clipped-Muon variant studied here: it applies singular-value clipping to the same Muon matrix, $D^{\mathrm{MuCon}}\_τ(B) = \operatorname{MClip}\_τ(B) = U \operatorname{diag}\bigl(\min\{σ\_i,τ\}\bigr) V^\top, \qquad τ> 0$. Thus, $\operatorname{MClip}\_τ$ denotes the mathematical clipping operator, while MuCon denotes the optimizer primitive that substitutes this clipped direction for Muon's polar direction. The Muon/MuCon scaling parameterization used in this work is called $\text{SpectralP}$: it is the hidden-matrix scaling recipe under which polar Muon or clipped MuCon directions are applied. The map $\operatorname{MClip}\_τ$ is the Frobenius projection onto the spectral-norm ball $\{X : \|X\|_2 \le τ\}$: it leaves singular values at or below $τ$ unchanged and modifies only the violating singular directions. This paper asks when the MuCon clipping step can be approximated without a full dense SVD. We record two exact identities, a polar/absolute-value formula and a scalar-root formulation leading to a rational Newton filter for the clipped positive-semidefinite factor, and identify the numerical obstruction common to both: singular values near the threshold make sign decisions and rational solves ill-conditioned. Matrix-function methods are therefore useful only when paired with stable polar/square-root primitives or explicit regularization near the clipping boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26459 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | MuCon: Clipped Muon Updates for LLM Training Yi, Albert Machine Learning Muon-style optimizers take a matrix-valued momentum or preconditioned update $B = U \operatorname{diag}(σ_1,\ldots,σ_r) V^\top$ and replace it with its canonical partial polar factor $\operatorname{Pol}(B) = U V^\top$. This maps every nonzero singular value to one. MuCon is the clipped-Muon variant studied here: it applies singular-value clipping to the same Muon matrix, $D^{\mathrm{MuCon}}\_τ(B) = \operatorname{MClip}\_τ(B) = U \operatorname{diag}\bigl(\min\{σ\_i,τ\}\bigr) V^\top, \qquad τ> 0$. Thus, $\operatorname{MClip}\_τ$ denotes the mathematical clipping operator, while MuCon denotes the optimizer primitive that substitutes this clipped direction for Muon's polar direction. The Muon/MuCon scaling parameterization used in this work is called $\text{SpectralP}$: it is the hidden-matrix scaling recipe under which polar Muon or clipped MuCon directions are applied. The map $\operatorname{MClip}\_τ$ is the Frobenius projection onto the spectral-norm ball $\{X : \|X\|_2 \le τ\}$: it leaves singular values at or below $τ$ unchanged and modifies only the violating singular directions. This paper asks when the MuCon clipping step can be approximated without a full dense SVD. We record two exact identities, a polar/absolute-value formula and a scalar-root formulation leading to a rational Newton filter for the clipped positive-semidefinite factor, and identify the numerical obstruction common to both: singular values near the threshold make sign decisions and rational solves ill-conditioned. Matrix-function methods are therefore useful only when paired with stable polar/square-root primitives or explicit regularization near the clipping boundary. |
| title | MuCon: Clipped Muon Updates for LLM Training |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.26459 |