From Spectral to Frobenius: When Muon Loses Its Spectral Implicit Bias
Prior work on implicit bias shows that gradient descent selects the Frobenius max-margin direction on linearly separable problems, whereas Fan et al. (2025) established spectral max-margin behavior for spectral-norm steepest descent methods, including idealized Muon with exact orthogonalization. In practice, however, Muon approximates orthogonalization using finite Newton-Schulz iterations and a positive stabilization constant. We prove that these implementation choices cause Muon to select the Frobenius rather than the spectral max-margin direction in linear classification with separable data. We also establish this behavior for both exponential moving average and Nesterov-style momentum...
| Authors | Yongtao Wu, Mahdi Mansouri, Zhenyu Zhu |
|---|---|
| Primary category | cs.LG |
| Published | 14 October 2026 |
| Last updated at source | 14 October 2026 |
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