Muon Optimizes Under Spectral Norm Constraints
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911183320121344 |
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| author | Chen, Lizhang Li, Jonathan Liu, Qiang |
| author_facet | Chen, Lizhang Li, Jonathan Liu, Qiang |
| contents | The pursuit of faster optimization algorithms remains an active and important research direction in deep learning. Recently, the Muon optimizer [JJB+24] has demonstrated promising empirical performance, but its theoretical foundation remains less understood. In this paper, we bridge this gap and provide a theoretical analysis of Muon by placing it within the Lion-$\mathcal{K}$ family of optimizers [CLLL24]. Specifically, we show that Muon corresponds to Lion-$\mathcal{K}$ when equipped with the nuclear norm, and we leverage the theoretical results of Lion-$\mathcal{K}$ to establish that Muon (with decoupled weight decay) implicitly solves an optimization problem that enforces a constraint on the spectral norm of weight matrices. This perspective not only demystifies the implicit regularization effects of Muon but also leads to natural generalizations through varying the choice of convex map $\mathcal{K}$, allowing for the exploration of a broader class of implicitly regularized and constrained optimization algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Muon Optimizes Under Spectral Norm Constraints Chen, Lizhang Li, Jonathan Liu, Qiang Machine Learning Optimization and Control The pursuit of faster optimization algorithms remains an active and important research direction in deep learning. Recently, the Muon optimizer [JJB+24] has demonstrated promising empirical performance, but its theoretical foundation remains less understood. In this paper, we bridge this gap and provide a theoretical analysis of Muon by placing it within the Lion-$\mathcal{K}$ family of optimizers [CLLL24]. Specifically, we show that Muon corresponds to Lion-$\mathcal{K}$ when equipped with the nuclear norm, and we leverage the theoretical results of Lion-$\mathcal{K}$ to establish that Muon (with decoupled weight decay) implicitly solves an optimization problem that enforces a constraint on the spectral norm of weight matrices. This perspective not only demystifies the implicit regularization effects of Muon but also leads to natural generalizations through varying the choice of convex map $\mathcal{K}$, allowing for the exploration of a broader class of implicitly regularized and constrained optimization algorithms. |
| title | Muon Optimizes Under Spectral Norm Constraints |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2506.15054 |