Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908306497339392 |
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| author | Kovalev, Dmitry |
| author_facet | Kovalev, Dmitry |
| contents | Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization Kovalev, Dmitry Machine Learning Optimization and Control Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models. |
| title | Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2503.12645 |