Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization

• URL: http://arxiv.org/abs/2503.12645v2
• Date: Tue, 08 Apr 2025 16:47:42 GMT
• Title: Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization
• Authors: Dmitry Kovalev,
• Abstract summary: We provide a theoretical analysis of motivated matrixization.<n>In particular, we show that the non-Euclisky trust-region method can be seen as a special case.<n>Our findings provide an explanation for several practical observations.
• Score: 19.574602844234814
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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