Why Muon Outperforms Adam: A Curvature Perspective
Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find t...
| Authors | Shuche Wang, Fengzhuo Zhang, Jiaxiang Li et al. |
|---|---|
| Primary category | cs.LG |
| Published | 3 June 2026 |
| Last updated at source | 3 June 2026 |
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