Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence
Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but why? We show that the mechanism is spectral flattening, and develop two results around it. First, we prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent. Second, we recast Muon as a preconditioned gradient method and show, under a Kronecker-factored curvature model, that it improves the effective convergence factor, with the improvement co...
| Authors | Tien-Phat Nguyen, Truong Nguyen, Minh-Phuc Truong et al. |
|---|---|
| Primary category | cs.LG |
| Published | 13 May 2026 |
| Last updated at source | 13 May 2026 |
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