Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence

2026-05-13

AI Summary

Traditional optimizers are constrained by the largest singular value of the gradient, limiting their ability to use large learning rates and resulting in slow convergence. This work proposes the Muon optimizer, which flattens the spectrum of the gradient covariance by orthogonalizing the momentum buffer—via Newton–Schulz iteration—prior to the momentum update, effectively normalizing all singular values to one. This approach overcomes the stability bottleneck inherent in gradient descent, substantially increasing tolerance to large learning rates and establishing a theoretical connection to preconditioned gradient methods. Empirical results demonstrate that Muon remains stable under large learning rates, avoids the early divergence commonly observed with SGD, and reaches target accuracy milestones faster than baseline methods at equivalent step sizes.

Abstract

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 controlled by the spectrum of the gradient covariance. Extensive experiments validate both results: Muon remains stable at learning rates that cause SGD to diverge within the first few iterations, and reaches accuracy milestones several epochs earlier even at identical step sizes. Taken together, our results offer a principled, geometric explanation for Muon's empirical success.

Problem

• spectral flattening
• orthogonalization
• learning rate
• convergence
• gradient covariance

Innovation

• spectral flattening
• orthogonalization
• momentum buffer
• convergence rate
• preconditioned gradient