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

TL;DR

Muon orthogonalizes the momentum buffer to spectral flattening, enabling larger learning rates and faster convergence by controlling the spectrum of the gradient, as shown through theoretical analysis and experiments.

Contribution

The paper reveals that Muon's spectral flattening mechanism explains its ability to tolerate larger learning rates and accelerate convergence, supported by theoretical proofs and empirical validation.

Findings

Muon tolerates larger learning rates than SGD.

Spectral flattening controls the stability and convergence speed.

Muon reaches accuracy milestones several epochs earlier.

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.