Constant Step Sizes for Muon-Type Spectral Optimizers on Strongly Convex Quadratics

Muon and related optimizers update a weight matrix with a spectrally transformed gradient. In this work, we study such spectral optimizers with a constant step size on strongly convex matrix quadratics, which isolate the effect of the update from the confounds of nonconvexity, noise, and step size schedules. We analyze an update using two quantities: the gain, which is the factor by which it amplifies the gradient along each singular direction, and the length, which is the norm of the update. These two quantities place the popular spectral updates in three regimes with different behavior: (i) with a bounded gain, the iterates behave like preconditioned gradient descent and converge for step sizes set by the largest curvature; (ii) with an unbounded gain but a vanishing length, they still converge, with the admissible step sizes set by the total curvature instead; and (iii) with a non-vanishing length, they do not converge for almost every initialization. Interestingly, the practical variants of Muon fall in all three regimes, depending only on how the update is normalized and scaled and not on the polynomial that approximates the polar factor. Experiments validate these findings.