Convergence of Practical Muon with Finite Newton-Schulz Iterations and Nesterov Momentum

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of $O \left(\varepsilon^{\frac{-(3α-2)}{(α-1)}} \right)$ for finding an $\varepsilon$-stationary point, where $α\in(1,2]$ denotes the heavy-tail index. For the inexact-polar setting with $σ_1=0$, we also provide guarantees that do not require prior knowledge of $α$. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

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.

Orthogonalized momentum updates, as used in Muon-style optimizers, have recently shown strong empirical stability in large-scale deep learning. However, most current orthogonalized methods are still paired with fixed, externally scheduled, or otherwise open-loop magnitude rules, so their scale is not directly calibrated from the realized optimization trajectory. Motivated by the closed-loop perspective behind Lipschitz-free and noise-adaptive methods, we propose OptMuon, a family of adaptive momentum orthogonalization methods for stochastic nonconvex optimization. OptMuon combines Muon-style polar-factor directions with a trajectory-dependent AdaGrad-Norm-type coefficient schedule, so that the update magnitude is determined by the observed gradient and momentum history rather than by a prescribed Lipschitz-dependent rule. The schedule does not use the smoothness constant, the variance level, or the bounded-gradient constant in parameter selection, and its running-maximum correction prevents isolated gradient spikes from causing excessive coefficient collapse. Under lower-boundedness, unbiased stochastic gradients with bounded variance, smoothness, and an almost-sure bounded stochastic-gradient condition, we prove two complementary expected-stationarity guarantees. OptMuon-A achieves the noise-adaptive rate $\tilde{\mathcal O}(T^{-1/2}+σ^{1/2}T^{-1/4})$ under average smoothness, while OptMuon-I achieves $\tilde{\mathcal O}(T^{-1/2}+σ^{1/3}T^{-1/3})$ under individual smoothness. In the zero-noise regime, both bounds automatically reduce to a nearly optimal deterministic first-order rate $\tilde{\mathcal O}(T^{-1/2})$ without manual hyperparameter retuning. These results show that closed-loop scalar adaptation can be combined with Muon-style momentum orthogonalization while retaining noise adaptivity and zero-noise optimality up to logarithmic factors.