Muon Converges under Heavy-Tailed Noise: Nonconvex Hölder-Smooth Empirical Risk Minimization

Muon is emerging as a promising alternative to AdamW for large-scale neural network training, yet theoretical understanding of its practical implementation remains incomplete, as existing analyses often simplify or omit two key components: (i) practical Newton--Schulz iterations with empirically tuned polynomial coefficients $(3.4445,-4.7750,2.0315)$; and (ii) decoupled weight decay for regularization. In this paper, we provide an optimization interpretation and establish convergence for practical Muon, jointly accounting for both components. Specifically, we interpret practical Muon as right-preconditioned optimization of the original loss with a dynamic weighted $\ell_2$ regularizer that vanishes as stationarity is approached, so that the optimization target remains the original objective. We then establish, to our best knowledge, the first convergence guarantee for practical Muon in the stochastic nonconvex setting, with an $\mathcal{O}(T^{-1/4})$ convergence rate in terms of the expected Frobenius norm of the gradient, improving the dimension dependence of the best known AdamW's convergence rate by a factor of $\sqrt{d}$, where $T$ is the iteration horizon and $d$ is the parameter dimension. Experiments further support the theoretical convergence results.

Muon and its variants have shown strong empirical performance in a variety of deep learning tasks. Existing convergence analyses of Muon rely on smoothness assumptions, though arguably the most successful function class for developing deep learning methods (such as AdaGrad, Shampoo, Schedule-Free and more) has been the class of convex and Lipschitz functions. In this paper we question whether the classical convex Lipschitz model is a useful one for understanding Muon. Our answer is no. We show that Muon does not converge on the class of convex and Lipschitz functions, regardless of the choice of learning rate schedule. We also show that error feedback restores convergence of Muon and all the non-Euclidean subgradient methods with momentum. However, this theoretical fix using error feedback degrades the performance of Muon in two representative settings for image classification (CIFAR-10) and language modeling (nanoGPT on FineWeb-Edu 10B). Our conclusion is that convex Lipschitz theory, despite having a prominent role in the design of practical methods for deep learning, is not the most suited one for Muon. This suggests that Muon's success must come from structure absent from this model, most plausibly related to smoothness.

Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice. In contrast, we study nonconvex stochastic optimization under the joint relaxation of these assumptions: objectives with $(L,s)$-Hölder continuous gradients, $s\in(0,1]$, and gradient noise satisfying only a bounded $α$-th moment condition for $α\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $α\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and Hölder smoothness simultaneously. Secondly, we analyze $δ$-regularized gradient clipping ($δ$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(α-1)/[(1+s)(2α-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $α\ge1+s$ while in the very heavy-tailed regime $α<1+s$, it has a convergence rate $O(T^{-2s(α-1)/[(α-1)+s(2α-1)]})$ --- the first convergence guarantee in this regime for any stochastic gradient based method.