Convergence guarantees for Muon: New parameter regimes and generalizations
Abstract
In this paper, we establish the first asymptotic convergence guarantees for the Muon algorithm through a more accurate proxy for the Newton-Schultz iteration than the typical matrix sign function. We prove that, for appropriate choices of hyperparameters, the iterates satisfy limk→∞|∇ f(xk)|=0, and, under a global Polyak-Łojasiewicz condition, that the sequence of function values converges linearly. The key insight is that the regularization, implicit in Muon's Newton-Schulz implementation, induces a bounded preconditioner, exposing Muon as a \emph{preconditioned Polyak heavy-ball} method and enabling a classical Lyapunov analysis. This observation naturally motivates applying the same preconditioning structure to the Nesterov gradient evaluation. We formalize this idea by introducing \emph{Muesterov}, a Nesterov-based variant of Muon, and prove that it enjoys the same convergence guarantees, extending the theoretical framework beyond the heavy-ball setting. Numerical experiments on a scalar cross-entropy problem corroborate the theory and illuminate the joint role of the learning rate and the Newton-Schulz regularizer in controlling convergence. Preliminary numerical simulations training the nanoGPT dataset provide intuition regarding the relevance of the observations in this paper to practical applications.
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BibTeX
@misc{oliveira2026convergence,
title = {{Convergence guarantees for Muon: New parameter regimes and generalizations}},
author = {Arthur C. B. de Oliveira and Dhruv D. Jatkar and Guilherme S. Vicinansa and Eduardo D. Sontag},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30546},
eprint = {2609.30546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30546},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Convergence guarantees for Muon: New parameter regimes and generalizations
AU - Arthur C. B. de Oliveira
AU - Dhruv D. Jatkar
AU - Guilherme S. Vicinansa
AU - Eduardo D. Sontag
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30546
UR - https://arxiv.org/abs/2609.30546
AB - In this paper, we establish the first asymptotic convergence guarantees for the Muon algorithm through a more accurate proxy for the Newton-Schultz iteration than the typical matrix sign function. We prove that, for appropriate choices of hyperparameters, the iterates satisfy lim_k→∞|∇ f(x_k)|=0, and, under a global Polyak-Łojasiewicz condition, that the sequence of function values converges linearly. The key insight is that the regularization, implicit in Muon's Newton-Schulz implementation, induces a bounded preconditioner, exposing Muon as a \emph{preconditioned Polyak heavy-ball} method and enabling a classical Lyapunov analysis. This observation naturally motivates applying the same preconditioning structure to the Nesterov gradient evaluation. We formalize this idea by introducing \emph{Muesterov}, a Nesterov-based variant of Muon, and prove that it enjoys the same convergence guarantees, extending the theoretical framework beyond the heavy-ball setting. Numerical experiments on a scalar cross-entropy problem corroborate the theory and illuminate the joint role of the learning rate and the Newton-Schulz regularizer in controlling convergence. Preliminary numerical simulations training the nanoGPT dataset provide intuition regarding the relevance of the observations in this paper to practical applications.
ER -