Convergence guarantees for Muon: New parameter regimes and generalizations
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 $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, 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...
| Authors | Arthur C. B. de Oliveira, Dhruv D. Jatkar, Guilherme S. Vicinansa et al. |
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| Primary category | math.NA |
| Published | 24 September 2026 |
| Last updated at source | 24 September 2026 |
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