Convergence guarantees for Muon: New parameter regimes and generalizations
Quick summary
arXiv:2609.30546v1 Announce Type: cross 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 $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, and, under a global Polyak-\L{}ojasiewicz 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 precondition
Key takeaways
- arXiv:2609.30546v1 Announce Type: cross 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 $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, and, under a global Polyak-\L{}ojasiewicz 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 precondition
Why it matters
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