# Convergence guarantees for Muon: New parameter regimes and generalizations

arXiv: https://arxiv.org/abs/2609.30546 · HTML: https://mlsanity.com/paper/2609.30546/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 18 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [A Note on the Convergence of Muon](https://mlsanity.com/paper/2502.02900/) (2025)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [Convergence of Muon with Newton-Schulz](https://mlsanity.com/paper/2601.19156/) (2026)
- [On the Convergence of Muon and Beyond](https://mlsanity.com/paper/2509.15816/) (2025)
- [Convergence Bound and Critical Batch Size of Muon Optimizer](https://mlsanity.com/paper/2507.01598/) (2025)

## Built on by (most cited first, 2 of 2)

- [Training-Loss Guarantees for Muon with Finite-Step Newton--Schulz Orthogonalization](https://mlsanity.com/paper/2610.03306/) (2026)
- [From Spectral to Frobenius: When Muon Loses Its Spectral Implicit Bias](https://mlsanity.com/paper/2610.20112/) (2026)

## Abstract (arXiv)

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→∞|nabla f(xk)|=0, and, under a global Polyak-Lojasiewicz 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 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 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.