# Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization

arXiv: https://arxiv.org/abs/2601.19400 · HTML: https://mlsanity.com/paper/2601.19400/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 24 references that are in the map.
- First posted on arXiv 27 Jan 2026.

## Builds on (most cited first, 14 of 14)

- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [A Note on the Convergence of Muon](https://mlsanity.com/paper/2502.02900/) (2025)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [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)
- [AdaGrad Meets Muon: Adaptive Stepsizes for Orthogonal Updates](https://mlsanity.com/paper/2509.02981/) (2025)
- [LiMuon: Light and Fast Muon Optimizer for Large Models](https://mlsanity.com/paper/2509.14562/) (2025)
- [Generalized Gradient Norm Clipping & Non-Euclidean (L0,L1)-Smoothness](https://mlsanity.com/paper/2506.01913/) (2025)
- [FedMuon: Accelerating Federated Learning with Matrix Orthogonalization](https://mlsanity.com/paper/2510.27403/) (2025)
- [Dion2: A Simple Method to Shrink Matrix in Muon](https://mlsanity.com/paper/2512.16928/) (2025)
- [Increasing Both Batch Size and Learning Rate Accelerates Stochastic Gradient Descent](https://mlsanity.com/paper/2409.08770/) (2024)
- [A Convergence Analysis of Adaptive Optimizers under Floating-point Quantization](https://mlsanity.com/paper/2510.21314/) (2025)
- [Faster Convergence of Riemannian Stochastic Gradient Descent with Increasing Batch Size](https://mlsanity.com/paper/2501.18164/) (2025)

## Built on by (most cited first, 9 of 9)

- [MuonEq: Balancing Before Orthogonalization with Lightweight Equilibration](https://mlsanity.com/paper/2603.28254/) (2026)
- [Insights on Muon from Simple Quadratics](https://mlsanity.com/paper/2602.11948/) (2026)
- [Muon Converges under Heavy-Tailed Noise: Nonconvex H"older-Smooth Empirical Risk Minimization](https://mlsanity.com/paper/2603.15059/) (2026)
- [LionMuon: Alternating Spectral and Sign Descent for Efficient Training](https://mlsanity.com/paper/2605.19811/) (2026)
- [Reassessing Muon for Matrix Factorization](https://mlsanity.com/paper/2607.13246/) (2026)
- [Muse: Representation Geometry of Muon Beyond Normalized Momentum](https://mlsanity.com/paper/2607.14536/) (2026)
- [MALT: Lightweight Curvature-Aware Muon via Diagonal Preconditioning](https://mlsanity.com/paper/2608.05088/) (2026)
- [Accelerating LMO-Based Optimization via Implicit Gradient Transport](https://mlsanity.com/paper/2605.05577/) (2026)
- [LionMuon: Alternating Spectral and Sign Descent for Efficient Training](https://mlsanity.com/paper/2609.35297/) (2026)

## Abstract (arXiv)

The Muon optimizer has recently attracted attention due to its orthogonalized first-order updates, and a deeper theoretical understanding of its convergence behavior is essential for guiding practical applications; however, existing convergence guarantees are either coarse or obtained under restrictive analytical settings. In this work, we establish sharper convergence guarantees for the Muon optimizer through a direct and simplified analysis that does not rely on restrictive assumptions on the update rule. Our results improve upon existing bounds by achieving faster convergence rates while covering a broader class of problem settings. These findings provide a more accurate theoretical characterization of Muon and offer insights applicable to a broader class of orthogonalized first-order methods.