# On the Convergence Analysis of Muon

arXiv: https://arxiv.org/abs/2505.23737 · HTML: https://mlsanity.com/paper/2505.23737/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 50 references that are in the map.
- First posted on arXiv 29 May 2025.

## Builds on (most cited first, 30 of 30)

- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [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)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond](https://mlsanity.com/paper/1611.07476/) (2016)
- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [An Alternative View: When Does SGD Escape Local Minima?](https://mlsanity.com/paper/1802.06175/) (2018)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [A Note on the Convergence of Muon](https://mlsanity.com/paper/2502.02900/) (2025)
- [A New Perspective on Shampoo's Preconditioner](https://mlsanity.com/paper/2406.17748/) (2024)
- [ASGO: Adaptive Structured Gradient Optimization](https://mlsanity.com/paper/2503.20762/) (2025)
- [Dissecting Hessian: Understanding Common Structure of Hessian in Neural Networks](https://mlsanity.com/paper/2010.04261/) (2020)
- [SGD Converges to Global Minimum in Deep Learning via Star-convex Path](https://mlsanity.com/paper/1901.00451/) (2019)
- [Structured Preconditioners in Adaptive Optimization: A Unified Analysis](https://mlsanity.com/paper/2503.10537/) (2025)
- [COSMOS: A Hybrid Adaptive Optimizer for Memory-Efficient Training of LLMs](https://mlsanity.com/paper/2502.17410/) (2025)
- [Tensor Normal Training for Deep Learning Models](https://mlsanity.com/paper/2106.02925/) (2021)
- [Towards Quantifying the Hessian Structure of Neural Networks](https://mlsanity.com/paper/2505.02809/) (2025)

## Built on by (most cited first, 5 of 5)

- [Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory](https://mlsanity.com/paper/2603.26554/) (2026)
- [Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers](https://mlsanity.com/paper/2605.18106/) (2026)
- [Muon Does Not Converge on Convex Lipschitz Functions](https://mlsanity.com/paper/2605.08980/) (2026)
- [Phases of Muon: When Muon Eclipses SignSGD](https://mlsanity.com/paper/2605.09552/) (2026)
- [PC Layer: Polynomial Weight Preconditioning for Improving LLM Pre-Training](https://mlsanity.com/paper/2606.06470/) (2026)

## Abstract (arXiv)

The majority of parameters in neural networks are naturally represented as matrices. However, most commonly used optimizers treat these matrix parameters as flattened vectors during optimization, potentially overlooking their inherent structural properties. Recently, an optimizer called Muon has been proposed, specifically designed to optimize matrix-structured parameters. Extensive empirical evidence shows that Muon can significantly outperform traditional optimizers when training neural networks. Nonetheless, the theoretical understanding of Muon's convergence behavior and the reasons behind its superior performance remain limited. In this work, we present a comprehensive convergence rate analysis of Muon and its comparison with Gradient Descent (GD). We characterize the conditions under which Muon can outperform GD. Our theoretical results reveal that Muon can benefit from the low-rank structure of Hessian matrices, a phenomenon widely observed in practical neural network training. Our experimental results support and corroborate the theoretical findings.