# Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence

arXiv: https://arxiv.org/abs/2605.13079 · HTML: https://mlsanity.com/paper/2605.13079/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 25 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 18 of 18)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Muon Optimizes Under Spectral Norm Constraints](https://mlsanity.com/paper/2506.15054/) (2025)
- [A Note on the Convergence of Muon](https://mlsanity.com/paper/2502.02900/) (2025)
- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [AdaMuon: Adaptive Muon Optimizer](https://mlsanity.com/paper/2507.11005/) (2025)
- [When do spectral gradient updates help in deep learning?](https://mlsanity.com/paper/2512.04299/) (2025)
- [An Exploration of Non-Euclidean Gradient Descent: Muon and its Many Variants](https://mlsanity.com/paper/2510.09827/) (2025)
- [Adam Improves Muon: Adaptive Moment Estimation with Orthogonalized Momentum](https://mlsanity.com/paper/2602.17080/) (2026)
- [Muon Optimizer Accelerates Grokking](https://mlsanity.com/paper/2504.16041/) (2025)
- [MuonAll: Muon Variant for Efficient Finetuning of Large Language Models](https://mlsanity.com/paper/2511.06086/) (2025)
- [Muon: Training and Trade-offs with Latent Attention and MoE](https://mlsanity.com/paper/2509.24406/) (2025)

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

- [The Spectral Dynamics and Noise Geometry of Muon](https://mlsanity.com/paper/2606.08388/) (2026)
- [A Note on Stability for Orthogonalized Matrix Momentum with Client Sampling](https://mlsanity.com/paper/2606.01720/) (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)
- [DGA-Muon: Decoupled Geometry-Aligned Adaptive Scaling for Muon](https://mlsanity.com/paper/2610.06578/) (2026)

## Abstract (arXiv)

Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but why? We show that the mechanism is spectral flattening, and develop two results around it. First, we prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent. Second, we recast Muon as a preconditioned gradient method and show, under a Kronecker-factored curvature model, that it improves the effective convergence factor, with the improvement controlled by the spectrum of the gradient covariance. Extensive experiments validate both results: Muon remains stable at learning rates that cause SGD to diverge within the first few iterations, and reaches accuracy milestones several epochs earlier even at identical step sizes. Taken together, our results offer a principled, geometric explanation for Muon's empirical success.