# Second-Order Muon Done Right: A Principled Marriage of Spectral Geometry and Curvature

arXiv: https://arxiv.org/abs/2608.09763 · HTML: https://mlsanity.com/paper/2608.09763/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 31 references that are in the map.
- First posted on arXiv 10 Aug 2026.

## Builds on (most cited first, 25 of 25)

- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [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)
- [Limitations of the Empirical Fisher Approximation for Natural Gradient Descent](https://mlsanity.com/paper/1905.12558/) (2019)
- [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)
- [A Distributed Data-Parallel PyTorch Implementation of the Distributed Shampoo Optimizer for Training Neural Networks At-Scale](https://mlsanity.com/paper/2309.06497/) (2023)
- [Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)](https://mlsanity.com/paper/2505.13416/) (2025)
- [Scalable Optimization in the Modular Norm](https://mlsanity.com/paper/2405.14813/) (2024)
- [Large-Scale Distributed Second-Order Optimization Using Kronecker-Factored Approximate Curvature for Deep Convolutional Neural Networks](https://mlsanity.com/paper/1811.12019/) (2018)
- [Purifying Shampoo: Investigating Shampoo's Heuristics by Decomposing its Preconditioner](https://mlsanity.com/paper/2506.03595/) (2025)
- [Convolutional Neural Network Training with Distributed K-FAC](https://mlsanity.com/paper/2007.00784/) (2020)
- [KAISA: An Adaptive Second-Order Optimizer Framework for Deep Neural Networks](https://mlsanity.com/paper/2107.01739/) (2021)
- [The Newton-Muon Optimizer](https://mlsanity.com/paper/2604.01472/) (2026)
- [The Potential of Second-Order Optimization for LLMs: A Study with Full Gauss-Newton](https://mlsanity.com/paper/2510.09378/) (2025)
- [Mousse: Rectifying the Geometry of Muon with Curvature-Aware Preconditioning](https://mlsanity.com/paper/2603.09697/) (2026)
- [MuonEq: Balancing Before Orthogonalization with Lightweight Equilibration](https://mlsanity.com/paper/2603.28254/) (2026)
- [Muon2: Boosting Muon via Adaptive Second-Moment Preconditioning](https://mlsanity.com/paper/2604.09967/) (2026)
- [What Really Matters in Matrix-Whitening Optimizers?](https://mlsanity.com/paper/2510.25000/) (2025)
- [FISMO: Fisher-Structured Momentum-Orthogonalized Optimizer](https://mlsanity.com/paper/2601.21750/) (2026)
- [Structured Inverse-Free Natural Gradient: Memory-Efficient & Numerically-Stable KFAC](https://mlsanity.com/paper/2312.05705/) (2023)
- [A Coordinate-Free Construction of Scalable Natural Gradient](https://mlsanity.com/paper/1808.10340/) (2018)
- [MALT: Lightweight Curvature-Aware Muon via Diagonal Preconditioning](https://mlsanity.com/paper/2608.05088/) (2026)

## Built on by (most cited first, 1 of 1)

- [Beyond the Matrix Sign: Quadratic Spectral Descent](https://mlsanity.com/paper/2609.07597/) (2026)

## Abstract (arXiv)

Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.