# Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

arXiv: https://arxiv.org/abs/2512.04632 · HTML: https://mlsanity.com/paper/2512.04632/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 38 references that are in the map.
- First posted on arXiv 4 Dec 2025.

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

- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [GLM-4.5: Agentic, Reasoning, and Coding (ARC) Foundation Models](https://mlsanity.com/paper/2508.06471/) (2025)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [Modular Duality in Deep Learning](https://mlsanity.com/paper/2410.21265/) (2024)
- [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)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [Skew Orthogonal Convolutions](https://mlsanity.com/paper/2105.11417/) (2021)
- [Almost-Orthogonal Layers for Efficient General-Purpose Lipschitz Networks](https://mlsanity.com/paper/2208.03160/) (2022)
- [Beyond the Ideal: Analyzing the Inexact Muon Update](https://mlsanity.com/paper/2510.19933/) (2025)

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

- [Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers](https://mlsanity.com/paper/2605.18106/) (2026)

## Abstract (arXiv)

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a pre-conditioning procedure that improves the initialization of the Newton--Schulz iterations while incurring negligible overhead. Furthermore, our pre-conditioning reduces the initial polar error and enables the removal of one Newton-Schulz iteration (out of the five iterations usually used in practice). The resulting implementation significantly reduces Muon's overhead. At the end-to-end training level, we observe consistent runtime improvements across speed-run and standard benchmarks, including ~3% reductions in training time on multiple fast training benchmarks, while matching reference performance on both language and vision tasks. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Beyond empirical gains, we provide theoretical insight into the geometry of the update and its potential robustness against feature collapse. Our code is publicly available on github, in optax and huggingface kernels.