# Variance-Adaptive Muon: Pre-Orthogonalization Variance Modulation for Efficient Language Model Pretraining

arXiv: https://arxiv.org/abs/2601.14603 · HTML: https://mlsanity.com/paper/2601.14603/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 44 references that are in the map.
- First posted on arXiv 21 Jan 2026.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [SuperGLUE: A Stickier Benchmark for General-Purpose Language Understanding Systems](https://mlsanity.com/paper/1905.00537/) (2019)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [RACE: Large-scale ReAding Comprehension Dataset From Examinations](https://mlsanity.com/paper/1704.04683/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [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)
- [DataComp-LM: In search of the next generation of training sets for language models](https://mlsanity.com/paper/2406.11794/) (2024)
- [ReCoRD: Bridging the Gap between Human and Machine Commonsense Reading Comprehension](https://mlsanity.com/paper/1810.12885/) (2018)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [NorMuon: Making Muon more efficient and scalable](https://mlsanity.com/paper/2510.05491/) (2025)
- [Benchmarking Optimizers for Large Language Model Pretraining](https://mlsanity.com/paper/2509.01440/) (2025)
- [AdaMuon: Adaptive Muon Optimizer](https://mlsanity.com/paper/2507.11005/) (2025)
- [PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective](https://mlsanity.com/paper/2505.21799/) (2025)
- [Cautious Optimizers: Improving Training with One Line of Code](https://mlsanity.com/paper/2411.16085/) (2024)

## Abstract (arXiv)

Optimizer design plays a central role in efficient language model pretraining, directly affecting optimization dynamics, convergence speed, and compute cost under fixed training budgets. Muon has emerged as a strong optimizer by orthogonalizing momentum updates, yielding a matrix-valued analogue of sign-based normalization. However, unlike Adam-style methods, Muon does not explicitly incorporate gradient-variance information into its updates. Motivated by Adam's variance-adaptive interpretation, we propose Muon-NSR and Muon-VS, two variance-adaptive Muon variants for language model pretraining. Muon-NSR applies noise-to-signal ratio (NSR) modulation before Newton--Schulz orthogonalization, whereas Muon-VS uses variance scaling (VS) without introducing any additional hyperparameters beyond those of Muon. Both methods preserve Muon's spectral normalization structure while requiring only one additional variance buffer. Experiments on Llama-style and GPT-2 pretraining across model scales from 125M to 1.2B parameters show that our methods improve over well-tuned Muon baselines and remain competitive with representative adaptive Muon-family baselines. On Llama-1.2B, Muon-VS achieves a 1.33× step-to-target speedup over a well-tuned Muon baseline, with Muon's final validation loss as the target. These results indicate that variance-adaptive modulation is a simple and effective mechanism for improving Muon-style optimizers in language model pretraining.