# Towards a Principled Muon under μmathsfP: Ensuring Spectral Conditions throughout Training

arXiv: https://arxiv.org/abs/2601.01306 · HTML: https://mlsanity.com/paper/2601.01306/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 49 of its 74 references that are in the map.
- First posted on arXiv 4 Jan 2026.

## Builds on (most cited first, 30 of 49)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [Kimi K2: Open Agentic Intelligence](https://mlsanity.com/paper/2507.20534/) (2025)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (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)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [Non-stochastic Best Arm Identification and Hyperparameter Optimization](https://mlsanity.com/paper/1502.07943/) (2015)
- [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)
- [Scaling Laws for Fine-Grained Mixture of Experts](https://mlsanity.com/paper/2402.07871/) (2024)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks](https://mlsanity.com/paper/2310.02244/) (2023)
- [Scaling Exponents Across Parameterizations and Optimizers](https://mlsanity.com/paper/2407.05872/) (2024)
- [Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)](https://mlsanity.com/paper/2505.13416/) (2025)
- [Benchmarking Optimizers for Large Language Model Pretraining](https://mlsanity.com/paper/2509.01440/) (2025)
- [MARS: Unleashing the Power of Variance Reduction for Training Large Models](https://mlsanity.com/paper/2411.10438/) (2024)

## Abstract (arXiv)

The μ-parameterization (μP) provides a principled foundation for large language model (LLM) training by prescribing width-independent learning dynamics, which in turn enables predictable scaling behavior and robust hyperparameter transfer across model sizes. A central requirement of μP is the satisfaction of certain spectral conditions on weight matrices, which ensure consistent feature learning and optimization behavior as model width grows. While these conditions are well understood in theory, guaranteeing their validity in practical training for matrix-based optimizers such as Muon is still under studied. Existing works that study Muon under μP exhibit important limitations: they either do not ensure that the spectral conditions hold throughout the entire training horizon, or require repeated spectral normalization (or Newton-Schulz iterations) applied to both weights and updates, leading to significant computational overhead and reduced practicality. In this work, we show how to reliably guarantee the spectral conditions required by μP for Muon during the entire training process. Our key insight is that for moderately large models, maintaining spectral control at the level of optimizer updates alone is sufficient to preserve μP-compatible scaling, eliminating the need for explicit spectral normalization of the weights. Based on this principle, we develop a variant of Muon, namely Muon++, that satisfies spectral condition throughout the training process. Our results bridge the gap between the theoretical promises of μP and the practical deployment of matrix-based optimizers in long-horizon training. We also take the first step towards an adaptive spectral condition by incorporating data-dependent effects, making it better suited for long-horizon LLM training.