# When Muon Optimizer Meets Adversarial Training: A Theoretical and Empirical Study

arXiv: https://arxiv.org/abs/2605.26929 · HTML: https://mlsanity.com/paper/2605.26929/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 53 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 30 of 41)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Fast is better than free: Revisiting adversarial training](https://mlsanity.com/paper/2001.03994/) (2020)
- [RobustBench: a standardized adversarial robustness benchmark](https://mlsanity.com/paper/2010.09670/) (2020)
- [Adversarial Training for Free!](https://mlsanity.com/paper/1904.12843/) (2019)
- [On Adaptive Attacks to Adversarial Example Defenses](https://mlsanity.com/paper/2002.08347/) (2020)
- [Feature Squeezing: Detecting Adversarial Examples in Deep Neural Networks](https://mlsanity.com/paper/1704.01155/) (2017)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Understanding and Improving Fast Adversarial Training](https://mlsanity.com/paper/2007.02617/) (2020)
- [Bag of Tricks for Adversarial Training](https://mlsanity.com/paper/2010.00467/) (2020)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation](https://mlsanity.com/paper/1705.08475/) (2017)
- [Towards Understanding Sharpness-Aware Minimization](https://mlsanity.com/paper/2206.06232/) (2022)
- [On the Convergence and Robustness of Adversarial Training](https://mlsanity.com/paper/2112.08304/) (2021)
- [Feature Purification: How Adversarial Training Performs Robust Deep Learning](https://mlsanity.com/paper/2005.10190/) (2020)
- [Improving DNN Robustness to Adversarial Attacks using Jacobian Regularization](https://mlsanity.com/paper/1803.08680/) (2018)
- [Muon Optimizes Under Spectral Norm Constraints](https://mlsanity.com/paper/2506.15054/) (2025)
- [Modular Duality in Deep Learning](https://mlsanity.com/paper/2410.21265/) (2024)

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

- [The Spectral Dynamics and Noise Geometry of Muon](https://mlsanity.com/paper/2606.08388/) (2026)

## Abstract (arXiv)

Adversarial training (AT) remains one of the most reliable empirical defenses against adversarial attacks. Its robustness critically depends on how the underlying min-max objective is optimized. In practice, Stochastic Gradient Descent (SGD) optimizer remains the default optimization choice for AT, whereas adaptive optimizers often improve standard training but may yield inferior robustness. Recently, the Muon optimizer, which orthogonalizes matrix-valued updates via an approximate polar decomposition, has achieved notable success in large-scale training at a memory cost comparable to SGD. This raises a security-relevant question: textitcan orthogonalized optimization improve AT under strong and heterogeneous threat models? Focusing on this problem, we conduct a comprehensive theoretical and empirical study. Theoretically, we show that Muon imposes a spectral-norm stability ceiling on matrix updates, limiting uncontrolled spectral growth in the training dynamics without explicitly shrinking the learned weights. Empirically, across five architectures and three ℓp threat models (ℓ_∞, ℓ1, ℓ2) and their union, Muon is competitive with SGD on CNNs and substantially outperforms AdamW on both CNNs and ViTs. These results identify optimizer geometry as a security-relevant factor in adversarial training, while clarifying the empirical regimes in which orthogonalized updates are beneficial. Overall, our findings highlight optimizer design as a security-critical component of AT.