Convergence of Practical Muon

arXiv:2609.33152 · cs.LG, math.OC · Submitted 2026-09-27 · Read on arXiv

cs.LG, math.OC

Submitted: 2026-09-27

Updated: 2026-09-27

Terminology

• Adam — 10× in this paper · explained in 2 episodes
• Alignment — 10× in this paper · explained in 1 episode
• Accuracy — 6× in this paper · explained in 1 episode
• Regularization — 6× in this paper · explained in 3 episodes
• Generative AI — 3× in this paper · explained in 2 episodes
• AdamW Optimizer — 2× in this paper · explained in 1 episode
• Associative Memory — 2× in this paper · explained in 2 episodes
• Convergence Guarantees — 2× in this paper · explained in 3 episodes
• Embeddings — 2× in this paper · explained in 2 episodes
• Generalization — 2× in this paper · explained in 13 episodes
• Transformer — 2× in this paper · explained in 2 episodes
• Anisotropy — 1× in this paper · explained in 3 episodes
• Data Augmentation — 1× in this paper · explained in 7 episodes
• Foundation Model — 1× in this paper · explained in 3 episodes
• Loss Function — 1× in this paper · explained in 2 episodes
• Quantization — 1× in this paper · explained in 12 episodes
• Scaling Laws — 1× in this paper · explained in 4 episodes
• Unified Framework — 1× in this paper · explained in 5 episodes

Sources

• Scale Weight Decay and Train Better
• Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition
• On MUON optimization: From non-convergence to an error analysis with Polar Express and the Newton-Schulz polynomial from implementations
• Rethinking Muon Beyond Pretraining: Spectral Failures and High-Pass Remedies for VLA and RLVR
• Accelerating Newton-Schulz Iteration for Orthogonalization via Chebyshev-type Polynomials
• GLM-5V-Turbo: Toward a Native Foundation Model for Multimodal Agents
• AMUSE: Anytime Muon with Stable Gradient Evaluation
• PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective
• Muon is Scalable for LLM Training
• Preconditioning Benefits of Spectral Orthogonalization in Muon
• Phases of Muon: When Muon Eclipses SignSGD
• Convergence Analysis of Muon-type Methods with Inexact LMO in the Degenerate Case
• Convergence Bound and Critical Batch Size of Muon Optimizer
• Practical Efficiency of Muon for Pretraining
• Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?
• Kimi K2: Open Agentic Intelligence
• Kimi K2.5: Visual Agentic Intelligence
• Why Muon Outperforms Adam: A Curvature Perspective
• DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence
• Qwen3 Technical Report

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