Spectral Allocation: Why Muon Outperforms Adam, and How to Improve Muon

arXiv:2608.25990 · cs.LG · Submitted 2026-08-26 · Read on arXiv

cs.LG

Submitted: 2026-08-26

Updated: 2026-08-26

Comments: 34 pages, 13 figures, 7 tables

Code: https://github.com/KellerJordan/modded-nanogpt

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Adam — 42× in this paper · explained in 2 episodes
• Transformer — 21× in this paper · explained in 2 episodes
• Alignment — 13× in this paper · explained in 1 episode
• Anisotropy — 7× in this paper · explained in 3 episodes
• Embeddings — 5× in this paper · explained in 2 episodes
• Large Language Model — 4× in this paper
• Benchmark — 3× in this paper · explained in 2 episodes
• Objective Function — 3× in this paper · explained in 2 episodes
• Ablation — 2× in this paper · explained in 1 episode
• Convergence Guarantees — 2× in this paper · explained in 3 episodes
• In International Conference on Machine Learning — 2× in this paper
• Singular Value Decomposition — 2× in this paper
• Singular Value Decomposition (SVD) — 2× in this paper · explained in 2 episodes
• Adaptivity — 1× in this paper · explained in 1 episode
• Calibration — 1× in this paper · explained in 7 episodes
• Clustering — 1× in this paper · explained in 2 episodes
• Computational overhead — 1× in this paper · explained in 2 episodes
• Core — 1× in this paper · explained in 2 episodes
• Distillation — 1× in this paper · explained in 2 episodes
• Exponential Moving Average (EMA) — 1× in this paper · explained in 2 episodes
• In International Conference on Learning Representations — 1× in this paper
• Large Language Model (LLM) — 1× in this paper · explained in 3 episodes
• Large Language Models (LLMs) — 1× in this paper · explained in 6 episodes
• Quantization — 1× in this paper · explained in 12 episodes
• Regularization — 1× in this paper · explained in 3 episodes
• Scaling Laws — 1× in this paper · explained in 4 episodes
• Spectral Geometry — 1× in this paper · explained in 2 episodes
• Stochastic Gradient Descent (SGD) — 1× in this paper · explained in 5 episodes
• Trace — 1× in this paper · explained in 2 episodes

Sources

• The Newton-Muon Optimizer
• Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data
• LiMuon: Light and Fast Muon Optimizer for Large Models
• Kimi K2: Open Agentic Intelligence
• Adam: A Method for Stochastic Optimization
• Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization
• Scalable Optimization in the Modular Norm
• PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective
• A Note on the Convergence of Muon
• NorMuon: Making Muon more efficient and scalable
• Muon is Scalable for LLM Training
• Spectral Scaling Laws of Muon
• The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale
• On the Convergence Analysis of Muon
• AdaMuon: Adaptive Muon Optimizer
• Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?
• A Spectral Condition for Feature Learning
• Mousse: Rectifying the Geometry of Muon with Curvature-Aware Preconditioning

Related papers

• Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation
• AIRL-S: Unifying Reinforcement Learning and Search-Based Test-Time Scaling via Adversarial Inverse Reinforcement Learning
• Transformers as Bayesian In-Context Experimenters: Smoothness-Adaptive Efficient ATE Estimation
• Convergence issues in Relational Concept Analysis based on AOC-posets
• Beliefs Beyond Posteriors: Local-Consistency Optimisation for Bayesian Neural Networks
• Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks