From SGD to Muon: Adaptive Optimization via Schatten-p Norms

arXiv:2605.19781 · cs.AI · Submitted 2026-05-19 · Read on arXiv

cs.AI

Submitted: 2026-05-19

Updated: 2026-10-09

Journal ref: Conference on Neural Information Processing Systems, Dec 2026, Sydney, Australia

Code: https://github.com/KellerJordan/Muon

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

Terminology

• Adam — 1× in this paper · explained in 2 episodes
• Linear Minimization Oracle — 1× in this paper
• Surrogate Model — 1× in this paper · explained in 2 episodes

Sources

• Dion: Distributed Orthonormalized Updates
• MuonEq: Balancing Before Orthogonalization with Lightweight Equilibration
• Training Verifiers to Solve Math Word Problems
• Insights on Muon from Simple Quadratics
• Accelerating Newton-Schulz Iteration for Orthogonalization via Chebyshev-type Polynomials
• Scaling Laws for Neural Language Models
• Adam: A Method for Stochastic Optimization
• NorMuon: Making Muon more efficient and scalable
• Muon is Scalable for LLM Training
• Decoupled Weight Decay Regularization
• Curvature-Informed SGD via General Purpose Lie-Group Preconditioners
• Delving into Muon and Beyond: Deep Analysis and Extensions
• AdaMuon: Adaptive Muon Optimizer
• Preconditioned Norms: A Unified Framework for Steepest Descent, Quasi-Newton and Adaptive Methods
• Qwen2 Technical Report

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