Muon Optimizer

Muon is a matrix-aware optimizer that accumulates momentum for weight matrices and approximately orthogonalizes each update, often with Newton–Schulz iterations, rather than scaling each parameter independently as Adam does. Research explores variants with curvature or variance preconditioning, different spectral weighting and normalization, and cheaper or distributed orthogonalization. Studies test Muon in language-model training, vision, recommendation, diffusion, and scientific machine learning, while theory examines convergence, noise, implicit bias, and generalization. A central open question is when its spectral updates improve learning enough to justify their cost: results depend on architecture, batch size, and training phase, and practical approximations can change both behavior and guarantees.