Muon outperforms Adam by reducing curvature penalty via lower Normalized Directional Sharpness, as shown via Taylor approximation on LLM training and proven on stylized quadratic problems with heterogeneous curvature.
arXiv preprint arXiv:2510.16981 , year=
9 Pith papers cite this work. Polarity classification is still indexing.
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MONA integrates Nesterov acceleration into Muon's orthogonalization framework, reporting better convergence than Muon and AdamW on MoE models up to 68B parameters trained on 1T tokens and SOTA fine-tuning results.
Orth-Dion uses QR factorization on the right factor instead of column normalization to eliminate the geometric mismatch in low-rank approximations of spectral optimizers like Muon, achieving O(sqrt(L_r/T)) rate under non-Euclidean smoothness.
Muon learns more robust and transferable features than Adam and SGD, shown via corruption robustness tests, transfer experiments, layer-wise probes, effective rank measurements, and a theoretical proof on margins in a multi-component classification problem.
Different Bregman divergences distribute unavoidable Kronecker approximation error differently across covariance spectrum, with top eigenspace better aligned to Hessian, motivating a subspace-aware optimizer using eigenvalue preconditioning on top subspace and adaptive isotropic acceleration on bott
SF-NorMuon is a new schedule-free spectral optimizer that closes the gap with tuned AdamW on 125M-772M parameter models across 1-8x Chinchilla horizons while providing stationarity guarantees.
Derives finite-round upper-tail guarantee on population-empirical gap for client-sampled orthogonalized matrix momentum under heterogeneous data, with Lipschitz condition on the orthogonalizer.
Muon+ adds one normalization step after polar orthogonalization in the Muon optimizer, yielding lower training and validation perplexity and faster pre-training across 60M-7B models.
citing papers explorer
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Why Muon Outperforms Adam: A Curvature Perspective
Muon outperforms Adam by reducing curvature penalty via lower Normalized Directional Sharpness, as shown via Taylor approximation on LLM training and proven on stylized quadratic problems with heterogeneous curvature.
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MONA: Muon Optimizer with Nesterov Acceleration for Scalable Language Model Training
MONA integrates Nesterov acceleration into Muon's orthogonalization framework, reporting better convergence than Muon and AdamW on MoE models up to 68B parameters trained on 1T tokens and SOTA fine-tuning results.
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Orth-Dion: Eliminating Geometric Mismatch in Distributed Low-Rank Spectral Optimization
Orth-Dion uses QR factorization on the right factor instead of column normalization to eliminate the geometric mismatch in low-rank approximations of spectral optimizers like Muon, achieving O(sqrt(L_r/T)) rate under non-Euclidean smoothness.
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Muon Learns More Robust and Transferable Features than Adam
Muon learns more robust and transferable features than Adam and SGD, shown via corruption robustness tests, transfer experiments, layer-wise probes, effective rank measurements, and a theoretical proof on margins in a multi-component classification problem.
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Rethinking Bregman Divergences in Kronecker-Factored Optimizers
Different Bregman divergences distribute unavoidable Kronecker approximation error differently across covariance spectrum, with top eigenspace better aligned to Hessian, motivating a subspace-aware optimizer using eigenvalue preconditioning on top subspace and adaptive isotropic acceleration on bott
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Anytime Training with Schedule-Free Spectral Optimization
SF-NorMuon is a new schedule-free spectral optimizer that closes the gap with tuned AdamW on 125M-772M parameter models across 1-8x Chinchilla horizons while providing stationarity guarantees.
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A Note on Stability for Orthogonalized Matrix Momentum with Client Sampling
Derives finite-round upper-tail guarantee on population-empirical gap for client-sampled orthogonalized matrix momentum under heterogeneous data, with Lipschitz condition on the orthogonalizer.
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MUON+: Towards More Effective Muon via One Additional Normalization Step for LLM Pre-training
Muon+ adds one normalization step after polar orthogonalization in the Muon optimizer, yielding lower training and validation perplexity and faster pre-training across 60M-7B models.
- LionMuon: Alternating Spectral and Sign Descent for Efficient Training