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.
Muon$^2$: Boosting Muon via Adaptive Second-Moment Preconditioning
5 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Muon has emerged as a promising optimizer for large-scale foundation model pre-training by exploiting the matrix structure of neural network updates through iterative orthogonalization. However, the orthogonalization quality of Muon hinges on the number of Newton--Schulz (NS) iterations performed, which poses efficiency challenges due to its non-trivial computation and communication cost. We propose Muon$^2$, an extension of Muon, to improve both quality and efficiency by applying Adam-style adaptive second-moment preconditioning before orthogonalization. Our key insight is that the core challenge of polar approximation in Muon lies in the ill-conditioned momentum matrix, of which the spectrum is substantially improved by Muon$^2$, leading to faster convergence toward a practically sufficient orthogonalization. We further characterize the practical orthogonalization quality via directional alignment, under which Muon$^2$ demonstrates dramatic improvement over Muon at each polar step. Across GPT, LLaMA, and Mixture-of-Experts pre-training experiments up to 13B parameters, Muon$^2$ (and its memory-efficient variant Muon$^2$-F that preserves most of its benefits) consistently outperforms Muon and its variants while reducing NS iterations by 40%, and saves up to 1/4 training time over Muon when achieving the same loss.
fields
cs.LG 5years
2026 5representative citing papers
PolarAdamW disentangles spectral control from gauge-equivariance in matrix optimizers, with experiments demonstrating their distinct roles on standard versus symmetry-aware neural networks.
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.
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.
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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.