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Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining

arXiv · AI, language, vision and robotics · article · Aug 28, 2026 · UTC

Pretraining accounts for a large fraction of the total computational cost in LLM training. However, noise-dominant gradients and the highly ill-conditioned loss landscape bring severe challenges. Although modern adaptive optimizers such as AdamW and Muon have achieved great success in large-scale pretraining, their reliance on gradient normalization offers limited mitigation of the ill-conditioned curvature. The progress along flat directions (eigen-directions of small eigenvalues), which dominates the final loss reduction, remains relatively slow. To enhance training dynamics along flat direc

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