Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining

• URL: http://arxiv.org/abs/2608.28442v1
• Date: Fri, 28 Aug 2026 15:28:05 GMT
• Title: Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining
• Abstract summary: We propose a curvature-conditioned multiscale momentum method with sphere constraints to enhance training dynamics along flat directions.<n>We show that the proposed method significantly accelerates Muon across diverse architectures (dense, MoE) and model sizes (0.12B--2.3B parameters)
• Score: 19.3401576040633
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 directions, we propose a curvature-conditioned multiscale momentum method with sphere constraints, delivering steady acceleration in LLM pretraining. This multiscale momentum, applied only along flat directions, pairs a slow-decay component for noise reduction with a fast-decay component for rapid curvature adaptation, harnessing their complementary strengths. Crucially, we employ a sphere constraint technique to prevent parameter inflation and excessively rapid effective learning rate decay that would otherwise arise from a naive combination. Extensive experiments show that the proposed method significantly accelerates Muon across diverse architectures (dense, MoE) and model sizes (0.12B--2.3B parameters). Theoretically, we verify the acceleration effect and provide insight into the design principles underlying the flat-direction multiscale momentum.

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