arXiv — Machine Learning · · 3 min read

Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining

Mirrored from arXiv — Machine Learning for archival readability. Support the source by reading on the original site.

Computer Science > Machine Learning

arXiv:2608.28442 (cs)
[Submitted on 28 Aug 2026]

Title:Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining

View a PDF of the paper titled Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining, by Shuchen Zhu and Yuxin Fang and Mingze Wang and Kun Yuan
View PDF
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.
Comments: 50 pages
Subjects: Machine Learning (cs.LG)
Cite as: arXiv:2608.28442 [cs.LG]
  (or arXiv:2608.28442v1 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2608.28442
arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Shuchen Zhu [view email]
[v1] Fri, 28 Aug 2026 15:28:05 UTC (4,808 KB)
Full-text links:

Access Paper:

    View a PDF of the paper titled Curvature-Conditioned Multiscale Momentum with Sphere Constraints for LLM Pretraining, by Shuchen Zhu and Yuxin Fang and Mingze Wang and Kun Yuan
  • View PDF
  • TeX Source

Current browse context:

cs.LG
< prev   |   next >
Change to browse by:
cs

References & Citations

Loading...

BibTeX formatted citation

loading...
Data provided by:

Bookmark

BibSonomy Reddit
Bibliographic Tools

Bibliographic and Citation Tools

Bibliographic Explorer Toggle
Bibliographic Explorer (What is the Explorer?)
Connected Papers Toggle
Connected Papers (What is Connected Papers?)
Litmaps Toggle
Litmaps (What is Litmaps?)
scite.ai Toggle
scite Smart Citations (What are Smart Citations?)
Code, Data, Media

Code, Data and Media Associated with this Article

alphaXiv Toggle
alphaXiv (What is alphaXiv?)
Links to Code Toggle
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub Toggle
DagsHub (What is DagsHub?)
GotitPub Toggle
Gotit.pub (What is GotitPub?)
Huggingface Toggle
Hugging Face (What is Huggingface?)
ScienceCast Toggle
ScienceCast (What is ScienceCast?)
Demos

Demos

Replicate Toggle
Replicate (What is Replicate?)
Spaces Toggle
Hugging Face Spaces (What is Spaces?)
Spaces Toggle
TXYZ.AI (What is TXYZ.AI?)
Related Papers

Recommenders and Search Tools

Link to Influence Flower
Influence Flower (What are Influence Flowers?)
Core recommender toggle
CORE Recommender (What is CORE?)
IArxiv recommender toggle
IArxiv Recommender (What is IArxiv?)
About arXivLabs

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Discussion (0)

Sign in to join the discussion. Free account, 30 seconds — email code or GitHub.

Sign in →

No comments yet. Sign in and be the first to say something.

More from arXiv — Machine Learning