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A Physical Response-and-Memory Model for Muon Optimization

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TL;DR - This paper models Muon optimization as a physical medium with memory, interpreting semi-orthogonalized updates as maximally dissipative responses under a safety constraint. It derives a two-timescale Bi-Maxwell optimizer that reaches a target loss in fewer steps on a public LLM training benchmark.

  • The model explains momentum as accumulated internal stress and its averaging window as a stress-relaxation timescale.
  • Because real media can relax at multiple rates, Bi-Maxwell replaces Muon’s single-timescale memory kernel with fast and slow components.
  • Measurements across eight independent training runs support the prediction that optimal memory length should increase as gradient directions change more slowly later in training.
  • Changing only the memory kernel to the two-timescale form reduced the steps needed to reach the benchmark’s target loss.

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A Physical Response-and-Memory Model for Muon Optimization

arXiv cs.LG Yinze Hu, Hongjun Xiang, Xingao Gong, Hongyu Yu 2026-08-24 arXiv:2608.22994
Public signals Semantic Scholar citations 0 · Semantic Scholar influential citations 0
Providers: Hugging Face · N/A OpenAlex · N/A Publisher · N/A Semantic Scholar · Citations 0 · Influential citations 0 X · N/A Fetched 2026-09-17 14:28:43.119346 UTC

TL;DR - This paper models Muon optimization as a physical medium with memory, interpreting semi-orthogonalized updates as maximally dissipative responses under a safety constraint. It derives a two-timescale Bi-Maxwell optimizer that reaches a target loss in fewer steps on a public LLM training benchmark.

  • The model explains momentum as accumulated internal stress and its averaging window as a stress-relaxation timescale.
  • Because real media can relax at multiple rates, Bi-Maxwell replaces Muon’s single-timescale memory kernel with fast and slow components.
  • Measurements across eight independent training runs support the prediction that optimal memory length should increase as gradient directions change more slowly later in training.
  • Changing only the memory kernel to the two-timescale form reduced the steps needed to reach the benchmark’s target loss.
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