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MUR: Momentum Uncertainty guided Reasoning for Large Language Models

Paper · 2026 · Hang Yan, Fangzhi Xu, Rongman Xu et al.

Summary

Test-time scaling (TTS) makes LLMs reason better but tends to cause overthinking: extra compute goes to every reasoning step, including trivial ones. The authors ask how to decide, without any additional training, which steps actually need extra compute. Their method, MUR (Momentum Uncertainty guided Reasoning), measures each step's uncertainty as the average negative log-likelihood of its tokens. It then keeps a running, exponentially weighted aggregate of these values, an idea borrowed from physical momentum and momentum-based gradient descent. A detector invokes a TTS method only when the current step looks unusually uncertain relative to this accumulated signal. A single hyperparameter γ ("γ-control") sets how strict the trigger is, and therefore how much compute is spent. The authors argue theoretically that the momentum estimate behaves like a low-pass filter that smooths decoding noise. Across four hard benchmarks and three Qwen3 sizes, MUR uses far fewer tokens than scaling every step, and accuracy slightly improves rather than drops. This makes TTS cheaper and easier to plug into existing pipelines.

Key points

  • Formula: momentum uncertainty is computed as M_t = αM_{t−1} + (1−α)m_t, with M_0 = 0. This gives more weight to recent steps and lets early ones fade.
  • Evaluation setup:
    • Benchmarks: MATH-500, AIME24, AIME25 and GPQA-diamond.
    • Backbones: Qwen3-1.7B, Qwen3-4B and Qwen3-8B.
    • MUR is paired with three TTS techniques: ϕ-decoding, Guided search, and an approach that uses an LLM to critique steps.
  • Main result: compared with Per-Step Scale, MUR cuts compute by over 45% on average while raising accuracy by 0.33–3.46%.
  • Step analysis:
    • Queries average 4.38–6.49 reasoning steps, but only 0.45–0.90 of them are scaled.
    • Harder benchmarks trigger more scaling.
    • Randomly scaling the same number of steps performs worse; MUR beats random scaling by 1.72% and Per-Step Scale by 1.53% on average.
  • γ-control and limitations:
    • Raising γ trades more tokens for accuracy; γ = ∞ reduces to Per-Step Scale.
    • MUR depends on the model's own probability calibration.
    • It is hard to apply to unstructured, non-stepwise outputs.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🎓National University of SingaporeSingapore · Singapore207 works · 39,576 citations🎓Shanghai Jiao Tong UniversityShanghai · China212 works · 19,544 citations🎓Nanyang Technological UniversitySingapore · Singapore184 works · 27,484 citations🔬Shanghai Artificial Intelligence LaboratoryShanghai · China152 works · 13,164 citations🎓Xi'an Jiaotong UniversityXi'an · China42 works · 36,431 citations🎓VinUniversityHanoi · Vietnam13 works · 56 citations

 Fields, topics and concepts

📚 Shelves 5The field of science this work belongs to (OpenAlex fields).Computer Science · 18,614 worksArtificial Intelligence · 17,965 worksArts and Humanities · 17,450 worksLanguage and Linguistics · 17,296 worksComputer Vision and Pattern Recognition · 3,486 works

🧭 Topics 2Research subjects and languages the work is about.Language models & pretraining · 3,894 worksReasoning · 1,685 works

Cite

Cite as Hang Yan, Fangzhi Xu, Rongman Xu, Yifei Li, Jian Zhang, Haoran Luo, Xiaobao Wu, Anh Tuan Luu, Haiteng Zhao, Qika Lin and Jun Liu. 2026. MUR: Momentum Uncertainty guided Reasoning for Large Language Models. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers).

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.18653/v1/2026.acl-long.1058

@inproceedings{yan2026mur,
  title = {{MUR: Momentum Uncertainty guided Reasoning for Large Language Models}},
  author = {Hang Yan, Fangzhi Xu, Rongman Xu, Yifei Li, Jian Zhang, Haoran Luo, Xiaobao Wu, Anh Tuan Luu, Haiteng Zhao, Qika Lin and Jun Liu},
  booktitle = {Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)},
  year = {2026},
  doi = {10.18653/v1/2026.acl-long.1058},
  url = {https://doi.org/10.18653/v1/2026.acl-long.1058}
}

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