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arXiv · 2507.14958

MUR: Momentum Uncertainty guided Reasoning for Large Language Models

Hang Yan·Fangzhi Xu·Rongman Xu·Yifei Li·Jian Zhang·Haoran Luo·Xiaobao Wu·Luu Anh Tuan·Haiteng Zhao·Qika Lin·Jun Liu
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Abstract

Large Language Models have achieved impressive performance on reasoning-intensive tasks, yet optimizing their reasoning efficiency remains an open challenge. While Test-Time Scaling (TTS) improves reasoning quality, it often leads to overthinking, wasting tokens on redundant computations. This work investigates how to efficiently and adaptively guide current model' test-time scaling without additional training. Inspired by the concept of momentum in physics, we propose Momentum Uncertainty-guided Reasoning (MUR), which dynamically allocates thinking budgets to critical reasoning steps by tracking and aggregating stepwise uncertainty over time. To support flexible inference-time control, we introduce gamma-control, a simple mechanism that tunes the reasoning budget via a single hyperparameter. We provide in-depth theoretical proof to support the superiority of MUR in terms of stability and biases. MUR is comprehensively evaluated against various TTS methods across four challenging benchmarks (MATH-500, AIME24, AIME25, and GPQA-diamond) using different sizes of recent Qwen3 models (1.7B, 4B, and 8B). Results demonstrate that MUR reduces computation by by over 45% on average while improving accuracy from 0.33 to 3.46%.

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Hang Yan, Fangzhi Xu, Rongman Xu, Yifei Li, Jian Zhang, Haoran Luo, Xiaobao Wu, Luu Anh Tuan, Haiteng Zhao, Qika Lin, Jun Liu. 2025-07-20. MUR: Momentum Uncertainty guided Reasoning for Large Language Models. https://arxiv.org/abs/2507.14958

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