MUR: Momentum Uncertainty guided Reasoning for Large Language Models
Authors: name: Hang Yan, name: Fangzhi Xu, name: Rongman Xu, name: Yifei Li, name: Jian Zhang, name: Haoran Luo, name: Xiaobao Wu, name: Luu Anh Tuan, name: Haiteng Zhao, name: Qika Lin, name: Jun Liu

Large Language Models (LLMs) 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 LLM 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 over 50% on average while improving accuracy by
0.62-3.37%.