MUR: Momentum Uncertainty guided Reasoning for Large Language Models
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%.
688332d5846d2e78b7548b6f
Hang Yan
688332d5846d2e78b7548b70
Fangzhi Xu
688332d5846d2e78b7548b71
Rongman Xu
688332d5846d2e78b7548b72
Yifei Li
688332d5846d2e78b7548b73
Jian Zhang
688332d5846d2e78b7548b74
Haoran Luo
688332d5846d2e78b7548b75
Xiaobao Wu
688332d5846d2e78b7548b76
Luu Anh Tuan
688332d5846d2e78b7548b77
Haiteng Zhao
688332d5846d2e78b7548b78
Qika Lin
688332d5846d2e78b7548b79
Jun Liu
MUR: Momentum Uncertainty guided Reasoning for Large Language Models
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%.
Momentum Uncertainty-guided Reasoning (MUR) dynamically allocates computational resources to improve reasoning efficiency and accuracy in Large Language Models without additional training.
Large Language Models Test-Time Scaling Momentum Uncertainty-guided Reasoning gamma-control MATH-500 AIME24 AIME25 GPQA-diamond Qwen3