# MUR: Momentum Uncertainty guided Reasoning for Large Language Models

arXiv: https://arxiv.org/abs/2507.14958 · HTML: https://mlsanity.com/paper/2507.14958/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 48 of its 63 references that are in the map.
- First posted on arXiv 20 Jul 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 48)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [SelfCheckGPT: Zero-Resource Black-Box Hallucination Detection for Generative Large Language Models](https://mlsanity.com/paper/2303.08896/) (2023)
- [Do NOT Think That Much for 2+3=? On the Overthinking of o1-Like LLMs](https://mlsanity.com/paper/2412.21187/) (2024)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Stop Overthinking: A Survey on Efficient Reasoning for Large Language Models](https://mlsanity.com/paper/2503.16419/) (2025)
- [INSIDE: LLMs' Internal States Retain the Power of Hallucination Detection](https://mlsanity.com/paper/2402.03744/) (2024)
- [TokenSkip: Controllable Chain-of-Thought Compression in LLMs](https://mlsanity.com/paper/2502.12067/) (2025)
- [Dynamic Early Exit in Reasoning Models](https://mlsanity.com/paper/2504.15895/) (2025)
- [Out-of-Distribution Detection and Selective Generation for Conditional Language Models](https://mlsanity.com/paper/2209.15558/) (2022)
- [Kernel Language Entropy: Fine-grained Uncertainty Quantification for LLMs from Semantic Similarities](https://mlsanity.com/paper/2405.20003/) (2024)
- [General-Reasoner: Advancing LLM Reasoning Across All Domains](https://mlsanity.com/paper/2505.14652/) (2025)
- [Quantifying Uncertainty in Answers from any Language Model and Enhancing their Trustworthiness](https://mlsanity.com/paper/2308.16175/) (2023)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Towards Thinking-Optimal Scaling of Test-Time Compute for LLM Reasoning](https://mlsanity.com/paper/2502.18080/) (2025)
- [Decomposing Uncertainty for Large Language Models through Input Clarification Ensembling](https://mlsanity.com/paper/2311.08718/) (2023)
- [GenPRM: Scaling Test-Time Compute of Process Reward Models via Generative Reasoning](https://mlsanity.com/paper/2504.00891/) (2025)
- [Think Only When You Need with Large Hybrid-Reasoning Models](https://mlsanity.com/paper/2505.14631/) (2025)
- [Sampling-Efficient Test-Time Scaling: Self-Estimating the Best-of-N Sampling in Early Decoding](https://mlsanity.com/paper/2503.01422/) (2025)

## Built on by (most cited first, 17 of 17)

- [Semantic-Space Exploration and Exploitation in RLVR for LLM Reasoning](https://mlsanity.com/paper/2509.23808/) (2025)
- [ReProbe: Efficient Test-Time Scaling of Multi-Step Reasoning by Probing Internal States of Large Language Models](https://mlsanity.com/paper/2511.06209/) (2025)
- [Neural Chain-of-Thought Search: Searching the Optimal Reasoning Path to Enhance Large Language Models](https://mlsanity.com/paper/2601.11340/) (2026)
- [Reliable Thinking with Images](https://mlsanity.com/paper/2602.12916/) (2026)
- [Overthinking Reduction with Decoupled Rewards and Curriculum Data Scheduling](https://mlsanity.com/paper/2509.25827/) (2025)
- [Putting on the Thinking Hats: A Survey on Chain of Thought Fine-tuning from the Perspective of Human Reasoning Mechanism](https://mlsanity.com/paper/2510.13170/) (2025)
- [Draft-Thinking: Learning Efficient Reasoning in Long Chain-of-Thought LLMs](https://mlsanity.com/paper/2603.00578/) (2026)
- [Learning Adaptive LLM Decoding](https://mlsanity.com/paper/2603.09065/) (2026)
- [Fast, Slow, and Tool-augmented Thinking for LLMs: A Review](https://mlsanity.com/paper/2508.12265/) (2025)
- [TRiMS: Real-Time Tracking of Minimal Sufficient Length for Efficient Reasoning via RL](https://mlsanity.com/paper/2603.17449/) (2026)
- [When Embedding-Based Defenses Fail: Rethinking Safety in LLM-Based Multi-Agent Systems](https://mlsanity.com/paper/2605.01133/) (2026)
- [MAXS: Meta-Adaptive Exploration with LLM Agents](https://mlsanity.com/paper/2601.09259/) (2026)
- [Test-time Scaling over Perception: Resolving the Grounding Paradox in Thinking with Images](https://mlsanity.com/paper/2604.11025/) (2026)
- [Dual-Dimensional Consistency: Balancing Budget and Quality in Adaptive Inference-Time Scaling](https://mlsanity.com/paper/2605.15100/) (2026)
- [Towards Efficient and Robust Linguistic Emotion Diagnosis for Mental Health via Multi-Agent Instruction Refinement](https://mlsanity.com/paper/2601.13481/) (2026)
- [Uncertainty-Aware Budget Allocation for Adaptive Test-Time Reasoning](https://mlsanity.com/paper/2605.26849/) (2026)
- [ThinkBooster: A Unified Framework for Seamless Test-Time Scaling of LLM Reasoning](https://mlsanity.com/paper/2606.06915/) (2026)

## Abstract (arXiv)

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%.