# Critical Tokens Matter: Token-Level Contrastive Estimation Enhances LLM's Reasoning Capability

arXiv: https://arxiv.org/abs/2411.19943 · HTML: https://mlsanity.com/paper/2411.19943/

## Key facts

- Cited by 83 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 50 of its 53 references that are in the map.
- First posted on arXiv 29 Nov 2024.

## 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 50)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Scaling Laws for Reward Model Overoptimization](https://mlsanity.com/paper/2210.10760/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [A General Theoretical Paradigm to Understand Learning from Human Preferences](https://mlsanity.com/paper/2310.12036/) (2023)
- [Open Problems and Fundamental Limitations of Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2307.15217/) (2023)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Contrastive Decoding: Open-ended Text Generation as Optimization](https://mlsanity.com/paper/2210.15097/) (2022)
- [RewardBench: Evaluating Reward Models for Language Modeling](https://mlsanity.com/paper/2403.13787/) (2024)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs](https://mlsanity.com/paper/2406.18629/) (2024)
- [Is DPO Superior to PPO for LLM Alignment? A Comprehensive Study](https://mlsanity.com/paper/2404.10719/) (2024)
- [Iterative Reasoning Preference Optimization](https://mlsanity.com/paper/2404.19733/) (2024)
- [Smaug: Fixing Failure Modes of Preference Optimisation with DPO-Positive](https://mlsanity.com/paper/2402.13228/) (2024)
- [Secrets of RLHF in Large Language Models Part I: PPO](https://mlsanity.com/paper/2307.04964/) (2023)
- [Towards Understanding Chain-of-Thought Prompting: An Empirical Study of What Matters](https://mlsanity.com/paper/2212.10001/) (2022)
- [ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models](https://mlsanity.com/paper/2310.10505/) (2023)

## Built on by (most cited first, 30 of 83)

- [Phi-4 Technical Report](https://mlsanity.com/paper/2412.08905/) (2024)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [CODI: Compressing Chain-of-Thought into Continuous Space via Self-Distillation](https://mlsanity.com/paper/2502.21074/) (2025)
- [A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?](https://mlsanity.com/paper/2503.24235/) (2025)
- [Agentic Reinforced Policy Optimization](https://mlsanity.com/paper/2507.19849/) (2025)
- [Inner Thinking Transformer: Leveraging Dynamic Depth Scaling to Foster Adaptive Internal Thinking](https://mlsanity.com/paper/2502.13842/) (2025)
- [A Comprehensive Survey of Reward Models: Taxonomy, Applications, Challenges, and Future](https://mlsanity.com/paper/2504.12328/) (2025)
- [Demystifying Reasoning Dynamics with Mutual Information: Thinking Tokens are Information Peaks in LLM Reasoning](https://mlsanity.com/paper/2506.02867/) (2025)
- [Distilling LLM Agent into Small Models with Retrieval and Code Tools](https://mlsanity.com/paper/2505.17612/) (2025)
- [Embarrassingly Simple Self-Distillation Improves Code Generation](https://mlsanity.com/paper/2604.01193/) (2026)
- [Parameters vs. Context: Fine-Grained Control of Knowledge Reliance in Language Models](https://mlsanity.com/paper/2503.15888/) (2025)
- [Spotlight on Token Perception for Multimodal Reinforcement Learning](https://mlsanity.com/paper/2510.09285/) (2025)
- [Rethinking On-Policy Self-Distillation for Thinking Models](https://mlsanity.com/paper/2607.05184/) (2026)
- [VCRL: Variance-based Curriculum Reinforcement Learning for Large Language Models](https://mlsanity.com/paper/2509.19803/) (2025)
- [Reinforcing Video Reasoning with Focused Thinking](https://mlsanity.com/paper/2505.24718/) (2025)
- [Estimating LLM Uncertainty with Evidence](https://mlsanity.com/paper/2502.00290/) (2025)
- [On the Position Bias of On-Policy Distillation](https://mlsanity.com/paper/2606.22600/) (2026)
- [Attention Illuminates LLM Reasoning: The Preplan-and-Anchor Rhythm Enables Fine-Grained Policy Optimization](https://mlsanity.com/paper/2510.13554/) (2025)
- [From Judgment to Interference: Early Stopping LLM Harmful Outputs via Streaming Content Monitoring](https://mlsanity.com/paper/2506.09996/) (2025)
- [Sparse but Critical: A Token-Level Analysis of Distributional Shifts in RLVR Fine-Tuning of LLMs](https://mlsanity.com/paper/2603.22446/) (2026)
- [AdaReasoner: Adaptive Reasoning Enables More Flexible Thinking in Large Language Models](https://mlsanity.com/paper/2505.17312/) (2025)
- [Group Critical-token Policy Optimization for Autoregressive Image Generation](https://mlsanity.com/paper/2509.22485/) (2025)
- [STEPS: Selective On-Policy Self-Distillation for Reasoning](https://mlsanity.com/paper/2605.10194/) (2026)
- [Focused-DPO: Enhancing Code Generation Through Focused Preference Optimization on Error-Prone Points](https://mlsanity.com/paper/2502.11475/) (2025)
- [Token-Importance Guided Direct Preference Optimization](https://mlsanity.com/paper/2505.19653/) (2025)
- [Dancing with Critiques: Enhancing LLM Reasoning with Stepwise Natural Language Self-Critique](https://mlsanity.com/paper/2503.17363/) (2025)
- [Learning to Focus: Causal Attention Distillation via Gradient-Guided Token Pruning](https://mlsanity.com/paper/2506.07851/) (2025)
- [SynAdapt: Learning Adaptive Reasoning in Large Language Models via Synthetic Continuous Chain-of-Thought](https://mlsanity.com/paper/2508.00574/) (2025)
- [LLM2: Let Large Language Models Harness System 2 Reasoning](https://mlsanity.com/paper/2412.20372/) (2024)

## Abstract (arXiv)

Mathematical reasoning tasks pose significant challenges for large language models (LLMs) because they require precise logical deduction and sequence analysis. In this work, we introduce the concept of critical tokens -- elements within reasoning trajectories that significantly influence incorrect outcomes. We present a novel framework for identifying these tokens through rollout sampling and demonstrate their substantial divergence from traditional error tokens. Through extensive experiments on datasets such as GSM8K and MATH500, we show that identifying and replacing critical tokens significantly improves model accuracy. We propose an efficient methodology for pinpointing these tokens in large-scale datasets using contrastive estimation and extend this framework to enhance model training processes with direct preference optimization (DPO). Experimental results on GSM8K and MATH500 benchmarks with the widely used models Llama-3 (8B and 70B) and Deepseek-math (7B) demonstrate the effectiveness of the proposed approach, cDPO. Our results underscore the potential of leveraging critical tokens to reduce errors in reasoning tasks, advancing the development of AI systems capable of robust logical deduction. Our code, annotated datasets, and trained models are available at https://github.com/chenzhiling9954/Critical-Tokens-Matter to support and encourage future research in this promising field.