# InfoDensity: Rewarding Information-Dense Traces for Efficient Reasoning

arXiv: https://arxiv.org/abs/2603.17310 · HTML: https://mlsanity.com/paper/2603.17310/

## Key facts

- Cited by 1 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 28 of its 47 references that are in the map.
- First posted on arXiv 18 Mar 2026.

## 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, 28 of 28)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Do NOT Think That Much for 2+3=? On the Overthinking of o1-Like LLMs](https://mlsanity.com/paper/2412.21187/) (2024)
- [L1: Controlling How Long A Reasoning Model Thinks With Reinforcement Learning](https://mlsanity.com/paper/2503.04697/) (2025)
- [Demystifying Long Chain-of-Thought Reasoning in LLMs](https://mlsanity.com/paper/2502.03373/) (2025)
- [Chain of Draft: Thinking Faster by Writing Less](https://mlsanity.com/paper/2502.18600/) (2025)
- [DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning](https://mlsanity.com/paper/2504.11456/) (2025)
- [O1-Pruner: Length-Harmonizing Fine-Tuning for O1-Like Reasoning Pruning](https://mlsanity.com/paper/2501.12570/) (2025)
- [Training Language Models to Reason Efficiently](https://mlsanity.com/paper/2502.04463/) (2025)
- [Dynamic Early Exit in Reasoning Models](https://mlsanity.com/paper/2504.15895/) (2025)
- [ThinkPrune: Pruning Long Chain-of-Thought of LLMs via Reinforcement Learning](https://mlsanity.com/paper/2504.01296/) (2025)
- [Reasoning Models Can Be Effective Without Thinking](https://mlsanity.com/paper/2504.09858/) (2025)
- [Learn to Reason Efficiently with Adaptive Length-based Reward Shaping](https://mlsanity.com/paper/2505.15612/) (2025)
- [Sample More to Think Less: Group Filtered Policy Optimization for Concise Reasoning](https://mlsanity.com/paper/2508.09726/) (2025)
- [Consistent Paths Lead to Truth: Self-Rewarding Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.08745/) (2025)
- [Making Slow Thinking Faster: Compressing LLM Chain-of-Thought via Step Entropy](https://mlsanity.com/paper/2508.03346/) (2025)
- [Towards Concise and Adaptive Thinking in Large Reasoning Models: A Survey](https://mlsanity.com/paper/2507.09662/) (2025)
- [AALC: Large Language Model Efficient Reasoning via Adaptive Accuracy-Length Control](https://mlsanity.com/paper/2506.20160/) (2025)
- [What Defines Good Reasoning in LLMs? Dissecting Reasoning Steps with Multi-Aspect Evaluation](https://mlsanity.com/paper/2510.20603/) (2025)
- [REA-RL: Reflection-Aware Online Reinforcement Learning for Efficient Reasoning](https://mlsanity.com/paper/2505.19862/) (2025)
- [Measuring Reasoning Utility in LLMs via Conditional Entropy Reduction](https://mlsanity.com/paper/2508.20395/) (2025)
- [Uncertainty Under the Curve: A Sequence-Level Entropy Area Metric for Reasoning LLM](https://mlsanity.com/paper/2508.20384/) (2025)
- [PEAR: Phase Entropy Aware Reward for Efficient Reasoning](https://mlsanity.com/paper/2510.08026/) (2025)

## Built on by (most cited first, 1 of 1)

- [Momentum for Reasoning: Dense Intrinsic Signals in Policy Optimization](https://mlsanity.com/paper/2606.08815/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) with extended reasoning capabilities often generate verbose and redundant reasoning traces, incurring unnecessary computational cost. While existing reinforcement learning approaches address this by optimizing final response length, they neglect the quality of intermediate reasoning steps, leaving models vulnerable to reward hacking. We argue that verbosity is not merely a length problem, but a symptom of poor intermediate reasoning quality. To investigate this, we conduct an empirical study tracking the per-token predictive entropy of large reasoning models across reasoning trajectories. We find that high-quality reasoning traces exhibit two consistent properties: low uncertainty convergence and fast uncertainty descent. These findings suggest that high-quality reasoning traces are informationally dense, that is, reasoning steps contribute to reaching a low uncertainty level relative to the total reasoning length. Motivated by this, we propose InfoDensity, a reward framework for RL training that captures both properties through a single suffix-max envelope of the entropy trajectory, weighted by a length scaling term that favors achieving equivalent quality more concisely. Experiments on mathematical and general reasoning benchmarks demonstrate that InfoDensity outperforms state-of-the-art baselines on the accuracy-efficiency trade-off.