InfoDensity: Rewarding Information-Dense Traces for Efficient Reasoning

TL;DR

This paper introduces InfoDensity, a reward framework for reinforcement learning that encourages information-dense reasoning traces in large language models, leading to more efficient and accurate reasoning with less verbosity.

Contribution

It proposes a novel reward method combining entropy-based and monotonicity measures to improve reasoning quality and efficiency in LLMs.

Findings

InfoDensity achieves comparable or better accuracy than state-of-the-art methods.

Models trained with InfoDensity use significantly fewer tokens.

The approach promotes concise, high-quality reasoning traces.

Abstract

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 conditional entropy of the answer distribution across reasoning steps. We find that high-quality reasoning traces exhibit two consistent properties: low uncertainty convergence and monotonic progress. These findings suggest that high-quality reasoning traces are informationally dense, that is, each step contributes meaningful entropy reduction relative to the total reasoning length. Motivated by this, we propose InfoDensity, a reward framework for RL training that combines an AUC-based reward and a monotonicity reward as a unified measure of reasoning quality, weighted by a length scaling term that favors achieving equivalent quality more concisely. Experiments on mathematical reasoning benchmarks demonstrate that InfoDensity matches or surpasses state-of-the-art baselines in accuracy while significantly reducing token usage, achieving a strong accuracy-efficiency trade-off.