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The paper introduces InfoDensity, a reinforcement learning reward framework designed to improve the efficiency of LLM reasoning by incentivizing information-dense reasoning traces. InfoDensity combines an AUC-based reward and a monotonicity reward to measure reasoning quality, scaled by a length term to favor conciseness. Experiments on mathematical reasoning benchmarks show that InfoDensity achieves comparable or better accuracy than state-of-the-art methods while significantly reducing token usage, demonstrating an improved accuracy-efficiency trade-off.
LLMs can achieve state-of-the-art reasoning accuracy with significantly fewer tokens by rewarding intermediate reasoning steps that maximize information gain and maintain monotonic progress.
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.
