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This paper proposes a reinforcement learning method for LLM arithmetic reasoning that converts sparse end-of-sequence rewards into denser token-level training signals using Bellman residual minimization, RLOO variance reduction, and hindsight relabeling. The evidence rests on a single benchmark (Countdown) and claims of "state-of-the-art" and "exceptional" efficiency, so readers should check the baseline setup and whether the gains hold on other reasoning tasks. For ML practitioners, the main question is whether the dense-reward approach generalizes beyond this toy-like setting.
Bellman-Consistent Trajectory RL for LLM Arithmetic Reasoning
arXiv:2610.21060v1 Announce Type: cross Abstract: To address sparse terminal rewards in LLM arithmetic reasoning, we propose a Bellman-consistent trajectory RL method. Our framework introduces Trajectory Bellman Residual Minimization (TBRM) to transform sparse rewards into dense token-level signals. We integrate a Reinforce Leave-One-Out (RLOO) group strategy to reduce gradient variance of Bellman residuals and use hindsight goal relabeling to accelerate intermediate learning. Evaluated on the Countdown task, our method alleviates optimization bottlenecks and stabilizes gradient variance significantly better than GRPO. The TBRM-RLOO framework boosts Pass@1 to 0.55--0.64 and achieves a state-of-the-art Pass@16 accuracy of 76.0--84.0\%, outperforming the SFT, IPO/DPO, and RLOO baselines. Our approach demonstrates exceptional single-sample efficiency and successfully circumvents standard performance plateaus via test-time scaling.
Abstract:To address sparse terminal rewards in LLM arithmetic reasoning, we propose a Bellman-consistent trajectory RL method. Our framework introduces Trajectory Bellman Residual Minimization (TBRM) to transform sparse rewards into dense token-level signals. We integrate a Reinforce Leave-One-Out (RLOO) group strategy to reduce gradient variance of Bellman residuals and use hindsight goal relabeling to accelerate intermediate learning. Evaluated on the Countdown task, our method alleviates optimization bottlenecks and stabilizes gradient variance significantly better than GRPO. The TBRM-RLOO framework boosts Pass@1 to 0.55--0.64 and achieves a state-of-the-art Pass@16 accuracy of 76.0--84.0\%, outperforming the SFT, IPO/DPO, and RLOO baselines. Our approach demonstrates exceptional single-sample efficiency and successfully circumvents standard performance plateaus via test-time scaling.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2610.21060 [cs.LG]
(or arXiv:2610.21060v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.21060 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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