# Cross-Rollout Bellman Closure for Long-Horizon Agentic Reinforcement Learning

arXiv: https://arxiv.org/abs/2609.35082 · HTML: https://mlsanity.com/paper/2609.35082/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 42 of its 48 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- LLM agents and tool use / [Reinforcement learning for LLM agents](https://mlsanity.com/agents/reinforcement-learning-for-llm-agents/)

## Builds on (most cited first, 30 of 42)

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- [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)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [Search-R1: Training LLMs to Reason and Leverage Search Engines with Reinforcement Learning](https://mlsanity.com/paper/2503.09516/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [ALFWorld: Aligning Text and Embodied Environments for Interactive Learning](https://mlsanity.com/paper/2010.03768/) (2020)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [Advantage-Weighted Regression: Simple and Scalable Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1910.00177/) (2019)
- [Group-in-Group Policy Optimization for LLM Agent Training](https://mlsanity.com/paper/2505.10978/) (2025)
- [AppWorld: A Controllable World of Apps and People for Benchmarking Interactive Coding Agents](https://mlsanity.com/paper/2407.18901/) (2024)
- [RAGEN: Understanding Self-Evolution in LLM Agents via Multi-Turn Reinforcement Learning](https://mlsanity.com/paper/2504.20073/) (2025)
- [ArCHer: Training Language Model Agents via Hierarchical Multi-Turn RL](https://mlsanity.com/paper/2402.19446/) (2024)
- [DigiRL: Training In-The-Wild Device-Control Agents with Autonomous Reinforcement Learning](https://mlsanity.com/paper/2406.11896/) (2024)
- [Agent Q: Advanced Reasoning and Learning for Autonomous AI Agents](https://mlsanity.com/paper/2408.07199/) (2024)
- [Multimodal Web Navigation with Instruction-Finetuned Foundation Models](https://mlsanity.com/paper/2305.11854/) (2023)
- [Fine-Tuning Large Vision-Language Models as Decision-Making Agents via Reinforcement Learning](https://mlsanity.com/paper/2405.10292/) (2024)
- [Back to Basics: Revisiting REINFORCE Style Optimization for Learning from Human Feedback in LLMs](https://mlsanity.com/paper/2402.14740/) (2024)

## Abstract (arXiv)

Group-based reinforcement learning such as GRPO trains LLM agents by comparing rollouts sampled for each task, without a learned critic. In long-horizon settings, these rollouts revisit shared anchor states, offering cross-rollout evidence for step-level credit. Ideally, step-level credit should incorporate evidence beyond the realized suffixes observed at an anchor while aggregating alternative continuations according to their empirical frequencies. Visit-local averaging pools realized suffix returns at shared anchors and respects observed frequencies, but does not recursively propagate evidence across rollouts, whereas shortest-path estimators have global reach but allow a rarely observed route to dominate an anchor's value. We introduce Cross-Rollout Bellman Closure (CRBC), which merges each rollout group into a finite empirical process with absorbing success and failure boundaries and evaluates its behavior-policy Bellman fixed point with one linear solve. This fixed point uses the same empirical action and transition frequencies to propagate evidence through shared anchors and aggregate alternative continuations. Backing up the resulting state values through observed transitions yields action values, whose gain over the corresponding state value provides step-level credit. A corresponding finite-depth family recovers visit-local return averaging at zero depth and converges to the exact closure as depth increases. The normalized closure credit is combined with the trajectory-level group advantage for policy optimization, without additional environment rollouts. Across ALFWorld, WebShop, and Sokoban benchmarks with multiple model scales, CRBC consistently improves final performance and learning efficiency. For example, CRBC outperforms the strongest evaluated baseline by 5.59 percentage points on ALFWorld with Qwen2.5-1.5B-Instruct.