# THOR: Tool-Integrated Hierarchical Optimization via RL for Mathematical Reasoning

arXiv: https://arxiv.org/abs/2509.13761 · HTML: https://mlsanity.com/paper/2509.13761/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 52 of its 60 references that are in the map.
- First posted on arXiv 17 Sep 2025.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 30 of 52)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [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)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Voyager: An Open-Ended Embodied Agent with Large Language Models](https://mlsanity.com/paper/2305.16291/) (2023)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Is Your Code Generated by ChatGPT Really Correct? Rigorous Evaluation of Large Language Models for Code Generation](https://mlsanity.com/paper/2305.01210/) (2023)
- [ZeRO: Memory Optimizations Toward Training Trillion Parameter Models](https://mlsanity.com/paper/1910.02054/) (2019)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Graph of Thoughts: Solving Elaborate Problems with Large Language Models](https://mlsanity.com/paper/2308.09687/) (2023)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models](https://mlsanity.com/paper/2401.01335/) (2024)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Generative Verifiers: Reward Modeling as Next-Token Prediction](https://mlsanity.com/paper/2408.15240/) (2024)
- [Search-o1: Agentic Search-Enhanced Large Reasoning Models](https://mlsanity.com/paper/2501.05366/) (2025)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)

## Built on by (most cited first, 9 of 9)

- [AgentEvolver: Towards Efficient Self-Evolving Agent System](https://mlsanity.com/paper/2511.10395/) (2025)
- [AgentMath: Empowering Mathematical Reasoning for Large Language Models via Tool-Augmented Agent](https://mlsanity.com/paper/2512.20745/) (2025)
- [MatchTIR: Fine-Grained Supervision for Tool-Integrated Reasoning via Bipartite Matching](https://mlsanity.com/paper/2601.10712/) (2026)
- [Unveiling the Cognitive Compass: Theory-of-Mind-Guided Multimodal Emotion Reasoning](https://mlsanity.com/paper/2602.00971/) (2026)
- [Reward Modeling for Reinforcement Learning-Based LLM Reasoning: Design, Challenges, and Evaluation](https://mlsanity.com/paper/2602.09305/) (2026)
- [DiG: Differential Grounding for Enhancing Fine-Grained Perception in Multimodal Large Language Model](https://mlsanity.com/paper/2512.12633/) (2025)
- [Neuro-Symbolic Financial Reasoning via Deterministic Fact Ledgers and Adversarial Low-Latency Hallucination Detector](https://mlsanity.com/paper/2603.04663/) (2026)
- [PEARL: Training Socratic Tutors with Pedagogically Aligned Reinforcement Learning](https://mlsanity.com/paper/2605.29582/) (2026)
- [RTPO: Reverse-Turn Policy Optimization for Stabilizing Agentic RL Training](https://mlsanity.com/paper/2608.18682/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) have made remarkable progress in mathematical reasoning, but still continue to struggle with high-precision tasks like numerical computation and formal symbolic manipulation. Integrating external tools has emerged as a promising approach to bridge this gap. Despite recent advances, existing methods struggle with three key challenges: constructing tool-integrated reasoning data, performing fine-grained optimization, and enhancing inference. To overcome these limitations, we propose THOR (Tool-Integrated Hierarchical Optimization via RL). First, we introduce TIRGen, a multi-agent based pipeline for constructing high-quality datasets of tool-integrated reasoning paths, aligning with the policy and generalizing well across diverse models. Second, to perform fine-grained hierarchical optimization, we introduce an RL strategy that jointly optimizes for both episode-level problem solving and step-level code generation. This is motivated by our key insight that the success of an intermediate tool call is a strong predictor of the final answer's correctness. Finally, THOR incorporates a self-correction mechanism that leverages immediate tool feedback to dynamically revise erroneous reasoning paths during inference. Our approach demonstrates strong generalization across diverse models, performing effectively in both reasoning and non-reasoning models. It further achieves state-of-the-art performance for models of a similar scale on multiple mathematical benchmarks, while also delivering consistent improvements on code benchmarks. Our code will be publicly available at https://github.com/JingMog/THOR.