# Building Math Agents with Multi-Turn Iterative Preference Learning

arXiv: https://arxiv.org/abs/2409.02392 · HTML: https://mlsanity.com/paper/2409.02392/

## Key facts

- Cited by 66 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 86 of its 113 references that are in the map.
- First posted on arXiv 4 Sep 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)
- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 30 of 86)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [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)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [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)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Scaling Laws for Reward Model Overoptimization](https://mlsanity.com/paper/2210.10760/) (2022)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [SimPO: Simple Preference Optimization with a Reference-Free Reward](https://mlsanity.com/paper/2405.14734/) (2024)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [A General Theoretical Paradigm to Understand Learning from Human Preferences](https://mlsanity.com/paper/2310.12036/) (2023)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [CRITIC: Large Language Models Can Self-Correct with Tool-Interactive Critiquing](https://mlsanity.com/paper/2305.11738/) (2023)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment](https://mlsanity.com/paper/2304.06767/) (2023)
- [PAL: Program-aided Language Models](https://mlsanity.com/paper/2211.10435/) (2022)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [ORPO: Monolithic Preference Optimization without Reference Model](https://mlsanity.com/paper/2403.07691/) (2024)
- [Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models](https://mlsanity.com/paper/2401.01335/) (2024)

## Built on by (most cited first, 30 of 66)

- [Training Language Models to Self-Correct via Reinforcement Learning](https://mlsanity.com/paper/2409.12917/) (2024)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [ProcessBench: Identifying Process Errors in Mathematical Reasoning](https://mlsanity.com/paper/2412.06559/) (2024)
- [AIMO-2 Winning Solution: Building State-of-the-Art Mathematical Reasoning Models with OpenMathReasoning dataset](https://mlsanity.com/paper/2504.16891/) (2025)
- [A Minimalist Approach to LLM Reasoning: from Rejection Sampling to Reinforce](https://mlsanity.com/paper/2504.11343/) (2025)
- [SWEET-RL: Training Multi-Turn LLM Agents on Collaborative Reasoning Tasks](https://mlsanity.com/paper/2503.15478/) (2025)
- [Agentic Reasoning and Tool Integration for LLMs via Reinforcement Learning](https://mlsanity.com/paper/2505.01441/) (2025)
- [MAPoRL: Multi-Agent Post-Co-Training for Collaborative Large Language Models with Reinforcement Learning](https://mlsanity.com/paper/2502.18439/) (2025)
- [Tree Search for LLM Agent Reinforcement Learning](https://mlsanity.com/paper/2509.21240/) (2025)
- [RRM: Robust Reward Model Training Mitigates Reward Hacking](https://mlsanity.com/paper/2409.13156/) (2024)
- [Self-rewarding correction for mathematical reasoning](https://mlsanity.com/paper/2502.19613/) (2025)
- [LLM Agents for Education: Advances and Applications](https://mlsanity.com/paper/2503.11733/) (2025)
- [Self-Challenging Language Model Agents](https://mlsanity.com/paper/2506.01716/) (2025)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [Offline Reinforcement Learning for LLM Multi-Step Reasoning](https://mlsanity.com/paper/2412.16145/) (2024)
- [From Lists to Emojis: How Format Bias Affects Model Alignment](https://mlsanity.com/paper/2409.11704/) (2024)
- [Entropy-Regularized Process Reward Model](https://mlsanity.com/paper/2412.11006/) (2024)
- [Magnet: Multi-turn Tool-use Data Synthesis and Distillation via Graph Translation](https://mlsanity.com/paper/2503.07826/) (2025)
- [A Survey of Direct Preference Optimization](https://mlsanity.com/paper/2503.11701/) (2025)
- [Sharp Analysis for KL-Regularized Contextual Bandits and RLHF](https://mlsanity.com/paper/2411.04625/) (2024)
- [Fine-Grained Preference Optimization Improves Spatial Reasoning in VLMs](https://mlsanity.com/paper/2506.21656/) (2025)
- [Regressing the Relative Future: Efficient Policy Optimization for Multi-turn RLHF](https://mlsanity.com/paper/2410.04612/) (2024)
- [RAG-RL: Advancing Retrieval-Augmented Generation via RL and Curriculum Learning](https://mlsanity.com/paper/2503.12759/) (2025)
- [Beyond Correctness: Harmonizing Process and Outcome Rewards through RL Training](https://mlsanity.com/paper/2509.03403/) (2025)
- [PAG: Multi-Turn Reinforced LLM Self-Correction with Policy as Generative Verifier](https://mlsanity.com/paper/2506.10406/) (2025)
- [StepWiser: Stepwise Generative Judges for Wiser Reasoning](https://mlsanity.com/paper/2508.19229/) (2025)
- [A Comprehensive Survey of Direct Preference Optimization: Datasets, Theories, Variants, and Applications](https://mlsanity.com/paper/2410.15595/) (2024)
- [Reinforce LLM Reasoning through Multi-Agent Reflection](https://mlsanity.com/paper/2506.08379/) (2025)
- [MathAgent: Leveraging a Mixture-of-Math-Agent Framework for Real-World Multimodal Mathematical Error Detection](https://mlsanity.com/paper/2503.18132/) (2025)
- [AURORA:Automated Training Framework of Universal Process Reward Models via Ensemble Prompting and Reverse Verification](https://mlsanity.com/paper/2502.11520/) (2025)

## Abstract (arXiv)

Recent studies have shown that large language models' (LLMs) mathematical problem-solving capabilities can be enhanced by integrating external tools, such as code interpreters, and employing multi-turn Chain-of-Thought (CoT) reasoning. While current methods focus on synthetic data generation and Supervised Fine-Tuning (SFT), this paper studies the complementary direct preference learning approach to further improve model performance. However, existing direct preference learning algorithms are originally designed for the single-turn chat task, and do not fully address the complexities of multi-turn reasoning and external tool integration required for tool-integrated mathematical reasoning tasks. To fill in this gap, we introduce a multi-turn direct preference learning framework, tailored for this context, that leverages feedback from code interpreters and optimizes trajectory-level preferences. This framework includes multi-turn DPO and multi-turn KTO as specific implementations. The effectiveness of our framework is validated through training of various language models using an augmented prompt set from the GSM8K and MATH datasets. Our results demonstrate substantial improvements: a supervised fine-tuned Gemma-1.1-it-7B model's performance increased from 77.5% to 83.9% on GSM8K and from 46.1% to 51.2% on MATH. Similarly, a Gemma-2-it-9B model improved from 84.1% to 86.3% on GSM8K and from 51.0% to 54.5% on MATH.