Key-Point-Driven Data Synthesis with its Enhancement on Mathematical Reasoning
Proposed Key-Point-Driven Data Synthesis (KPDDS), creating 800K+ math reasoning QA pairs, significantly boosting model performance.
Key Findings
Methodology
This paper introduces KPDDS, a framework combining knowledge extraction, topic co-occurrence modeling, and quality filtering. It extracts key points via GPT-4 from seed datasets, clusters and deduplicates them, then constructs a Topic Co-occurrence Probability Matrix (TCPM). Using TCPM, multiple topics are sampled probabilistically, and GPT-4 generates new questions based on selected key points. A scoring model filters high-quality questions, and multi-round voting ensures answer reliability. The process extends to large datasets like KPMath and KPMath-Plus, resulting in over 800K QA pairs, which fine-tune models like Qwen1.5-72B, achieving state-of-the-art results.
Key Results
- On GSM8K, the fine-tuned Qwen1.5-72B reaches 87.0% PASS@1, outperforming most 7B-70B models and approaching GPT-4.
- On MATH, the model achieves 58.3%, surpassing many competitors, with an average improvement of over 10%.
- KPMath-Plus dataset attains 79.4% average accuracy, validating the effectiveness of knowledge-driven synthesis.
Significance
This work addresses the scarcity of high-quality math reasoning data, providing a scalable, knowledge-driven method to generate diverse, authentic datasets. It enhances the reasoning capabilities of large models, impacting AI research, automated education, and intelligent tutoring systems, fostering broader adoption of AI in complex reasoning tasks.
Technical Contribution
The paper introduces a novel framework integrating automatic knowledge extraction, topic co-occurrence modeling, and multi-stage quality filtering. The TCPM guides multi-topic sampling, and voting mechanisms ensure answer accuracy, enabling scalable, high-quality data synthesis that surpasses previous rule-based or knowledge-base methods.
Novelty
This is the first systematic combination of key point extraction, topic co-occurrence modeling, and quality filtering to generate large-scale, high-quality math datasets. It moves beyond traditional rule-based or knowledge-base approaches, offering a scalable, knowledge-driven paradigm that significantly improves data diversity and authenticity.
Limitations
- The approach heavily relies on GPT-4, which is costly and may introduce biases or inaccuracies. Its scalability in resource-constrained environments is limited.
- Knowledge extraction and clustering may produce redundant or incomplete key points, affecting data diversity.
- Generating extremely complex or high-difficulty problems remains challenging due to model limitations.
Future Work
Future efforts will focus on reducing dependence on large models by developing autonomous knowledge bases, improving data filtering for higher quality, and extending the framework to multi-modal reasoning tasks. Exploring more efficient algorithms for knowledge extraction and question generation will also be prioritized.
AI Executive Summary
The rapid advancement of large language models (LLMs) such as GPT-4 has revolutionized natural language understanding and reasoning. However, their performance in complex mathematical reasoning remains constrained by the scarcity of high-quality, diverse training data. Existing datasets like MATH and GSM8K, while valuable, are limited in scale and often rely on manual annotation, which hampers their ability to cover the full spectrum of reasoning challenges. Synthetic data generation offers a promising solution, but traditional methods depend heavily on rule-based systems or knowledge bases, which lack authenticity and scalability.
This paper introduces the Key-Point-Driven Data Synthesis (KPDDS) framework, a knowledge-centric approach that systematically extracts key points from authentic datasets using GPT-4. These key points are clustered and analyzed through a Topic Co-occurrence Probability Matrix (TCPM), which models the relationships between different mathematical topics. Guided by TCPM, multiple topics are probabilistically sampled, and GPT-4 generates new questions based on selected key points. A multi-stage quality filtering process, including model scoring and voting, ensures the generated questions are both diverse and reliable.
Leveraging this framework, the authors constructed the KPMath dataset with over 800,000 high-quality QA pairs, and further extended it to the comprehensive KPMath-Plus dataset by integrating additional reasoning-intensive corpora. Fine-tuning the Qwen1.5-72B model on this dataset resulted in state-of-the-art performance, achieving 87.0% PASS@1 on GSM8K and 58.3% on MATH, surpassing existing models in the 7B-70B range and approaching GPT-4’s capabilities.
This work significantly advances the field by demonstrating how knowledge-driven synthetic data can effectively enhance mathematical reasoning in large models. It opens new avenues for scalable, authentic data generation, with broad implications for AI research, automated education, and intelligent tutoring systems. Future directions include reducing reliance on large models for data generation, improving the diversity and complexity of generated problems, and extending the framework to multi-modal reasoning tasks, promising a new era of AI-powered mathematical understanding.
Deep Analysis
Background
近年来，随着GPT-4等大模型的崛起，推动了自然语言理解和推理能力的快速发展。数学推理作为评估模型推理能力的重要指标，吸引了大量研究关注。早期数据集如MATH、GSM8K提供了基础训练资源，但存在规模不足、题目多为人工标注、难以覆盖复杂推理场景的问题。近年来，合成数据的方法逐渐兴起，试图通过自动生成题目提升数据多样性，但多依赖规则或知识库，缺乏真实性。现有研究如Chain-of-Thought prompting、程序化答案等虽有效，但仍面临数据质量和规模的瓶颈。本文在此背景下，提出基于知识驱动的合成框架，以解决数据不足和多样性不足的问题，推动模型推理能力的实质性提升。
Core Problem
当前数学推理模型的性能受限于训练数据的质量和规模。虽然存在一些合成数据方法，但多依赖规则或知识库，缺乏真实性和多样性，导致模型泛化能力不足。此外，缺乏系统的知识提取与问题生成机制，难以有效覆盖复杂推理场景。如何高效生成高质量、多样化的数学题目，成为提升模型推理能力的关键。本文旨在通过知识驱动的合成框架，系统挖掘真实题库中的知识点，结合主题共现模型，自动生成符合逻辑且多样化的题目，为模型提供丰富的训练资源。
Innovation
本研究的核心创新在于提出结合知识提取、主题共现建模与质量筛选的KPDDS框架。首先，通过GPT-4自动提取题库中的主题和关键点，建立MPKP数据集。其次，利用TCPM模型分析主题的共现关系，生成主题共现概率矩阵，指导多主题采样。再次，采用GPT-4进行题目生成，并通过模型评分和投票机制筛选高质量题目，提升数据的真实性和多样性。最后，将生成的题目用于微调大模型，显著提升其数学推理能力。这一流程突破了传统纯规则或知识库依赖的局限，提供了可扩展的知识驱动数据合成新范式。
Methodology
- �� 知识提取：利用GPT-4从真实题库中自动提取主题和关键点，进行去重和聚类，构建MPKP数据集。
- �� TCPM构建：分析题库中主题的共现关系，生成主题共现概率矩阵，指导多主题采样。
- �� 问题生成：基于TCPM采样多个主题，从题库中随机选题，结合关键点，利用GPT-4生成新题。
- �� 质量评估：采用GPT-4模型对生成题目进行打分，筛选得分高于0.85的题目。
- �� 答案生成：通过多轮采样和投票机制，生成多样化答案，确保答案的正确性和一致性。
- �� 数据集构建：将高质量题目与答案整合，形成KPMath和KPMath-Plus，作为模型训练基础。
Experiments
在GSM8K和MATH数据集上进行微调，采用PASS@1作为性能指标。模型训练细节包括：使用BFloat16、DeepSpeed ZeRO Stage3、Flash-Attention 2，最大序列长度2048。对比多种模型（如GPT-4、LLaMA-2、Mistral）和不同数据集（KPMath、KPMath-Plus），验证合成数据的有效性。通过消融实验分析TCPM、质量筛选等关键环节的贡献。评估模型在多场景、多难度任务中的表现，确保泛化能力。
Results
微调Qwen1.5-72B模型在GSM8K达87.0%的PASS@1，优于大部分同规模模型，接近GPT-4。MATH测试集达58.3%，显著优于未增强模型。结合多源数据，KPMath-Plus实现79.4%的平均准确率，验证知识驱动合成的有效性。实验还显示，TCPM引导的多主题采样和质量筛选机制是提升数据质量的关键因素。这些结果表明，系统化的知识驱动合成方法能有效增强模型推理能力，具有广泛的应用潜力。
Applications
该方法可用于自动生成数学竞赛题、智能辅导系统和教育内容开发。只需提供基础题库和关键知识点，即可快速扩充高质量训练数据，降低人工成本。长远来看，知识驱动的数据合成有望推动自主学习系统的发展，实现个性化教育和智能评测，促进数学教育的普及与创新。
Limitations & Outlook
目前依赖GPT-4等大模型，成本较高且存在偏差问题。知识提取和聚类可能引入冗余或遗漏，影响数据多样性。生成题目的复杂度受模型能力限制，难以涵盖所有高难度场景。未来需优化算法，提高效率，降低成本，并探索自主知识库的构建以减少对大模型的依赖。
Plain Language Accessible to non-experts
想象你在准备一场盛大的厨艺比赛，想要做出各种不同的菜肴。你有很多食谱（题库），每个食谱都包含一些关键的步骤（知识点）和材料（主题）。为了做出新菜，你会从这些食谱中提取重要的步骤，分析哪些步骤经常一起出现（主题共现），然后用这些信息设计新的菜谱。你还会用厨师的建议（模型评分）判断新菜谱是否合理，最后由厨师投票决定哪些菜谱可以用。这样一来，你就能不断创造出丰富多样的菜肴，供大家品尝，既保证质量，又能不断创新。
ELI14 Explained like you're 14
想象你在学校参加厨艺比赛，你想做出很多不同的菜肴。你有一些很棒的食谱，但数量不够，不能让每个人都满意。于是，你决定用一种聪明的方法，先找出这些食谱中最重要的步骤，比如“切菜”、“调味”、“烹饪时间”。然后，你分析哪些步骤经常一起出现，比如“切菜”和“调味”常常搭配。接着，你用这些信息，自己设计新的菜谱，把这些关键步骤组合在一起，确保新菜肴合理又有趣。最后，你请老师（模型）尝试做这些菜，老师会给出评分，你只留下评分高的菜谱。这样，你就能不断创造出新颖又好吃的菜肴，赢得比赛！
Glossary
Key Point（关键点）
在题目中起到核心作用的知识或方法，帮助理解和解决问题。
用于提取和指导题目生成的关键知识元素。
Topic Co-occurrence Probability Matrix（主题共现概率矩阵）
描述不同数学主题在题库中同时出现频率的矩阵，用于指导多主题题目的生成。
模型分析题目主题关系的重要工具。
PASS@1
模型在一次尝试中正确回答问题的比例，用于衡量推理性能。
评估模型数学推理能力的标准指标。
KPMath
由知识驱动合成方法生成的数学问答数据集，包含超过80万对问答。
模型训练和性能评估的重要数据资源。
GPT-4
由OpenAI开发的先进大模型，具备强大的自然语言理解和生成能力。
用于知识提取、题目生成和质量评估。
Open Questions Unanswered questions from this research
- 1 如何进一步降低生成数据的偏差和重复率？
- 2 能否实现完全自主的知识库构建以减少对大模型的依赖？
- 3 在更复杂的数学推理场景中，模型的表现如何提升？
Applications
Immediate Applications
智能教育平台
利用合成数据丰富题库，提升智能辅导系统的推理能力，支持个性化学习路径。
自动题库生成
为教师和考试机构自动生成多样化高质量数学题，节省人工成本，提升效率。
Long-term Vision
自主学习系统
构建具备自主知识更新和推理能力的智能系统，实现个性化、普及化的数学教育。
Abstract
Large language models (LLMs) have shown great potential in complex reasoning tasks, yet their performance is often hampered by the scarcity of high-quality and reasoning-focused training datasets. Addressing this challenge, we propose Key-Point-Driven Data Synthesis (KPDDS), a novel data synthesis framework that synthesizes question-answer pairs by leveraging key points and exemplar practices from authentic data sources. KPDDS ensures the generation of novel questions with rigorous quality control and substantial scalability. As a result, we present KPMath, an extensive synthetic dataset tailored for mathematical reasoning, comprising over 800K question-answer pairs. Utilizing KPMath and augmenting it with additional reasoning-intensive corpora, we create the comprehensive KPMath-Plus dataset. The Qwen1.5-72B model, fine-tuned on KPMath-Plus, achieves 87.0% PASS@1 accuracy on GSM8K and 58.3% on MATH, surpassing competitors in the 7B to 70B range and best commercial models like GPT-4 across multiple math reasoning datasets.