JiuZhang3.0: Efficiently Improving Mathematical Reasoning by Training Small Data Synthesis Models

Mathematical reasoning is an important capability of large language models~(LLMs) for real-world applications.To enhance this capability, existing work either collects large-scale math-related texts for pre-training, or relies on stronger LLMs (\eg GPT-4) to synthesize massive math problems. Both types of work generally lead to large costs in training or synthesis.To reduce the cost, based on open-source available texts, we propose an efficient way that trains a small LLM for math problem synthesis, to efficiently generate sufficient high-quality pre-training data.To achieve it, we create a dataset using GPT-4 to distill its data synthesis capability into the small LLM.Concretely, we craft a set of prompts based on human education stages to guide GPT-4, to synthesize problems covering diverse math knowledge and difficulty levels.Besides, we adopt the gradient-based influence estimation method to select the most valuable math-related texts.The both are fed into GPT-4 for creating the knowledge distillation dataset to train the small LLM.We leverage it to synthesize 6 million math problems for pre-training our JiuZhang3.0 model. The whole process only needs to invoke GPT-4 API 9.3k times and use 4.6B data for training.Experimental results have shown that JiuZhang3.0 achieves state-of-the-art performance on several mathematical reasoning datasets, under both natural language reasoning and tool manipulation settings.Our code and data will be publicly released in \url{https://github.com/RUCAIBox/JiuZhang3.0}.