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

TL;DR

JiuZhang3.0 introduces an efficient method to train a small language model for math problem synthesis by distilling GPT-4's capabilities, enabling high-quality data generation with reduced costs and achieving state-of-the-art reasoning performance.

Contribution

The paper presents a cost-effective approach to train a small LLM for math problem synthesis using knowledge distillation from GPT-4 and data selection techniques.

Findings

JiuZhang3.0 generates 6 million high-quality math problems for pre-training.

It achieves state-of-the-art results on multiple mathematical reasoning benchmarks.

The method significantly reduces reliance on large-scale data and expensive model training.

Abstract

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, which only needs to invoke GPT-4 API 9.3k times and pre-train on 4.6B data. 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}.