Advancing Math Reasoning in Language Models: The Impact of Problem-Solving Data, Data Synthesis Methods, and Training Stages

Mathematical reasoning remains a challenging area for large language models (LLMs), prompting the development of math-specific LLMs such as LLEMMA, DeepSeekMath, and Qwen2-Math, among others. These models typically follow a two-stage training paradigm: pre-training with math-related corpora and posttraining with problem datasets for supervised fine-tuning (SFT). Despite these efforts, the improvements in mathematical reasoning achieved through continued pre-training (CPT) are often less significant compared to those obtained via SFT. We investigate three primary research questions: (1) Can problem-solving data enhance the model's mathematical reasoning capabilities more effectively than general mathematical corpora during CPT? (2) Are synthetic data from the same source equally effective, and which synthesis methods are most efficient? Our findings indicate that problem-solving data significantly enhances the model's mathematical capabilities compared to general mathematical corpora. We also identify effective data synthesis methods, demonstrating that the tutorship amplification synthesis method achieves the best performance. Furthermore, while SFT facilitates instruction-following abilities, it underperforms compared to CPT with the same data, which can be partially attributed to its poor learning capacity for more challenging problem-solving data. To address the challenge of insufficient mathematical reasoning capabilities in large language models (LLMs), various math-specific LLMs are developed. These models generally follow a common training paradigm. During the pre-training stage, math-related corpora are filtered from extensive internet data to augment the model's mathematical knowledge. Work done during Zui Chen's internship at Guangdong Institute of Smart Education, Jinan University, Guangzhou, China. Model is available at https://huggingface.co/ai4ed/MathGPT-8B This enables the models to follow instructions and produce outputs in the desired format. Recently, there is a growing focus on constructing preference datasets for the solution process to perform Step-DPO (Lai et al., 2024) or online-RLHF (Dong et al., 2024). These approaches aim to obtain more accurate reasoning pathways, thereby significantly enhancing the mathematical reasoning capabilities of the models.