BackMATH: Towards Backward Reasoning for Solving Math Problems Step by Step

Abstract

Large language models (LLMs) have achieved impressive results in reasoning, particularly in multi-step reasoning tasks. However, when faced with more complex mathematical problems, the performance of LLMs drops significantly. To address this issue, in this paper, we propose a backward reasoning dataset, BackMATH-Data. The dataset comprises approximately 14K backward reasoning problems and 100K reasoning steps. It follows a result-oriented approach, to construct backward reasoning problems by swapping the reasoning results with specific solving conditions in the original problems.Additionally, we introduce Backward-reasoning Process-supervision Reward Model (BackPRM) and BackMATH-LLM. BackPRM supervises the quality of the generated backward reasoning problems, while BackMATH-LLM is designed for mathematical reasoning. BackMATH-LLM is fine-tuned and enhanced through reinforcement learning by supervising the quality of backward reasoning problems and by providing feedback on reasoning steps, thereby improving the mathematical reasoning capabilities of LLMs.Extensive experiments demonstrate that our model achieves an accuracy of 68.1% on the GSM8K dataset and 21.9% on the MATH dataset, exceeding the SOTA by 1.6% and 2.1% respectively.