Flow of Reasoning: Training LLMs for Divergent Reasoning with Minimal Examples

The ability to generate diverse solutions to a given problem is a hallmark of human creativity. This divergent reasoning is also crucial for machines, enhancing their robustness and enabling them to assist humans in many applications such as scientific discovery. However, existing approaches to multi-step reasoning with large language models (LLMs) have mostly focused only on reasoning accuracy, without further discovering more diverse valid solutions. For example, supervised fine-tuning improves reasoning quality but requires vast labeled data, while reward-maximizing reinforcement learning finds top-reward solutions while neglecting the solution diversity. To fill this gap, we propose Flow of Reasoning (FOR), an efficient diversity-seeking LLM finetuning method aimed at improving reasoning quality and diversity with minimal data. FOR formulates multi-step LLM reasoning as a Markovian flow on a DAG-structured reasoning graph. This formulation allows us to incorporate and adapt principled GFlowNet approaches, for finetuning LLMs to sample divergent paths with probabilities proportional to the (unnormalized) reward of target problems. Extensive experiments show that, with limited training examples (e.g., 15 examples), FOR enables the discovery of diverse, creative, high-quality solutions, greatly outperforming a wide range of existing inference and training methods across six challenging reasoning tasks, including BlocksWorld (embodied reasoning), Game24 (math puzzle solving), Rubik's Cube (spatial reasoning), 1D-ARC (abstraction reasoning), GSM8k (math reasoning), and Pron-toQA (logical reasoning). Code is available at https://github.com/Yu-Fangxu/FoR .