Enhancing Mathematical Reasoning in Language Models Through Focused Differentiation Training

Enhancing the mathematical capabilities of large language models (LLMs) is crucial for applications requiring precise and rigorous mathematical reasoning. Current models, even when trained with methods like Direct Preference Optimization (DPO), often struggle to effectively differentiate between correct and erroneous mathematical responses, especially when errors occur in multi-step solutions. Traditional approaches focusing on token or logit-level analysis fail to capture the nuanced semantic differences in mathematical reasoning. To address this challenge, we propose leveraging the rich semantic information embedded in the hidden state space of LLMs. Our novel approach, Focused Differentiation Training (FDT), fine-tunes the model by emphasizing the differences between the hidden states of correct and incorrect responses, rather than their common features. Unlike other methods that detect errors at the token or logits level and often rely on human input or more powerful models, our approach enhances mathematical reasoning capabilities using only the model's inherent abilities. This methodology promotes a more accurate alignment with mathematical correctness, thereby improving the model's ability to evaluate and generate precise mathematical responses. Experimental results demonstrate that our algorithm substantially outperforms traditional alignment methods in mathematical tasks, offering a robust solution for enhancing the mathematical reasoning capabilities of language models.