A Compact Model for Mathematics Problem Representations Distilled from BERT

Abstract

Large language models (LLMs) have made significant advancements in math problem solving, but their large size and high latency render them impractical for real-world applications in intelligent mathematics solvers. Recently, task-agnostic compact models have been developed to replace LLMs in general natural language processing tasks. However, these models often struggle to acquire sufficient math-related knowledge from LLMs, leading to unsatisfactory performance in solving math word problems (MWPs). To develop a specialized compact model for representing MWPs, we develop the knowledge distillation (KD) technique to extract mathematical semantics knowledge from the large pre-trained model BERT. Effective knowledge types and distillation strategies are explored through extensive experiments. Our KD algorithm employs multi-knowledge distillation to extract fundamental knowledge from hidden states in the middle to lower layers, while also incorporating knowledge of mathematical relations and symbol constraints from higher-layer outputs and math decoder outputs, by leveraging bottleneck networks. Pre-training tasks on MWP datasets, such as masked language modeling and part-of-speech tagging, are also utilized to enhance the generalization of the compact model for MWP understanding. Additionally, a simple parameter mixing strategy is employed to prevent catastrophic forgetting of acquired knowledge. Our findings indicate that our approach can reduce the size of a BERT model by 10% while retaining approximately 95% of its performance on MWP datasets, outperforming the mainstream BERT-based task-agnostic compact models. The efficacy of each component has been validated through ablation studies.