Theorem Prover as a Judge for Synthetic Data Generation

Abstract: The demand for synthetic data in mathematical reasoning has increased due to its potential to enhance the mathematical capabilities of LLMs. However, ensuring the validity of intermediate reasoning steps remains a significant challenge, affecting data quality. While formal verification via theorem provers effectively validates LLM reasoning, the autoformalisation of mathematical proofs remains error-prone. In response, we introduce iterative autoformalisation, an approach that iteratively refines theorem prover formalisation to mitigate errors, thereby increasing the execution rate on the Lean prover from 60% to 87%. Building upon that, we introduce Theorem Prover as a Judge (TP-as-a-Judge), a method that employs theorem prover formalisation to rigorously assess LLM intermediate reasoning, effectively integrating autoformalisation with synthetic data generation. Finally, we present Reinforcement Learning from Theorem Prover Feedback (RLTPF), a framework that replaces human annotation with theorem prover feedback in Reinforcement Learning from Human Feedback (RLHF). Across multiple LLMs, applying TP-as-a-Judge and RLTPF improves benchmarks with only 3,508 samples, achieving 5.56% accuracy gain on Mistral-7B for MultiArith, 6.00% on Llama-2-7B for SVAMP, and 3.55% on Llama-3.1-8B for AQUA.