DeepSeek-Prover: Advancing Theorem Proving in LLMs through Large-Scale Synthetic Data (2405.14333v1)

Abstract: Proof assistants like Lean have revolutionized mathematical proof verification, ensuring high accuracy and reliability. Although LLMs show promise in mathematical reasoning, their advancement in formal theorem proving is hindered by a lack of training data. To address this issue, we introduce an approach to generate extensive Lean 4 proof data derived from high-school and undergraduate-level mathematical competition problems. This approach involves translating natural language problems into formal statements, filtering out low-quality statements, and generating proofs to create synthetic data. After fine-tuning the DeepSeekMath 7B model on this synthetic dataset, which comprises 8 million formal statements with proofs, our model achieved whole-proof generation accuracies of 46.3% with 64 samples and 52% cumulatively on the Lean 4 miniF2F test, surpassing the baseline GPT-4 at 23.0% with 64 samples and a tree search reinforcement learning method at 41.0%. Additionally, our model successfully proved 5 out of 148 problems in the Lean 4 Formalized International Mathematical Olympiad (FIMO) benchmark, while GPT-4 failed to prove any. These results demonstrate the potential of leveraging large-scale synthetic data to enhance theorem-proving capabilities in LLMs. Both the synthetic dataset and the model will be made available to facilitate further research in this promising field.

Improving Theorem Proving in AI with Synthetic Data

Introduction

In the world of mathematics, verifying proofs can be tedious and prone to errors. Enter automated theorem proving (ATP), a solution that leverages AI to verify mathematical proofs efficiently. A paper presents a novel method to enhance the performance of LLMs in formal theorem proving. This method involves generating extensive proof data from high-school and undergraduate-level mathematical problems. The authors have introduced an innovative strategy to create a synthetic dataset and fine-tune a model called DeepSeekMath. Let's dive into the nitty-gritty of their approach and its implications.

Generating Formal Proof Data

One of the main challenges in training LLMs for theorem proving is the scarcity of formal proof data. Unlike coding, where vast repositories of Python and Java code exist, formal mathematical proofs are less common. To counter this, the authors devised a method to convert informal mathematical problems into formal statements using the Lean 4 proof assistant.

Quality Assurance

To ensure high-quality data, the researchers set up a multi-step process:

1. Initial Translation: Translate natural language problems into formal statements.
2. Filtering: Use a quality scoring model to discard simple or invalid statements.
3. Proof Generation: Generate proofs for these statements and validate them using Lean 4.

This iterative process helps refine the model, making it stronger and more accurate in subsequent iterations.

Scaling Up

The generation of formal proofs requires exploring vast search spaces, often leading to inefficiencies. To tackle this, the authors propose proving both the statement and its negation in parallel. This approach helps quickly identify and discard unprovable statements, enhancing the efficiency of the proof generation process.

Experimental Results

The efficacy of this approach was tested on two benchmarks:

• miniF2F: A dataset of 488 problems.
• FIMO: A benchmark with 148 problems derived from the International Mathematical Olympiad (IMO).

The results were impressive. DeepSeekMath achieved:

• 46.3% whole-proof generation accuracy on miniF2F, compared to GPT-4's 23.0%.
• It proved 5 out of 148 problems on the FIMO benchmark, where GPT-4 proved none.

These substantial improvements suggest that leveraging large-scale synthetic data can significantly enhance the theorem-proving capabilities of LLMs.

Implications and Future Directions

The implications of this research are quite exciting:

• Practical Applications: Enhanced ATP could streamline peer-review processes in mathematics, making it easier to verify complex proofs quickly and accurately.
• Theoretical Advances: On the theoretical front, this method opens avenues for better understanding and developing advanced AI models capable of tackling even more complex mathematical problems.

Looking forward, future developments might include:

• Extending this approach to a wider variety of mathematical problems.
• Experimenting with different proof assistants and verification systems.
• Exploring the applicability of this method to other domains requiring formal verification, such as software engineering.

Conclusion

This innovative approach to generating and utilizing synthetic proof data presents significant advancements in the field of automated theorem proving. By fine-tuning models on large-scale, high-quality synthetic datasets, researchers have achieved state-of-the-art performance, paving the way for future developments in AI-driven formal reasoning. Keep an eye on this space—it's only going to get more interesting!