Key takeaways
- OpenAI's neural theorem prover demonstrated significant success in solving high-school Olympiad problems.
- The prover uses a language model to generate proofs, continuously improving its performance.
- Challenges in formal mathematics, like infinite action space, are addressed through innovative training approaches.
- The system shows potential for deep learning models in mathematical reasoning, though consistency remains an issue.
OpenAI has built a neural theorem prover for Lean that was able to solve a range of problems from AMC12 and AIME competitions as well as problems from the IMO.
The prover uses a language model to find proofs of formal statements. Whenever a new proof is found, the prover uses this as training data which improves the neural network and enables it to constantly find solutions to more complex problems.
Formal mathematics involves two major challenges that make a simple application of reinforcement learning unlikely to succeed: Infinite action space and Lack of self-play.
To overcome the infinite action space challenge, the natural theorem prover uses sampling actions from a language model as it searches for proof. Language models have the capacity to generate tactic calls and mathematical terms often required as arguments. To address the lack of self-play, the creators replaced unsupervised curriculum with an auxiliary set of problem statements of varying difficulty. The training procedure was able to solve a curriculum of increasingly difficult problems when provided with a varied set of difficulties in auxiliary problems.
Although, the neural theorem prover was able to generate exciting results proving the potential deep learning models capability of non-trivial mathematical reasoning when interacting with a formal system. But, the system is still far away from providing consistent results when compared with the performance of the best students who participate in these challenging olympiad problems.
