#openai #math #imo
Formal mathematics is a challenging area for both humans and machines. For humans, formal proofs require very tedious and meticulous specifications of every last detail and results in very long, overly cumbersome and verbose outputs. For machines, the discreteness and sparse reward nature of the problem presents a significant problem, which is classically tackled by brute force search, guided by a couple of heuristics. Previously, language models have been employed to better guide these proof searches and delivered significant improvements, but automated systems are still far from usable. This paper introduces another concept: An expert iteration procedure is employed to iteratively produce more and more challenging, but solvable problems for the machine to train on, which results in an automated curriculum, and a final algorithm that performs well above the previous models. OpenAI used this method to even solve two problems of the international math olympiad, which was previously infeasible for AI systems.
OUTLINE:
0:00 – Intro
2:35 – Paper Overview
5:50 – How do formal proofs work?
9:35 – How expert iteration creates a curriculum
16:50 – Model, data, and training procedure
25:30 – Predicting proof lengths for guiding search
29:10 – Bootstrapping expert iteration
34:10 – Experimental evaluation & scaling properties
40:10 – Results on synthetic data
44:15 – Solving real math problems
47:15 – Discussion & comments
Paper:
miniF2F benchmark:
Abstract:
We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically outperforms proof search only. We also observe that when applied to a collection of formal statements of sufficiently varied difficulty, expert iteration is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs. Finally, by applying this expert iteration to a manually curated set of problem statements, we achieve state-of-the-art on the miniF2F benchmark, automatically solving multiple challenging problems drawn from high school olympiads.
Authors: Stanislas Polu, Jesse Michael Han, Kunhao Zheng, Mantas Baksys, Igor Babuschkin, Ilya Sutskever
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