AlphaGeometry: Solving olympiad geometry without human demonstrations (Paper Explained)

#deepmind #alphageometry #llm

AlphaGeometry is a combination of a symbolic solver and a large language model by Google DeepMind that tackles IMO geometry questions without any human-generated trainind data.

OUTLINE: 0:00 - Introduction 1:30 - Problem Statement 7:30 - Core Contribution: Synthetic Data Generation 9:30 - Sampling Premises 13:00 - Symbolic Deduction 17:00 - Traceback 19:00 - Auxiliary Construction 25:20 - Experimental Results 32:00 - Problem Representation 34:30 - Final Comments

Paper: https://www.nature.com/articles/s41586-023-06747-5

Abstract: Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1,2,3,4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

Authors: Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, He He & Thang Luong

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