Discovering Hidden Algebraic Structures via Transformers with Rank-Aware Beam GRPO

TL;DR

This paper explores the use of transformers for discovering algebraic structures, specifically multivariate polynomial decomposition, introducing a new synthetic data pipeline, a rank-aware reinforcement learning method, and demonstrating improved accuracy and efficiency.

Contribution

It introduces BGRPO, a rank-aware reinforcement learning approach, and a synthetic data pipeline for training transformers on complex algebraic tasks, advancing symbolic reasoning capabilities.

Findings

Finetuning with BGRPO improves accuracy and reduces inference compute by 75%.

Transformer models outperform Mathematica in polynomial simplification tasks.

The synthetic data pipeline enables controlled complexity for training and evaluation.

Abstract

Recent efforts have extended the capabilities of transformers in logical reasoning and symbolic computations. In this work, we investigate their capacity for non-linear latent pattern discovery in the context of functional decomposition, focusing on the challenging algebraic task of multivariate polynomial decomposition. This problem, with widespread applications in science and engineering, is proved to be NP-hard, and demands both precision and insight. Our contributions are threefold: First, we develop a synthetic data generation pipeline providing fine-grained control over problem complexity. Second, we train transformer models via supervised learning and evaluate them across four key dimensions involving scaling behavior and generalizability. Third, we propose Beam Grouped Relative Policy Optimization (BGRPO), a rank-aware reinforcement learning method suitable for hard algebraic problems. Finetuning with BGRPO improves accuracy while reducing beam width by up to half, resulting in approximately 75% lower inference compute. Additionally, our model demonstrates competitive performance in polynomial simplification, outperforming Mathematica in various cases.