Learning selection strategies in Buchberger's algorithm
This work addresses a key bottleneck in symbolic computation algorithms used by systems like Mathematica and Maple, though it's an incremental improvement focused on specific polynomial types.
The authors tackled the problem of improving Buchberger's algorithm for polynomial equation systems by using reinforcement learning to optimize S-pair selection, resulting in trained models that outperform state-of-the-art heuristics in certain domains by reducing the total number of polynomial additions.
Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger's algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger's algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. We then study how the difficulty of the problem depends on the choices of domain and distribution of polynomials, about which little is known. Finally, we train a policy model using proximal policy optimization (PPO) to learn S-pair selection strategies for random systems of binomial equations. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.