Learning Selection Strategies in Buchberger’s Algorithm

Dylan Peifer, Michael Stillman, Daniel Halpern-Leistner
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:7575-7585, 2020.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-peifer20a, title = {Learning Selection Strategies in Buchberger’s Algorithm}, author = {Peifer, Dylan and Stillman, Michael and Halpern-Leistner, Daniel}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {7575--7585}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/peifer20a/peifer20a.pdf}, url = {https://proceedings.mlr.press/v119/peifer20a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Learning Selection Strategies in Buchberger’s Algorithm %A Dylan Peifer %A Michael Stillman %A Daniel Halpern-Leistner %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-peifer20a %I PMLR %P 7575--7585 %U https://proceedings.mlr.press/v119/peifer20a.html %V 119 %X 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.
APA
Peifer, D., Stillman, M. & Halpern-Leistner, D.. (2020). Learning Selection Strategies in Buchberger’s Algorithm. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:7575-7585 Available from https://proceedings.mlr.press/v119/peifer20a.html.

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