Using Machine Learning to Improve Cylindrical Algebraic Decomposition

TL;DR

This paper explores using support vector machines to optimize choices in Cylindrical Algebraic Decomposition, significantly outperforming traditional heuristics in variable ordering and preconditioning decisions.

Contribution

It introduces the first application of machine learning to symbolic computation, specifically for improving CAD variable ordering and preconditioning strategies.

Findings

Machine learning outperforms heuristics in CAD variable ordering.

Support vector machines effectively identify when preconditioning benefits CAD.

The approach enhances efficiency in computational algebraic geometry tasks.

Abstract

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in the size of the input, which is often encountered in practice. It has been observed that for many problems a change in algorithm settings or problem formulation can cause huge differences in runtime costs, changing problem instances from intractable to easy. A number of heuristics have been developed to help with such choices, but the complicated nature of the geometric relationships involved means these are imperfect and can sometimes make poor choices. We investigate the use of machine learning (specifically support vector machines) to make such choices instead. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we apply it in two case studies: the first to select between heuristics for choosing a CAD variable ordering; the second to identify when a CAD problem instance would benefit from Groebner Basis preconditioning. These appear to be the first such applications of machine learning to Symbolic Computation. We demonstrate in both cases that the machine learned choice outperforms human developed heuristics.