A machine learning based software pipeline to pick the variable ordering for algorithms with polynomial inputs

TL;DR

This paper presents a machine learning pipeline that optimizes variable ordering in algorithms for polynomial systems, improving performance without affecting correctness, demonstrated through CAD variable ordering experiments.

Contribution

It introduces a novel ML-based software pipeline for selecting variable orderings in polynomial algorithms, adaptable to other decision points in mathematical software.

Findings

Effective ML models for variable ordering selection

New feature generation techniques for ML in mathematical software

Open-source code for the pipeline available online

Abstract

We are interested in the application of Machine Learning (ML) technology to improve mathematical software. It may seem that the probabilistic nature of ML tools would invalidate the exact results prized by such software, however, the algorithms which underpin the software often come with a range of choices which are good candidates for ML application. We refer to choices which have no effect on the mathematical correctness of the software, but do impact its performance. In the past we experimented with one such choice: the variable ordering to use when building a Cylindrical Algebraic Decomposition (CAD). We used the Python library Scikit-Learn (sklearn) to experiment with different ML models, and developed new techniques for feature generation and hyper-parameter selection. These techniques could easily be adapted for making decisions other than our immediate application of CAD variable ordering. Hence in this paper we present a software pipeline to use sklearn to pick the variable ordering for an algorithm that acts on a polynomial system. The code described is freely available online.