The Spark Randomizer: a learned randomized framework for computing Gr\"obner bases

TL;DR

This paper introduces a novel framework combining violator spaces and machine learning to efficiently compute Gr"obner bases using a Clarkson-style randomized algorithm, achieving expected linear runtime.

Contribution

It formulates the Gr"obner basis computation as a violator space problem and integrates machine learning predictions with randomized sampling for improved efficiency.

Findings

Achieves expected linear runtime for Gr"obner basis computation

Introduces a violator operator framework for minimal Gr"obner bases

Combines machine learning with randomized algorithms for algebraic computations

Abstract

We define a violator operator which captures the definition of a minimal Gr\"obner basis of an ideal. This construction places the problem of computing a Gr\"obner basis within the framework of violator spaces, introduced in 2008 by G{\"a}rtner, Matou{\v{s}}ek, R{\"u}st, and {\v{S}}kovro{\v{n}} in a different context. The key aspect which we use is their successful utilization of a Clarkson-style fast sampling algorithm from geometric optimization. Using the output of a machine learning algorithm, we combine the prediction of the size of a minimal Gr\"obner basis of an ideal with the Clarkson-style biased random sampling method to compute a Gr\"obner basis in expected runtime linear in the size of the violator space.