The Spark Randomizer: a learned randomized framework for computing Gröbner bases

Abstract

We define a violator operator which captures the definition of a minimal Gröbner basis of an ideal. This construction places the problem of computing a Gröbner basis within the framework of violator spaces, introduced in 2008 by G{ä}rtner, Matou{š}ek, R{ü}st, and {Š}kovro{ň} 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 Gröbner basis of an ideal with the Clarkson-style biased random sampling method to compute a Gröbner basis in expected runtime linear in the size of the violator space.