Using Machine Learning to Improve Cylindrical Algebraic Decomposition

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 Gröbner 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.1 principally {= 0, > 0, < 0}, but the Boolean combinations allow { = 0, ≥ 0, ≤ 0} as well 2 The feature set they used for their support vector machine was seeded by Table 1.

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

“…-The same authors studied a different choice related to CAD (whether to precondition the input with Gröbner bases) in [35] [36], again finding that a support vector machine could make the choice more accurately than the human-made heuristic (if features of the Gröbner Basis could be used). -At MACIS 2016 there was a study applying a support vector machine to decide the order of sub-formulae solving for a QE procedure [40].…”

Deep Learning Gauss–Manin Connections

“…However, directly predicting the solution to a problem is not the only way one can apply deep learning methods to mathematical computations. Instead, deep learning methods can be used to assist a more robust method by providing dynamically generated heuristics, thereby improving performance while preserving reliability-see for instance [19]. This is our approach here, see Remark 4.1.…”

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

“…The first application of ML to the problem was in 2014 when a support vector machine was trained to choose which of these heuristics to follow [20], [19]. The machine learned choice did significantly better than any one heuristic overall.…”