Applying Machine Learning to the Problem of Choosing a Heuristic to Select the Variable Ordering for Cylindrical Algebraic Decomposition

Huang, Zongyan; England, Matthew; Wilson, David J.; Davenport, James H.; Paulson, Lawrence C.; Bridge, James P.

doi:10.1007/978-3-319-08434-3_8

Cited by 46 publications

(62 citation statements)

References 50 publications

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“…Of the two human-made heuristics, Brown performed far worse than sotd. This is the opposite of the findings in [29], [23], [25] for 3-variable problems. This is not necessarily in conflict: the added information taken by sotd will grow in size exponentially with the variables, and thus we would expect the predictive information it carries to be more valuable.…”

Section: Comparison Of Brown and Sotd On The 4-variable Datasetcontrasting

confidence: 95%

“…In [23] we also considered a more diverse selection of ML methods than [29]. We experimented with four common ML classifiers: K−Nearest Neighbours (KNN); Multi-Layer Perceptron (MLP); Decision Tree (DT); and Support Vector Machine (SVM) with RBF kernel, all using the same set of 11 features from [29].…”

Section: Results From Cicm 2019mentioning

confidence: 99%

“…We next considered how to extract further information from the input data. The 11 features used in [29], [23] were inspired by Brown's heuristic [9] (e.g. measures of variable degree and frequency of occurrence).…”

Section: Results From Sc-square 2019mentioning

confidence: 99%

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Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

Florescu

England

2020

Lecture Notes in Computer Science

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Our topic is the use of machine learning to improve software by making choices which do not compromise the correctness of the output, but do affect the time taken to produce such output. We are particularly concerned with computer algebra systems (CASs), and in particular, our experiments are for selecting the variable ordering to use when performing a cylindrical algebraic decomposition of n-dimensional real space with respect to the signs of a set of polynomials.In our prior work we explored the different ML models that could be used, and how to identify suitable features of the input polynomials. In the present paper we both repeat our prior experiments on problems which have more variables (and thus exponentially more possible orderings), and examine the metric which our ML classifiers targets. The natural metric is computational runtime, with classifiers trained to pick the ordering which minimises this. However, this leads to the situation were models do not distinguish between any of the non-optimal orderings, whose runtimes may still vary dramatically. In this paper we investigate a modification to the cross-validation algorithms of the classifiers so that they do distinguish these cases, leading to improved results.

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Section: Comparison Of Brown and Sotd On The 4-variable Datasetcontrasting

confidence: 95%

Section: Results From Cicm 2019mentioning

confidence: 99%

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Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

Florescu

England

2020

Lecture Notes in Computer Science

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“…Removing all quantifiers gave a corresponding problem set for evaluating CAD alone. In Huang et al [4] the problems were split into training, evaluation and test sets but here we report on the performance of the heuristics for all problems. In each case the heuristic's selections were compared according to the number of cells produced (as opposed to computation time: so the experiment concerns the CAD theory rather than just the Qepcad implementation).…”

Section: Experiments and Resultsmentioning

confidence: 99%

A Comparison of Three Heuristics to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Huang

England

Wilson

et al. 2015

ACM Commun. Comput. Algebra

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Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples. BackgroundA cylindrical algebraic decomposition (CAD) dissects R n into cells, each described by polynomial relations and arranged cylindrically so that the projection of any two cells into lower coordinates is equal or disjoint. CAD is a key tool, both for its original motivation of quantifier elimination (QE) over real-closed fields and many other applications discovered since. For a more detailed introduction see Huang et al. [4] and the references within. When using CAD there may be a choice for the variable ordering and it is well known that this can have a great effect on the tractability of a problem. We recently tested the following three heuristics for picking this ordering.Brown: Suggested by Brown [2], this heuristic chooses a variable ordering according to three criteria on the input system. We start with the first and break ties with successive ones. The advice is to eliminate a variable first if: 1. it has lower overall degree in the input; 2. it has lower (maximum) total degree of those terms in the input in which it occurs; 3. there is a smaller number of terms in the input which contain the variable.sotd: Suggested by Dolzmann et al. [3], this heuristic constructs the full set of projection polynomials for each ordering and selects the ordering whose set has the lowest sum of total degree for each of the monomials in each of the polynomials.ndrr: Suggested by Bradford et al.[1], this heuristic also constructs the full projection set, and selects the one with the lowest number of distinct real roots of the univariate polynomials.1

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“…Hence the great interest in variable order selection methods for CAD [DSS04, EBDW14,HEW+14] to name a few. An example which needs 2 2 O(n) cells for all possible variable orders is also produced in [BD07], along with another which needs 2 2 O(n) cells in one order, but a constant number in another.…”

Section: Lower Boundsmentioning

confidence: 99%

Automated Deduction in Geometry

Botana¹,

Recio²

2015

Lecture Notes in Computer Science

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Applying Machine Learning to the Problem of Choosing a Heuristic to Select the Variable Ordering for Cylindrical Algebraic Decomposition

Cited by 46 publications

References 50 publications

Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

A Comparison of Three Heuristics to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Automated Deduction in Geometry

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