Learning a performance metric of Buchberger's algorithm
This work provides a proof of concept for using machine learning to estimate the complexity of Buchberger's algorithm, which could help in quickly assessing difficulty and optimizing implementations, though it is incremental as it builds on prior reinforcement learning approaches.
The paper tackled the problem of predicting the number of polynomial additions in Buchberger's algorithm for Gröbner basis computation, showing that a multiple linear regression model and a simple recursive neural network can make somewhat accurate predictions, outperforming uninformed and intuitive commutative algebra models.
What can be (machine) learned about the complexity of Buchberger's algorithm? Given a system of polynomials, Buchberger's algorithm computes a Gröbner basis of the ideal these polynomials generate using an iterative procedure based on multivariate long division. The runtime of each step of the algorithm is typically dominated by a series of polynomial additions, and the total number of these additions is a hardware independent performance metric that is often used to evaluate and optimize various implementation choices. In this work we attempt to predict, using just the starting input, the number of polynomial additions that take place during one run of Buchberger's algorithm. Good predictions are useful for quickly estimating difficulty and understanding what features make Gröbner basis computation hard. Our features and methods could also be used for value models in the reinforcement learning approach to optimize Buchberger's algorithm introduced in [Peifer, Stillman, and Halpern-Leistner, 2020]. We show that a multiple linear regression model built from a set of easy-to-compute ideal generator statistics can predict the number of polynomial additions somewhat well, better than an uninformed model, and better than regression models built on some intuitive commutative algebra invariants that are more difficult to compute. We also train a simple recursive neural network that outperforms these linear models. Our work serves as a proof of concept, demonstrating that predicting the number of polynomial additions in Buchberger's algorithm is a feasible problem from the point of view of machine learning.