Improving multivariate Horner schemes with Monte Carlo tree search

TL;DR

This paper applies Monte Carlo tree search to optimize multivariate Horner schemes, significantly reducing polynomial evaluation costs beyond traditional greedy methods.

Contribution

It introduces Monte Carlo tree search as a novel approach for optimizing multivariate Horner schemes, outperforming existing greedy algorithms.

Findings

Horner schemes optimized with Monte Carlo tree search can halve evaluation costs.

The method outperforms traditional greedy schemes in several cases.

Cost reductions up to a factor of two were observed.

Abstract

Optimizing the cost of evaluating a polynomial is a classic problem in computer science. For polynomials in one variable, Horner's method provides a scheme for producing a computationally efficient form. For multivariate polynomials it is possible to generalize Horner's method, but this leaves freedom in the order of the variables. Traditionally, greedy schemes like most-occurring variable first are used. This simple textbook algorithm has given remarkably efficient results. Finding better algorithms has proved difficult. In trying to improve upon the greedy scheme we have implemented Monte Carlo tree search, a recent search method from the field of artificial intelligence. This results in better Horner schemes and reduces the cost of evaluating polynomials, sometimes by factors up to two.