Improving multivariate Horner schemes with Monte Carlo tree search
This work addresses a classic computational efficiency problem in computer science, offering incremental improvements for polynomial evaluation in fields like scientific computing.
The paper tackled the problem of optimizing the evaluation cost of multivariate polynomials by improving upon traditional greedy Horner schemes, resulting in better schemes that sometimes reduce evaluation costs by factors up to two.
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.