Investigations with Monte Carlo Tree Search for Finding Better Multivariate Horner Schemes

After a computer chess program had defeated the human World Champion in 1997, many researchers turned their attention to the oriental game of Go. It turned out that the minimax approach, so successful in chess, did not work in Go. Instead, after some ten years of intensive research, a new method was developed: MCTS (Monte Carlo Tree Search), with promising results. MCTS works by averaging the results of random play-outs. At first glance it is quite surprising that MCTS works so well. However, deeper analysis revealed the reasons. The success of MCTS in Go caused researchers to apply the method to other domains. In this article we report on experiments with MCTS for finding improved orderings for multivariate Horner schemes, a basic method for evaluating polynomials. We report on initial results, and continue with an investigation into two parameters that guide the MCTS search. Horner's rule turns out to be a fruitful testbed for MCTS, allowing easy experimentation with its parameters. The results reported here provide insight into how and why MCTS works. It will be interesting to see if these insights can be transferred to other domains, for example, back to Go.