Combining Simulated Annealing and Monte Carlo Tree Search for Expression Simplification

TL;DR

This paper introduces SA-UCT, a dynamic selection criterion for Monte Carlo Tree Search, improving expression simplification by reducing sensitivity to parameters and enhancing results in computer algebra applications.

Contribution

It proposes SA-UCT, a novel dynamic exploration-exploitation strategy for MCTS, tailored for expression simplification, and demonstrates its effectiveness on large expressions.

Findings

SA-UCT widens the optimal initial $C_p$ interval.

The method achieves over tenfold improvement in results.

It reduces parameter sensitivity in MCTS for algebraic expression simplification.

Abstract

In many applications of computer algebra large expressions must be simplified to make repeated numerical evaluations tractable. Previous works presented heuristically guided improvements, e.g., for Horner schemes. The remaining expression is then further reduced by common subexpression elimination. A recent approach successfully applied a relatively new algorithm, Monte Carlo Tree Search (MCTS) with UCT as the selection criterion, to find better variable orderings. Yet, this approach is fit for further improvements since it is sensitive to the so-called exploration-exploitation constant $C_p$ and the number of tree updates $N$. In this paper we propose a new selection criterion called Simulated Annealing UCT (SA-UCT) that has a dynamic exploration-exploitation parameter, which decreases with the iteration number $i$ and thus reduces the importance of exploration over time. First, we provide an intuitive explanation in terms of the exploration-exploitation behavior of the algorithm. Then, we test our algorithm on three large expressions of different origins. We observe that SA-UCT widens the interval of good initial values $C_p$ where best results are achieved. The improvement is large (more than a tenfold) and facilitates the selection of an appropriate $C_p$.