Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics

TL;DR

This paper advances symbolic regression by formulating it as a nonconvex MINLP, introducing new cuts and heuristics to improve solution quality and efficiency, and comparing these methods with existing approaches.

Contribution

It extends the MINLP formulation for symbolic regression with new cuts and proposes a heuristic for iterative expression tree construction.

Findings

The new cuts improve MINLP solution times.

The heuristic effectively builds accurate expression trees.

Comparative results favor the proposed methods over existing ones.

Abstract

In this paper we consider the problem of learning a regression function without assuming its functional form. This problem is referred to as symbolic regression. An expression tree is typically used to represent a solution function, which is determined by assigning operators and operands to the nodes. The symbolic regression problem can be formulated as a nonconvex mixed-integer nonlinear program (MINLP), where binary variables are used to assign operators and nonlinear expressions are used to propagate data values through nonlinear operators such as square, square root, and exponential. We extend this formulation by adding new cuts that improve the solution of this challenging MINLP. We also propose a heuristic that iteratively builds an expression tree by solving a restricted MINLP. We perform computational experiments and compare our approach with a mixed-integer program-based method and a neural-network-based method from the literature.