Using differential evolution for symbolic regression and numerical constant creation

One problem that has plagued Genetic Programming (GP) and its derivatives is numerical constant creation. Given a mathematical formula expressed as a tree structure, the leaf nodes are either variables or constants. Such constants are usually unknown in Symbolic Regression (SR) problems, and GP, as well as many of its derivatives, lack the ability to precisely approximate these values. This is due to the inherently discrete encoding of GP-like methods which are more suited to combinatorial searches than real-valued optimization tasks. Previously, several attempts have been made to resolve this issue, and the dominant solutions have been to either embed a real-valued local optimizer or to develop additional numerically oriented operators. In this paper, an entirely new approach is proposed for constant creation. Through the adoption of a robust, real-valued optimization algorithm known as Differential Evolution (DE), constants and GP-like programs will be simultaneously evolved in such a way that the values of the leaf nodes will be approximated as the tree structure is itself changing. Experimental results from several SR benchmarks are presented and analyzed. The results demonstrate the feasibility of the proposed algorithm and suggest that exotic or computationally expensive methods are not necessary for successful constant creation.