A Functional Analysis Approach to Symbolic Regression

arXiv (Cornell University)

Abstract

Symbolic regression (SR) poses a significant challenge for randomized search heuristics due to its reliance on the synthesis of expressions for input-output mappings. Although traditional genetic programming (GP) algorithms have achieved success in various domains, they exhibit limited performance when tree-based representations are used for SR. To address these limitations, we introduce a novel SR approach called Fourier Tree Growing (FTG) that draws insights from functional analysis. This new perspective enables us to perform optimization directly in a different space, thus avoiding intricate symbolic expressions. Our proposed algorithm exhibits significant performance improvements over traditional GP methods on a range of classical one-dimensional benchmarking problems. To identify and explain limiting factors of GP and FTG, we perform experiments on a large-scale polynomials benchmark with high-order polynomials up to degree 100. To the best of the authors' knowledge, this work represents the pioneering application of functional analysis in addressing SR problems. The superior performance of the proposed algorithm and insights into the limitations of GP open the way for further advancing GP for SR and related areas of explainable machine learning.

Authors 6

1. Kirill Antonov Aachen

   Leiden University · University of Applied Sciences Leiden

   Affiliation as printed

   LIACS , Leiden University Leiden , The Netherlands

2. Roman Kalkreuth

   Centre National de la Recherche Scientifique · Sorbonne Université · Université Paris 1 Panthéon-Sorbonne

   Affiliation as printed

   CNRS , LIP6 , Sorbonne Université Paris , France

3. Kaifeng Yang

   University of Applied Sciences Upper Austria

   Affiliation as printed

   University of Applied Sciences Upper Austria Hagenger , Austria

4. Thomas Bäck
5. Bas van Stein Aachen

   Leiden University · University of Applied Sciences Leiden

   Affiliation as printed

   LIACS , Leiden University Leiden , The Netherlands

6. Anna V. Kononova Aachen

   Leiden University · University of Applied Sciences Leiden

   Affiliation as printed

   LIACS , Leiden University Leiden , The Netherlands