Contemporary Symbolic Regression Methods and their Relative Performance (2107.14351v1)

Abstract: Many promising approaches to symbolic regression have been presented in recent years, yet progress in the field continues to suffer from a lack of uniform, robust, and transparent benchmarking standards. In this paper, we address this shortcoming by introducing an open-source, reproducible benchmarking platform for symbolic regression. We assess 14 symbolic regression methods and 7 machine learning methods on a set of 252 diverse regression problems. Our assessment includes both real-world datasets with no known model form as well as ground-truth benchmark problems, including physics equations and systems of ordinary differential equations. For the real-world datasets, we benchmark the ability of each method to learn models with low error and low complexity relative to state-of-the-art machine learning methods. For the synthetic problems, we assess each method's ability to find exact solutions in the presence of varying levels of noise. Under these controlled experiments, we conclude that the best performing methods for real-world regression combine genetic algorithms with parameter estimation and/or semantic search drivers. When tasked with recovering exact equations in the presence of noise, we find that deep learning and genetic algorithm-based approaches perform similarly. We provide a detailed guide to reproducing this experiment and contributing new methods, and encourage other researchers to collaborate with us on a common and living symbolic regression benchmark.

• The paper introduces a comprehensive SR benchmarking platform that evaluates 14 SR methods alongside 7 ML techniques across 252 diverse datasets.
• The paper shows that GP-based methods, particularly Operon and FEAT, balance predictive accuracy with model simplicity on real-world data.
• The paper highlights method sensitivities to noise and suggests integrating semantic search with gradient-based optimization for further performance improvements.

An Expert Analysis of the Paper "Contemporary Symbolic Regression Methods and their Relative Performance"

Symbolic regression (SR) serves as a significant component in the ML landscape due to its ability to yield interpretable models in the form of mathematical expressions. Despite advancements over years in SR techniques, there is a lack of established benchmarking practices for method evaluation. The authors of this paper address this fundamental gap by proposing a robust benchmarking platform for SR. This platform assesses a variety of SR methods alongside machine learning approaches on a diverse array of regression problems.

Overview of Methods and Benchmarking

The paper engages with fourteen distinct SR methods and seven ML techniques, rigorously evaluating them on 252 datasets comprising real-world scenarios and synthetic ground-truth benchmarks. This comprehensive setup allows for a multifaceted performance assessment, focusing not only on accuracy but also on model interpretability through complexity analysis.

The SR methods benchmarked derive from distinct algorithmic philosophies, including traditional genetic programming (GP) approaches, deep learning methodologies, and novel Bayesian frameworks. Particularly notable methods include AFP_FE and Operon, with the latter being highlighted for its advantageous performance on black-box regression tasks. By employing both traditional and state-of-the-art symbolic methods, alongside recognized ML algorithms such as Gradient Boosted Trees and Random Forests, the paper situates itself to offer a holistic view of current SR capabilities.

Key Findings

The thorough experimental analysis conducted reveals that GP-based methods, particularly those fine-tuned to exploit semantic search enhancements or incorporating parameter optimization (e.g., Operon and FEAT), outperform other approaches significantly on real-world data while balancing complexity with accuracy. This highlights the efficacy of combining evolutionary methods with local search for optimizing constant parameters. The Operon method notably provided robust performance and model simplicity compared to competitive ML models such as XGBoost.

On synthetic datasets with defined model solutions, the paper observes a divergence in method efficacy. AIFeynman emerges as a dominant player in identifying exact solutions with minimal noise, showcasing its capacity for function discovery given certain problem structures aligning with its design. However, performance drops with increased noise, where other methods such as DSR and some GP approaches show resilience.

Implications and Future Directions

The established benchmark provides a dependable foundation for SR evaluation, fostering future advancements and discussions around SR methodologies. The openness and extensibility of the benchmark platform encourage ongoing contributions that could reflect SR's progress through comprehensive, standard evaluations. It also underscores the need for SR research to focus on real-world applicability, emphasizing predictive accuracy and simplicity.

Future work could explore optimization within combinatorial SR approaches, especially under noise-influenced conditions. The paper hints at vast areas for potential improvement through cited weaknesses and mismatches in SR effectiveness between synthetic and real-world data scenarios. Furthermore, combining SR methods based on complementary strengths, such as incorporating semantic-driven selections with gradient-based optimization, could lead to novel synergies yielding enhanced performance.

In summary, the paper provides an insightful consolidation of current SR strategies, assessing them comprehensively across a diverse problem spectrum. The results guide practitioners and researchers towards methodologies balancing predictive capability with interpretability—an increasingly critical requirement in high-stakes real-world applications.