A Comparative Study of Model Selection Criteria for Symbolic Regression

TL;DR

This paper systematically compares model selection criteria for symbolic regression, finding that MDL and BIC most effectively identify models with low test error and true underlying functions.

Contribution

It provides a comprehensive empirical evaluation of popular selection criteria like AIC, BIC, MDL, and others across multiple datasets in symbolic regression.

Findings

MDL consistently finds models with lowest test error and shortest length.

BIC and MDL have highest probability of selecting ground-truth expressions.

No single criterion outperforms others across all datasets.

Abstract

Effective model selection is critical in symbolic regression (SR) to identify mathematical expressions that balance accuracy and complexity, and have low expected error on unseen data. Many modern implementations of genetic programming (GP) for SR generate a set of Pareto optimal candidate solutions, but reliable automatic selection of solutions that generalize well remains an open issue. Current literature offers various information-theoretic and Bayesian approaches, yet comprehensive comparisons of their performance across different data regimes are limited. This study presents a systematic empirical comparison of widely used selection criteria: the Akaike information criterion (AIC), the corrected AIC (AICc), the Bayesian information criterion (BIC), minimum description length (MDL), as well as Efron's bootstrap estimate for the in-sample prediction error on seven synthetic datasets with Gaussian noise. We rank candidate expressions generated by perturbing ground-truth functions to assess generalization error and selection probability of the ground-truth expression. Our findings reveal that MDL consistently identifies models with the lowest test error and the shortest length across most datasets. While no single criterion dominates all results, MDL and BIC produced the highest probability of selecting the ground-truth expressions.