Mattia Billa

Ali Soltani, Gabriel Kronberger, Fabricio Olivetti de Franca et al.

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.

Mattia Billa, Giovanni Orlandi, Veronica Guidetti et al.

In recent years, Machine Learning (ML) has seen widespread adoption across a broad range of sectors, including high-stakes domains such as healthcare, finance, and law. This growing reliance has raised increasing concerns regarding model interpretability and accountability, particularly as legal and regulatory frameworks place tighter constraints on using black-box models in critical applications. Although interpretable ML has attracted substantial attention, systematic evaluations of inherently interpretable models, especially for tabular data, remain relatively scarce and often focus primarily on aggregated performance outcomes. To address this gap, we present a large-scale comparative evaluation of 16 inherently interpretable methods, ranging from classical linear models and decision trees to more recent approaches such as Explainable Boosting Machines (EBMs), Symbolic Regression (SR), and Generalized Optimal Sparse Decision Trees (GOSDT). Our study spans 216 real-world tabular datasets and goes beyond aggregate rankings by stratifying performance according to structural dataset characteristics, including dimensionality, sample size, linearity, and class imbalance. In addition, we assess training time and robustness under controlled distributional shifts. Our results reveal clear performance hierarchies, especially for regression tasks, where EBMs consistently achieve strong predictive accuracy. At the same time, we show that performance is highly context-dependent: SR and Interpretable Generalized Additive Neural Networks (IGANNs) perform particularly well in non-linear regimes, while GOSDT models exhibit pronounced sensitivity to class imbalance. Overall, these findings provide practical guidance for practitioners seeking a balance between interpretability and predictive performance, and contribute to a deeper empirical understanding of interpretable modeling for tabular data.