Interpretability in Symbolic Regression: a benchmark of Explanatory Methods using the Feynman data set

TL;DR

This work introduces iirsBenchmark, a public framework to benchmark explanatory methods for regression with a focus on symbolic regression (SR). By leveraging the Feynman data set of 100 physics equations, the authors compare SR methods (ITEA, Operon) against white-box and black-box regressors using a suite of local and global explainers (SHAP, LIME, Partial Effects, Integrated Gradients, etc.) and robust quality metrics (stability, fidelity, cosine similarity, NMSE). Key findings show SR models can achieve competitive accuracy while yielding explanations that are as robust as those for some traditional models, with Partial Effects and SHAP often providing the most stable explanations; Integrated Gradients can be unstable on tree ensembles. The study emphasizes that explanation quality correlates with model quality and that trusting explanations requires assessing their robustness, advocating for broader benchmarks and dataset coverage. The framework and insights provide a practical pathway to integrate interpretability with SR in scientific discovery contexts.

Abstract

In some situations, the interpretability of the machine learning models plays a role as important as the model accuracy. Interpretability comes from the need to trust the prediction model, verify some of its properties, or even enforce them to improve fairness. Many model-agnostic explanatory methods exists to provide explanations for black-box models. In the regression task, the practitioner can use white-boxes or gray-boxes models to achieve more interpretable results, which is the case of symbolic regression. When using an explanatory method, and since interpretability lacks a rigorous definition, there is a need to evaluate and compare the quality and different explainers. This paper proposes a benchmark scheme to evaluate explanatory methods to explain regression models, mainly symbolic regression models. Experiments were performed using 100 physics equations with different interpretable and non-interpretable regression methods and popular explanation methods, evaluating the performance of the explainers performance with several explanation measures. In addition, we further analyzed four benchmarks from the GP community. The results have shown that Symbolic Regression models can be an interesting alternative to white-box and black-box models that is capable of returning accurate models with appropriate explanations. Regarding the explainers, we observed that Partial Effects and SHAP were the most robust explanation models, with Integrated Gradients being unstable only with tree-based models. This benchmark is publicly available for further experiments.

Figures (10)

• Figure 1: Operon GP-NLS expanded tree (new nodes in gray, original tree nodes in a lighter color). The offset node $w_0$ is summed with the original tree, now scaled with the scale node $a$. Every feature has a coefficient associated with it ($w_1, w_2$). The original coefficient --- a fixed value --- is transformed into a free parameter $w_3$. The Levenberg-Marquardt algorithm will find optimal values for all free parameters in the expression tree ($\{w_0, a, w_1, w_2, w_3\}$)
• Figure 2: Diagram illustrating the relationships between the black-box with its predictions and the explainer with its explanations. The ML prediction model is used as an input together with the training data. Then, it generates feature importance explainers to help understand the model. Adapted from avaliacaoDaInterpretabilidade.
• Figure 3: Scheme of the whole benchmark process to generate the results. The Feynman equations are used to generate the train and test data. Then the regression methods are fine-adjusted through a gridsearch process to finally be trained and used as input to feature importance methods.
• Figure 4: Joint distribution of the number of features and the number of nodes required to build the expression tree for the iirsBenchmark data sets. The lighter portion of the histograms represents the GP benchmark.
• Figure 5: MAE (smaller is better) and NMSE (smaller is better) boxplots for all regressors, vertically ordered from the best to worst median values. The Critical Diagram below each plot indicates the absence of statistical significance between groups connected by a horizontal bar.

Theorems & Definitions (16)

• Definition 1: Local and Global feature Importance Explanations
• Definition 2: Permutation Importance -- Global Importance Explanation
• Definition 3: LIME -- Local Importance Explanation
• Definition 4: ELA -- Local Importance Explanation
• Definition 5: SHAP -- Local Importance Explanation
• Definition 6: SHAP -- Global Importance Explanation
• Definition 7: SAGE global explanation
• Definition 8: Morris Sensitivity -- Global Feature Explanation
• Definition 9: Integrated Gradients -- Local Feature Explanation
• Definition 10: Partial Effects -- Local Feature Explanation