Split Regression Modeling

In the statistical literature, sparse modeling is the standard approach to achieve improvements in prediction tasks and interpretability. Alternatively, in the seminal paper "Statistical Modeling: The Two Cultures," Breiman (2001) advocated for the adoption of algorithmic approaches to generate ensembles to achieve superior prediction accuracy than single-model methods at the cost of loss of interpretability. In a recent important and critical paper, Rudin (2019) argued that blackbox algorithmic approaches should be avoided for high-stakes decisions and that the tradeoff between accuracy and interpretability is a myth. In response to this recent change in philosophy, we generalize best subset selection (BSS) to best split selection (BSpS), a data-driven approach aimed at finding the optimal split of predictor variables among the models of an ensemble. The proposed methodology results in an ensemble of sparse and diverse models that provide possible mechanisms that explain the relationship between the predictors and the response. The high computational cost of BSpS motivates the need for computational tractable ways to approximate the exhaustive search, and we benchmark one such recent proposal by Christidis et al. (2020) based on a multi-convex relaxation. Our objective with this article is to motivate research in this new exciting field with great potential for data analysis tasks for high-dimensional data.