Optimal Risk Scores for Continuous Predictors
Cristina Molero-Río, Claudia D'Ambrosio
TL;DR
This work tackles learning interpretable risk scores when predictors are continuous, by casting the problem as a mixed-integer nonlinear optimization that minimizes logistic loss under sparsity constraints. It extends prior binary-input risk-score formulations to continuous predictors by learning thresholds and introducing linearizations (bilinear and Fortet-based) to maintain tractability. Computational experiments on synthetic data reveal a severe scalability gap for state-of-the-art solvers, motivating a simple, effective matheuristic that delivers fast, competitive feasible solutions with high predictive performance. The approach promises interpretable, threshold-tuned risk scores suitable for high-stakes domains, and it outlines clear directions for scaling to larger datasets via MILO reformulations and enhanced modeling techniques.
Abstract
In this paper, we propose a novel Mixed-Integer Non-Linear Optimization formulation to construct a risk score, where we optimize the logistic loss with sparsity constraints. Previous approaches are typically designed to handle binary datasets, where continuous predictor variables are discretized in a preprocessing step by using arbitrary thresholds, such as quantiles. In contrast, we allow the model to decide for each continuous predictor variable the particular threshold that is critical for prediction. The usefulness of the resulting optimization problem is tested in synthetic datasets.