Buffered AUC maximization for scoring systems via mixed-integer optimization

TL;DR

The paper addresses the need for interpretable scoring systems that retain strong discriminatory power. It introduces a mixed-integer linear optimization framework that directly maximizes a convex surrogate of $AUC$, $bAUC$, while enforcing group sparsity and small integer coefficients via a RiskSLIM-inspired formulation reformulated as MILO. Key contributions include a novel group-sparsity constraint for limiting questions, a MILO representation that handles the non-differentiable $bAUC$ objective, and comprehensive experiments on four real-world datasets showing superior $AUC$ performance in three of four cases compared to baselines, with manageable computation times. The work advances the design of interpretable, high-performing scoring systems suitable for high-stakes decision-making, and points to future work on leveraging full datasets and pursuing direct $AUC$ maximization.

Abstract

A scoring system is a linear classifier composed of a small number of explanatory variables, each assigned a small integer coefficient. This system is highly interpretable and allows predictions to be made with simple manual calculations without the need for a calculator. Several previous studies have used mixed-integer optimization (MIO) techniques to develop scoring systems for binary classification; however, they have not focused on directly maximizing AUC (i.e., area under the receiver operating characteristic curve), even though AUC is recognized as an essential evaluation metric for scoring systems. Our goal herein is to establish an effective MIO framework for constructing scoring systems that directly maximize the buffered AUC (bAUC) as the tightest concave lower bound on AUC. Our optimization model is formulated as a mixed-integer linear optimization (MILO) problem that maximizes bAUC subject to a group sparsity constraint for limiting the number of questions in the scoring system. Computational experiments using publicly available real-world datasets demonstrate that our MILO method can build scoring systems with superior AUC values compared to the baseline methods based on regularization and stepwise regression. This research contributes to the advancement of MIO techniques for developing highly interpretable classification models.

Figures (1)

• Figure 1: Relationship between $y = \mathbb{I}_{\mathbb{R}_{+}}(x)$ and $y = \frac{1}{\gamma}[x + \gamma]_{+}$ for $\gamma \in \{\frac{1}{2}, 1, 2\}$