Risk scores are simple classification models that let users quickly assess risk by adding, subtracting, and multiplying a few small numbers. These models are widely used for high-stakes applications in healthcare and criminology, but are difficult to create because they need to be risk-calibrated, sparse, use small integer coefficients, and obey operational constraints. In this paper, we present a new approach to learn risk scores that are fully optimized for feature selection, integer coefficients, and operational constraints. We formulate the risk score problem as a mixed integer nonlinear program, and present a new cutting plane algorithm to efficiently recover its optimal solution while avoiding the stalling behavior of existing cutting plane algorithms in non-convex settings. We pair our algorithm with specialized techniques to generate feasible solutions, narrow the optimality gap, and reduce data-related computation. The resulting approach can learn optimized risk scores in a way that scales linearly in the number of samples, provides a proof of optimality, and accommodates complex operational constraints. We illustrate the benefits of this approach through extensive numerical experiments.
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