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The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation. Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship. [ABSTRACT FROM AUTHOR]

Many real-world analytics problems involve two significant challenges: prediction and optimization. Because of the typically complex nature of each challenge, the standard paradigm is predict-then-optimize. By and large, machine learning tools are intended to minimize prediction error and do not account for how the predictions will be used in the downstream optimization problem. In contrast, we propose a new and very general framework, called Smart "Predict, then Optimize" (SPO), which directly leverages the optimization problem structure—that is, its objective and constraints—for designing better prediction models. A key component of our framework is the SPO loss function, which measures the decision error induced by a prediction. Training a prediction model with respect to the SPO loss is computationally challenging, and, thus, we derive, using duality theory, a convex surrogate loss function, which we call the SPO+ loss. Most importantly, we prove that the SPO+ loss is statistically consistent with respect to the SPO loss under mild conditions. Our SPO+ loss function can tractably handle any polyhedral, convex, or even mixed-integer optimization problem with a linear objective. Numerical experiments on shortest-path and portfolio-optimization problems show that the SPO framework can lead to significant improvement under the predict-then-optimize paradigm, in particular, when the prediction model being trained is misspecified. We find that linear models trained using SPO+ loss tend to dominate random-forest algorithms, even when the ground truth is highly nonlinear. This paper was accepted by Yinyu Ye, optimization. Supplemental Material: Data and the online appendix are available at https://doi.org/10.1287/mnsc.2020.3922 [ABSTRACT FROM AUTHOR]