Risk Bounds and Calibration for a Smart Predict-then-Optimize Method

• URL: http://arxiv.org/abs/2108.08887v1
• Date: Thu, 19 Aug 2021 19:25:46 GMT
• Title: Risk Bounds and Calibration for a Smart Predict-then-Optimize Method
• Authors: Heyuan Liu, Paul Grigas
• Abstract summary: We show that the SPO+ loss can minimize low excess true risk with high probability. We show that the results can be strengthened substantially when the feasible region is a level set of bounds.
• Score: 2.28438857884398
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The predict-then-optimize framework is fundamental in practical stochastic decision-making problems: first predict unknown parameters of an optimization model, then solve the problem using the predicted values. A natural loss function in this setting is defined by measuring the decision error induced by the predicted parameters, which was named the Smart Predict-then-Optimize (SPO) loss by Elmachtoub and Grigas [arXiv:1710.08005]. Since the SPO loss is typically nonconvex and possibly discontinuous, Elmachtoub and Grigas [arXiv:1710.08005] introduced a convex surrogate, called the SPO+ loss, that importantly accounts for the underlying structure of the optimization model. In this paper, we greatly expand upon the consistency results for the SPO+ loss provided by Elmachtoub and Grigas [arXiv:1710.08005]. We develop risk bounds and uniform calibration results for the SPO+ loss relative to the SPO loss, which provide a quantitative way to transfer the excess surrogate risk to excess true risk. By combining our risk bounds with generalization bounds, we show that the empirical minimizer of the SPO+ loss achieves low excess true risk with high probability. We first demonstrate these results in the case when the feasible region of the underlying optimization problem is a polyhedron, and then we show that the results can be strengthened substantially when the feasible region is a level set of a strongly convex function. We perform experiments to empirically demonstrate the strength of the SPO+ surrogate, as compared to standard $\ell_1$ and squared $\ell_2$ prediction error losses, on portfolio allocation and cost-sensitive multi-class classification problems.

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