The Perils of Learning Before Optimizing

TL;DR

This paper analyzes the limitations of traditional two-stage prediction-then-optimization methods and demonstrates how end-to-end learning can adaptively improve performance, especially with multiple prediction targets and stochastic data.

Contribution

It characterizes when end-to-end learning outperforms two-stage methods and provides explicit constructions showing potential unbounded gaps, with practical implications for real-world applications.

Findings

End-to-end learning can adaptively optimize prediction targets in stochastic settings.

Two-stage methods may perform unboundedly worse than end-to-end in certain scenarios.

Simulations show significant potential improvements in real-world applications.

Abstract

Formulating real-world optimization problems often begins with making predictions from historical data (e.g., an optimizer that aims to recommend fast routes relies upon travel-time predictions). Typically, learning the prediction model used to generate the optimization problem and solving that problem are performed in two separate stages. Recent work has showed how such prediction models can be learned end-to-end by differentiating through the optimization task. Such methods often yield empirical improvements, which are typically attributed to end-to-end making better error tradeoffs than the standard loss function used in a two-stage solution. We refine this explanation and more precisely characterize when end-to-end can improve performance. When prediction targets are stochastic, a two-stage solution must make an a priori choice about which statistics of the target distribution to model-we consider expectations over prediction targets-while an end-to-end solution can make this choice adaptively. We show that the performance gap between a two-stage and end-to-end approach is closely related to the price of correlation concept in stochastic optimization and show the implications of some existing POC results for the predict-then-optimize problem. We then consider a novel and particularly practical setting, where multiple prediction targets are combined to obtain each of the objective function's coefficients. We give explicit constructions where (1) two-stage performs unboundedly worse than end-to-end; and (2) two-stage is optimal. We use simulations to experimentally quantify performance gaps and identify a wide range of real-world applications from the literature whose objective functions rely on multiple prediction targets, suggesting that end-to-end learning could yield significant improvements.