Smart "Predict, then Optimize"

Abstract: Many real-world analytics problems involve two significant challenges: prediction and optimization. Due to 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, i.e., 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.

• The paper presents the SPO loss function that quantifies decision error by directly linking predictions with optimization outcomes.
• It introduces the convex SPO+ loss as a surrogate to overcome non-convexity challenges, enabling efficient training via linear programming duality.
• Numerical experiments on shortest path and portfolio optimization demonstrate that SPO+ can outperform traditional models even with simple linear predictors.

Smart "Predict, then Optimize" Framework

The paper introduces the Smart "Predict, then Optimize" (SPO) framework, a methodology designed to tackle the predict-then-optimize problem prevalent in operations research and data-driven decision-making. Traditional approaches decouple prediction from optimization, typically leading to suboptimal decisions when predictions are flawed. The SPO framework aims to bridge this gap by developing predictive models that incorporate the structure of the underlying optimization problem directly into the learning process.

SPO Loss Function

A central concept in the SPO framework is the SPO loss function, which quantifies decision error rather than merely prediction error. This loss function measures the discrepancy between the cost of an actual decision made using predicted parameters and the optimal decision that would be made if the true parameters were known. The challenge is its non-convexity and lack of continuity in predictions due to ties in decision-making, leading to potential computational intractability.

To handle these challenges, the paper introduces a convex surrogate, the SPO+ loss function. This convex approximation is derived using duality theory and is shown to be statistically consistent with the original non-convex SPO loss under mild conditions. The SPO+ loss function enables efficient training of prediction models by leveraging the optimization problem's structure, thus directly minimizing decision error.

Implementation Considerations

The practical implementation revolves around training predictive models with respect to the SPO+ loss. This involves solving an empirical risk minimization problem, which can be reformulated for tractable optimization using linear programming duality. Practical use of SPO+ incorporates stochastic gradient descent and other optimization techniques suitable for large-scale applications.

The paper highlights numerical experiments on problems such as shortest path and portfolio optimization, demonstrating that the SPO framework provides significant improvements, especially under model misspecification where traditional methods falter. A key insight is that even simple linear models trained with SPO+ loss can outperform more complex algorithms like random forests, particularly when the ground truth exhibits nonlinearity.

Future Directions and Implications

The SPO framework has significant potential for enhancing the integration of machine learning and optimization. By explicitly considering how predictions influence downstream decisions, this approach offers a more robust methodology for real-world decision-making. Future work could explore its application to broader classes of optimization problems, including those with non-linear objectives and constraints, as well as integrating robustness into the decision-making framework.

In conclusion, the SPO framework offers a novel perspective for predictive modeling by standardizing integration with decision-making through optimization. Its practical applicability, especially in operations-driven contexts, marks a meaningful advancement in prescriptive analytics.