Decision-Focused Learning via Tangent-Space Projection of Prediction Error

2026-05-02

AI Summary

This work addresses the high computational cost or objective bias in traditional decision-focused learning, where regret gradients rely on differentiable solvers or surrogate losses. The authors propose Projected Error as Regret Gradient (PEAR), a method that derives a closed-form geometric expression for regret gradients by projecting prediction errors onto the tangent space of active constraints and incorporating local curvature—eliminating the need for iterative solving or auxiliary optimization. PEAR constructs a reduced-dimensional linear system based on the active constraint set to enable efficient gradient computation. Experiments demonstrate that PEAR significantly outperforms existing baselines on both linear programming and real-world quadratic programming tasks, achieving superior performance in decision quality, computational efficiency, and robustness under varying constraints.

Abstract

Decision-Focused Learning (DFL) trains predictors to improve downstream decision quality, but computing regret gradients typically requires differentiating through solvers or relying on surrogate losses, which can be computationally expensive or deviate from the true objective. We show that, under standard regularity with locally stable active constraints, the regret gradient admits a closed-form geometric characterization, equivalent to the prediction error projected onto the tangent space of active constraints, scaled by local curvature. This reveals that regret gradients can be obtained by filtering decision-irrelevant components from the MSE gradient, providing a simpler and more direct alternative to existing approaches. Based on this, we propose PEAR (Projected Error As Regret-gradient), which computes regret gradients via a reduced linear system over active constraints, avoiding differentiation through solver iterations or additional optimization solves. Experiments on LP benchmarks and a real-world QP task show that PEAR achieves the best decision quality among all baselines while being the most computationally efficient, with gains that persist under constraint shifts.

Problem

• Decision-Focused Learning
• regret gradient
• computational efficiency
• solver differentiation
• prediction error

Innovation

• Decision-Focused Learning
• Regret Gradient
• Tangent-Space Projection
• Active Constraints
• PEAR