The Curvature of Regret in Contextual Linear Optimization

2026-10-01

AI Summary

This study addresses the challenge in decision-focused learning for linear optimization, where optimizer discontinuity causes cost errors to degrade decision quality. We reveal that nonsmooth losses exhibit local quadratic smoothness under distributional averaging. By integrating normal cone geometry with weak convergence theory, we derive closed-form curvature solutions and feasible-set-dependent matrix measures, proposing a curvature approximation algorithm requiring only a single projection. Theoretically, we validate the quadratic smoothing properties and weak convergence behavior. Empirically, our method reduces regret by 30.8% compared to uniform allocation baselines on battery arbitrage tasks, substantially improving end-to-end decision-making performance.

Abstract

Decision-focused learning for linear optimization is complicated by the discontinuity of the optimizer, where small cost errors may leave the decision unchanged or move it to a different vertex. We show that this non-smooth pointwise behavior becomes locally quadratic after averaging over the data distribution, and we derive the curvature in closed form, specifically, a matrix-valued measure supported on the walls of the normal fan. This measure depends only on the feasible set, with the data distribution entering only as a weight. We then offer a tractable approximation for this curvature, computable with just one projection to the feasible set. We prove that the approximation weakly converges to the true population curvature. We offer one application of our findings, a decision-aware scenario generation method for expected-cost linear optimization. Our experiments test the quadratic and weak convergence laws and show a 30.8% regret improvement over uniform allocation on battery arbitrage.

Problem

• decision-focused learning
• linear optimization
• regret curvature
• optimizer discontinuity
• normal fan

Innovation

• Decision-focused learning
• Curvature of regret
• Normal fan
• Weak convergence
• Scenario generation