arXiv：cs.LG（机器学习，全量分类）· Konstantinos Ziliaskopoulos, Alexander Vinel, Alice E. Smith·· 14 小时前AI 评分36
上下文线性优化中遗憾值的曲率研究
The Curvature of Regret in Contextual Linear Optimization
AI 导读
针对线性优化决策聚焦学习中优化器不连续的问题，研究者证明这种非光滑的逐点行为在数据分布平均后局部呈二次型，并给出了闭式曲率——一种支撑在法扇壁上的矩阵值测度，仅依赖可行集。该曲率可用一次到可行集的投影进行可计算近似，并证明其弱收敛于真实总体曲率。在电池套利实验中，该方法相比均匀分配取得 30.8% 的遗憾值改进。
正文
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.
| Comments: | 4 pages main body plus appendix, 3 figures. Accepted to the NeurIPS 2026 Workshop on MLxOR |
| Subjects: | Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.01980 [cs.LG] |
| (or arXiv:2610.01980v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.01980 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Konstantinos Ziliaskopoulos [view email]
[v1]
Thu, 1 Oct 2026 16:25:09 UTC (76 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org