预测-然后-优化学习中的几何识别
Geometric Identification in Predict-Then-Optimize Learning
AI总结：
本文在预测-然后-优化学习中，通过刻画SPO+替代模型的风险等式集，提出在中心对称及选择平衡条件下识别均值类与商数报告的几何条件，并验证于多个数据集。
AI中文摘要：
决策聚焦的替代模型可以在不识别商数报告的情况下恢复下游决策。我们刻画了凸Smart Predict-then-Optimize替代模型（SPO+）总体风险的等式集。在中心对称条件下，当且仅当每个非零有效位移使旧优化器以正概率离开平移后的最优面时，居中均值类是唯一的贝叶斯最小化器。该条件将面穿越与所选预言机分歧区分开来，并给出定量的局部强制性。在没有对称性的情况下，严格的穿越本身不一定能识别均值；带反射穿越的选择平衡恢复了商数报告的识别，条件版本将结果扩展到可测预测器。这些是总体层面的陈述，不提供有限样本报告恢复或通用转移遗憾保证。闭式机制再现了解析恒等式和速率。投资组合、完整矩阵KuaiRec和能源/存储研究衡量了预测保真度、平移遗憾和拟合报告几何。一个已知数据生成过程（DGP）的伴随研究保留了其应用几何，同时隔离了条件均值恢复和穿越，而不测试原始观测假设。
英文摘要：
Decision-focused surrogates can recover downstream decisions without identifying the quotient report. We characterize the equality set of the convex Smart Predict-then-Optimize surrogate (SPO+) population risk. Under central symmetry, the centered mean class is the unique Bayes minimizer exactly when every nonzero effective displacement makes the old optimizer leave the shifted optimal face with positive probability. This condition separates face crossing from selected-oracle disagreement and gives quantitative local coercivity. Without symmetry, strict crossing alone need not identify the mean; selection balance with reflected crossing restores quotient-report identification, and conditional versions extend the result to measurable predictors. These are population statements, without finite-sample report-recovery or generic transfer-regret guarantees. Closed-form mechanisms reproduce the analytic identities and rates. Portfolio, complete-matrix KuaiRec, and Energy/Storage studies measure predictive fidelity, shifted regret, and fitted-report geometry. A known data-generating process (DGP) companion retains their application geometries while isolating conditional-mean recovery and crossing, without testing the original observational assumptions.
