改进分片共形回归的区间体积，提升预测精度。
On Volume Minimization in Conformal Regression
- 通过最小化校准阶段的体积损失优化预测区间长度。
- 提出EffOrt方法，使基础预测函数选择更优，降低区间体积。
- 引入自适应版本Ad-EffOrt，区间大小随特征值动态调整。
我们研究分片共形回归中的体积最优性问题，该问题在覆盖控制之外仍缺乏深入理解。利用校准步骤可视为经验体积最小化问题的性质，我们首次推导出经典分片方法所得区间的过剩体积损失的有限样本上界。该指标衡量了分片方法结果与理想最短预言区间的长度差异。随后，我们提出EffOrt方法，通过修改学习阶段，使基础预测函数的选择以最小化返回区间的长度为目标。理论分析揭示了学习与校准步骤之间的联系，尤其是基础预测器函数类选择的影响。此外，我们引入Ad-EffOrt，一种可自适应协变量取值的扩展方法，可生成大小随输入变化的区间。最后，我们对方法的实证性能与鲁棒性进行了评估。
原文摘要 · Abstract (English)
We study the question of volume optimality in split conformal regression, a topic still poorly understood in comparison to coverage control. Using the fact that the calibration step can be seen as an empirical volume minimization problem, we first derive a finite-sample upper-bound on the excess volume loss of the interval returned by the classical split method. This important quantity measures the difference in length between the interval obtained with the split method and the shortest oracle prediction interval. Then, we introduce EffOrt, a methodology that modifies the learning step so that the base prediction function is selected in order to minimize the length of the returned intervals. In particular, our theoretical analysis of the excess volume loss of the prediction sets produced by EffOrt reveals the links between the learning and calibration steps, and notably the impact of the choice of the function class of the base predictor. We also introduce Ad-EffOrt, an extension of the previous method, which produces intervals whose size adapts to the value of the covariate. Finally, we evaluate the empirical performance and the robustness of our methodologies.
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