Conformal prediction is a generic methodology for finite-sample valid distribution-free prediction. This technique has garnered a lot of attention in the literature partly because it can be applied with any machine learning algorithm that provides point predictions to yield valid prediction regions. Of course, the efficiency (width/volume) of the resulting prediction region depends on the performance of the machine learning algorithm. In the context of point prediction, several techniques (such as cross-validation) exist to select one of many machine learning algorithms for better performance. In contrast, such selection techniques are seldom discussed in the context of set prediction (or prediction regions). In this paper, we consider the problem of obtaining the smallest conformal prediction region given a family of machine learning algorithms. We provide two general-purpose selection algorithms and consider coverage as well as width properties of the final prediction region. The first selection method yields the smallest width prediction region among the family of conformal prediction regions for all sample sizes but only has an approximate coverage guarantee. The second selection method has a finite sample coverage guarantee but only attains close to the smallest width. The approximate optimal width property of the second method is quantified via an oracle inequality. As an illustration, we consider the use of aggregation of non-parametric regression estimators in the split conformal method with the absolute residual conformal score.

翻译：共形预测是一种通用方法论，用于实现有限样本有效的无分布预测。该技术之所以受到文献广泛关注，部分原因在于它能与任何提供点预测的机器学习算法结合，从而生成有效的预测区域。当然，最终预测区域的效率（宽度/体积）取决于机器学习算法的性能。在点预测领域，已存在多种技术（如交叉验证）可从多个机器学习算法中选取性能更优者。然而，这类选取技术在集合预测（或预测区域）背景下鲜有讨论。本文针对给定家族机器学习算法，研究如何获取最小共形预测区域的问题。我们提出了两种通用型选取算法，并分析了最终预测区域的覆盖域与宽度特性。第一种选取方法能在所有样本量下生成共形预测区域族中宽度最小的预测区域，但仅能保证近似覆盖域。第二种选取方法具有有限样本覆盖域保证，但仅能达到接近最小宽度的水平。通过最优性不等式量化了第二种方法的近似最优宽度特性。作为示例，我们展示了在分裂共形预测方法中结合绝对残差共形分数时，对非参数回归估计量的聚合应用。