Conformal prediction is a statistical tool for producing prediction regions for machine learning models that are valid with high probability. A key component of conformal prediction algorithms is a \emph{non-conformity score function} that quantifies how different a model's prediction is from the unknown ground truth value. Essentially, these functions determine the shape and the size of the conformal prediction regions. While prior work has gone into creating score functions that produce multi-model prediction regions, such regions are generally too complex for use in downstream planning and control problems. We propose a method that optimizes parameterized \emph{shape template functions} over calibration data, which results in non-conformity score functions that produce prediction regions with minimum volume. Our approach results in prediction regions that are \emph{multi-modal}, so they can properly capture residuals of distributions that have multiple modes, and \emph{practical}, so each region is convex and can be easily incorporated into downstream tasks, such as a motion planner using conformal prediction regions. Our method applies to general supervised learning tasks, while we illustrate its use in time-series prediction. We provide a toolbox and present illustrative case studies of F16 fighter jets and autonomous vehicles, showing an up to $68\%$ reduction in prediction region area compared to a circular baseline region.

翻译：共形预测是一种统计工具，可为机器学习模型生成具有高概率有效性的预测区域。共形预测算法的关键组成部分是\emph{非共形评分函数}，该函数量化模型预测与未知真实值之间的差异程度。本质上，这些函数决定了共形预测区域的形状和大小。尽管已有研究致力于创建能够产生多模态预测区域的评分函数，但此类区域通常过于复杂，难以用于下游规划与控制问题。我们提出一种方法，通过在标定数据上优化参数化的\emph{形状模板函数}，从而得到能产生最小体积预测区域的非共形评分函数。我们的方法生成的预测区域具有\emph{多模态}特性，能够准确捕捉多峰分布的残差；同时具备\emph{实用性}，每个区域均为凸集，可轻松整合至下游任务（例如使用共形预测区域的运动规划器）。本方法适用于一般监督学习任务，本文以时间序列预测为例进行说明。我们提供了工具箱，并通过F16战斗机和自动驾驶车辆的案例研究进行展示：与圆形基准区域相比，预测区域面积最高可减少$68\%$。