Conformal prediction provides a principled framework for constructing predictive sets with finite-sample validity. While much of the focus has been on univariate response variables, existing multivariate methods either impose rigid geometric assumptions or rely on flexible but computationally expensive approaches that do not explicitly optimize prediction set volume. We propose an optimization-driven framework based on a novel loss function that directly learns minimum-volume covering sets while ensuring valid coverage. This formulation naturally induces a new nonconformity score for conformal prediction, which adapts to the residual distribution and covariates. Our approach optimizes over prediction sets defined by arbitrary norm balls, including single and multi-norm formulations. Additionally, by jointly optimizing both the predictive model and predictive uncertainty, we obtain prediction sets that are tight, informative, and computationally efficient, as demonstrated in our experiments on real-world datasets.

翻译：共形预测为构建具有有限样本有效性的预测集提供了一个原则性框架。尽管现有研究多聚焦于单变量响应变量，但现有的多变量方法要么施加严格的几何假设，要么依赖灵活但计算成本高昂的方法，这些方法并未显式优化预测集体积。我们提出了一种基于新颖损失函数的优化驱动框架，该损失函数直接学习最小体积覆盖集，同时确保有效覆盖。这种公式自然为共形预测引入了一种新的非一致性得分，该得分可适应残差分布和协变量。我们的方法在由任意范数球定义的预测集上进行优化，包括单范数和多范数形式。此外，通过联合优化预测模型和预测不确定性，我们获得了紧致、信息丰富且计算高效的预测集，这一点在真实世界数据集上的实验中得到了验证。