There has been a recent interest in understanding and characterizing the sample complexity of list learning tasks, where the learning algorithm is allowed to make a short list of $k$ predictions, and we simply require one of the predictions to be correct. This includes recent works characterizing the PAC sample complexity of standard list classification and online list classification. Adding to this theme, in this work, we provide a complete characterization of list PAC regression. We propose two combinatorial dimensions, namely the $k$-OIG dimension and the $k$-fat-shattering dimension, and show that they optimally characterize realizable and agnostic $k$-list regression respectively. These quantities generalize known dimensions for standard regression. Our work thus extends existing list learning characterizations from classification to regression.

翻译：近期，理解并刻画列表学习任务的样本复杂度引起了广泛关注。在此类任务中，学习算法被允许生成一个包含 $k$ 个预测的短列表，我们仅要求其中一个预测正确即可。这包括近期刻画标准列表分类和在线列表分类的PAC样本复杂度的研究工作。延续这一主题，本文对列表PAC回归进行了完整刻画。我们提出了两个组合维度，即 $k$-OIG 维度和 $k$-fat-shattering 维度，并证明它们分别最优地刻画了可实现和不可知情况下的 $k$-列表回归。这些量是标准回归中已知维度的推广。因此，我们的工作将现有的列表学习刻画从分类领域扩展到了回归领域。