A classical result in learning theory shows the equivalence of PAC learnability of binary hypothesis classes and the finiteness of VC dimension. Extending this to the multiclass setting was an open problem, which was settled in a recent breakthrough result characterizing multiclass PAC learnability via the DS dimension introduced earlier by Daniely and Shalev-Shwartz. In this work we consider list PAC learning where the goal is to output a list of $k$ predictions. List learning algorithms have been developed in several settings before and indeed, list learning played an important role in the recent characterization of multiclass learnability. In this work we ask: when is it possible to $k$-list learn a hypothesis class? We completely characterize $k$-list learnability in terms of a generalization of DS dimension that we call the $k$-DS dimension. Generalizing the recent characterization of multiclass learnability, we show that a hypothesis class is $k$-list learnable if and only if the $k$-DS dimension is finite.

翻译：学习理论中的一个经典结果表明，二元假设类的PAC可学习性与VC维的有限性等价。将该结果扩展到多类设置曾是一个开放问题，近期一项突破性的研究通过Daniely和Shalev-Shwartz之前引入的DS维度，对多类PAC可学习性进行了表征，从而解决了该问题。本文研究列表PAC学习，其目标是输出一个包含$k$个预测的列表。列表学习算法已在多种场景中得到开发，并且列表学习在多类可学习性的最新表征中发挥了重要作用。本文提出的问题是：何时可以对假设类进行$k$列表学习？我们通过将DS维度推广为$k$维DS维度，完全刻画了$k$列表可学习性。将多类可学习性的最新表征进行推广，我们证明假设类是$k$列表可学习的当且仅当$k$维DS维度有限。