In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced a unifying framework to study the learnability of classes of partial concepts. One of the central questions studied in their work is whether the learnability of a partial concept class is always inherited from the learnability of some ``extension'' of it to a total concept class. They showed this is not the case for PAC learning but left the problem open for the stronger notion of online learnability. We resolve this problem by constructing a class of partial concepts that is online learnable, but no extension of it to a class of total concepts is online learnable (or even PAC learnable).

翻译：在近期的一篇文章中，Alon、Hanneke、Holzman和Moran（FOCS '21）提出了一个统一框架来研究部分概念类的可学习性。他们工作中研究的核心问题之一是：部分概念类的可学习性是否总是继承自其某个“扩展”为整体概念类的可学习性。研究表明，对于PAC学习而言，这一结论并不成立，但对于更强的在线学习概念，该问题仍悬而未决。我们通过构造一个在线可学习的部分概念类解决了这一问题，但该类的任何扩展为整体概念类后既不可在线学习（甚至也不可PAC学习）。