List learning is a variant of supervised classification where the learner outputs multiple plausible labels for each instance rather than just one. We investigate classical principles related to generalization within the context of list learning. Our primary goal is to determine whether classical principles in the PAC setting retain their applicability in the domain of list PAC learning. We focus on uniform convergence (which is the basis of Empirical Risk Minimization) and on sample compression (which is a powerful manifestation of Occam's Razor). In classical PAC learning, both uniform convergence and sample compression satisfy a form of `completeness': whenever a class is learnable, it can also be learned by a learning rule that adheres to these principles. We ask whether the same completeness holds true in the list learning setting. We show that uniform convergence remains equivalent to learnability in the list PAC learning setting. In contrast, our findings reveal surprising results regarding sample compression: we prove that when the label space is $Y=\{0,1,2\}$, then there are 2-list-learnable classes that cannot be compressed. This refutes the list version of the sample compression conjecture by Littlestone and Warmuth (1986). We prove an even stronger impossibility result, showing that there are $2$-list-learnable classes that cannot be compressed even when the reconstructed function can work with lists of arbitrarily large size. We prove a similar result for (1-list) PAC learnable classes when the label space is unbounded. This generalizes a recent result by arXiv:2308.06424.

翻译：列表学习是监督分类的一种变体，其中学习器为每个实例输出多个可能的标签，而非单一标签。我们研究列表学习背景下与泛化相关的经典原理。主要目标是确定PAC设置中的经典原理在列表PAC学习领域是否仍适用。我们聚焦于一致收敛（经验风险最小化的基础）和样本压缩（奥卡姆剃刀的有力体现）。在经典PAC学习中，一致收敛和样本压缩均满足某种"完备性"：只要一个类是可学习的，它也能通过遵循这些原理的学习规则进行学习。我们探究列表学习设置中是否同样具有这种完备性。我们证明，在列表PAC学习设置中，一致收敛仍与可学习性等价。然而，关于样本压缩的研究结果却出人意料：我们证明当标签空间为$Y=\{0,1,2\}$时，存在无法压缩的2-列表可学习类。这反驳了Littlestone和Warmuth（1986）提出的样本压缩猜想的列表版本。我们进一步证明了更强的不可实现性结果：即使重构函数可以处理任意大规模列表，仍存在无法压缩的$2$-列表可学习类。对于标签空间无界的情况，我们证明了（1-列表）PAC可学习类的类似结果，这推广了arXiv:2308.06424的最新发现。