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，而非仅一个。我们在列表学习的背景下研究经典泛化原理。我们的主要目标是确定PAC框架中的经典原理在列表PAC学习领域中是否仍然适用。我们重点关注一致收敛（这是经验风险最小化的基础）和样本压缩（这是奥卡姆剃刀原理的有力体现）。在经典PAC学习中，一致收敛和样本压缩均满足一种“完备性”：只要一个类别是可学习的，它就可以通过遵循这些原理的学习规则来学习。我们探讨在列表学习设置中是否同样存在这种完备性。我们证明在列表PAC学习设置中，一致收敛仍然等价于可学习性。相比之下，我们的研究结果在样本压缩方面揭示了令人惊讶的结论：我们证明当标签空间为$Y=\{0,1,2\}$时，存在可进行2-列表学习的类别无法被压缩。这推翻了Littlestone和Warmuth（1986）提出的样本压缩猜想的列表版本。我们证明了一个更强的不可能性结果：存在可进行$2$-列表学习的类别，即使重构函数可以处理任意大小的列表，也无法被压缩。对于标签空间无界情况下的（1-列表）PAC可学习类别，我们也证明了类似的结果。这推广了arXiv:2308.06424的最新研究成果。