In this paper, we consider the problem of replicable realizable PAC learning. We construct a particularly hard learning problem and show a sample complexity lower bound with a close to $(\log|H|)^{3/2}$ dependence on the size of the hypothesis class $H$. Our proof uses several novel techniques and works by defining a particular Cayley graph associated with $H$ and analyzing a suitable random walk on this graph by examining the spectral properties of its adjacency matrix. Furthermore, we show an almost matching upper bound for the lower bound instance, meaning if a stronger lower bound exists, one would have to consider a different instance of the problem.

翻译：本文研究可复现可实现PAC学习问题。我们构造了一个特别困难的学习问题，并证明了样本复杂度下界与假设类$H$规模之间存在接近$(\log|H|)^{3/2}$的依赖关系。我们的证明采用多项创新技术：通过定义与$H$相关的特定凯莱图，分析该图上的合适随机游走过程，并考察其邻接矩阵的谱特性。此外，我们针对该下界实例给出了近乎匹配的上界，这意味着若存在更强的下界，则必须考虑该问题的不同实例。