We provide estimates on the fat-shattering dimension of aggregation rules of real-valued function classes. The latter consists of all ways of choosing $k$ functions, one from each of the $k$ classes, and computing a pointwise function of them, such as the median, mean, and maximum. The bound is stated in terms of the fat-shattering dimensions of the component classes. For linear and affine function classes, we provide a considerably sharper upper bound and a matching lower bound, achieving, in particular, an optimal dependence on $k$. Along the way, we improve several known results in addition to pointing out and correcting a number of erroneous claims in the literature.

翻译：本文对实值函数类聚合规则的脂肪破碎维度进行了估计。聚合规则包含从每个类别中选取 $k$ 个函数，并通过逐点计算（如中位数、均值和最大值）得到结果的所有方式。所得界限以各组件类的脂肪破碎维度表示。针对线性和仿射函数类，我们给出了显著更紧的上界及匹配的下界，特别地实现了对 $k$ 的最优依赖关系。在此过程中，我们改进了若干已知结果，同时指出并纠正了文献中的多处错误论断。