Fat-Shattering Dimension of k-fold Aggregations

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.