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Fat-Shattering Dimension of $k$-fold Aggregations. (arXiv:2110.04763v2 [math.FA] UPDATED)

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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.

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