Fat-shattering dimension of k-fold maxima

Idan Attias, Aryeh Kontorovich

Abstract:

We provide improved estimates on the fat-shattering dimension of the k-fold maximum of
real-valued function classes. The latter consists of all ways of choosing k functions, one
from each of the k classes, and computing their pointwise 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 point out
and correct a number of erroneous claims in the literature.