Rademacher averages and phase transitions in Glivenko-Cantelli classes

We introduce a new parameter which may replace the fat-shattering dimension. Using this parameter we are able to provide improved complexity estimates for the agnostic learning problem with respect to any norm. Moreover, we show that if fat ( ) = ( ) then displays a clear phase transition which occurs at = 2. The phase transition appears in the sample complexity estimates, covering numbers estimates, and in the growth rate of the Rademacher averages associated with the class. As a part of our discussion, we prove the best known estimates on the covering numbers of a class when considered as a subset of spaces. We also estimate the fat-shattering dimension of the convex hull of a given class. Both these estimates are given in terms of the fat-shattering dimension of the original class.