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Rademacher averages and phase transitions in Glivenko-Cantelli classes

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1 Author(s)
S. Mendelson ; Comput. Sci. Lab., Australian Nat. Univ., Canberra, ACT, Australia

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 Lp norm. Moreover, we show that if fatε(F) = O(ε-p) then F displays a clear phase transition which occurs at p=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 Lp 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

Published in:

IEEE Transactions on Information Theory  (Volume:48 ,  Issue: 1 )