A note on different covering numbers in learning theory

The covering number of a set F in the space of continuous functions on a compact set X plays an important role in learning theory. In this paper, we study the relation between this covering number and its discrete version, obtained by replacing X with a finite subset. We formally show that when F is made of smooth functions, the discrete covering number is close to its continuous counterpart. In particular, we illustrate this result in the case that F is a ball in a reproducing kernel Hilbert space. (C) 2003 Published by Elsevier Science (USA).