E-norm and its application to learning theory

Abstract

We investigate connections between an important parameter in the theory of Banach spaces called the l-norm, and two properties of classes of functions which are essential in Learning Theory – the uniform law of large numbers and the Vapnik–Chervonenkis (VC) dimension. We show that if the l-norm of a set of functions is bounded in some sense, then the set satisfies the uniform law of large numbers. Applying this result, we show that if X is a Banach space which has a nontrivial type, then the unit ball of its dual satisfies the uniform law of large numbers. Next, we estimate the l-norm of a set of {0,1}-functions in terms of its VC dimension. Finally, we present a `Gelfand number' like estimate of certain classes of functions. We use this estimate to formulate a learning rule, which may be used to approximate functions from the unit balls of several Banach spaces.