Entropy numbers of convex hulls and an application to learning algorithms

Given a positive sequence a = (an) ∈ lp,q, for 0 < p < 2 and 0 < q ≤ ∞, and a finite set A = {x1, . . . , xm} ⊂ l2 with |xi| ≤ a for all i = 1, . . . ,m we prove ‖(en(acoA))‖p,q ≤ cp,q √ log(m + 1) ‖a‖p,q , where en(acoA) is the n th dyadic entropy number of the absolutely convex hull acoA of A and cp,q > 0 is a suitable constant only depending on p and q. Moreover we show that this is asymptotically optimal in m for the most interesting case q = ∞. As an application we give an upper bound for the so-called growth function which is of special interest in the theory of learning algorithms. AMS classification: 41A46, 68T05