Entropy numbers of convex hulls and an application to learning algorithms

Abstract.

Given a positive sequence $a = (a_n) \in \ell_{p,q}$, for 0 < p < 2 and $0 < q \leq \infty$, and a finite set $A = \{x_1, \dots , x_m\} \subset \ell_2$ with $|x_i| \leq a$ for all $i = 1, \dots , m$ we prove¶¶$\|(e_{n}(\textrm{aco}A))\|_{p,q} \leq c_{p,q} \sqrt{\textrm{log}(m + 1)}\,\, \|a\|_{p,q},$¶¶where $e_{n}(\textrm{aco}A$ is the n ^th dyadic entropy number of the absolutely convex hull acoA of A and c _p,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 $q = \infty$.¶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.