Uniform Entropy Numbers

Abstract

In Section 2.5.1 the empirical process was shown to converge weakly for indexing sets F satisfying a uniform entropy condition. In particular, if

$$ s\mathop u\limits_Q p\log N\left( {\varepsilon \parallel F{\parallel _{Q,2}},F,\mathop L\nolimits_2 \left( Q \right)} \right) \leqslant K{\left( {\frac{1}{\varepsilon }} \right)^{2 - \delta }} $$

for some δ >0, then the entropy integral (2.5.1) converges and F is a Donsker class for any probability measure P such that P*F ² < ∞, provided measurability conditions are met. Many classes of functions satisfy this condition and often even the much stronger condition

$$ s\mathop u\limits_Q pN\left( {\varepsilon \parallel F{\parallel _{Q,2}},F,{L_2}\left( Q \right)} \right) \leqslant K{\left( {\frac{1}{\varepsilon }} \right)^v},0 < \varepsilon < 1 $$

for some number V. In this chapter this is shown for classes satisfying certain combinatorial conditions. For classes of sets, these were first studied by Vapnik and Červonenkis, whence the name VC-classes. In the second part of this chapter, VC-classes of functions are defined in terms of VC-classes of sets. The remainder of this chapter considers operations on classes that preserve entropy properties, such as taking convex hulls.