Weak Convergence and Empirical Processes pp 134-153 | Cite as

# Uniform Entropy Numbers

Chapter

## Abstract

In Section 2.5.1 the empirical process was shown to converge weakly for indexing sets for some δ >0, then the entropy integral (2.5.1) converges and for some number

*F*satisfying a uniform entropy condition. In particular, if$$
s\mathop u\limits_Q p\log N\left( {\varepsilon \parallel F{\parallel _{Q,2}},F,\mathop L\nolimits_2 \left( Q \right)} \right) \leqslant K{\left( {\frac{1}{\varepsilon }} \right)^{2 - \delta }}
$$

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

*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.## Keywords

Probability Measure Convex Hull Empirical Process Symmetric Convex Entropy Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1996