Epsilon Nets

Motivation and Background

Epsilon nets were introduced by Haussler and Welz (1987), and their usefulness for computational learning theory has been discovered by Blumer et al. (1989).

Let X ≠ ∅ be any learning domain and let $\mathcal{C}\subseteq \wp (X)$ be any nonempty concept class. For the sake of simplicity, we also use $\mathcal{C}$ here as hypothesis space. In order to guarantee that all probabilities considered below do exist, we restrict ourselves to well-behaved concept classes (see PAC Learning).

Furthermore, let D be any arbitrarily fixed probability distribution over the learning domain X and let $c \in \mathcal{C}$ be any fixed concept.

A hypothesis $h \in \mathcal{C}$ is said to be bad for c iff

$$\displaystyle\begin{array}{rcl} d(c,h) =\sum \limits _{x\in c\bigtriangleup h}D(x)>\varepsilon . \end{array}$$

Furthermore, we use

$$\displaystyle\begin{array}{rcl} \Delta (c) = _{\mathit{df }}\{h\,\bigtriangleup \,c\mid h \in \mathcal{C}\} \end{array}$$

to denote the...