Shattering Coefficient

Synonyms

Growth function

Definition

The shattering coefficient $S_{\mathcal{F}}(n)$ is a function that measures the size of a function class $\mathcal{F}$ when its functions $f : \mathcal{X} \rightarrow \mathbb{R}$ are restricted to sets of points $\mathbf{x} = (x_{1},\ldots,x_{n}) \in \mathcal{X}^{n}$ of size n. Specifically, for each $n \in \mathbb{N}$ the shattering coefficient is the maximum size of the set of vectors $\mathcal{F}_{\mathbf{x}} =\{ (f(x_{1}),.\ldots,f(x_{n})) : f \in \mathcal{F}\}\subset \mathbb{R}^{n}$ that can be realized for some choice of $\mathbf{x} \in \mathcal{X}^{n}$. That is,

$$\displaystyle\begin{array}{rcl} S_{\mathcal{F}}(n) =\sup \limits _{\mathbf{x}\in \mathcal{X}^{n}}\vert \mathcal{F}_{x}\vert . \end{array}$$

The shattering coefficient of a hypothesis class $\mathcal{H}$ is used in generalization bounds as an analogue to the class’s size in the finite case.