Growth function explained

The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of function. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods.The term 'growth function' was coined by Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties.It is a basic concept in machine learning.^[1]

Definitions

Set-family definition

Let

H be a set family (a set of sets) and C a set. Their intersection is defined as the following set-family:

$$H\backslash cap\; C\; :=\; \backslash$$

The intersection-size (also called the index) of

H with respect to C is |H\cap C| . If a set C _{m} has m elements then the index is at most 2 ^{m} . If the index is exactly 2^m then the set C is said to be shattered by H , because $$H\backslash cap\; C$$ contains all the subsets of C, i.e.:

$$|H\backslash cap\; C|=2^$$

,

The growth function measures the size of

H\cap C as a function of |C| . Formally:

$$\backslash operatorname(H,m)\; :=\; \backslash max\_\; |H\backslash cap\; C|$$

Hypothesis-class definition

Equivalently, let

H be a hypothesis-class (a set of binary functions) and C a set with m elements. The restriction of H to C is the set of binary functions on C that can be derived from H :

$$H\_\; :=\; \backslash$$The growth function measures the size of

H _{C} as a function of |C| :

$$\backslash operatorname(H,m)\; :=\; \backslash max\_\; |H\_C|$$

Examples

1. The domain is the real line

R . The set-family H contains all the half-lines (rays) from a given number to positive infinity, i.e., all sets of the form \{x > x _{0} \mid x\in R \} for some x _{0\in} R . For any set C of m real numbers, the intersection H\cap C contains m +1 sets: the empty set, the set containing the largest element of C , the set containing the two largest elements of C , and so on. Therefore: \operatorname{Growth}(H,m) = m +1 . The same is true whether H contains open half-lines, closed half-lines, or both.

2. The domain is the segment

[ 0 , 1 ] . The set-family H contains all the open sets. For any finite set C of m real numbers, the intersection H\cap C contains all possible subsets of C . There are 2 ^{m} such subsets, so \operatorname{Growth}(H,m) =2 ^{m} .

3. The domain is the Euclidean space

R ^{n} . The set-family H contains all the half-spaces of the form: x ⋅ \phi \geq 1 , where \phi is a fixed vector.Then \operatorname{Growth}(H,m) = \operatorname{Comp}(n,m) ,where Comp is the number of components in a partitioning of an n-dimensional space by m hyperplanes.

4. The domain is the real line

R . The set-family H contains all the real intervals, i.e., all sets of the form \{x \in [x _{0,x} _{1]} | x\in R \} for some x _{0,x} _{1\in} R . For any set C of m real numbers, the intersection H\cap C contains all runs of between 0 and m consecutive elements of C . The number of such runs is {m +1 \choose 2 } +1 , so \operatorname{Growth}(H,m) = {m +1 \choose 2 } +1 .

Polynomial or exponential

The main property that makes the growth function interesting is that it can be either polynomial or exponential - nothing in-between.

The following is a property of the intersection-size:

• If, for some set

C _{m} of size m , and for some number n\leq m , |H\cap C _{m|} \geq \operatorname{Comp}(n,m) -

• then, there exists a subset

C _{n\subseteq} C _{m} of size n such that |H\cap C _{n|} = 2 ^{n} .

This implies the following property of the Growth function.For every family

H there are two cases:

• The exponential case:

\operatorname{Growth}(H,m) = 2 ^{m} identically.

• The polynomial case:

\operatorname{Growth}(H,m) is majorized by \operatorname{Comp}(n,m) \leq m ^{n+1} , where n is the smallest integer for which \operatorname{Growth}(H,n) < 2 ^{n} .

Other properties

Trivial upper bound

For any finite

H :\operatorname{Growth}(H,m) \leq |H| since for every C , the number of elements in H\cap C is at most |H| . Therefore, the growth function is mainly interesting when H is infinite.

Exponential upper bound

For any nonempty

H :\operatorname{Growth}(H,m) \leq 2 ^{m} I.e, the growth function has an exponential upper-bound.

We say that a set-family

H shatters a set C if their intersection contains all possible subsets of C , i.e. H\cap C = 2 ^{C} .If H shatters C of size m , then \operatorname{Growth}(H,C) =2 ^{m} , which is the upper bound.

Cartesian intersection

Define the Cartesian intersection of two set-families as:

H _{1otimes} H _{2} : = \{h _{1\cap} h _{2} \mid h _{1\in} H _{1,} h _{2\in} H _{2\}} .Then:^[1] \operatorname{Growth}(H _{1otimes} H _{2,} m) \leq \operatorname{Growth}(H _{1,} m) ⋅ \operatorname{Growth}(H _{2,} m)

Union

For every two set-families:^[1]

\operatorname{Growth}(H _{1\cup} H _{2,} m) \leq \operatorname{Growth}(H _{1,} m) + \operatorname{Growth}(H _{2,} m)

VC dimension

The VC dimension of

H is defined according to these two cases:

• In the polynomial case,

\operatorname{VCDim}(H) = n -1 = the largest integer d for which \operatorname{Growth}(H,d) = 2 ^{d} .

• In the exponential case

\operatorname{VCDim}(H) = inf ty .

So

\operatorname{VCDim}(H)\geq d if-and-only-if \operatorname{Growth}(H,d) =2 ^{d} .

The growth function can be regarded as a refinement of the concept of VC dimension. The VC dimension only tells us whether

\operatorname{Growth}(H,d) is equal to or smaller than 2 ^{d} , while the growth function tells us exactly how \operatorname{Growth}(H,m) changes as a function of m .

Another connection between the growth function and the VC dimension is given by the Sauer–Shelah lemma:

If

\operatorname{VCDim}(H) = d , then:

for all

m : \operatorname{Growth}(H,m)\leq \sum ^{d} _{i =0} {m\choose i} In particular,

for all

m>d +1 : \operatorname{Growth}(H,m)\leq (em/d) ^{d} = O(m ^{d)}

so when the VC dimension is finite, the growth function grows polynomially with

m .This upper bound is tight, i.e., for all m>d there exists H with VC dimension d such that:^[1] \operatorname{Growth}(H,m) = \sum ^{d} _{i =0} {m\choose i}

Entropy

While the growth-function is related to the maximum intersection-size,the entropy is related to the average intersection size:

\operatorname{Entropy}(H, m) = E _{|C _{m|=m}} [ log _{2(|H\cap} C _{m|)]} The intersection-size has the following property. For every set-family H :|H\cap (C _{1} \cup C _{2)|} \leq |H\cap C _{1| ⋅} |H \cap C _{2|} Hence:\operatorname{Entropy}(H, m _{1+m} _{2) \leq \operatorname{Entropy}(H,} m _{1)} + \operatorname{Entropy}(H, m _{2)} Moreover, the sequence \operatorname{Entropy}(H, m)/m converges to a constant c\in [ 0 , 1 ] when m\to inf ty .

Moreover, the random-variable

log _{2{|H\cap} C _{m|/m}} is concentrated near c .

Applications in probability theory

Let

\Omega be a set on which a probability measure \Pr is defined. Let H be family of subsets of \Omega (= a family of events).

Suppose we choose a set

C _{m} that contains m elements of \Omega ,where each element is chosen at random according to the probability measure P , independently of the others (i.e., with replacements). For each event h\in H , we compare the following two quantities:

• Its relative frequency in

C _{m} , i.e., |h\cap C _{m|/m} ;

• Its probability

\Pr[h] .We are interested in the difference, D(h,C _{m)} : = | |h\cap C _{m|/m} - \Pr[h] | . This difference satisfies the following upper bound:

$$\backslash Pr\backslash left[\backslash forall\; h\backslash in\; H:\; D(h,C\_m)\; \backslash leq\; \backslash sqrt\{8(\backslash ln\backslash operatorname\{Growth\}(H,\; 2m)\; +\; \backslash ln(4/\backslash delta))\; \backslash over\; m\}\; \backslash right]~~~~>~~~~1\; -\; \backslash delta$$which is equivalent to:

$$\backslash Pr\backslash big[\backslash forall\; h\backslash in\; H:\; D(h,C\_m)\; \backslash leq\; \backslash varepsilon\backslash big]~~~~>~~~~1\; -\; 4\backslash cdot\; \backslash operatorname(H,\; 2m)\backslash cdot\; \backslash exp(-\backslash varepsilon^2\backslash cdot\; m/8)$$In words: the probability that for all events in

H , the relative-frequency is near the probability, is lower-bounded by an expression that depends on the growth-function of H .

A corollary of this is that, if the growth function is polynomial in

m (i.e., there exists some n such that \operatorname{Growth}(H,m)\leq m ^{n+1} ), then the above probability approaches 1 as m\to inf ty . I.e, the family H enjoys uniform convergence in probability.

Notes and References

1. , especially Section 3.2