§ VC dimension

Consider a ground set $X$. Let the space of all possible binary classifications be the function space $C \equiv \{ f \mid f : X \rightarrow \pm 1 \}$. Now, a hypothesis class $H$ is a subset of $C$. For example, some model such as "return $+1$ if a point is inside a region, $-1$ otherwise" is a subset of the full class $C$.
The VC dimension of $H$ measures how good $H$ is generating different classification. We first need the notion of shattering to define this.
A subset $S \subseteq X$ of the ground set shatters a hypothesis class $H$ if the function $act_S$ has full range, where $act_S$ is defined as:

That is, the hypothesis class $H$ can classify all the subsets of $S$.
Now the VC dimension of the hypothesis class $H$ of a ground set $X$ is the size of largest possible $S \subseteq X$ such that $S$ is shattered by $H$.

§ Correct interpretation

• We need just one set $S$ of size $n$ to be shattered by $H$. We get to pick the set $S$.

§ Subtletly 1:

• We do not need all sets of size $n$ to be shattered by $H$.

We can have the case where:

• All sets of size 3 are shattered by H
• Only one set of size 4 is shattered by H. All other sets of size 4 are not.
• Only one size of size 5 is shattered by H. All other sets of size 5 are not.
• No set of size 6 is shattered by H.

In this case, the VC dimension of $H$ is 5, not 3 .

§ Subtletly 2:

We cannot have the case where:

• All sets of size 3 are shattered by H
• No set of size 4 is shattered by H
• Some set of size 5 is shattered by H

For contradiction, let $S$ be the set of size $5$ that is shattered by $H$. Let $T \subsetneq S$, $|T| = 4$. Now, $H$ shatters $T$ since $H$ shatters $S$. Hence, Some set of size 4 has been shattered. Contradiction, since we assumed that no set of size 4 is shattered by $H$.
So, to prove that sets of size $(\geq n)$ cannot be shattered, it suffices to prove that sets of size equal to $n$ cannot be shattered.

§ Growth of number of sets shattered in $|S|$ for $S \subseteq X$ for a fixed $H$.

If we fix a hypothesis class $H$ for $X$, and we want to understand how $H$ varies over subsets of $X$, the idea is this:
Let $S$ be a set that is the maximum sized set that is shattered by $X$. ie, $|S| = Vcdim(H)$ and $H$ shatters $S$.
Now, the idea is this:

• For subsets $T \subseteq S$, $|act_T(H)| = 2^{|T|}$ -- exponential.
• For subpersets $S \subsetneq Sup$, $|act_{Sup}(H) = Comb(|Sup|, |S)$ -- polynomial.

We can show that this exponential/polynomial behaviour happens in general for $S \subseteq X$.