(a) Let $\mathcal{F}$ be a family of functions $f: \mathcal{X} \rightarrow\{0,1\}$. What does it mean for $x_{1: n} \in \mathcal{X}^{n}$ to be shattered by $\mathcal{F}$ ? Define the shattering coefficient $s(\mathcal{F}, n)$ and the $V C$ dimension $\operatorname{VC}(\mathcal{F})$ of $\mathcal{F}$

Let

$$\mathcal{A}=\left\{\prod_{j=1}^{d}\left(-\infty, a_{j}\right]: a_{1}, \ldots, a_{d} \in \mathbb{R}\right\}$$

and set $\mathcal{F}=\left\{\mathbf{1}_{A}: A \in \mathcal{A}\right\}$. Compute $\operatorname{VC}(\mathcal{F})$.

(b) State the Sauer-Shelah lemma.

(c) Let $\mathcal{F}_{1}, \ldots, \mathcal{F}_{r}$ be families of functions $f: \mathcal{X} \rightarrow\{0,1\}$ with finite VC dimension $v \geqslant 1$. Now suppose $x_{1: n}$ is shattered by $\cup_{k=1}^{r} \mathcal{F}_{k}$. Show that

$$2^{n} \leqslant r(n+1)^{v} .$$

Conclude that for $v \geqslant 3$,

$$\mathrm{VC}\left(\cup_{k=1}^{r} \mathcal{F}_{k}\right) \leqslant 4 v \log _{2}(2 v)+2 \log _{2}(r)$$

[You may use without proof the fact that if $x \leqslant \alpha+\beta \log _{2}(x+1)$ with $\alpha>0$ and $\beta \geqslant 3$, then $x \leqslant 4 \beta \log _{2}(2 \beta)+2 \alpha$ for $x \geqslant 1$.]

(d) Now let $\mathcal{B}$ be the collection of subsets of $\mathbb{R}^{p}$ of the form of a product $\prod_{j=1}^{p} A_{j}$ of intervals $A_{j}$, where exactly $d \in\{1, \ldots, p\}$ of the $A_{j}$ are of the form $\left(-\infty, a_{j}\right]$ for $a_{j} \in \mathbb{R}$ and the remaining $p-d$ intervals are $\mathbb{R}$. Set $\mathcal{G}=\left\{\mathbf{1}_{B}: B \in \mathcal{B}\right\}$. Show that when $d \geqslant 3$,

$$\mathrm{VC}(\mathcal{G}) \leqslant 2 d\left[2 \log _{2}(2 d)+\log _{2}(p)\right]$$