PLEASE DO NOT USE CHATGPT$/$AI$!($I$\text{'}$ll know if you do$)$

Let $x=\{-1,1{\}}^d$ be the feature space, $Y=\{-1,1\}$ be the outcome

space, and $f^{*}$:$xY$ be the unknown target function. There are samples of training

data $S=\{(x_1,y_1),$dots,$(x_n,y_n)\}$ with $y_i=f^{*}(x_i)$ and distinct features. For the testing data,

the feature $x_{\text{t}\text{e}\text{s}\text{t}}{}_{u}x$ is sampled uniformly at random and $y_{\text{t}\text{e}\text{s}\text{t}}=f^{*}(x_{\text{t}\text{e}\text{s}\text{t}}).$

$($a$)$ Consider the target $f^{*}(x)=1$ for all $x$ and training set $S=((x_1,y_1),$dots,$(x_n,y_n))$ that

has distinct feature. If $g(x)=1$ when xinS and $-1$ otherwise, compute the training error

$\frac{1}{n}{\displaystyle \underset{i=1}{\overset{?}{}}},$dots,$n1[g(x_i)f^{*}(x_i)]$ and testing error $E_{x_{\text{t}\text{e}\text{s}\text{t}{}_u}x}[1[f^{*}(x_{\text{t}\text{e}\text{s}\text{t}})g(x_{\text{t}\text{e}\text{s}\text{t}})]].$

$($b$)$ We say $f$ is compatible with a training data $S$ if $f(x_i)=y_i$ for all $i.$ How many possible

$f$:$xY$ is compatible with $S.$

$($c$)$ Given a fixed $g$ and training data $S,$ we sample all compatible classifier uniformly at

random as the target function $f^{*}.$ If the training error of $g$ is $0\mathrm{.}1,$ what is the expected

testing error $E_{f^{*}}[E_{x_{\text{t}\text{e}\text{s}\text{t}}{}_{u}x}[1[f^{*}(x_{\text{t}\text{e}\text{s}\text{t}})g(x_{\text{t}\text{e}\text{s}\text{t}})]]]?$ Given $d,n,$ and training data $S,$ what

is the smallest possible testing error for any deterministic $g?$