$(1$ point$)$ Cross validation plays an important role in hyperparameter tuning and model

evaluation in machine learning. Suppose that we are given some i$.$i$.$d observations

$\{(x_i,y_i){\}}_i=1^{100}$ drawn from some unknown probability distribution $$ and that we are

interested learning through the following empirical risk minimization

hat$(f)=argmi{n}_{\text{f}\text{i}\text{n}\text{H}}\frac{1}{100}{\displaystyle \underset{i=1}{\overset{100}{}}}l(y_i-f(x_i))+||f||,$

where $>0$ is a regularization parameter, $H$ is a hypothesis space and $l$:$R{R}_{+}$

denotes a loss function. Answer the following questions:

$(1).$ Suppose that $$ is the only tuning parameter we have and we perform a five$-$fold

cross validation to tune this parameter. The candidate set of $$ is $\{{}_1,{}_2,$cdots,${}_9\},$

and the error criterion is the mean squared error. Express this cross validation

process mathematically.

$(2).$ Suppose that the response variable is binary$-$valued and takes on only the values

$\{2,3\}.$ Specifically, we observed that $80$ of the $100$ observations are labeled as $3.$

In this case, which cross valiation method would you choose to preform cross

validation? Please explain your reasoning.