$[$Adaboost$]$ Consider the labeled data points in Figure $1,$ where $\text{'}+\text{'}$ and $\text{'}-\text{'}$ indicate class labels. We will use AdaBoost with Separating Hyperplane to train a classifier for the $\text{'}+\text{'}$ and $\text{'}-\text{'}$ labels. Each boosting iteration will select a horizontal or vertical Separating Hyperplane: a vertical or horizontal line that would split the space into half$-$spaces with a goal of minimizing the weighted training error. Breaking ties by choosing $\text{'}+\text{'}.$ All of the data points start with uniform weights. Please display your answers for $($a$),($b$),($d$)$ and $($e$)$ in a single figure. would choose. Label the decision boundary as $(3),$ also indicate the $\text{'}+\text{'}/\text{'}-\text{'}$ sides

of this boundary.

$($f$)(7$ points$)$ Assuming that a "New Data point" is given $($shown in the graph below$),$

using your classifier built from decision boundaries $(1),(2)$ and $(3)$ to predict the

class label for the new data point. Provide your final classifier along with the class

label. Show your work.

rigure $1$: $1$ ne uriginal Data Uoservations.

$($a$)(4$ points$)$ In Figure $1,$ draw a decision boundary corresponding to the first decision stump that the algorithm would choose $($the decision boundary should be either a vertical or horizontal straight line$).$ Label the decision boundary as $(1),$ also indicate the $\text{'}+\text{'}/\text{'}-\text{'}$ sides of this boundary.

$($b$)(3$ points$)$ Circle the point$($s$)$ that have the highest weight after the first boosting iteration. Also, report the value of the highest weight and show your calculations.

$($c$)(6$ points$)$ After the labels have been re$-$weighted in the first boosting iteration, what is the weighted error of the decision boundary $(1)?$

$($d$)(4$ points$)$ Draw the decision boundary corresponding to the second decision stump that the algorithm would choose. Label the decision boundary as $(2),$ also indicate the $\text{'}+\text{'}/\text{'}-\text{'}$ sides of this boundary.

$($e$)(6$ points$)$ Next, compute the weighted error of the decision boundary $(2)$ and draw a decision boundary corresponding to the third decision stump that the algorithm