Problem $4(15$ points$)$ Consider the AdaBoost algorithm we discussed in the class ${?}^1.$ AdaBoost is an example of ensemble classifiers where the weights in next round are decided based on the training error of the weak classifier learned on the current weighted training set. We wish to run the AdaBoost on the dataset provided in Table $1.$

$\backslash$table$[[$Instance$,$Color,Size,Shape,Edible?$],[$D$1,$Yellow,Small,Round,Yes$],[$D$2,$Yellow,Small,Round,No$],[$D$3,$Green,Small,Irregular,Yes$],[$D$4,$Green,Large,Irregular,No$],[$D$5,$Yellow,Large,Round,Yes$],[$D$6,$Yellow,Small,Round,Yes$],[$D$7,$Yellow,Small,Round,Yes$],[$D$8,$Yellow,Small,Round,Yes$],[$D$9,$Green,Small,Round,No$],[$D$10,$Yellow,Large,Round,No$],[$D$11,$Yellow,Large,Round,Yes$],[$D$12,$Yellow,Large,Round,No$],[$D$13,$Yellow,Large,Round,No$],[$D$14,$Yellow,Large,Round,No$],[$D$15,$Yellow,Small,Irregular,Yes$],[$D$16,$Yellow,Large,Irregular,Yes$]]$

Table $1$: Mushroom data with $16$ instances, three categorical features, and binary labels.

a$)$ Assume we choose the following decision stump $f_1($a shallow tree with a single decision node$),$ as the first predictor $($i$.$e$.,$ when training instances are weighted uniformly$)$:What would be the weight of f$\_(1)$ in final ensemble classifier $($i$.$e$.,$ alpha$\_(1)$ in $\{$:f$($x$)=$sum$\_($i$=1)^($K$)$alpha$\_($i$)$f$\_($i$)($x$))?$b$)$ After computing f$\_(1),$ we proceed to next round of AdaBoost. We begin by recomputing data weights depending on the error of f$\_(1)$ and whether a point was $($mis$)$classified by f$\_(1).$ What is the weight of each instance in second boosting iteration, i$.$e$.,$ after the points have been re$-$weighted? Please note that the weights across the training set are to be uniformly initialized.c$)$ In AdaBoost, would you stop the iteration if the error rate of the current weak classifier on the weighted training data is $0?$