$[6$ pts$]$ Suppose $S=\{(x_i,y_i){\}}_i=1^{n}sub{R}^d\{-1,+1\}$ is a linearly separable training dataset. We saw

in that there exists a $w^{**}in{R}^d$ such that $y_i($:$w^{**},x_i$:$)>1$ for all iin$[n].$ Recall that the Perceptron

algorithm outputs a hyperplane that separates the positive and negative examples $($i$.$e$.,y_i($:$w,x_i$:$)>0,$

for all iin$[n].$

$($a$)$ Devise a new algorithm called MARGIN$-$PERCEPTRON that outputs a widehat$(w)$ that separates the positive

and negative examples by a margin, that is$,y_i($:$(widehat(w)),x_i$:$)1$ for all iin$[n].$

$($b$)$ Suppose, as in class, that $Rma{x}_i||x|{|}_2$ and $Bmin\{\begin{array}{c}||w|{|}_2\end{array}$ : for all $\{$:$y_i($:$w,x_i$:$)1\}.$ Show using

the technique we used in class to show that MARGIN$-$PERCEPTRON in at most $B^2(R^2+2)$ steps. $[6$ pts$]$ Suppose $S=\{(x_i,y_i){\}}_i=1^{n}sub{R}^d\{-1,+1\}$ is a linearly separable training dataset. We saw

in that there exists a $w^{**}in{R}^d$ such that $y_i($:$w^{**},x_i$:$)>1$ for all iin$[n].$ Recall that the Perceptron

algorithm outputs a hyperplane that separates the positive and negative examples $($i$.$e$.,y_i($:$w,x_i$:$)>0,$

for all iin$[n].$

$($a$)$ Devise a new algorithm called MARGIN$-$PERCEPTRON that outputs a widehat$(w)$ that separates the positive

and negative examples by a margin, that is$,y_i($:$(widehat(w)),x_i$:$)1$ for all iin$[n].$

$($b$)$ Suppose, as in class, that $Rma{x}_i||x|{|}_2$ and $Bmin\{\begin{array}{c}||w|{|}_2\end{array}$ : for all $\{$:$y_i($:$w,x_i$:$)1\}.$ Show using

the technique we used in class to show that MARGIN$-$PERCEPTRON in at most $B^2(R^2+2)$ steps.