fit.

$[3$ pts$]$ Suppose we modify the Perceptron algorithm as follows: In the update step, instead of per$-$

forming $w^{(t+1)}=w^{(t)}+y_i{x}_i,$ whenever we make a mistake, we instead perform $w^{(t+1)}=w^{(t)}+y_i{x}_i,$

for some $>0$;$$ is sometimes referred to as the learning rate or the step size. Show that this modified

Perceptron will perform the same number of iterations as the original Perceptron we studied in class,

and that it will converge to a vector that points in the same direction as the output of the vanilla

Perceptron.

Hint: What can you say about the relationship between the signs of $($:$w,x$:$)$ and $($:$w,x$:$)?$