1. Suppose we modify the Perceptron algorithm as follows: In the update step, instead of performing w(t+1) =w(t)+yixi whenever wemake a mistake, we perform w(t+1) = w(t) + ηyixi for some η > 0. Prove that the modified Perceptron will perform the same number of iterations as the vanilla Perceptron and will converge to a vector that points to the same direction as the output of the vanilla Perceptron.
2. In this problem, we will get bounds on the VC-dimension of the class of (closed) balls in Rd, that is, Bd = {Bv,r : v ∈ Rd ,r > 0}, where Bv,r (x) = * 1 if x−v ≤r 0 otherwise .
