2 [A better output Perceptron algorithm guarantee] In class, we saw that when the training sample 8' is linearly separable with a maximum margin 7 > 0, then the Perceptron algorithm run cyclically over S is guaranteed to converge after T g (Rh/)2 updates, where R is the radius of the sphere containing the sample points. This does not guarantee however that the hyperplane solution returned by Perceptron, i.e. wT achieves a margin close to 7. (i) Show an example training dataset S in R2 that has margin 7, and an order of updates made by the Perceptron algorithm where the hyperplane solution returned has arbitrarily bad margin on S. (ii) Consider the following modification to the perceptron algorithm: Modified Perceptron Algorithm Input: training dataset S = (Xi, yi)i=1,...,n Output: learned vector w - Initialize wo := 0, t := 0 - while there exists an example (x, y) E S, such that 2y(wt . x)