Consider applying the perceptron algorithm (through the origin) on a dataset {(xi , yi)} n i=1. the algorithm starts with one data point and then cycles through the data until it makes no further mistakes. the algorithm starts with w = 0. due to the ambiguity of which side of the decision boundary a point is on when it is on the decision boundary itself, when wt xi = 0, the algorithm will regard this as making a mistake on point i. (a) the dataset is x1 = [1, 1], x2 = [0, −1], and x3 = [−2, 1] with labels y1 = 1, y2 = −1, and y3 = 1. how many mistakes will the algorithm make when it starts cycling through the points, starting from x1? what about if it starts from x2 instead? (b) now assume that x1 is [1, 0] instead of [1, 1] while x2, x3 remain unchanged from (a). how many mistakes will the algorithm make when it starts from x1? what if x3 = [−10, 1] and x1, x2 remain unchanged from (a) (again starting from x1 and x1 is [1, 1])

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