Question

Warm-Up: Subgradients & More (15 pts) 1. Recall that a function f:R" + R is convex if for all 2, Y ER" and le (0,1), \f (2) + (1 - 1)f(y) = f(2x + (1 - 1)y). Using this definition, show that (a) f(3) = wfi (2) is a convex function for x ER" whenever fi: R → R is a convex function and w > 0 (b) f(x) = f1(x) + f2(2) is a convex function for x ER" whenever fi : RM + R and $2:R" + R are convex functions (c) f(0) = max{fi(2), f2(2)} is a convex function for IER" whenever fi: R" → R and $2: R" + R are convex functions 2. Compute a subgradient at the specified points for each of the following convex functions. (a) f(x) = max{22 – 2x, |Z|} at x = 0 and x = -2. (b) g(x) = max{(x - 1), (x - 2)2} at x = 1.5 and x = 0. Problem 1: Perceptron Learning (30 pts) Consider the data set (perceptron.data) attached to this homework. This data file consists of M data elements of the form (x"), cm), c),x"), y(m)) where x"),..., LT) ER define a data point in R4 and y(m) € (-1,1} is the corresponding class label. In class, we saw how to use the perceptron algorithm to minimize the following objective. M max{0,-y(m). (w7x(m) + b)} m=1 In this problem, you are to implement the perceptron algorithm using two different subgradient descent strategies. For each strategy below, report the number of iterations that it takes to find a perfect classifier for the data, the values of w and b for the first three iterations, and the final

Answer 1

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