Cs 270 -homework 3

(1) Consider the randomized multiplicative weights algorithm on the alternatingly correct pair of experts A, B, with update (1 −). Without loss of generality, assume that A is correct on even days, and B on odd days. At iteration t, the loss vector l(t) = [l A (t), l B (t)] is given by l(t) = [1, 0] if t is odd [0, 1] if t is even therefore the wight updates are given by w(t + 1) = [w A (t)(1 −), w B (t)] if t is odd [w A (t), w B (t)(1 −)] if t is even by induction, stating from the uniform weights w(1) = [1, 1], we have w(2k + 1) = [(1 −) k , (1 −) k ] w(2k) = [(1 −) k , (1 −) k−1 ] Let L(t) be the (random) loss incurred by the algorithm at iteration t. Then the expected loss at iteration t is E[L(t)] = w A (t)l A (t) + w B (t)l B (t) w A (t) + w B (t) =    (1−) k 2(1−) k = 1 2 if t = 2k + 1 (1−) k−1 (1−) k +(1−) k−1 = 1 2− if t = 2k Finally, the expected loss over T iterations is T t=1 E[L(t)] = T /2 1 2 + T /2 1 2 − since 1/2 ≤ 1/(2 −), an upper bound on the expected loss is simply given by T t=1 E[L(t)] ≤ T 2 − For the case = 1 2 , the expectation becomes T t=1 E[L(t)] = T /2 1 2 + T /2 2 3 1