1. Prove that the function h given in Equation (10.5) equals the piece-wise constant function defined according to the same thresholds as h.

2. We have informally argued that the AdaBoost algorithm uses the weighting mechanism to “force” the weak learner to focus on the problematic examples in the next iteration. In this question we will find some rigorous justification for this argument. Show that the error of ht w.r.t. the distribution D(t+1) is exactly 1/2. That is, show that for every t ∈ [T ] _m i=1 D(t+1) I 1[yi _=ht (xi )] = 1/2.

3. In this exercise we discuss the VC-dimension of classes of the form L(B,T). We proved an upper bound of O(dT log(dT)), where d = VCdim(B). Here we wish to prove an almost matching lower bound. However, that will not be the case for all classes B.