Activized Learning with Uniform Classification Noise: Supplementary Material

This document provides specifications of the estimators used in Subroutine 1, along with a formal proof of Lemma 1. 1. Specification of Estimators Following (Hanneke, 2009; 2012), we specify the estimators P̂ used in the algorithm as follows. For convenience, we suppose we have access to two independent sequences W1 = {w1, w2, . . .} and W2 = {w′ 1, w 2, . . .} of independent Ddistributed random variables, with (W1,W2) independent of Z . Such sequences could easily be taken from the unlabeled data sequence in a preprocessing step, in which case we interpret the {Xi}i=1 sequence referenced in the algorithms as those points remaining in the pool after extracting the sequences W1 and W2. Fix any H ⊆ C and m ∈ N. For any k ∈ N, define S(H) = {S ∈ X k−1 : H shatters S}. For any (x, y) ∈ X × {−1,+1}, define Γ̂ m (x, y,W2,H) = 1⋂ h∈H{h(x)} (y) and ∆̂ m (x,W2,H) = 1S1(H)(x). For any k ∈ {2, . . . , d+ 1}, ∀i ∈ N, let S (k) i = {w 1+(i−1)(k−1), . . . , w i(k−1)}; then let M (k) m (H) = max 