Active Learning of Linear Separators

We focus on binary classification problems; that is, we consider the problem of predicting a binary label y based on its corresponding input vector x. As in the standard machine learning formulation, we assume that the data points (x, y) are drawn from an unknown underlying distribution DXY over X × Y ; X is called the instance space and Y is the label space. We assume that Y = {±1} and X = Rd; we also denote the marginal distribution over X by D. Let C be the class of linear separators through the origin, that is C = {sign(w · x) : w ∈ Rd, ||w|| = 1}. To keep the notation simple, we sometimes refer to a weight vector and the linear classifier with that weight vector interchangeably. Our goal is to output a hypothesis function w ∈ C of small error, where err(w) = errDXY (w) = P(x,y)∼DXY [sign(w · x) 6= y]. Recall that in (pool-based) active learning, a set of labeled examples (x1, y1) . . . (xm, ym) is drawn i.i.d. from DXY ; the learning algorithm is permitted direct access to the sequence of xi values (unlabeled data points), but has to make a label request to obtain the label yi of example xi. The hope is that we can output a classifier of small error by using many fewer label requests than in passive learning by actively directing the queries to informative examples (while keeping the number of unlabeled examples polynomial). For added generality, we also consider the selective sampling active learning model, where the algorithm visits the unlabeled data points xi in sequence, and, for each i, makes a decision on whether or not to request the label yi based only on the previouslyobserved xj values (j ≤ i) and corresponding requested labels, and never changes this decision once made. Our upper and lower bounds will apply to both selective sampling and pool-based active learning.