  We study online active learning for classifying streaming instances within
the framework of statistical learning theory. At each time, the learner either
queries the label of the current instance or predicts the label based on past
seen examples. The objective is to minimize the number of queries while
constraining the number of prediction errors over a horizon of length $T$. We
develop a disagreement-based online learning algorithm for a general hypothesis
space and under the Tsybakov noise. We show that the proposed algorithm has a
label complexity of $O(dT^{\frac{2-2\alpha}{2-\alpha}}\log^2 T)$ under a
constraint of bounded regret in terms of classification errors, where $d$ is
the VC dimension of the hypothesis space and $\alpha$ is the Tsybakov noise
parameter. We further establish a matching (up to a poly-logarithmic factor)
lower bound, demonstrating the order optimality of the proposed algorithm. We
address the tradeoff between label complexity and regret and show that the
algorithm can be modified to operate at a different point on the tradeoff
curve.
