We construct and analyze active learning algorithms for the problem of binary classification with abstention, in which the learner has an additional option to withhold its decision on certain points in the input space. We consider this problem in the fixed-cost setting, where the learner incurs a cost $\lambda \in (0, 1/2)$ every time the abstain option is invoked. Our proposed algorithm can work with the three most commonly used active learning query models, namely, membership-query, pool-based, and stream-based models. We obtain a high probability upper-bound on the excess risk of our algorithm, and establish its minimax near-optimality by deriving matching lower-bound (modulo polylogarithmic factors). Since our algorithm relies on the knowledge of the smoothness parameters of the regression function, we also describe a new strategy to adapt to these unknown parameters in a data-driven manner under an additional quality assumption. We show that using this strategy our algorithm achieves the same performance in terms of excess risk as their counterparts with the knowledge of the smoothness parameters. We end the paper with a discussion about the extension of our results to the setting of bounded rate of abstention.
