  We construct and analyze active learning algorithms for the problem of binary
classification with abstention. We consider three abstention settings:
\emph{fixed-cost} and two variants of \emph{bounded-rate} abstention, and for
each of them propose an active learning algorithm. All the proposed algorithms
can work in the most commonly used active learning models, i.e.,
\emph{membership-query}, \emph{pool-based}, and \emph{stream-based} sampling.
We obtain upper-bounds on the excess risk of our algorithms in a general
non-parametric framework and establish their minimax near-optimality by
deriving matching lower-bounds. Since our algorithms rely on the knowledge of
some smoothness parameters of the regression function, we then describe a new
strategy to adapt to these unknown parameters in a data-driven manner. Since
the worst case computational complexity of our proposed algorithms increases
exponentially with the dimension of the input space, we conclude the paper with
a computationally efficient variant of our algorithm whose computational
complexity has a polynomial dependence over a smaller but rich class of
learning problems.
