  Active Learning (AL) is a learning task that requires learners interactively
query the labels of the sampled unlabeled instances to minimize the training
outputs with human supervisions. In theoretical study, learners approximate the
version space which covers all possible classification hypothesis into a
bounded convex body and try to shrink the volume of it into a half-space by a
given cut size. However, only the hypersphere with finite VC dimensions has
obtained formal approximation guarantees that hold when the classes of
Euclidean space are separable with a margin. In this paper, we approximate the
version space to a structured {hypersphere} that covers most of the hypotheses,
and then divide the available AL sampling approaches into two kinds of
strategies: Outer Volume Sampling and Inner Volume Sampling. After providing
provable guarantees for the performance of AL in version space, we aggregate
the two kinds of volumes to eliminate their sampling biases via finding the
optimal inscribed hyperspheres in the enclosing space of outer volume. To touch
the version space from Euclidean space, we propose a theoretical bridge called
Volume-based Model that increases the `sampling target-independent'. In
non-linear feature space, spanned by kernel, we use sequential optimization to
globally optimize the original space to a sparse space by halving the size of
the kernel space. Then, the EM (Expectation Maximization) model which returns
the local center helps us to find a local representation. To describe this
process, we propose an easy-to-implement algorithm called Volume-based AL
(VAL).
