A Structured Perspective of Volumes on Active Learning
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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).
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