  How should we gather information to make effective decisions? We address
Bayesian active learning and experimental design problems, where we
sequentially select tests to reduce uncertainty about a set of hypotheses.
Instead of minimizing uncertainty per se, we consider a set of overlapping
decision regions of these hypotheses. Our goal is to drive uncertainty into a
single decision region as quickly as possible.
  We identify necessary and sufficient conditions for correctly identifying a
decision region that contains all hypotheses consistent with observations. We
develop a novel Hyperedge Cutting (HEC) algorithm for this problem, and prove
that is competitive with the intractable optimal policy. Our efficient
implementation of the algorithm relies on computing subsets of the complete
homogeneous symmetric polynomials. Finally, we demonstrate its effectiveness on
two practical applications: approximate comparison-based learning and active
localization using a robot manipulator.
