  We consider the problem of active learning in the context of spatial sampling
for level set estimation (LSE), where the goal is to localize all regions where
a function of interest lies above/below a given threshold as quickly as
possible. We present a finite-horizon search procedure to perform LSE in one
dimension while optimally balancing both the final estimation error and the
distance traveled for a fixed number of samples. A tuning parameter is used to
trade off between the estimation accuracy and distance traveled. We show that
the resulting optimization problem can be solved in closed form and that the
resulting policy generalizes existing approaches to this problem. We then show
how this approach can be used to perform level set estimation in higher
dimensions under the popular Gaussian process model. Empirical results on
synthetic data indicate that as the cost of travel increases, our method's
ability to treat distance nonmyopically allows it to significantly improve on
the state of the art. On real air quality data, our approach achieves roughly
one fifth the estimation error at less than half the cost of competing
algorithms.
