  We study the worst-case adaptive optimization problem with budget constraint
that is useful for modeling various practical applications in artificial
intelligence and machine learning. We investigate the near-optimality of greedy
algorithms for this problem with both modular and non-modular cost functions.
In both cases, we prove that two simple greedy algorithms are not near-optimal
but the best between them is near-optimal if the utility function satisfies
pointwise submodularity and pointwise cost-sensitive submodularity
respectively. This implies a combined algorithm that is near-optimal with
respect to the optimal algorithm that uses half of the budget. We discuss
applications of our theoretical results and also report experiments comparing
the greedy algorithms on the active learning problem.
