  Solving stochastic optimization problems under partial observability, where
one needs to adaptively make decisions with uncertain outcomes, is a
fundamental but notoriously difficult challenge. In this paper, we introduce
the concept of adaptive submodularity, generalizing submodular set functions to
adaptive policies. We prove that if a problem satisfies this property, a simple
adaptive greedy algorithm is guaranteed to be competitive with the optimal
policy. In addition to providing performance guarantees for both stochastic
maximization and coverage, adaptive submodularity can be exploited to
drastically speed up the greedy algorithm by using lazy evaluations. We
illustrate the usefulness of the concept by giving several examples of adaptive
submodular objectives arising in diverse applications including sensor
placement, viral marketing and active learning. Proving adaptive submodularity
for these problems allows us to recover existing results in these applications
as special cases, improve approximation guarantees and handle natural
generalizations.
