  We propose a new concept named adaptive submodularity ratio to study the
greedy policy for sequential decision making. While the greedy policy is known
to perform well for a wide variety of adaptive stochastic optimization problems
in practice, its theoretical properties have been analyzed only for a limited
class of problems. We narrow the gap between theory and practice by using
adaptive submodularity ratio, which enables us to prove approximation
guarantees of the greedy policy for a substantially wider class of problems.
Examples of newly analyzed problems include important applications such as
adaptive influence maximization and adaptive feature selection. Our adaptive
submodularity ratio also provides bounds of adaptivity gaps. Experiments
confirm that the greedy policy performs well with the applications being
considered compared to standard heuristics.
