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.
. Another attempt to relax adaptive submodularity is presented in @cite_2 . They introduced as follows: Analogous to our adaptive submodularity ratio, one can readily see that @math -weak adaptive submodularity is equivalent to the adaptive submodularity. In general, however, there is a difference between the two notions; the adaptive submodularity ratio can be bounded from below by @math , implying that it is more demanding to bound the value of @math than that of the adaptive submodularity ratio. We provide a proof in subsec:proof-yong . studied a problem called and gave a bound of @math , but some vital assumptions seem to have been missed. In sec:counter-yong , we provide a problem instance in which their bound does not hold. We also present instances of adaptive influence maximization and adaptive feature selection for which our framework provides strictly better approximation ratios than those obtained with the weak adaptive submodularity in subsec:comparison-yong-infmax,subsec:comparison-yong-feature .
@cite_2
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In this paper, we consider adaptive decision-making problems for stochastic state estimation with partial observations. First, we introduce the concept of weak adaptive submodularity, a generalization of adaptive submodularity, which has found great success in solving challenging adaptive state estimation problems. Then, for the problem of active diagnosis, i.e., discrete state estimation via active sensing, we show that an adaptive greedy policy has a near-optimal performance guarantee when the reward function possesses this property. We further show that the reward function for group-based active diagnosis, which arises in applications such as medical diagnosis and state estimation with persistent sensor faults, is also weakly adaptive submodular. Finally, in experiments of state estimation for an aircraft electrical system with persistent sensor faults, we observe that an adaptive greedy policy performs equally well as an exhaustive search.