Partial-Monotone Adaptive Submodular Maximization

Abstract

Many sequential decision making problems, including pool-based active learning and adaptive viral marketing, can be formulated as an adaptive submodular maximization problem. The input of this problem is a group of items and each item has a random state whose realization is unknown initially, one must pick an item before observing its realization. The objective is to adaptively select a group of items to maximize an adaptive submodular utility function, which is defined over items and states. Most of existing studies on adaptive submodular optimization focus on either monotone case or non-monotone case. Specifically, if the utility function is monotone and adaptive submodular, Golovin and Krause (2011) developed a greedy policy that achieves a (1 − 1/e) approximation ratio subject to a cardinality constraint. If the utility function is non-monotone and adaptive submodular, Tang (2021a) showed that a random greedy policy achieves a 1/e approximation ratio subject to a cardinality constraint. In this work, we aim to generalize the above mentioned results by studying the partial-monotone adaptive submodular maximization problem. To this end, we introduce the notation of adaptive monotonicity ratio m ∈ [0, 1] to measure the degree of monotonicity of a function. Our main result is to show that a random greedy policy achieves an approximation ratio of m(1 − 1/e) + (1 − m)(1/e) if the utility function is m-adaptive monotone and adaptive submodular. Notably this result recovers the aforementioned (1 − 1/e) and 1/e approximation ratios when m = 0 and m = 1, respectively. We further extend our results to consider a knapsack constraint. We show that a sampling-based policy achieves an approximation ratio of (m + 1)/10 if the utility function is m-adaptive monotone and adaptive submodular. One important implication of our results is that even for a non-monotone utility function, we still can achieve an approximation ratio close to (1 − 1/e) if this function is "close" to a monotone function. This leads to improved performance bounds for many machine learning applications whose utility functions are almost adaptive monotone.