Beyond pointwise submodularity: Non-monotone adaptive submodular maximization in linear time

Abstract

In this paper, we study the non-monotone adaptive submodular maximization problem subject to a knapsack constraint. The input of our problem is a set of items, where each item has a particular state drawn from a known prior distribution. However, the state of an item is initially unknown, one must select an item in order to reveal the state of that item. Moreover, each item has a fixed cost. There is a utility function which is defined over items and states. Our objective is to sequentially select a group of items to maximize the expected utility subject to a knapsack constraint. Although the cardinality-constrained, as well as the more general matroid-constrained, adaptive submodular maximization has been well studied in the literature, whether there exists a constant approximation solution for the knapsack-constrained adaptive submodular maximization problem remains an open problem. We fill this gap by proposing the first constant approximation solution. In particular, our main contribution is to develop a sampling-based randomized algorithm that achieves a 1 10 approximation for maximizing an adaptive submodular function subject to a knapsack constraint.

Non-monotone Adaptive Submodular Meta-Learning

“…Remark 2: We next show that the training objective in Problem (3.2) does not satisfy the property of diminishing returns as specified in (3.1), making the existing results on adaptive submodular maximization [3,13] not applicable to our setting. Recall that we must select the first group of at most l items before observing the incoming task.…”

“…We assume that each task can be represented as an adaptive submodular function. Thus, our study is related to non-monotone adaptive submodular maximization [13,14]. Our study is also closely related to batch model active learning [3], where the selection is performed in batches.…”

“…It starts with a empty set, and in each round, it selects an item that maximizes the marginal utility on top of the current partial realization. For the non-monotone case, [13] develop a randomized policy that achieves a 1/e approximation ratio. Their policy starts with an empty set, and in each round, it selects an item uniformly at random from a set of k items (which may contain some dummy items whose marginal utility is zero on top of any partial realization) that have the largest marginal utility on top of the current partial realization.…”

“…Proof: Recall that in each of the first l rounds of π r , it selects an item uniformly at random from a set of l items. According to Lemma 1 in [13], if f i is adaptive submodular, we have…”

“…Clearly, f i avg (π b+ k−1 ) = f i avg (π b ) for any i ∈ M . According to Theorem 1 in[13], we can lower bound the performance of π b+ as follows:f i avg (π b+ ) ≥ 1 e f i avg (π o ) (4.11)due to f i is adaptive submodular and π o is a feasible adaptive policy that selects at most k items. Assume e k is the last item added to the solution by π b+ , we have∆ i (e k | ψ b k−1 ) ≤ ∆ i (e k | ∅) for any ψ b k−1 due to f i is adaptive submodular and ∅ ⊆ ψ b k−1 .…”