Worst-Case Adaptive Submodular Cover

In this paper, we study the adaptive submodular cover problem under the worst-case setting. This problem generalizes many previously studied problems, namely, the pool-based active learning and the stochastic submodular set cover. The input of our problem is a set of items (e.g., medical tests) and each item has a random state (e.g., the outcome of a medical test), whose realization is initially unknown. One must select an item at a fixed cost in order to observe its realization. There is a utility function which maps a subset of items and their states to a non-negative real number. We aim to sequentially select a group of items to achieve a “target value” while minimizing the maximum cost across realizations (a.k.a. worst-case cost). To facilitate our study, we assume that the utility function is worst-case submodular, a property that is commonly found in many machine learning applications. With this assumption, we develop a tight (log(Q/η) + 1)-approximation policy, where Q is the “target value” and η is the smallest difference between Q and any achievable utility value Q̂ < Q. We also study a worst-case maximum-coverage problem, a dual problem of the minimum-cost-cover problem, whose goal is to select a group of items to maximize its worst-case utility subject to a budget constraint. To solve this problem, we develop a (1-1/e)/2-approximation solution.