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 an 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 \emph{worst-case submodular}, a property that is commonly found in many machine learning applications. With this assumption, we develop a tight $(\log (Q/\eta)+1)$-approximation policy, where $Q$ is the ``target value'' and $\eta$ is the smallest difference between $Q$ and any achievable utility value $\hat{Q}<Q$. 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.

翻译：本文研究了最坏情况下的自适应子模覆盖问题。该问题推广了许多先前研究的问题，即基于池的主动学习和随机子模集覆盖。我们的问题输入是一组物品（例如，医学检测），每个物品具有一个随机状态（例如，医学检测的结果），其实现初始未知。必须以固定成本选择一个物品才能观察其实现。存在一个效用函数，该函数将物品子集及其状态映射为非负实数。我们旨在顺序选择一组物品以实现“目标值”，同时最小化跨实现的最高成本（即最坏情况成本）。为便于研究，我们假设效用函数是“最坏情况子模”的，这是许多机器学习应用中常见的性质。在此假设下，我们开发了一个紧的$(\log (Q/\eta)+1)$-近似策略，其中$Q$是“目标值”，$\eta$是$Q$与任何可达效用值$\hat{Q}<Q$之间的最小差值。我们还研究了最坏情况最大覆盖问题，这是最小成本覆盖问题的对偶问题，其目标是在预算约束下选择一组物品以最大化其最坏情况效用。为解决此问题，我们开发了一个$(1-1/e)/2$-近似解。