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:在本文中, 我们研究适应性亚模块在最坏情况下的覆盖问题。 这个问题概括了许多先前研究过的问题, 即以池为基础的积极学习和软化子模块覆盖。 我们问题的投入是一组项目( 如医学测试), 每个项目都有随机状态( 例如医学测试的结果), 其实现最初未知。 一个人必须选择一个固定成本的项目, 以观察其实现情况。 有一种实用功能, 将一组项目及其状态映射成一个非负真实数字。 我们的目标是按顺序选择一组项目, 以达到“ 目标值”, 同时将实现的最大成本降到最低成本 (a. k. a. 最差的成本) 。 为了方便我们的研究, 我们假设该功能的功能功能功能是\ emph{ worst- case copulate 。 在许多机器学习应用中通常发现的一种属性。 根据这个假设, 我们开发了一个非常紧的 $( Q/\\ deta)+1 美元- a a a asseral legal- legal legal legal legal legal- legal- pal- legal legal legal legal legal leg) a we. we. we. we. leglegleglegal legleglegal legal legal legal leg leg legal legal legal legal lex legal legal 和 legal lex legal_ legal_ legal_ legal_ legal_ lex_ legal_ legal_ legal_ legal_ lex a a a a a to a a a le le legal legal_ legald legal_ legal_ lex lex a a a lex le le le le le le le le le le lex lex lex lex le le le le le le le