Stochastic optimization is one of the central problems in Machine Learning and Theoretical Computer Science. In the standard model, the algorithm is given a fixed distribution known in advance. In practice though, one may acquire at a cost extra information to make better decisions. In this paper, we study how to buy information for stochastic optimization and formulate this question as an online learning problem. Assuming the learner has an oracle for the original optimization problem, we design a $2$-competitive deterministic algorithm and a $e/(e-1)$-competitive randomized algorithm for buying information. We show that this ratio is tight as the problem is equivalent to a robust generalization of the ski-rental problem, which we call super-martingale stopping. We also consider an adaptive setting where the learner can choose to buy information after taking some actions for the underlying optimization problem. We focus on the classic optimization problem, Min-Sum Set Cover, where the goal is to quickly find an action that covers a given request drawn from a known distribution. We provide an $8$-competitive algorithm running in polynomial time that chooses actions and decides when to buy information about the underlying request.

翻译：随机优化是机器学习和理论计算机科学中的核心问题之一。在标准模型中，算法需预先获知一个固定分布。然而在实践中，人们可能通过支付成本获取额外信息以做出更优决策。本文研究如何为随机优化购买信息，并将该问题形式化为一个在线学习问题。假设学习器拥有原始优化问题的预言机，我们设计了两个用于购买信息的算法：一个2-竞争比的确定性算法，以及一个$e/(e-1)$-竞争比的随机化算法。我们证明该竞争比是紧的，因为该问题等价于滑雪租赁问题的一个鲁棒推广形式，我们称之为超鞅停止问题。此外，我们还考虑自适应设置：学习器可在对底层优化问题采取某些操作后选择购买信息。我们聚焦于经典优化问题——最小和集合覆盖，其目标是快速找到一个动作，以覆盖从已知分布中抽取的给定请求。我们提出一个运行于多项式时间内的8-竞争比算法，该算法可选择动作，并决定何时购买关于底层请求的信息。