Two fundamental models in online decision making are that of competitive analysis and that of optimal stopping. In the former the input is produced by an adversary, while in the latter the algorithm has full distributional knowledge of the input. In recent years, there has been a lot of interest in bridging these two models by considering data-driven or sample-based versions of optimal stopping problems. In this paper, we study such a version of the classic single selection optimal stopping problem, as introduced by Kaplan et al. [2020]. In this problem a collection of arbitrary non-negative numbers is shuffled in uniform random order. A decision maker gets to observe a fraction $p\in [0,1)$ of the numbers and the remaining are revealed sequentially. Upon seeing a number, she must decide whether to take that number and stop the sequence, or to drop it and continue with the next number. Her goal is to maximize the expected value with which she stops. On one end of the spectrum, when $p=0$, the problem is essentially equivalent to the secretary problem and thus the optimal algorithm guarantees a reward within a factor $1/e$ of the expected maximum value. We develop an approach, based on the continuous limit of a factor revealing LP, that allows us to obtain the best possible rank-based (ordinal) algorithm for any value of $p$. Notably, we prove that as $p$ approaches 1, our guarantee approaches 0.745, matching the best possible guarantee for the i.i.d. prophet inequality. This implies that there is no loss by considering this more general combinatorial version without full distributional knowledge. Furthermore, we prove that this convergence is very fast. Along the way we show that the optimal rank-based algorithm takes the form of a sequence of thresholds $t_1,t_2,\ldots$ such that at time $t_i$ the algorithm starts accepting values which are among the top $i$ values seen so far.

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样本驱动的最优停止：从秘书问题到 iid 先知不等式

在线决策中的两个基本模型是竞争分析模型和最优停止模型。在前者中，输入是由对手产生的，而在后者中，算法具有输入的完整分布知识。近年来，人们对通过考虑数据驱动或基于样本的最优停止问题的版本来桥接这两个模型产生了很大的兴趣。在本文中，我们研究了 Kaplan 等人介绍的经典单选最优停止问题的一个版本。[2020]。在这个问题中，一组任意非负数以统一的随机顺序打乱。决策者可以观察数字中的一小部分 $p\in [0,1)$，其余部分依次显示。看到一个数字，她必须决定是否取那个数字并停止序列，或放弃它并继续下一个数字。她的目标是最大化她停止的期望值。在频谱的一端，当 $p=0$ 时，问题本质上等同于秘书问题，因此最优算法保证在预期最大值的 $1/e$ 因子内的奖励。我们开发了一种基于揭示 LP 的因子的连续限制的方法，该方法使我们能够为 $p$ 的任何值获得最佳可能的基于等级（序数）的算法。值得注意的是，我们证明，随着 $p$ 接近 1，我们的保证接近 0.745，与 iid 先知不等式的最佳保证相匹配。这意味着在没有完整分布式知识的情况下考虑这个更通用的组合版本不会有任何损失。此外，我们证明这种收敛速度非常快。