Sample-driven optimal stopping: From the secretary problem to the i.i.d. prophet inequality
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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∈ [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,… 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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