[论文解读] Prophet secretary through blind strategies

José Correa, Raimundo Saona+1 more|arXiv (Cornell University)|Jan 6, 2019

Auction Theory and Applications被引用 19

一句话总结

本文提出了盲分位数策略——一类多阈值算法，其基于实例的固定函数设定非递增阈值——用于 Prophet Secretary 问题。通过利用 Schur 凸性精确分析停止时间分布，作者实现了 0.669 的近似比，优于先前结果，并为该策略类别建立了 0.675 的理论上限。

ABSTRACT

In the classic prophet inequality, a problem in optimal stopping theory, samples from independent random variables (possibly differently distributed) arrive online. A gambler that knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet, that sees all the realizations in advance, gets. In the late seventies, Krengel and Sucheston, and Garling [16], established that this worst case ratio is a constant and that 1/2 is the best possible such constant. In the last decade the theory of prophet inequalities has resurged as an important problem due to its connections to posted price mechanisms, frequently used in online sales. A particularly interesting variant is the so-called Prophet Secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms are known to achieve a constant of 1 − 1/e and very recently this barrier was slightly improved by Azar et al. [3].In this paper we derive a way of analyzing multi-threshold strategies that basically sets a nonincreasing sequence of thresholds to be applied at different times. The gambler will thus stop the first time a sample surpasses the corresponding threshold. Specifically we consider a class of very robust strategies that we call blind quantile strategies. These constitute a clever generalization of single threshold strategies and consist in fixing a function which is used to define a sequence of thresholds once the instance is revealed. Our main result shows that these strategies can achieve a constant of 0.669 in the Prophet Secretary problem, improving upon the best known result of Azar et al. [3], and even that of Beyhaghi et al. [4] that works in the case the gambler can select the order of the samples. The crux of the analysis is a very precise analysis of the underlying stopping time distribution for the gambler's strategy that is inspired by the theory of Schur convex functions. We further prove that our family of blind strategies cannot lead to a constant better than 0.675.Finally we prove that no nonadaptive algorithm for the gambler can achieve a constant better than 0.732, which also improves upon a recent result of Azar et al. [3]. Here, a nonadaptive algorithm is an algorithm whose decision to stop can depend on the index of the random variable being sampled, on the value sampled, and on the time, but not on the history that has been observed.

研究动机与目标

• 通过稳健的非自适应策略，将 Prophet Secretary 问题的近似比超越已知的 1−1/e 门槛。
• 通过仅依赖于实例而非历史记录的盲分位数函数，建立分析多阈值策略的一般框架。
• 为 Prophet Secretary 设置下的盲策略和非自适应算法建立更紧的性能上界。
• 利用 Schur 凸性理论工具分析这些策略的停止时间分布，以获得精确的性能保证。

提出的方法

• 提出盲分位数策略，通过在实例实现后对固定函数进行应用，定义非递增的阈值序列。
• 利用 Schur 凸函数技术分析赌徒的停止时间分布，以界定期望收益相对于预言者（prophet）的比值。
• 通过随机优势论证，将策略的期望值与最优离线基准进行比较，推导出性能保证。
• 通过刻画该策略类别的最坏情况实例，建立可实现近似比的上界。
• 利用非自适应算法的约束，证明任何非自适应策略的性能上限为 0.732。

实验结果

研究问题

• RQ1 多阈值盲策略能否在 Prophet Secretary 问题中实现优于 1−1/e 的近似比？
• RQ2 盲分位数策略的最优可实现性能是多少？其近似比存在哪些理论限制？
• RQ3 这些策略的停止时间分布如何影响其相对于预言者的期望性能？
• RQ4 在 Prophet Secretary 问题中，任何非自适应算法能达到的最佳近似比是多少？
• RQ5 Schur 凸性理论能否被有效用于分析和界定在线停止规则的性能？

主要发现

• 所提出的盲分位数策略在 Prophet Secretary 问题中实现了 0.669 的近似比，优于先前已知的最佳结果。
• 分析结果为任何盲分位数策略的性能建立了 0.675 的理论上限，表明该方法在该类别中几乎最优。
• 论文证明，任何非自适应算法都无法实现优于 0.732 的比值，优于 Azar 等人 [3] 的近期结果。
• 该方法的核心贡献在于利用 Schur 凸性对停止时间分布进行精确分析，从而实现更紧的性能界。
• 结果表明，稳健的非自适应策略无需依赖历史观测即可实现强大的近似保证。