arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1910.08693cs.LGstat.ML
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基于离线数据的在线定价:相变与平方反比律

Online Pricing with Offline Data: Phase Transition and Inverse Square Law

The Hong Kong Polytechnic University(香港理工大学);Massachusetts Institute of Technology(麻省理工学院)

Jinzhi Bu, David Simchi-Levi, Yunzong Xu

摘要:

本文研究了在动态定价的背景下,预先存在的离线数据对在线学习的影响。研究了销售期为$T$的单产品动态定价问题。每个时期的需求量由产品的价格决定,根据一个参数未知的线性需求模型。我们假设在销售期开始之前,卖家已经有了一些预先存在的离线数据。离线数据集包含$n$样本,每个样本都是由历史价格和相关联的需求观测组成的输入-输出对。卖方希望利用预先存在的离线数据和连续的在线数据来最小化在线学习过程的遗憾。我们的特点的联合效应的大小,位置和分散的离线数据上的最佳遗憾的在线学习过程。具体地,离线数据的大小、位置和离散度分别通过历史样本的数量$n$、平均历史价格和最优价格之间的距离$δ$以及历史价格的标准偏差$σ$来测量。我们证明了最优遗憾是$\widetilde Θ\left(\sqrt{T}\wedge \frac{T}{(n\wedge T)δ^2+nσ^2}\right)$,并设计了一个基于“乐观面对不确定性”原则的学习算法,其遗憾是最优的对数因子。我们的研究结果揭示了令人惊讶的转型的最佳后悔率相对于离线数据的大小,我们称之为相变。此外,我们的研究结果表明,离线数据的位置和分散性也对最佳后悔有内在的影响,我们通过平方反比定律量化这种影响。

英文摘要:

This paper investigates the impact of pre-existing offline data on online learning, in the context of dynamic pricing. We study a single-product dynamic pricing problem over a selling horizon of $T$ periods. The demand in each period is determined by the price of the product according to a linear demand model with unknown parameters. We assume that before the start of the selling horizon, the seller already has some pre-existing offline data. The offline data set contains $n$ samples, each of which is an input-output pair consisting of a historical price and an associated demand observation. The seller wants to utilize both the pre-existing offline data and the sequential online data to minimize the regret of the online learning process. We characterize the joint effect of the size, location and dispersion of the offline data on the optimal regret of the online learning process. Specifically, the size, location and dispersion of the offline data are measured by the number of historical samples $n$, the distance between the average historical price and the optimal price $δ$, and the standard deviation of the historical prices $σ$, respectively. We show that the optimal regret is $\widetilde Θ\left(\sqrt{T}\wedge \frac{T}{(n\wedge T)δ^2+nσ^2}\right)$, and design a learning algorithm based on the "optimism in the face of uncertainty" principle, whose regret is optimal up to a logarithmic factor. Our results reveal surprising transformations of the optimal regret rate with respect to the size of the offline data, which we refer to as phase transitions. In addition, our results demonstrate that the location and dispersion of the offline data also have an intrinsic effect on the optimal regret, and we quantify this effect via the inverse-square law.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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