Online Second Price Auction with Semibandit Feedback Under the NonStationary Setting
Abstract
In this paper, we study the nonstationary online second price auction problem. We assume that the seller is selling the same type of items in $T$ rounds by the second price auction, and she can set the reserve price in each round. In each round, the bidders draw their private values from a joint distribution unknown to the seller. Then, the seller announced the reserve price in this round. Next, bidders with private values higher than the announced reserve price in that round will report their values to the seller as their bids. The bidder with the highest bid larger than the reserved price would win the item and she will pay to the seller the price equal to the secondhighest bid or the reserve price, whichever is larger. The seller wants to maximize her total revenue during the time horizon $T$ while learning the distribution of private values over time. The problem is more challenging than the standard online learning scenario since the private value distribution is nonstationary, meaning that the distribution of bidders' private values may change over time, and we need to use the \emph{nonstationary regret} to measure the performance of our algorithm. To our knowledge, this paper is the first to study the repeated auction in the nonstationary setting theoretically. Our algorithm achieves the nonstationary regret upper bound $\tilde{\mathcal{O}}(\min\{\sqrt{\mathcal S T}, \bar{\mathcal{V}}^{\frac{1}{3}}T^{\frac{2}{3}}\})$, where $\mathcal S$ is the number of switches in the distribution, and $\bar{\mathcal{V}}$ is the sum of total variation, and $\mathcal S$ and $\bar{\mathcal{V}}$ are not needed to be known by the algorithm. We also prove regret lower bounds $\Omega(\sqrt{\mathcal S T})$ in the switching case and $\Omega(\bar{\mathcal{V}}^{\frac{1}{3}}T^{\frac{2}{3}})$ in the dynamic case, showing that our algorithm has nearly optimal \emph{nonstationary regret}.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.05949
 Bibcode:
 2019arXiv191105949Z
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Computer Science and Game Theory;
 Statistics  Machine Learning
 EPrint:
 Accepted to AAAI20