Online Second Price Auction with Semi-bandit Feedback Under the Non-Stationary Setting

In this paper, we study the non-stationary 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 second-highest 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 non-stationary, meaning that the distribution of bidders' private values may change over time, and we need to use the \emph{non-stationary regret} to measure the performance of our algorithm. To our knowledge, this paper is the first to study the repeated auction in the non-stationary setting theoretically. Our algorithm achieves the non-stationary 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 $Ω(\sqrt{\mathcal S T})$ in the switching case and $Ω(\bar{\mathcal{V}}^{\frac{1}{3}}T^{\frac{2}{3}})$ in the dynamic case, showing that our algorithm has nearly optimal \emph{non-stationary regret}.