  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
$\Omega(\sqrt{\mathcal S T})$ 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{non-stationary regret}.
