A finite-horizon variant of the quickest change detection problem is investigated, which is motivated by a change detection problem that arises in piecewise stationary bandits. The goal is to minimize the \emph{latency}, which is smallest threshold such that the probability that the detection delay exceeds the threshold is below a desired low level, while controlling the false alarm probability to a desired low level. When the pre- and post-change distributions are unknown, two tests are proposed as candidate solutions. These tests are shown to attain order optimality in terms of the horizon. Furthermore, the growth in their latencies with respect to the false alarm probability and late detection probability satisfies a property that is desirable in regret analysis for piecewise stationary bandits. Numerical results are provided to validate the theoretical performance results.