  We discuss a multiple-play multi-armed bandit (MAB) problem in which several
arms are selected at each round. Recently, Thompson sampling (TS), a randomized
algorithm with a Bayesian spirit, has attracted much attention for its
empirically excellent performance, and it is revealed to have an optimal regret
bound in the standard single-play MAB problem. In this paper, we propose the
multiple-play Thompson sampling (MP-TS) algorithm, an extension of TS to the
multiple-play MAB problem, and discuss its regret analysis. We prove that MP-TS
for binary rewards has the optimal regret upper bound that matches the regret
lower bound provided by Anantharam et al. (1987). Therefore, MP-TS is the first
computationally efficient algorithm with optimal regret. A set of computer
simulations was also conducted, which compared MP-TS with state-of-the-art
algorithms. We also propose a modification of MP-TS, which is shown to have
better empirical performance.
