  We study the asymptotic performance of the Thompson sampling algorithm in the
batched multi-armed bandit setting where the time horizon $T$ is divided into
batches, and the agent is not able to observe the rewards of her actions until
the end of each batch. We show that in this batched setting, Thompson sampling
achieves the same asymptotic performance as in the case where instantaneous
feedback is available after each action, provided that the batch sizes increase
subexponentially. This result implies that Thompson sampling can maintain its
performance even if it receives delayed feedback in $\omega(\log T)$ batches.
We further propose an adaptive batching scheme that reduces the number of
batches to $\Theta(\log T)$ while maintaining the same performance. Although
the batched multi-armed bandit setting has been considered in several recent
works, previous results rely on tailored algorithms for the batched setting,
which optimize the batch structure and prioritize exploration in the beginning
of the experiment to eliminate suboptimal actions. We show that Thompson
sampling, on the other hand, is able to achieve a similar asymptotic
performance in the batched setting without any modifications.
