  How can we make use of information parallelism in online decision making
problems while efficiently balancing the exploration-exploitation trade-off? In
this paper, we introduce a batch Thompson Sampling framework for two canonical
online decision making problems, namely, stochastic multi-arm bandit and linear
contextual bandit with finitely many arms. Over a time horizon $T$, our
\textit{batch} Thompson Sampling policy achieves the same (asymptotic) regret
bound of a fully sequential one while carrying out only $O(\log T)$ batch
queries. To achieve this exponential reduction, i.e., reducing the number of
interactions from $T$ to $O(\log T)$, our batch policy dynamically determines
the duration of each batch in order to balance the exploration-exploitation
trade-off. We also demonstrate experimentally that dynamic batch allocation
dramatically outperforms natural baselines such as static batch allocations.
