  We present simple and efficient algorithms for the batched stochastic
multi-armed bandit and batched stochastic linear bandit problems. We prove
bounds for their expected regrets that improve over the best-known regret
bounds for any number of batches. In particular, our algorithms in both
settings achieve the optimal expected regrets by using only a logarithmic
number of batches. We also study the batched adversarial multi-armed bandit
problem for the first time and find the optimal regret, up to logarithmic
factors, of any algorithm with predetermined batch sizes.
