  We study the regret of Thompson sampling (TS) algorithms for exponential
family bandits, where the reward distribution is from a one-dimensional
exponential family, which covers many common reward distributions including
Bernoulli, Gaussian, Gamma, Exponential, etc. We propose a Thompson sampling
algorithm, termed ExpTS, which uses a novel sampling distribution to avoid the
under-estimation of the optimal arm. We provide a tight regret analysis for
ExpTS, which simultaneously yields both the finite-time regret bound as well as
the asymptotic regret bound. In particular, for a $K$-armed bandit with
exponential family rewards, ExpTS over a horizon $T$ is sub-UCB (a strong
criterion for the finite-time regret that is problem-dependent), minimax
optimal up to a factor $\sqrt{\log K}$, and asymptotically optimal, for
exponential family rewards. Moreover, we propose ExpTS$^+$, by adding a greedy
exploitation step in addition to the sampling distribution used in ExpTS, to
avoid the over-estimation of sub-optimal arms. ExpTS$^+$ is an anytime bandit
algorithm and achieves the minimax optimality and asymptotic optimality
simultaneously for exponential family reward distributions. Our proof
techniques are general and conceptually simple and can be easily applied to
analyze standard Thompson sampling with specific reward distributions.
