  We derive an algorithm that achieves the optimal (within constants)
pseudo-regret in both adversarial and stochastic multi-armed bandits without
prior knowledge of the regime and time horizon. The algorithm is based on
online mirror descent (OMD) with Tsallis entropy regularization with power
$\alpha=1/2$ and reduced-variance loss estimators. More generally, we define an
adversarial regime with a self-bounding constraint, which includes stochastic
regime, stochastically constrained adversarial regime (Wei and Luo), and
stochastic regime with adversarial corruptions (Lykouris et al.) as special
cases, and show that the algorithm achieves logarithmic regret guarantee in
this regime and all of its special cases simultaneously with the adversarial
regret guarantee.} The algorithm also achieves adversarial and stochastic
optimality in the utility-based dueling bandit setting. We provide empirical
evaluation of the algorithm demonstrating that it significantly outperforms
UCB1 and EXP3 in stochastic environments. We also provide examples of
adversarial environments, where UCB1 and Thompson Sampling exhibit almost
linear regret, whereas our algorithm suffers only logarithmic regret. To the
best of our knowledge, this is the first example demonstrating vulnerability of
Thompson Sampling in adversarial environments. Last, but not least, we present
a general stochastic analysis and a general adversarial analysis of OMD
algorithms with Tsallis entropy regularization for $\alpha\in[0,1]$ and explain
the reason why $\alpha=1/2$ works best.
