We develop a novel and generic algorithm for the adversarial multi-armed bandit problem (or more generally the combinatorial semi-bandit problem). When instantiated differently, our algorithm achieves various new data-dependent regret bounds improving previous work. Examples include: 1) a regret bound depending on the variance of only the best arm; 2) a regret bound depending on the first-order path-length of only the best arm; 3) a regret bound depending on the sum of the first-order path-lengths of all arms as well as an important negative term, which together lead to faster convergence rates for some normal form games with partial feedback; 4) a regret bound that simultaneously implies small regret when the best arm has small loss {it and} logarithmic regret when there exists an arm whose expected loss is always smaller than those of other arms by a fixed gap (e.g. the classic i.i.d. setting). In some cases, such as the last two results, our algorithm is completely parameter-free. The main idea of our algorithm is to apply the optimism and adaptivity techniques to the well-known Online Mirror Descent framework with a special log-barrier regularizer. The challenges are to come up with appropriate optimistic predictions and correction terms in this framework. Some of our results also crucially rely on using a sophisticated increasing learning rate schedule.
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