  We develop a new approach to obtaining high probability regret bounds for
online learning with bandit feedback against an adaptive adversary. While
existing approaches all require carefully constructing optimistic and biased
loss estimators, our approach uses standard unbiased estimators and relies on a
simple increasing learning rate schedule, together with the help of
logarithmically homogeneous self-concordant barriers and a strengthened
Freedman's inequality.
  Besides its simplicity, our approach enjoys several advantages. First, the
obtained high-probability regret bounds are data-dependent and could be much
smaller than the worst-case bounds, which resolves an open problem asked by Neu
(2015). Second, resolving another open problem of Bartlett et al. (2008) and
Abernethy and Rakhlin (2009), our approach leads to the first general and
efficient algorithm with a high-probability regret bound for adversarial linear
bandits, while previous methods are either inefficient or only applicable to
specific action sets. Finally, our approach can also be applied to learning
adversarial Markov Decision Processes and provides the first algorithm with a
high-probability small-loss bound for this problem.
