  We consider the problem of adversarial bandit convex optimization, that is,
online learning over a sequence of arbitrary convex loss functions with only
one function evaluation for each of them. While all previous works assume known
and homogeneous curvature on these loss functions, we study a heterogeneous
setting where each function has its own curvature that is only revealed after
the learner makes a decision. We develop an efficient algorithm that is able to
adapt to the curvature on the fly. Specifically, our algorithm not only
recovers or \emph{even improves} existing results for several homogeneous
settings, but also leads to surprising results for some heterogeneous settings
-- for example, while Hazan and Levy (2014) showed that
$\widetilde{O}(d^{3/2}\sqrt{T})$ regret is achievable for a sequence of $T$
smooth and strongly convex $d$-dimensional functions, our algorithm reveals
that the same is achievable even if $T^{3/4}$ of them are not strongly convex,
and sometimes even if a constant fraction of them are not strongly convex. Our
approach is inspired by the framework of Bartlett et al. (2007) who studied a
similar heterogeneous setting but with stronger gradient feedback. Extending
their framework to the bandit feedback setting requires novel ideas such as
lifting the feasible domain and using a logarithmically homogeneous
self-concordant barrier regularizer.
