  We study bandit convex optimization methods that adapt to the norm of the
comparator, a topic that has only been studied before for its full-information
counterpart. Specifically, we develop convex bandit algorithms with regret
bounds that are small whenever the norm of the comparator is small. We first
use techniques from the full-information setting to develop comparator-adaptive
algorithms for linear bandits. Then, we extend the ideas to convex bandits with
Lipschitz or smooth loss functions, using a new single-point gradient estimator
and carefully designed surrogate losses.
