  Lipschitz bandits is a prominent version of multi-armed bandits that studies
large, structured action spaces such as the [0,1] interval, where similar
actions are guaranteed to have similar rewards. A central theme here is the
adaptive discretization of the action space, which gradually ``zooms in'' on
the more promising regions thereof. The goal is to take advantage of ``nicer''
problem instances, while retaining near-optimal worst-case performance. While
the stochastic version of the problem is well-understood, the general version
with adversarial rewards is not. We provide the first algorithm for adaptive
discretization in the adversarial version, and derive instance-dependent regret
bounds. In particular, we recover the worst-case optimal regret bound for the
adversarial version, and the instance-dependent regret bound for the stochastic
version. Further, an application of our algorithm to dynamic pricing (where a
seller repeatedly adjusts prices for a product) enjoys these regret bounds
without any smoothness assumptions.
