  We introduce the problem of regret minimization in Adversarial Dueling
Bandits. As in classic Dueling Bandits, the learner has to repeatedly choose a
pair of items and observe only a relative binary `win-loss' feedback for this
pair, but here this feedback is generated from an arbitrary preference matrix,
possibly chosen adversarially. Our main result is an algorithm whose $T$-round
regret compared to the \emph{Borda-winner} from a set of $K$ items is
$\tilde{O}(K^{1/3}T^{2/3})$, as well as a matching $\Omega(K^{1/3}T^{2/3})$
lower bound. We also prove a similar high probability regret bound. We further
consider a simpler \emph{fixed-gap} adversarial setup, which bridges between
two extreme preference feedback models for dueling bandits: stationary
preferences and an arbitrary sequence of preferences. For the fixed-gap
adversarial setup we give an $\smash{ \tilde{O}((K/\Delta^2)\log{T}) }$ regret
algorithm, where $\Delta$ is the gap in Borda scores between the best item and
all other items, and show a lower bound of $\Omega(K/\Delta^2)$ indicating that
our dependence on the main problem parameters $K$ and $\Delta$ is tight (up to
logarithmic factors).
