  A version of the dueling bandit problem is addressed in which a Condorcet
winner may not exist. Two algorithms are proposed that instead seek to minimize
regret with respect to the Copeland winner, which, unlike the Condorcet winner,
is guaranteed to exist. The first, Copeland Confidence Bound (CCB), is designed
for small numbers of arms, while the second, Scalable Copeland Bandits (SCB),
works better for large-scale problems. We provide theoretical results bounding
the regret accumulated by CCB and SCB, both substantially improving existing
results. Such existing results either offer bounds of the form $O(K \log T)$
but require restrictive assumptions, or offer bounds of the form $O(K^2 \log
T)$ without requiring such assumptions. Our results offer the best of both
worlds: $O(K \log T)$ bounds without restrictive assumptions.
