  Research on the multi-armed bandit problem has studied the trade-off of
exploration and exploitation in depth. However, there are numerous applications
where the cardinal absolute-valued feedback model (e.g. ratings from one to
five) is not suitable. This has motivated the formulation of the duelling
bandits problem, where the learner picks a pair of actions and observes a noisy
binary feedback, indicating a relative preference between the two. There exist
a multitude of different settings and interpretations of the problem for two
reasons. First, due to the absence of a total order of actions, there is no
natural definition of the best action. Existing work either explicitly assumes
the existence of a linear order, or uses a custom definition for the winner.
Second, there are multiple reasonable notions of regret to measure the
learner's performance. Most prior work has been focussing on the
$\textit{strong regret}$, which averages the quality of the two actions picked.
This work focusses on the $\textit{weak regret}$, which is based on the quality
of the better of the two actions selected. Weak regret is the more appropriate
performance measure when the pair's inferior action has no significant
detrimental effect on the pair's quality.
  We study the duelling bandits problem in the adversarial setting. We provide
an algorithm which has theoretical guarantees in both the utility-based
setting, which implies a total order, and the unrestricted setting. For the
latter, we work with the $\textit{Borda winner}$, finding the action maximising
the probability of winning against an action sampled uniformly at random. The
thesis concludes with experimental results based on both real-world data and
synthetic data, showing the algorithm's performance and limitations.
