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Research on the multi-armed bandit problem has studied the trade-off ofexploration and exploitation in depth. However, there are numerous applicationswhere the cardinal absolute-valued feedback model (e.g. ratings from one tofive) is not suitable. This has motivated the formulation of the duellingbandits problem, where the learner picks a pair of actions and observes a noisybinary feedback, indicating a relative preference between the two. There exista multitude of different settings and interpretations of the problem for tworeasons. First, due to the absence of a total order of actions, there is nonatural definition of the best action. Existing work either explicitly assumesthe existence of a linear order, or uses a custom definition for the winner.Second, there are multiple reasonable notions of regret to measure thelearner'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 qualityof the better of the two actions selected. Weak regret is the more appropriateperformance measure when the pair's inferior action has no significantdetrimental effect on the pair's quality.  We study the duelling bandits problem in the adversarial setting. We providean algorithm which has theoretical guarantees in both the utility-basedsetting, which implies a total order, and the unrestricted setting. For thelatter, we work with the $\textit{Borda winner}$, finding the action maximisingthe probability of winning against an action sampled uniformly at random. Thethesis concludes with experimental results based on both real-world data andsynthetic data, showing the algorithm's performance and limitations.

Duelling Bandits with Weak Regret in Adversarial Environments