# Duelling Bandits with Weak Regret in Adversarial Environments

## TL;DR

This paper addresses the duelling bandits problem in adversarial environments, focusing on weak regret and proposing an algorithm with theoretical guarantees for both utility-based and unrestricted settings, supported by experiments.

## Contribution

It introduces a novel algorithm for adversarial duelling bandits that minimizes weak regret, applicable to both total order and non-order scenarios, with proven theoretical bounds.

## Key findings

- Algorithm achieves sublinear weak regret in adversarial settings
- Effective in both utility-based and unrestricted models
- Experimental results validate theoretical guarantees

## Abstract

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.