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Dueling bandits are widely used to model preferential feedback prevalent inmany applications such as recommendation systems and ranking. In this paper, westudy the Borda regret minimization problem for dueling bandits, which aims toidentify the item with the highest Borda score while minimizing the cumulativeregret. We propose a rich class of generalized linear dueling bandit models,which cover many existing models. We first prove a regret lower bound of order$\Omega(d^{2/3} T^{2/3})$ for the Borda regret minimization problem, where $d$is the dimension of contextual vectors and $T$ is the time horizon. To attainthis lower bound, we propose an explore-then-commit type algorithm for thestochastic setting, which has a nearly matching regret upper bound$\tilde{O}(d^{2/3} T^{2/3})$. We also propose an EXP3-type algorithm for theadversarial linear setting, where the underlying model parameter can change ateach round. Our algorithm achieves an $\tilde{O}(d^{2/3} T^{2/3})$ regret,which is also optimal. Empirical evaluations on both synthetic data and asimulated real-world environment are conducted to corroborate our theoreticalanalysis.

Borda Regret Minimization for Generalized Linear Dueling Bandits