An Asymptotically Optimal Batched Algorithm for the Dueling Bandit Problem

We study the $K$-armed dueling bandit problem, a variation of the traditional multi-armed bandit problem in which feedback is obtained in the form of pairwise comparisons. Previous learning algorithms have focused on the $\textit{fully adaptive}$ setting, where the algorithm can make updates after every comparison. The "batched" dueling bandit problem is motivated by large-scale applications like web search ranking and recommendation systems, where performing sequential updates may be infeasible. In this work, we ask: $\textit{is there a solution using only a few adaptive rounds that matches the asymptotic regret bounds of the best sequential algorithms for $K$-armed dueling bandits?}$ We answer this in the affirmative $\textit{under the Condorcet condition}$, a standard setting of the $K$-armed dueling bandit problem. We obtain asymptotic regret of $O(K^2\log2(K)) + O(K\log(T))$ in $O(\log(T))$ rounds, where $T$ is the time horizon. Our regret bounds nearly match the best regret bounds known in the fully sequential setting under the Condorcet condition. Finally, in computational experiments over a variety of real-world datasets, we observe that our algorithm using $O(\log(T))$ rounds achieves almost the same performance as fully sequential algorithms (that use $T$ rounds).