The Sampling Complexity of Condorcet Winner Identification in Dueling Bandits

We study best-arm identification in large-scale stochastic dueling bandits under the sole assumption that a Condorcet winner exists, i.e., an arm that wins each noisy pairwise comparison with probability at least $1/2$. We introduce a new identification procedure that exploits the full gap matrix $\Delta_{i,j}=q_{i,j}-\tfrac12$ (where $q_{i,j}$ is the probability that arm $i$ beats arm $j$), rather than only the gaps between the Condorcet winner and the other arms. We derive high-probability, instance-dependent sample-complexity guarantees that (up to logarithmic factors) improve the best known ones by leveraging informative comparisons beyond those involving the winner. We complement these results with matching lower bounds that establish the optimality of our procedures in all regimes. Overall, our results reveal the general form of the sampling complexity, characterized by a trade-off between the cost of locating informative entries and the verification cost required to achieve the desired confidence. In particular, this complexity drastically differs from what is suggested by pure asymptotic results or by procedures that are tailored to Strongly Stochastic Transitive models.