The Sampling Complexity of Condorcet Winner Identification in Dueling Bandits

We study $N$-armed stochastic dueling bandits under the Condorcet-winner assumption, where three widely adopted objectives are considered: best-arm identification (BAI), weak regret, and strong regret. We propose Tree-Guided Identify-Then-Exploit (TG-ITE), the first unified framework to tackle all these objectives to our knowledge. Without requiring stronger assumptions, we propose a shared tree-guided identification approach to find a high-confidence incumbent within $O(N)$ comparisons. We further propose varied exploitation strategies to utilize this warm-start stage to optimize the specific objectives at hand. This methodology enables our approach to (1) achieve $O(N)$ sample complexity in BAI without commonly adopted stronger assumptions; (2) build the first winner-stays-style algorithm to achieve $O(N)$ weak regret; (3) enjoy the same $O(N \log T)$ guarantee as specialized strong-regret approaches; (4) realize the joint optimization of BAI and weak regret with $O(N)$ guarantees for both, eliminating the sub-optimal gap of $O(\log N)$ in the existing approach. Our results provide evidence that the trade-off between BAI and regret minimization is relatively benign in dueling bandits.

Multi-dueling bandits, where a learner selects $m \geq 2$ arms per round and observes only the winner, arise naturally in many applications including ranking and recommendation systems, yet a fundamental question has remained open: can a single algorithm perform optimally in both stochastic and adversarial environments, without knowing which regime it faces? We answer this affirmatively, providing the first best-of-both-worlds algorithms for multi-dueling bandits under both Condorcet and Borda objectives. For the Condorcet setting, we propose $\texttt{MetaDueling}$, a black-box reduction that converts any dueling bandit algorithm into a multi-dueling bandit algorithm by transforming multi-way winner feedback into an unbiased pairwise signal. Instantiating our reduction with $\texttt{Versatile-DB}$ yields the first best-of-both-worlds algorithm for multi-dueling bandits: it achieves $O(\sqrt{KT})$ pseudo-regret against adversarial preferences and the instance-optimal $O\left(\sum_{i \neq a^\star} \frac{\log T}{Δ_i}\right)$ pseudo-regret under stochastic preferences, both simultaneously and without prior knowledge of the regime. For the Borda setting, we propose $\texttt{SA-MiDEX}$, a stochastic-and-adversarial algorithm that achieves $O\left(K^2 \log KT + K \log^2 T + \sum_{i: Δ_i^{\mathrm{B}} > 0} \frac{K\log KT}{(Δ_i^{\mathrm{B}})^2}\right)$ regret in stochastic environments and $O\left(K \sqrt{T \log KT} + K^{1/3} T^{2/3} (\log K)^{1/3}\right)$ regret against adversaries, again without prior knowledge of the regime. We complement our upper bounds with matching lower bounds for the Condorcet setting. For the Borda setting, our upper bounds are near-optimal with respect to the lower bounds (within a factor of $K$) and match the best-known results in the literature.

We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.