Best-of-Both-Worlds Multi-Dueling Bandits: Unified Algorithms for Stochastic and Adversarial Preferences under Condorcet and Borda Objectives

TL;DR

This paper introduces the first unified algorithms for multi-dueling bandits that perform optimally in both stochastic and adversarial environments under Condorcet and Borda objectives, without prior knowledge of the environment.

Contribution

It proposes MetaDueling and SA-MiDEX algorithms that achieve best-of-both-worlds regret bounds for multi-dueling bandits under two different preference models.

Findings

MetaDueling converts any dueling bandit algorithm into a multi-dueling bandit algorithm.

The combined algorithms achieve near-optimal regret bounds in both stochastic and adversarial settings.

Matching lower bounds are provided for the Condorcet setting, confirming near-optimality.

Abstract

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 $O\left(\sum_{i \neq a^\star} \frac{\log T}{\Delta_i}\right)$ 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 $O\left(K^2 \log KT + K \log^2 T + \sum_{i: \Delta_i^{\mathrm{B}} > 0} \frac{K\log KT}{(\Delta_i^{\mathrm{B}})^2}\right)$ 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.