We study for the first time, stochastic dueling bandits over continuous action spaces with Lipschitz structure, where feedback is purely comparative. While dueling bandits and Lipschitz bandits have been studied separately, their combination has remained unexplored. We propose the first algorithm for Lipschitz dueling bandits, using round-based exploration and recursive region elimination guided by an adaptive reference arm. We develop new analytical tools for relative feedback and prove a regret bound of $\tilde O\left(T^{\frac{d_z+1}{d_z+2}}\right)$, where $d_z$ is the zooming dimension of the near-optimal region. Further, our algorithm takes only logarithmic space in terms of the total time horizon, best achievable by any bandit algorithm over a continuous action space.

翻译：我们首次研究了具有Lipschitz结构的连续动作空间上的随机对偶赌博机问题，其反馈为纯比较形式。尽管对偶赌博机与Lipschitz赌博机已被分别研究，但二者的结合尚未被探索。我们提出了首个针对Lipschitz对偶赌博机的算法，该算法采用基于轮次的探索策略，并通过自适应参考臂递归消除区域。我们发展了针对相对反馈的新分析工具，并证明了遗憾界 $\tilde O\left(T^{\frac{d_z+1}{d_z+2}}\right)$，其中 $d_z$ 为近最优区域的缩放维度。此外，我们的算法在总时间范围上仅需对数空间复杂度，这是连续动作空间上任何赌博机算法所能达到的最优结果。