Lipschitz Dueling Bandits over Continuous Action Spaces

TL;DR

This paper introduces the first algorithm for Lipschitz dueling bandits over continuous action spaces with purely comparative feedback, achieving near-optimal regret bounds and logarithmic space complexity.

Contribution

It combines Lipschitz and dueling bandit frameworks, developing new analytical tools and an adaptive algorithm with proven regret bounds.

Findings

Proposed the first algorithm for Lipschitz dueling bandits.

Achieved a regret bound of  O(T^{(d_z+1)/(d_z+2)}) with near-optimal performance.

Algorithm uses only logarithmic space relative to the time horizon.

Abstract

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.