Lipschitz Bandits with Batched Feedback

In this paper, we study Lipschitz bandit problems with batched feedback, where the expected reward is Lipschitz and the reward observations are communicated to the player in batches. We introduce a novel landscape-aware algorithm, called Batched Lipschitz Narrowing (BLiN), that optimally solves this problem. Specifically, we show that for a <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-step problem with Lipschitz reward of zooming dimension <inline-formula> <tex-math notation="LaTeX">$d_{z}$ </tex-math></inline-formula>, our algorithm achieves theoretically optimal (up to logarithmic factors) regret rate <inline-formula> <tex-math notation="LaTeX">$\widetilde {\mathcal {O}}\left ({T^{\frac {d_{z}+1}{d_{z}+2}}}\right)$ </tex-math></inline-formula> using only <inline-formula> <tex-math notation="LaTeX">$\mathcal {O} \left ({\log \log T}\right)$ </tex-math></inline-formula> batches. We also provide complexity analysis for this problem. Our theoretical lower bound implies that <inline-formula> <tex-math notation="LaTeX">$\Omega (\log \log T)$ </tex-math></inline-formula> batches are necessary for any algorithm to achieve the optimal regret. Thus, BLiN achieves optimal regret rate (up to logarithmic factors) using minimal communication.