Near-Optimal Regret for Distributed Adversarial Bandits: A Black-Box Approach

Abstract: We study distributed adversarial bandits, where $N$ agents cooperate to minimize the global average loss while observing only their own local losses. We show that the minimax regret for this problem is $\tildeΘ(\sqrt{(ρ<sup>{-1/2}+K/N)T})$, where $T$ is the horizon, $K$ is the number of actions, and $ρ$ is the spectral gap of the communication matrix. Our algorithm, based on a novel black-box reduction to bandits with delayed feedback, requires agents to communicate only through gossip. It achieves an upper bound that significantly improves over the previous best bound $\tilde{O}(ρ<sup>{-1/3}(KT)<sup>{2/3})$ of Yi and Vojnovic (2023). We complement this result with a matching lower bound, showing that the problem's difficulty decomposes into a communication cost $ρ<sup>{-1/4}\sqrt{T}$ and a bandit cost $\sqrt{KT/N}$. We further demonstrate the versatility of our approach by deriving first-order and best-of-both-worlds bounds in the distributed adversarial setting. Finally, we extend our framework to distributed linear bandits in $R<sup>d$, obtaining a regret bound of $\tilde{O}(\sqrt{(ρ<sup>{-1/2}+1/N)dT})$, achieved with only $O(d)$ communication cost per agent and per round via a volumetric spanner.