Individual Regret in Cooperative Stochastic Multi-Armed Bandits
Idan Barnea, Tal Lancewicki, Yishay Mansour
We study the regret in stochastic Multi-Armed Bandits (MAB) with multiple agents that communicate over an arbitrary connected communication graph.We show an individual regret bound of ${O}(\mathcal{R} / m + A^2   + A \sqrt{\log T})$ and a nearly matching lower bound.Here $A$ is the number of actions, $T$ the time horizon, $m$ the number of agents, and $\mathcal{R} = \sum_{\Delta_i > 0}\log(T)/\Delta_i$ is the optimal single agent regret, where $\Delta_i$ is the sub-optimality gap of action $i$.Our work is the first to show an individual regret bound in cooperative stochastic MAB that is independent of the graph's diameter.When considering communication networks there are additional considerations beyond regret, such as message size and number of communication rounds.First, we show that our regret bound holds even if we restrict the messages to be of logarithmic size.Second, for logarithmic number ofcommunication rounds, we obtain a regret bound of ${O}(\mathcal{R} / m+A \log T)$.