Delay and Cooperation in Nonstochastic Bandits

We study networks of communicating learning agents that cooperate to solve a common nonstochastic bandit problem. Agents use an underlying communication network to get messages about actions selected by other agents, and drop messages that took more than d hops to arrive, where d is a delay parameter. We introduce Exp3-Coop, a cooperative version of the Exp3 algorithm and prove that with K actions and N agents the average per-agent regret after T rounds is at most of order \sqrt\left(d+1 + \fracKN\alpha_≤d\right)(T\ln K), where \alpha_≤d is the independence number of the d-th power of the communication graph G. We then show that for any connected graph, for d=\sqrtK the regret bound is K^1/4\sqrtT, strictly better than the minimax regret \sqrtKT for noncooperating agents. More informed choices of d lead to bounds which are arbitrarily close to the full information minimax regret \sqrtT\ln K when G is dense. When G has sparse components, we show that a variant of Exp3-Coop, allowing agents to choose their parameters according to their centrality in G, strictly improves the regret. Finally, as a by-product of our analysis, we provide the first characterization of the minimax regret for bandit learning with delay.