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 dd hops to arrive, where dd is a delay parameter. We introduce Exp3-Coop, a cooperative version of the Exp3 algorithm and prove that with KK actions and NN agents the average per-agent regret after TT rounds is at most of order (d+1+KN\u3b1 64d)(TlnK) 1a(d+1+KN\u3b1 64d)(Tln\u2061K), where \u3b1 64d\u3b1 64d is the independence number of the dd-th power of the communication graph GG. We then show that for any connected graph, for d=K 12 1ad=K the regret bound is K1/4T 1aK1/4T, strictly better than the minimax regret KT 12 1aKT for noncooperating agents. More informed choices of dd lead to bounds which are arbitrarily close to the full information minimax regret TlnK 12 1aTln\u2061K when GG is dense. When GG has sparse components, we show that a variant of Exp3-Coop, allowing agents to choose their parameters according to their centrality in GG, 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