Online learning with graph-structured feedback against adaptive adversaries

We derive upper and lower bounds for the policy regret of T-round online learning problems with graph-structured feedback, where the adversary is nonoblivious but assumed to have a bounded memory. We obtain upper bounds of widetilde O(T2/3) and widetilde O(T3/4) for strongly-observable and weakly-observable graphs, respectively, based on analyzing a variant of the Exp3 algorithm. When the adversary is allowed a bounded memory of size 1, we show that a matching lower bound of widetildeOmega(T2/3) is achieved in the case of full-information feedback. We also study the particular loss structure of an oblivious adversary with switching costs, and show that in such a setting, non-revealing strongly-observable feedback graphs achieve a lower bound of widetildeOmega(T2/3), as well.