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 O~(T
^2/3
) and Õ(T
^3/4
) 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 Ω̃(T
^2/3
) 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 Ω̃(T
^2/3
), as well.