  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
$\widetilde O(T^{2/3})$ and $\widetilde O(T^{3/4})$ 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 $\widetilde\Omega(T^{2/3})$ 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 $\widetilde\Omega(T^{2/3})$, as well.
