  We study online learning problems in which the learner has extra knowledge
about the adversary's behaviour, i.e., in game-theoretic settings where
opponents typically follow some no-external regret learning algorithms. Under
this assumption, we propose two new online learning algorithms, Accurate Follow
the Regularized Leader (AFTRL) and Prod-Best Response (Prod-BR), that
intensively exploit this extra knowledge while maintaining the no-regret
property in the worst-case scenario of having inaccurate extra information.
Specifically, AFTRL achieves $O(1)$ external regret or $O(1)$ \emph{forward
regret} against no-external regret adversary in comparison with $O(\sqrt{T})$
\emph{dynamic regret} of Prod-BR. To the best of our knowledge, our algorithm
is the first to consider forward regret that achieves $O(1)$ regret against
strategic adversaries. When playing zero-sum games with Accurate Multiplicative
Weights Update (AMWU), a special case of AFTRL, we achieve \emph{last round
convergence} to the Nash Equilibrium. We also provide numerical experiments to
further support our theoretical results. In particular, we demonstrate that our
methods achieve significantly better regret bounds and rate of last round
convergence, compared to the state of the art (e.g., Multiplicative Weights
Update (MWU) and its optimistic counterpart, OMWU).
