  We consider a variant of the classical online linear optimization problem in
which at every step, the online player receives a "hint" vector before choosing
the action for that round. Rather surprisingly, it was shown that if the hint
vector is guaranteed to have a positive correlation with the cost vector, then
the online player can achieve a regret of $O(\log T)$, thus significantly
improving over the $O(\sqrt{T})$ regret in the general setting. However, the
result and analysis require the correlation property at \emph{all} time steps,
thus raising the natural question: can we design online learning algorithms
that are resilient to bad hints?
  In this paper we develop algorithms and nearly matching lower bounds for
online learning with imperfect directional hints. Our algorithms are oblivious
to the quality of the hints, and the regret bounds interpolate between the
always-correlated hints case and the no-hints case. Our results also
generalize, simplify, and improve upon previous results on optimistic regret
bounds, which can be viewed as an additive version of hints.
