  Online learning algorithms are designed to learn even when their input is
generated by an adversary. The widely-accepted formal definition of an online
algorithm's ability to learn is the game-theoretic notion of regret. We argue
that the standard definition of regret becomes inadequate if the adversary is
allowed to adapt to the online algorithm's actions. We define the alternative
notion of policy regret, which attempts to provide a more meaningful way to
measure an online algorithm's performance against adaptive adversaries.
Focusing on the online bandit setting, we show that no bandit algorithm can
guarantee a sublinear policy regret against an adaptive adversary with
unbounded memory. On the other hand, if the adversary's memory is bounded, we
present a general technique that converts any bandit algorithm with a sublinear
regret bound into an algorithm with a sublinear policy regret bound. We extend
this result to other variants of regret, such as switching regret, internal
regret, and swap regret.
