Bandits with switching costs: T^2/3 regret

We study the adversarial multi-armed bandit problem in a setting where the player incurs a unit cost each time he switches actions. We prove that the player's T-round minimax regret in this setting is ⊖̃(T ^2/3), thereby closing a fundamental gap in our understanding of learning with bandit feedback. In the corresponding full-information version of the problem, the minimax regret is known to grow at a much slower rate of ⊖ (√T). The difference between these two rates provides the first indication that learning with bandit feedback can be significantly harder than learning with fullinformation feedback (previous results only showed a different dependence on the number of actions, but not on T.) In addition to characterizing the inherent dificulty of the multi-armed bandit problem with switching costs, our results also resolve several other open problems in online learning. One direct implication is that learning with bandit feedback against bounded-memory adaptive adversaries has a minimax regret of ⊖̃ (T^2/3). Another implication is that the minimax regret of online learning in adversarial Markov decision processes (MDPs) is ⊖̃ (T^2/3). The key to all of our results is a new randomized construction of a multi-scale random walk, which is of independent interest and likely to prove useful in additional settings.