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 $\widetilde{\Theta}(T^{2/3})$, 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 $\Theta(\sqrt{T})$. The difference between these two rates provides the \emph{first} indication that learning with bandit feedback can be significantly harder than learning with full-information feedback (previous results only showed a different dependence on the number of actions, but not on $T$.) In addition to characterizing the inherent difficulty 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 $\widetilde{\Theta}(T^{2/3})$. Another implication is that the minimax regret of online learning in adversarial Markov decision processes (MDPs) is $\widetilde{\Theta}(T^{2/3})$. 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.