  We consider the non-stochastic Multi-Armed Bandit problem in a setting where
there is a fixed and known metric on the action space that determines a cost
for switching between any pair of actions. The loss of the online learner has
two components: the first is the usual loss of the selected actions, and the
second is an additional loss due to switching between actions. Our main
contribution gives a tight characterization of the expected minimax regret in
this setting, in terms of a complexity measure $\mathcal{C}$ of the underlying
metric which depends on its covering numbers. In finite metric spaces with $k$
actions, we give an efficient algorithm that achieves regret of the form
$\widetilde{O}(\max\{\mathcal{C}^{1/3}T^{2/3},\sqrt{kT}\})$, and show that this
is the best possible. Our regret bound generalizes previous known regret bounds
for some special cases: (i) the unit-switching cost regret
$\widetilde{\Theta}(\max\{k^{1/3}T^{2/3},\sqrt{kT}\})$ where
$\mathcal{C}=\Theta(k)$, and (ii) the interval metric with regret
$\widetilde{\Theta}(\max\{T^{2/3},\sqrt{kT}\})$ where $\mathcal{C}=\Theta(1)$.
For infinite metrics spaces with Lipschitz loss functions, we derive a tight
regret bound of $\widetilde{\Theta}(T^{\frac{d+1}{d+2}})$ where $d \ge 1$ is
the Minkowski dimension of the space, which is known to be tight even when
there are no switching costs.
