  We consider the problem of combining and learning over a set of adversarial
bandit algorithms with the goal of adaptively tracking the best one on the fly.
The CORRAL algorithm of Agarwal et al. (2017) and its variants (Foster et al.,
2020a) achieve this goal with a regret overhead of order
$\widetilde{O}(\sqrt{MT})$ where $M$ is the number of base algorithms and $T$
is the time horizon. The polynomial dependence on $M$, however, prevents one
from applying these algorithms to many applications where $M$ is poly$(T)$ or
even larger. Motivated by this issue, we propose a new recipe to corral a
larger band of bandit algorithms whose regret overhead has only
\emph{logarithmic} dependence on $M$ as long as some conditions are satisfied.
As the main example, we apply our recipe to the problem of adversarial linear
bandits over a $d$-dimensional $\ell_p$ unit-ball for $p \in (1,2]$. By
corralling a large set of $T$ base algorithms, each starting at a different
time step, our final algorithm achieves the first optimal switching regret
$\widetilde{O}(\sqrt{d S T})$ when competing against a sequence of comparators
with $S$ switches (for some known $S$). We further extend our results to linear
bandits over a smooth and strongly convex domain as well as unconstrained
linear bandits.
