  We study small-loss bounds for adversarial multi-armed bandits with graph
feedback, that is, adaptive regret bounds that depend on the loss of the best
arm or related quantities, instead of the total number of rounds. We derive the
first small-loss bound for general strongly observable graphs, resolving an
open problem of Lykouris et al. (2018). Specifically, we develop an algorithm
with regret $\mathcal{\tilde{O}}(\sqrt{\kappa L_*})$ where $\kappa$ is the
clique partition number and $L_*$ is the loss of the best arm, and for the
special case of self-aware graphs where every arm has a self-loop, we improve
the regret to $\mathcal{\tilde{O}}(\min\{\sqrt{\alpha T}, \sqrt{\kappa L_*}\})$
where $\alpha \leq \kappa$ is the independence number. Our results
significantly improve and extend those by Lykouris et al. (2018) who only
consider self-aware undirected graphs.
  Furthermore, we also take the first attempt at deriving small-loss bounds for
weakly observable graphs. We first prove that no typical small-loss bounds are
achievable in this case, and then propose algorithms with alternative
small-loss bounds in terms of the loss of some specific subset of arms. A
surprising side result is that $\mathcal{\tilde{O}}(\sqrt{T})$ regret is
achievable even for weakly observable graphs as long as the best arm has a
self-loop.
  Our algorithms are based on the Online Mirror Descent framework but require a
suite of novel techniques that might be of independent interest. Moreover, all
our algorithms can be made parameter-free without the knowledge of the
environment.
