  We consider the problem of online combinatorial optimization under
semi-bandit feedback, where a learner has to repeatedly pick actions from a
combinatorial decision set in order to minimize the total losses associated
with its decisions. After making each decision, the learner observes the losses
associated with its action, but not other losses. For this problem, there are
several learning algorithms that guarantee that the learner's expected regret
grows as $\widetilde{O}(\sqrt{T})$ with the number of rounds $T$. In this
paper, we propose an algorithm that improves this scaling to
$\widetilde{O}(\sqrt{{L_T^*}})$, where $L_T^*$ is the total loss of the best
action. Our algorithm is among the first to achieve such guarantees in a
partial-feedback scheme, and the first one to do so in a combinatorial setting.
