First-order regret bounds for combinatorial semi-bandits

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. The performance of the learner is measured in terms of its total expected regret against the best fixed action after $T$ rounds. In this paper, we propose a computationally efficient algorithm that improves existing worst-case guarantees of $\widetilde{O}(\sqrt{T})$ 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.