First-order regret bounds for combinatorial semi-bandits
This work addresses regret minimization for learners in combinatorial semi-bandit scenarios, representing an incremental improvement with a novel regret bound.
The paper tackles the problem of online combinatorial optimization under semi-bandit feedback by proposing an algorithm that reduces the expected regret scaling from $\widetilde{O}(\sqrt{T})$ to $\widetilde{O}(\sqrt{L_T^*})$, where $L_T^*$ is the total loss of the best action, achieving this in a combinatorial setting for the first time.
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.