Combinatorial Bandits with Relative Feedback

We consider combinatorial online learning with subset choices when only relative feedback information from subsets is available, instead of bandit or semi-bandit feedback which is absolute. Specifically, we study two regret minimisation problems over subsets of a finite ground set \([n]\), with subset-wise relative preference information feedback according to the Multinomial logit choice model. In the first setting, the learner can play subsets of size bounded by a maximum size and receives top-\(m\) rank-ordered feedback, while in the second setting the learner can play subsets of a fixed size \(k\) with a full subset ranking observed as feedback. For both settings, we devise instance-dependent and order-optimal regret algorithms with regret \(O(\frac{n}{m} \ln T)\) and \(O(\frac{n}{k} \ln T)\), respectively. We derive fundamental limits on the regret performance of online learning with subset-wise preferences, proving the tightness of our regret guarantees. Our results also show the value of eliciting more general top-\(m\) rank-ordered feedback over single winner feedback (\(m=1\)). Our theoretical results are corroborated with empirical evaluations.