  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.
