  We introduce the \emph{Correlated Preference Bandits} problem with random
utility-based choice models (RUMs), where the goal is to identify the best item
from a given pool of $n$ items through online subsetwise preference feedback.
We investigate whether models with a simple correlation structure, e.g. low
rank, can result in faster learning rates. While we show that the problem can
be impossible to solve for the general `low rank' choice models, faster
learning rates can be attained assuming more structured item correlations. In
particular, we introduce a new class of \emph{Block-Rank} based RUM model,
where the best item is shown to be $(\epsilon,\delta)$-PAC learnable with only
$O(r \epsilon^{-2} \log(n/\delta))$ samples. This improves on the standard
sample complexity bound of $\tilde{O}(n\epsilon^{-2} \log(1/\delta))$ known for
the usual learning algorithms which might not exploit the item-correlations ($r
\ll n$). We complement the above sample complexity with a matching lower bound
(up to logarithmic factors), justifying the tightness of our analysis.
Surprisingly, we also show a lower bound of
$\Omega(n\epsilon^{-2}\log(1/\delta))$ when the learner is forced to play just
duels instead of larger subsetwise queries. Further, we extend the results to a
more general `\emph{noisy Block-Rank}' model, which ensures robustness of our
techniques. Overall, our results justify the advantage of playing subsetwise
queries over pairwise preferences $(k=2)$, we show the latter provably fails to
exploit correlation.
