Exploiting Correlation to Achieve Faster Learning Rates in Low-Rank Preference Bandits

Abstract: 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.