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

TL;DR

This paper introduces the Correlated Preference Bandits problem with low-rank models, showing that exploiting structured item correlations enables faster learning of the best item through subsetwise preference feedback, with theoretical guarantees.

Contribution

The paper proposes a new Block-Rank RUM model and provides tight sample complexity bounds for learning the best item, demonstrating the advantage of subsetwise queries over pairwise preferences.

Findings

Faster learning rates are achievable with structured low-rank models.

Subsetwise queries outperform pairwise preferences in exploiting correlations.

Matching lower bounds justify the sample complexity results.

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.