Active Ranking with Subset-wise Preferences

We consider the problem of probably approximately correct (PAC) ranking $n$ items by adaptively eliciting subset-wise preference feedback. At each round, the learner chooses a subset of $k$ items and observes stochastic feedback indicating preference information of the winner (most preferred) item of the chosen subset drawn according to a Plackett-Luce (PL) subset choice model unknown a priori. The objective is to identify an $ε$-optimal ranking of the $n$ items with probability at least $1 - δ$. When the feedback in each subset round is a single Plackett-Luce-sampled item, we show $(ε, δ)$-PAC algorithms with a sample complexity of $O\left(\frac{n}{ε^2} \ln \frac{n}δ \right)$ rounds, which we establish as being order-optimal by exhibiting a matching sample complexity lower bound of $Ω\left(\frac{n}{ε^2} \ln \frac{n}δ \right)$---this shows that there is essentially no improvement possible from the pairwise comparisons setting ($k = 2$). When, however, it is possible to elicit top-$m$ ($\leq k$) ranking feedback according to the PL model from each adaptively chosen subset of size $k$, we show that an $(ε, δ)$-PAC ranking sample complexity of $O\left(\frac{n}{m ε^2} \ln \frac{n}δ \right)$ is achievable with explicit algorithms, which represents an $m$-wise reduction in sample complexity compared to the pairwise case. This again turns out to be order-wise unimprovable across the class of symmetric ranking algorithms. Our algorithms rely on a novel {pivot trick} to maintain only $n$ itemwise score estimates, unlike $O(n^2)$ pairwise score estimates that has been used in prior work. We report results of numerical experiments that corroborate our findings.