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 $\epsilon$-optimal ranking of
the $n$ items with probability at least $1 - \delta$. When the feedback in each
subset round is a single Plackett-Luce-sampled item, we show $(\epsilon,
\delta)$-PAC algorithms with a sample complexity of
$O\left(\frac{n}{\epsilon^2} \ln \frac{n}{\delta} \right)$ rounds, which we
establish as being order-optimal by exhibiting a matching sample complexity
lower bound of $\Omega\left(\frac{n}{\epsilon^2} \ln \frac{n}{\delta}
\right)$---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 $(\epsilon, \delta)$-PAC
ranking sample complexity of $O\left(\frac{n}{m \epsilon^2} \ln
\frac{n}{\delta} \right)$ 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.

本研究旨在通过自适应挑选子集并收集偏好反馈，在 Plackett-Luce 模型下解决 PAC 排名问题，提出了新的 pivot trick 技巧，从而实现了在一定概率下识别 n 个项目的 ε- 最优排名，（m-1）/m 降低的样本复杂度和对称排名算法的阶无法提高的。