  This paper explores the adaptive (active) PAC (probably approximately
correct) top-$k$ ranking (i.e., top-$k$ item selection) and total ranking
problems from $l$-wise ($l\geq 2$) comparisons under the multinomial logit
(MNL) model. By adaptively choosing sets to query and observing the noisy
output of the most favored item of each query, we want to design ranking
algorithms that recover the top-$k$ or total ranking using as few queries as
possible. For the PAC top-$k$ ranking problem, we derive a lower bound on the
sample complexity (aka number of queries), and propose an algorithm that is
sample-complexity-optimal up to an $O(\log(k+l)/\log{k})$ factor. When $l=2$
(i.e., pairwise comparisons) or $l=O(poly(k))$, this algorithm matches the
lower bound. For the PAC total ranking problem, we derive a tight lower bound,
and propose an algorithm that matches the lower bound. When $l=2$, the MNL
model reduces to the popular Plackett-Luce (PL) model. In this setting, our
results still outperform the state-of-the-art both theoretically and
numerically. We also compare our algorithms with the state-of-the-art using
synthetic data as well as real-world data to verify the efficiency of our
algorithms.
