Sample Complexity Bounds for Active Ranking from Multi-wise Comparisons

We study the sample complexity (i.e., the number of comparisons needed) bounds for actively ranking a set of $n$ items from multi-wise comparisons. Here, a multi-wise comparison takes $m$ items as input and returns a (noisy) result about the best item (the winner feedback) or the order of these items (the full-ranking feedback). We consider two basic ranking problems: top-$k$ items selection and full ranking. Unlike previous works that study ranking from multi-wise comparisons, in this paper, we do not require any parametric model or assumption and work on the fundamental setting where each comparison returns the correct result with probability $1$ or a certain probability larger than $\frac{1}{2}$. This paper helps understand whether and to what degree utilizing multi-wise comparisons can reduce the sample complexity for the ranking problems compared to ranking from pairwise comparisons. Specifically, under the winner feedback setting, one can reduce the sample complexity for top-$k$ selection up to an $m$ factor and that for full ranking up to a $\log{m}$ factor. Under the full-ranking feedback setting, one can reduce the sample complexity for top-$k$ selection up to an $m$ factor and that for full ranking up to an $m\log{m}$ factor. We also conduct numerical simulations to confirm our theoretical results.