A Nearly Instance Optimal Algorithm for Top-k Ranking under the Multinomial Logit Model

Abstract

We study the active learning problem of top-k ranking from multi-wise comparisons under the popular multinomial logit model. Our goal is to identify the topk items with high probability by adaptively querying sets for comparisons and observing the noisy output of the most preferred item from each comparison. To achieve this goal, we design a new active ranking algorithm without using any information about the underlying items' preference scores. We also establish a matching lower bound on the sample complexity even when the set of preference scores is given to the algorithm. These two results together show that the proposed algorithm is nearly instance optimal (similar to instance optimal [12], but up to polylog factors). Our work extends the existing literature on rank aggregation in three directions. First, instead of studying a static problem with fixed data, we investigate the top-k ranking problem in an active learning setting. Second, we show our algorithm is nearly instance optimal, which is a much stronger theoretical guarantee. Finally, we extend the pairwise comparison to the multi-wise comparison, which has not been fully explored in ranking literature.

Asymptotically Optimal Sequential Design for Rank Aggregation

“…Rank aggregation has been an active research problem in recent years (see, e.g., [16,18,19,24,25,31,46,51] and references therein) that finds many applications to social choice, tournament play, search rankings, advertisement placement, etc. With the advent of crowdsourcing services, one can easily ask crowd workers to conduct comparisons among a few objects in an online fashion at a low cost [15,17].…”

“…Finally, (18) in step 3 of the algorithm has a closed-form solution, obtained by by writing down the KKT condition. That is,…”

From PAC to Instance-Optimal Sample Complexity in the Plackett-Luce Model

“…However above has O n k many optimization variables (precisely E θ 1 [N S ]s), so we instead solve the dual LP to reach the desired bound. Lastly the Ω n k ln 1 δ term in the lower bound arises as any learning algorithm must at least test each item a constant number of times via k-wise subset plays before judging it optimality which is the bare minimum sample complexity the learner has to incur Chen et al [2018]. The complete proof is given in Appendix C.1.…”

“…This is a basic search problem motivated by applications in recommender systems and information retrieval Hofmann et al [2013], Radlinski et al [2008], crowdsourced ranking Chen et al [2013], tournament design Graepel and Herbrich [2006], etc. It has received recent attention in the online learning community, primarily under the rubric of dueling bandits (e.g., Yue et al [2012] and online ranking in the Plackett-Luce (PL) discrete choice model Chen et al [2018], Saha and Gopalan [2019], Ren et al [2018].…”

“…to achieve (0, δ)-PAC objective). Finally to get the additional n k log 1 δ term we appeal to the lower bound argument provided inChen et al [2018] (see their Thm. B.9) for the 0, 1 8 -PAC Now as derived in (5) in the proof of Thm.…”