Minimax-optimal Inference from Partial Rankings

This paper studies the problem of inferring a global preference based on the partial rankings provided by many users over different subsets of items according to the Plackett-Luce model. A question of particular interest is how to optimally assign items to users for ranking and how many item assignments are needed to achieve a target estimation error. For a given assignment of items to users, we first derive an oracle lower bound of the estimation error that holds even for the more general Thurstone models. Then we show that the Cram\'er-Rao lower bound and our upper bounds inversely depend on the spectral gap of the Laplacian of an appropriately defined comparison graph. When the system is allowed to choose the item assignment, we propose a random assignment scheme. Our oracle lower bound and upper bounds imply that it is minimax-optimal up to a logarithmic factor among all assignment schemes and the lower bound can be achieved by the maximum likelihood estimator as well as popular rank-breaking schemes that decompose partial rankings into pairwise comparisons. The numerical experiments corroborate our theoretical findings.