Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence
This provides theoretical guidelines for designing comparison experiments in domains like preference elicitation, though it is incremental as it refines existing minimax bounds.
The paper tackles the problem of estimating latent quality scores from pairwise comparisons, establishing sharp minimax error bounds that depend on the comparison graph's topology and showing that ordinal and cardinal models have similar error scaling up to constants.
Data in the form of pairwise comparisons arises in many domains, including preference elicitation, sporting competitions, and peer grading among others. We consider parametric ordinal models for such pairwise comparison data involving a latent vector $w^* \in \mathbb{R}^d$ that represents the "qualities" of the $d$ items being compared; this class of models includes the two most widely used parametric models--the Bradley-Terry-Luce (BTL) and the Thurstone models. Working within a standard minimax framework, we provide tight upper and lower bounds on the optimal error in estimating the quality score vector $w^*$ under this class of models. The bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via its Laplacian spectrum. Thus, in settings where the subset of pairs may be chosen, our results provide principled guidelines for making this choice. Finally, we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre-factors.