Rate-Optimal Rank Aggregation with Private Pairwise Rankings
Shirong Xu Will Wei Sun Guang Cheng
In various real-world scenarios like recommender systems and political
surveys, pairwise rankings are commonly collected and utilized for rank
aggregation to obtain an overall ranking of items. However, preference rankings
can reveal individuals' personal preferences, underscoring the need to protect
them before releasing for downstream analysis. In this paper, we address the
challenge of preserving privacy while ensuring the utility of rank aggregation
based on pairwise rankings generated from the Bradley-Terry-Luce (BTL) model.
Using the randomized response mechanism to perturb raw pairwise rankings is a
common privacy protection strategy used in practice, but a critical challenge
arises because the privatized rankings no longer adhere to the BTL model,
resulting in significant bias in downstream rank aggregation tasks. Motivated
from this, we propose a debiased randomized response mechanism to protect the
raw pairwise rankings, ensuring consistent estimation of true preferences and
rankings in downstream rank aggregation. Theoretically, we offer insights into
the relationship between overall privacy guarantees and estimation errors from
private ranking data, and establish minimax rates for estimation errors. This
enables the determination of optimal privacy guarantees that balance
consistency in rank aggregation with robust privacy protection. We also
investigate convergence rates of expected ranking errors for partial and full
ranking recovery, quantifying how privacy protection influences the
specification of top-$K$ item sets and complete rankings. Our findings are
validated through extensive simulations and a real application.