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Improved Differentially Private Algorithms for Rank Aggregation

Quentin Hillebrand, Pasin Manurangsi, Vorapong Suppakitpaisarn, Phanu Vajanopath

TL;DR

This work studies rank aggregation under differential privacy, focusing on two canonical objectives: footrule and Kemeny rankings. It introduces the first DP algorithm for footrule ranking with an optimal multiplicative factor and tight additive error bounds under central, approximate DP and local DP, yielding near-optimal results by leveraging a private release of per-item costs via a modified binary tree mechanism. The authors then translate footrule guarantees into 2-approximation results for Kemeny ranking and present two DP-PTAS approaches that substantially improve additive error exponents, including a large-n unbiasedness-based scheme and a small-n bucketing strategy that reduces the problem to private two-way marginals. Collectively, the paper advances the privacy-utility tradeoffs for both central and local DP in rank aggregation, offering practical polynomial-time algorithms with provable guarantees and highlighting open questions, particularly around LDP lower bounds and further tightening of PTAS additive terms. The techniques blend private median estimation, interval-based data release, and WFAS-based reductions to achieve state-of-the-art DP performance for rank aggregation tasks.

Abstract

Rank aggregation is a task of combining the rankings of items from multiple users into a single ranking that best represents the users' rankings. Alabi et al. (AAAI'22) presents differentially-private (DP) polynomial-time approximation schemes (PTASes) and $5$-approximation algorithms with certain additive errors for the Kemeny rank aggregation problem in both central and local models. In this paper, we present improved DP PTASes with smaller additive error in the central model. Furthermore, we are first to study the footrule rank aggregation problem under DP. We give a near-optimal algorithm for this problem; as a corollary, this leads to 2-approximation algorithms with the same additive error as the $5$-approximation algorithms of Alabi et al. for the Kemeny rank aggregation problem in both central and local models.

Improved Differentially Private Algorithms for Rank Aggregation

TL;DR

This work studies rank aggregation under differential privacy, focusing on two canonical objectives: footrule and Kemeny rankings. It introduces the first DP algorithm for footrule ranking with an optimal multiplicative factor and tight additive error bounds under central, approximate DP and local DP, yielding near-optimal results by leveraging a private release of per-item costs via a modified binary tree mechanism. The authors then translate footrule guarantees into 2-approximation results for Kemeny ranking and present two DP-PTAS approaches that substantially improve additive error exponents, including a large-n unbiasedness-based scheme and a small-n bucketing strategy that reduces the problem to private two-way marginals. Collectively, the paper advances the privacy-utility tradeoffs for both central and local DP in rank aggregation, offering practical polynomial-time algorithms with provable guarantees and highlighting open questions, particularly around LDP lower bounds and further tightening of PTAS additive terms. The techniques blend private median estimation, interval-based data release, and WFAS-based reductions to achieve state-of-the-art DP performance for rank aggregation tasks.

Abstract

Rank aggregation is a task of combining the rankings of items from multiple users into a single ranking that best represents the users' rankings. Alabi et al. (AAAI'22) presents differentially-private (DP) polynomial-time approximation schemes (PTASes) and -approximation algorithms with certain additive errors for the Kemeny rank aggregation problem in both central and local models. In this paper, we present improved DP PTASes with smaller additive error in the central model. Furthermore, we are first to study the footrule rank aggregation problem under DP. We give a near-optimal algorithm for this problem; as a corollary, this leads to 2-approximation algorithms with the same additive error as the -approximation algorithms of Alabi et al. for the Kemeny rank aggregation problem in both central and local models.

Paper Structure

This paper contains 60 sections, 39 theorems, 43 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Footrule Ranking
  4. Kemeny Ranking
  5. Technical Overview
  6. Algorithms for Footrule Ranking
  7. $(2,\beta)$-approximation algorithm for the Kemeny rank aggregation
  8. $(1+\xi,\beta)$-approximation algorithm for the Kemeny rank aggregation
  9. Large $n$: Leveraging Unbiasedness
  10. Small $n$: Reduction to 2-Way Marginal.
  11. Preliminaries
  12. Rank Aggregation
  13. Differential Privacy
  14. DP Approximate Median
  15. Modified Binary Tree Mechanism
  16. ...and 45 more sections

Key Result

Theorem 3.1

There is an efficient algorithm for ApxMed that is $\beta$-accurate with

Theorems & Definitions (59)

  • Definition 1.1: $(\varepsilon, \delta)$-DP
  • Definition 2.1: $\varepsilon$-LDP
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Definition 5.2: Weighted Feedback Arc Set (WFAS)
  • Definition 5.3: Bounded WFAS Instance
  • Theorem 5.4: MathieuS07
  • ...and 49 more