Federated Aggregation of Mallows Rankings: A Comparative Analysis of Borda and Lehmer Coding

TL;DR

This work proposes Federated Rank Aggregation (FRA) to privately learn a global ranking from distributed, non-shared data under the Mallows model, using two mechanisms: Borda scoring and Lehmer codes. It provides rigorous sample- and communication-complexity guarantees for each approach, with precise conditions on the number of rankings per client and the total communication cost, along with secure-aggregation-compatible quantization strategies. The authors prove concentration and recovery results for both methods and validate them on synthetic Mallows data and real-world datasets (Sushi, Jester, TCGA, and Maine elections), showing near-centralized performance, especially for Borda FRA. Collectively, the paper advances principled FRA with provable guarantees and practical privacy-preserving mechanisms, enabling scalable, distributed rank aggregation in privacy-sensitive domains.

Abstract

Rank aggregation combines multiple ranked lists into a consensus ranking. In fields like biomedical data sharing, rankings may be distributed and require privacy. This motivates the need for federated rank aggregation protocols, which support distributed, private, and communication-efficient learning across multiple clients with local data. We present the first known federated rank aggregation methods using Borda scoring and Lehmer codes, focusing on the sample complexity for federated algorithms on Mallows distributions with a known scaling factor $φ$ and an unknown centroid permutation $σ_0$. Federated Borda approach involves local client scoring, nontrivial quantization, and privacy-preserving protocols. We show that for $φ\in [0,1)$, and arbitrary $σ_0$ of length $N$, it suffices for each of the $L$ clients to locally aggregate $\max\{C_1(φ), C_2(φ)\frac{1}{L}\log \frac{N}δ\}$ rankings, where $C_1(φ)$ and $C_2(φ)$ are constants, quantize the result, and send it to the server who can then recover $σ_0$ with probability $\geq 1-δ$. Communication complexity scales as $NL \log N$. Our results represent the first rigorous analysis of Borda's method in centralized and distributed settings under the Mallows model. Federated Lehmer coding approach creates a local Lehmer code for each client, using a coordinate-majority aggregation approach with specialized quantization methods for efficiency and privacy. We show that for $φ+φ^2<1+φ^N$, and arbitrary $σ_0$ of length $N$, it suffices for each of the $L$ clients to locally aggregate $\max\{C_3(φ), C_4(φ)\frac{1}{L}\log \frac{N}δ\}$ rankings, where $C_3(φ)$ and $C_4(φ)$ are constants. Clients send truncated Lehmer coordinate histograms to the server, which can recover $σ_0$ with probability $\geq 1-δ$. Communication complexity is $\sim O(N\log NL\log L)$.

Figures (12)

• Figure 1: Overview of our proposed FRA algorithms based on normalized Borda scores (top) and Lehmer codes (bottom). In Borda's algorithm, each client computes the average of local permutations and nonuniformly quantizes their values to arrive at a vector representation that is not necessarily a permutation. The server aggregates the quantized messages into a ranking. In the algorithm based on Lehmer codes, each client computes the Lehmer encoding of each local permutation and their entry-wise majority. Then, each client encodes the truncated histogram of the entry-wise majority of the Lehmer codes, guided by concentration results. The server computes the overall entry-wise majority of the client codes and performs Lehmer decoding.
• Figure 2: Quantization bins (bars) and centroids (crosses) for Borda aggregates, and for different values of $\phi$, computed via Monte Carlo methods with $100,000$ samples.
• Figure 3: Comparison of the performance of various centralized and FRA methods (centralized results are depicted by dashed, while federated results are indicated by solid lines): (a-c) plots of the average Kendall $\tau$ distances between $\sigma_0$ and its estimate, for different values of $\phi$. (d-e) show the performance of our algorithms on Sushi preference, Jester, and prioritized cancer gene expression data, respectively. Here, the performance is evaluated using the Kemeny objective normalized by the permutation length. Results pertaining to the minimum weight bipartite matching aggregation algorithm are also shown dwork2001rank.
• Figure 4: Comparison of the performance of various centralized and FRA methods (centralized results are depicted by dashed, while federated results are indicated by solid lines). (a) Prioritized cancer gene expression data from TCGA for LUSC. (b) Ranked-choice voting data from the primary election for the Governor of Maine in 2018. Here, the performance is evaluated using the Kemeny objective normalized by the permutation length. Results pertaining to the minimum weight bipartite matching aggregation algorithm are shown as well.
• Figure 5: Comparison of the performance of various centralized and FRA methods (centralized results are depicted by dashed, while federated results are indicated by solid lines). Plots show the average Kendall $\tau$ distances between $\sigma_0$ and its estimate, for different values of $\phi$.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 3
• Theorem 4
• proof
• proof
• Remark 1