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

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 $\phi$ and an unknown centroid permutation $\sigma_0$. Federated Borda approach involves local client scoring, nontrivial quantization, and privacy-preserving protocols. We show that for $\phi \in [0,1)$, and arbitrary $\sigma_0$ of length $N$, it suffices for each of the $L$ clients to locally aggregate $\max\{C_1(\phi), C_2(\phi)\frac{1}{L}\log \frac{N}{\delta}\}$ rankings, where $C_1(\phi)$ and $C_2(\phi)$ are constants, quantize the result, and send it to the server who can then recover $\sigma_0$ with probability $\geq 1-\delta$. 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 $\phi+\phi^2<1+\phi^N$, and arbitrary $\sigma_0$ of length $N$, it suffices for each of the $L$ clients to locally aggregate $\max\{C_3(\phi), C_4(\phi)\frac{1}{L}\log \frac{N}{\delta}\}$ rankings, where $C_3(\phi)$ and $C_4(\phi)$ are constants. Clients send truncated Lehmer coordinate histograms to the server, which can recover $\sigma_0$ with probability $\geq 1-\delta$. Communication complexity is $\sim O(N\log NL\log L)$.