When Is Distributed Nonlinear Aggregation Private? Optimality and Information-Theoretical Bounds

TL;DR

The paper addresses privacy risks in distributed nonlinear aggregation under joint honest-but-curious corruption and eavesdropping, where nonlinear operators such as maxima, medians, and top-$K$ values pose intrinsic privacy challenges. It introduces a unified information-theoretic framework to derive fundamental leakage bounds and demonstrates that, with randomized initialization and careful parameter design, PDMM-based algorithms can asymptotically attain these optimal bounds while preserving exact aggregation accuracy. The results reveal how network topology and algorithmic choices (e.g., step-size and initialization scale) shape observed leakage, and provide concrete guidelines for privacy-preserving nonlinear aggregation. The work offers both theoretical benchmarks and practical mechanisms that inform secure design of distributed systems employing nonlinear aggregation.

Abstract

Nonlinear aggregation is central to modern distributed systems, yet its privacy behavior is far less understood than that of linear aggregation. Unlike linear aggregation where mature mechanisms can often suppress information leakage, nonlinear operators impose inherent structural limits on what privacy guarantees are theoretically achievable when the aggregate must be computed exactly. This paper develops a unified information-theoretic framework to characterize privacy leakage in distributed nonlinear aggregation under a joint adversary that combines passive (honest-but-curious) corruption and eavesdropping over communication channels. We cover two broad classes of nonlinear aggregates: order-based operators (maximum/minimum and top-$K$) and robust aggregation (median/quantiles and trimmed mean). We first derive fundamental lower bounds on leakage that hold without sacrificing accuracy, thereby identifying the minimum unavoidable information revealed by the computation and the transcript. We then propose simple yet effective privacy-preserving distributed algorithms, and show that with appropriate randomized initialization and parameter choices, our proposed approaches can attach the derived optimal bounds for the considered operators. Extensive experiments validate the tightness of the bounds and demonstrate that network topology and key algorithmic parameters (including the stepsize) govern the observed leakage in line with the theoretical analysis, yielding actionable guidelines for privacy-preserving nonlinear aggregation.

Figures (13)

• Figure 1: A schematic illustration of the leakage of $s_i$ during the $x$-update of the median consensus protocol.
• Figure 2: Output accuracy (MSE) of maximum and median consensus under the proposed approach, the non-private counterpart, and the existing DP-based approaches.
• Figure 3: Maximum consensus: NMI $\frac{I(S_i;X_i^{(t)})}{I(S_i;S_i)}$ under different initialization settings. Values in parentheses indicate the empirical leakage probability of each node over $1000$ Monte Carlo trials.
• Figure 4: Medain consensus: NMI $\frac{I(S_i;X_i^{(t)})}{I(S_i;S_i)}$ under different initializations. Values in parentheses indicate the empirical leakage probability of each node over $1000$ Monte Carlo trials.
• Figure 5: NMI $\frac{I(S_i;X_i^{(t)})}{I(S_i;S_i)}$ under ring topology and complete topology for both maximum and median consensus.

Theorems & Definitions (13)

• Proposition 1: Unavoidable leakage for maximum consensus
• Proposition 2: Unavoidable leakage for median consensus
• proof
• Theorem 1
• proof
• Lemma 1: Achievability of the Ideal-World Privacy Bound
• proof
• Remark 1
• Theorem 2
• proof