Decentralized Differentially Private Power Method

Abstract: We propose a novel Decentralized Differentially Private Power Method (D-DP-PM) for performing Principal Component Analysis (PCA) in networked multi-agent settings. Unlike conventional decentralized PCA approaches where each agent accesses the full n-dimensional sample space, we address the challenging scenario where each agent observes only a subset of dimensions through row-wise data partitioning. Our method ensures $(\epsilon,\delta)$-Differential Privacy (DP) while enabling collaborative estimation of global eigenvectors across the network without requiring a central aggregator. We achieve this by having agents share only local embeddings of the current eigenvector iterate, leveraging both the inherent privacy from random initialization and carefully calibrated Gaussian noise additions. We prove that our algorithm satisfies the prescribed $(\epsilon,\delta)$-DP guarantee and establish convergence rates that explicitly characterize the impact of the network topology. Our theoretical analysis, based on linear dynamics and high-dimensional probability theory, provides tight bounds on both privacy and utility. Experiments on real-world datasets demonstrate that D-DP-PM achieves superior privacy-utility tradeoffs compared to naive local DP approaches, with particularly strong performance in moderate privacy regimes ($\epsilon\in[2, 5]$). The method converges rapidly, allowing practitioners to trade iterations for enhanced privacy while maintaining competitive utility.