Interpolation-Based Optimization for Enforcing lp-Norm Metric Differential Privacy in Continuous and Fine-Grained Domains
Chenxi Qiu
TL;DR
The work tackles enforcing $(\epsilon,d_p)$-mDP in continuous and fine-grained spaces by introducing an interpolation-based framework that learns perturbations at anchor points and extends them to the rest of the domain via log-convex interpolation. It establishes a rigorous theory for one-dimensional and multi-dimensional interpolation with dimension-wise budget composition, and introduces Anchor Perturbation Optimization (APO) and its linear surrogate Approx-APO to achieve scalable utility optimization under privacy constraints. Empirically, the approach achieves zero mDP violations and competitive utility on real road-network datasets (Rome, NYC, London), while offering substantial efficiency gains over full discretization-based LP methods. The method provides a principled, practical pathway for private data release in continuous spatial domains and sets the stage for scalable extensions to higher-dimensional settings.
Abstract
Metric Differential Privacy (mDP) generalizes Local Differential Privacy (LDP) by adapting privacy guarantees based on pairwise distances, enabling context-aware protection and improved utility. While existing optimization-based methods reduce utility loss effectively in coarse-grained domains, optimizing mDP in fine-grained or continuous settings remains challenging due to the computational cost of constructing dense perterubation matrices and satisfying pointwise constraints. In this paper, we propose an interpolation-based framework for optimizing lp-norm mDP in such domains. Our approach optimizes perturbation distributions at a sparse set of anchor points and interpolates distributions at non-anchor locations via log-convex combinations, which provably preserve mDP. To address privacy violations caused by naive interpolation in high-dimensional spaces, we decompose the interpolation process into a sequence of one-dimensional steps and derive a corrected formulation that enforces lp-norm mDP by design. We further explore joint optimization over perturbation distributions and privacy budget allocation across dimensions. Experiments on real-world location datasets demonstrate that our method offers rigorous privacy guarantees and competitive utility in fine-grained domains, outperforming baseline mechanisms. in high-dimensional spaces, we decompose the interpolation process into a sequence of one-dimensional steps and derive a corrected formulation that enforces lp-norm mDP by design. We further explore joint optimization over perturbation distributions and privacy budget allocation across dimensions. Experiments on real-world location datasets demonstrate that our method offers rigorous privacy guarantees and competitive utility in fine-grained domains, outperforming baseline mechanisms.