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.

翻译：度量差分隐私（mDP）通过依据成对距离调整隐私保证，将局部差分隐私（LDP）推广至更一般情形，从而支持上下文感知的保护并提升数据效用。现有基于优化的方法在粗粒度域中能有效降低效用损失，但在细粒度或连续设置下优化 mDP 仍面临挑战，主要源于构建稠密扰动矩阵与满足逐点约束的计算开销。本文提出一种基于插值的框架，用于在此类域中优化 lp 范数 mDP。我们的方法在一组稀疏的锚点上优化扰动分布，并通过对数凸组合对非锚点位置的分布进行插值，该插值过程可证明保持 mDP。为解决高维空间中朴素插值可能导致的隐私违反问题，我们将插值过程分解为一系列一维步骤，并推导出一种经修正的公式，该公式通过设计确保 lp 范数 mDP 成立。我们进一步探索了跨维度的扰动分布与隐私预算分配的联合优化。在真实世界位置数据集上的实验表明，本方法在细粒度域中提供了严格的隐私保证与具有竞争力的数据效用，性能优于基线机制。