使用重建攻击的私有图优化问题下界
Lower Bounds for Private Graph Optimization Problems using Reconstruction Attacks
- University of Waterloo（滑铁卢大学）
- BARC, University of Copenhagen（哥本哈根大学 BARC 中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对差分隐私下的图优化问题，利用重建攻击证明最小生成树和完美匹配的紧下界，并推广到稀疏图和层次聚类，改进了现有结果。
AI中文摘要：
本文研究了在差分隐私（DP）下的基本图优化问题，并展示了新的、基于重建的下界。我们考虑一个图 $G = (V, E, \vec{w})$，其中顶点集 $V$ 和边集 $E$ 是公开的，而权重 $\mathbf{w}:E\rightarrow \mathbb{R}$ 必须在 $\ell_1$ 相邻关系下保持差分隐私。对于发布最小权重生成树和最小权重完美匹配的问题，我们在具有 $n$ 个顶点和 $m>2n$ 条边的最坏情况图上展示了新的、紧的误差界 $\Omega(n\cdot\log(m/n)/\epsilon)$。上界是已知的纯 DP 算法，而新的下界即使在近似 $(\varepsilon,\delta)$-DP 下也成立，只要 $\delta \leq (n/m)^{\Omega(1)}$。我们的下界改进了 Sealfon (PODS~'16) 的 $\Omega(n/\epsilon)$ 下界。在 $\ell_1$ 相邻关系下，近似 DP 不会降低 MST 的误差，这一事实与 Pagh 等人 (PODS~'25) 的最新上界形成对比，后者表明在 $\ell_\infty$ 相邻关系下近似 DP 允许更好的误差。超越最坏情况图，我们为具有扩展性质的大类稀疏图给出了下界。我们展示了对于任何最小割至少为 $\Omega(\log(n))$ 的图，最小生成树的下界为 $\Omega(n / \epsilon)$。最后，我们考虑了在 Dasgupta 代价函数 (STOC~'16) 下的私有层次聚类问题，并展示了第一个以平衡割最小权重为参数的近似 DP 下界。这将 Deng 等人 (ICLR~'25) 的下界推广到一般图和近似 DP。
英文摘要：
This paper studies fundamental graph optimization problems under differential privacy (DP) and shows new, reconstruction-based lower bounds. We consider a graph $G = (V, E, \vec{w})$ where the vertex set $V$ and edges $E$ are public and the weights $\mathbf{w}:E\rightarrow \mathbb{R}$ must be kept differentially private under an $\ell_1$ neighboring relation. For the problems of releasing a minimum-weight spanning tree and a minimum-weight perfect matching, we show new, tight error bounds of $Ω(n\cdot\log(m/n)/ε)$ on worst-case graphs with $n$ vertices and $m>2n$ edges. The upper bounds are known pure DP algorithms while the new lower bound holds even under approximate $(\varepsilon,δ)$-DP as long as $δ\leq (n/m)^{Ω(1)}$. Our lower bounds improve the $Ω(n/ε)$ lower bounds of Sealfon (PODS~'16). The fact that approximate DP does not reduce error for MST under the $\ell_1$ neighboring relation contrasts with the recent upper bound of Pagh et al. (PODS~'25) which shows that approximate DP allows much better error under the $\ell_\infty$ neighboring relation. Going beyond worst-case graphs, we give lower bounds for large families of sparse graphs with expansion properties. We show a lower bound of $Ω(n / ε)$ for the minimum spanning tree for any graph where the minimum cut is at least $Ω(\log(n))$. Finally, we consider the problem of private hierarchical clustering under Dasgupta's cost function (STOC~'16) and show the first approximate DP lower bound parameterized by the minimum weight of a balanced cut. This extends lower bounds of Deng et al. (ICLR~'25) to general graphs and to approximate DP.
