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arXiv 2408.06997cs.DScs.CRcs.LG
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更快的私有最小生成树

Faster Private Minimum Spanning Trees

BARC, University of Copenhagen(哥本哈根大学BARC研究院)

Rasmus Pagh, Lukas Retschmeier

AI总结:

针对边权重差分隐私下的最小生成树发布问题,提出一种结合权重离散化分组与亚线性Report-Noisy-Max模拟的新算法,在保证效用的同时大幅提升稠密图运行效率。

摘要:

受聚类和合成数据生成应用的驱动,我们研究在边权重差分隐私约束下发布最小生成树(MST)的问题:具有n个顶点和m条边的图拓扑G=(V,E)是公开的,权重矩阵$\boldsymbol{W}\boldsymbol{\u2208}\boldsymbol{\u211D}^{n \times n}$是私有的,我们希望在ρ-零集中差分隐私(ρ-zero-concentrated differential privacy)下发布近似MST。若两个权重矩阵的每个元素差异不超过$Δ_\u221e$,则它们被视为相邻的,即我们考虑$\u2113_\u221e$相邻关系。\n现有的私有MST算法要么向$\boldsymbol{W}$的每个元素添加噪声并通过后处理估计MST,要么在特定MST算法执行期间就地向权重添加噪声。将后处理方法与高效MST算法结合使用时,在稠密图上需要$O(n^2)$时间,但会导致MST权重产生量级为$O(n^2\boldsymbol{\u006cog} n)$的加性误差。就地算法能提供渐近更优的效用,但现有就地算法在稠密图上的运行时间为$O(n^3)$。\n我们的主要结果是一种新的差分隐私MST算法,在隐私参数ρ固定的情况下,其效用与现有就地方法相当,同时运行时间为$O(m + n^{3/2}\boldsymbol{\u006cog} n)$。我们算法的技术核心是一种高效的亚线性时间Report-Noisy-Max(报告噪声最大值)模拟,其工作原理是将所有边权重离散化为$Δ_\u221e$的倍数,并将权重相同的边分组。具体而言,我们提出了一种数据结构,允许我们在$O(\boldsymbol{\u221an} \boldsymbol{\u006cog} n)$时间内,从最多$O(n^2)$条割边中采样出一条带噪声的最小权重边。实验评估支持我们的结论:我们的算法在效用或运行时间上均显著优于现有算法。

英文摘要:

Motivated by applications in clustering and synthetic data generation, we consider the problem of releasing a minimum spanning tree (MST) under edge-weight differential privacy constraints where a graph topology $G=(V,E)$ with $n$ vertices and $m$ edges is public, the weight matrix $\vec{W}\in \mathbb{R}^{n \times n}$ is private, and we wish to release an approximate MST under $ρ$-zero-concentrated differential privacy. Weight matrices are considered neighboring if they differ by at most $Δ_\infty$ in each entry, i.e., we consider an $\ell_\infty$ neighboring relationship. Existing private MST algorithms either add noise to each entry in $\vec{W}$ and estimate the MST by post-processing or add noise to weights in-place during the execution of a specific MST algorithm. Using the post-processing approach with an efficient MST algorithm takes $O(n^2)$ time on dense graphs but results in an additive error on the weight of the MST of magnitude $O(n^2\log n)$. In-place algorithms give asymptotically better utility, but the running time of existing in-place algorithms is $O(n^3)$ for dense graphs. Our main result is a new differentially private MST algorithm that matches the utility of existing in-place methods while running in time $O(m + n^{3/2}\log n)$ for fixed privacy parameter $ρ$. The technical core of our algorithm is an efficient sublinear time simulation of Report-Noisy-Max that works by discretizing all edge weights to a multiple of $Δ_\infty$ and forming groups of edges with identical weights. Specifically, we present a data structure that allows us to sample a noisy minimum weight edge among at most $O(n^2)$ cut edges in $O(\sqrt{n} \log n)$ time. Experimental evaluations support our claims that our algorithm significantly improves previous algorithms either in utility or running time.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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