通过谱放大器对图谱和割进行隐私近似
Private Approximation of Graph Spectra and Cuts via Spectral Amplifiers
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中文总结 AI 辅助
研究边级差分隐私下发布近似输入图割大小的合成图问题，给出多项式时间算法改进先前界，主要技术是新的私有谱原语及边敏感终端割预言机，降低了私有割发布误差。
中文摘要 AI 辅助
我们研究了在边级差分隐私下发布一个近似输入图所有割大小的合成图的问题。若坚持纯加法误差，最优最坏情况误差为$\widetilde\Theta(n^{3/2})$。若允许小的乘法松弛，信息论指数时间机制能实现近线性加法误差，但已知最好的多项式时间算法误差大得多。我们给出一个多项式时间$(\varepsilon,\delta)$-差分隐私算法，对每个$n$顶点无加权图$G$，输出一个非负加权合成图$\widetilde G$，使得大概率下每个割$S\subseteq V(G)$满足\[ |w_G(S)-w_{\widetilde G}(S)| \le \gamma w_G(S)+\widetilde O_{\varepsilon,\delta,\gamma}(n^{13/12+o(1)}). \]这改进了Aamand等人（ICML 2025）之前混合乘法/加法私有割近似的多项式时间最坏情况界$\widetilde O(n^{5/4+o(1)})$。主要技术要素是一组新的有界度图的私有谱原语，其中一个在估计最大度为$d$的图的拉普拉斯矩阵时给出谱误差$\widetilde O_{\delta}((nd)^{1/4}/\sqrt\varepsilon)$，首次在高度情况下超越标准的$\min\{2d,\widetilde O_{\delta}(\sqrt{n}/\varepsilon)\}$基线。我们还为下游割近似开发了一个对$n$和$d$有更尖锐误差依赖的原语。结合一个新的边敏感终端割预言机，其在有$M$条边的图上有加法误差$\widetilde O(n+(n^2M)^{1/3})$，得到最终最坏情况$\widetilde O(n^{13/12+o(1)})$的私有割发布误差。
英文摘要
We study the problem of releasing a synthetic graph that approximates the sizes of all cuts of an input graph under edge-level differential privacy. If one insists on purely additive error, the optimal worst-case error is $\widetildeΘ(n^{3/2})$. If one allows a small multiplicative slack, an information-theoretic exponential-time mechanism achieves nearly linear additive error, but the best known polynomial-time algorithms have substantially larger error. We give a polynomial-time $(\varepsilon,δ)$-differentially private algorithm which, for every $n$-vertex unweighted graph $G$, outputs a non-negative weighted synthetic graph $\widetilde G$ such that, with high probability, every cut $S\subseteq V(G)$ satisfies \[ |w_G(S)-w_{\widetilde G}(S)| \le γw_G(S)+\widetilde O_{\varepsilon,δ,γ}(n^{13/12+o(1)}). \] This improves the previous polynomial-time worst-case bound $\widetilde O(n^{5/4+o(1)})$ of Aamand et al. (ICML 2025) for mixed multiplicative/additive private cut approximation. The main technical ingredient is a new set of private spectral primitives for bounded-degree graphs, one of them gives spectral error $\widetilde O_δ((nd)^{1/4}/\sqrt\varepsilon)$ in estimating the graph Laplacian for graphs of maximum degree $d$, being the first to beat the standard $\min\{2d,\widetilde O_δ(\sqrt{n}/\varepsilon)\}$ baseline in the high-degree regime. We further develop a primitive with a sharper error dependence on $n$ and $d$ for the downstream cut approximation. Combined with a new edge-sensitive terminal cut oracle with additive error $\widetilde O(n+(n^2M)^{1/3})$ on graphs with $M$ edges, this yields the final worst-case $\widetilde O(n^{13/12+o(1)})$ private cut-release error.
