AI 中文总结
该研究针对局部差分隐私下依赖高斯数据的谱密度估计，填补了极小极大速率的上下界差距，还将相关工具应用于固定滞后自协方差估计等问题。
AI 中文摘要
我们研究局部差分隐私（Local Differential Privacy, LDP）下估计中心化平稳高斯过程依赖结构的基础问题，在此设定中，谱密度表征数据的依赖结构，是待估计的量。我们的主要贡献是弥合了极小极大速率已知下界与上界之间的开放α²与α⁴的差距。具体而言，我们建立了一个极小极大下界，表明在Sobolev型谱密度类上，高隐私 regime下的有效样本量为Nα⁴，而非独立观测时通常出现的Nα²。这一额外的隐私成本由观测之间的时间依赖而非其边际分布导致，证明依赖于私有化依赖高斯观测的收缩界。我们的第二个贡献是一个匹配的上界，消除了前人工作中存在的多对数损失。我们未将通用私有化方案应用于经典估计量，而是构造了一个特定于问题的过程，达到我们下界确定的速率。除了弥合谱密度估计的差距，我们还将为该问题开发的工具应用于几个相关问题：（i）弥合固定滞后自协方差估计的对数差距；（ii）表明α⁴成本在每个远离零的有界谱密度附近局部出现；（iii）确立经典渐近与独立高斯实验的等价性在LDP下通常不成立。
英文摘要
We study the fundamental problem of estimating the dependence structure of a centered stationary Gaussian process under local differential privacy (LDP). In this setting, the spectral density characterizes the dependence structure of the data and is the quantity to be estimated. Our main contribution is to close the open $α^2$-versus-$α^4$ gap between the previously known lower and upper bounds on the minimax rate. Specifically, we establish a minimax lower bound showing that, over Sobolev-type classes of spectral densities, the effective sample size in the high-privacy regime is $Nα^4$, rather than the usual $Nα^2$ arising for independent observations. This additional privacy cost is caused by the temporal dependence between the observations rather than by their marginal distributions. The proof relies on a contraction bound for privatized dependent Gaussian observations. Our second contribution is a matching upper bound, free of the polylogarithmic losses present in previous work. Rather than applying a generic privatization scheme to classical estimators, we construct a problem-specific procedure attaining the rate identified by our lower bound. Beyond closing the gaps in spectral density estimation, we apply the tools developed for this problem to several related questions. We (i) close the logarithmic gap for fixed-lag autocovariance estimation, (ii) show that the $α^4$ cost arises locally around every spectral density bounded away from zero, and (iii) establish that classical asymptotic equivalence with an independent Gaussian experiment generally fails under LDP.
Comments55 pages
