Nonparametric spectral density estimation using interactive mechanisms under local differential privacy

TL;DR

This work tackles nonparametric spectral density estimation for centered stationary Gaussian time series under $\alpha$-local differential privacy. It introduces sequentially interactive privacy mechanisms that leverage past privatized data to improve information flow, achieving faster rates for estimating a single covariance coefficient, a fixed-frequency spectral density, and the full spectrum over Hölder and Sobolev classes. Key results show that interactive mechanisms yield rates scaling with $n$ and $\alpha$ as $(n\alpha^2)^{-1}$ (pointwise $\sigma_j$) and $(n\alpha^2)^{-{2s}/{(2s+1)}}$ or $(n\alpha^2)^{-{2s}/{(2s+2)}}$ for spectral densities, while non-interactive methods remain limited to $(n\alpha^4)$-type rates; a lower bound confirms optimality of the noninteractive rate for certain targets. The paper also develops a new information-theoretic inequality for dependent data under privacy and constructs a bona-fide locally private covariance estimator via a Toeplitz-structure approach, highlighting the practical impact for private covariance analysis in time-series settings.

Abstract

We address the problem of nonparametric estimation of the spectral density for a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: estimating a single covariance coefficient, estimating the spectral density at a fixed frequency, and estimating the entire spectral density function. Our approach achieves faster rates through a two-stage process: we apply first the Laplace mechanism to the truncated value and then use the former privatized sample to gain knowledge on the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our estimators improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $α$, since the pointwise rates depend on $nα^2$ instead of $nα^4$. Moreover, we show that the rate $(nα^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these estimators to provide a bona-fide locally differentially private covariance matrix estimator.

Theorems & Definitions (10)

• Proposition 1
• Lemma 1
• Theorem 1
• Lemma 2
• Theorem 2
• Lemma 3
• Theorem 3
• Corollary 4
• Lemma 4
• Theorem 5