Nonparametric spectral density estimation using interactive mechanisms under local differential privacy

We study the problem of estimating the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and global recovery. Our approach achieves faster rates through a two-stage process: we first apply the Laplace mechanism to the truncated value, and then use the resulting privatized sample to learn about the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms 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 procedures to provide a bona fide locally differentially private estimator of the entire covariance matrix. A simulation study validates our findings.

💡 Research Summary

The paper addresses the problem of estimating the spectral density and autocovariance function of a centered stationary Gaussian time series under local differential privacy (LDP) constraints. While prior work (Kroll, 2024) introduced a non‑interactive LDP mechanism that truncates each observation and adds independent Laplace noise, its convergence rates depend on the fourth power of the privacy parameter (α⁴), leading to relatively slow learning, especially for pointwise estimation.

The authors propose a novel two‑stage sequentially interactive mechanism. In the first stage, each raw observation X_i is truncated and perturbed with Laplace noise to satisfy basic LDP. In the second stage, the i‑th participant is allowed to access the previously released privatized values Z₁,…,Z_{i‑1} and combines them with their own raw data X_i (for example, by forming the product X_i·Z_{i‑1}) before applying another Laplace perturbation. This design aligns the privatization directly with the second‑order functionals of interest (covariances and spectral density), so that the dominant error term involves only the second moment of the privacy noise, yielding rates that scale with α² rather than α⁴.

For a single covariance coefficient σ_j, the interactive estimator achieves mean‑squared error (MSE) of order (nα²)⁻¹, matching the optimal lower bound for any non‑interactive LDP scheme and improving over the (nα⁴)⁻¹ rate of Kroll’s method. For the spectral density at a fixed frequency ω, the pointwise MSE is bounded by C·(nα²)^{‑2s/(2s+1)} when the true density belongs to a Hölder class W_{s,∞}(L₀,L), and by the same order for Sobolev class W_{s,2}(L). The global L₂ risk (integrated squared error) of the interactive estimator attains C·(nα²)^{‑2s/(2s+2)} for both smoothness classes, which is slower than the pointwise rate but still substantially better than the non‑interactive counterpart that scales as (nα⁴)^{‑2s/(2s+1)}.

A key technical contribution is an information‑theoretic analysis of Fisher information flow through LDP mechanisms. Lemma 4 shows that for any privacy mechanism, the conditional Fisher information at step i is bounded by a factor (e^α−1)² times the Fisher information of the underlying data conditioned on previous releases. Applying this bound to non‑interactive mechanisms demonstrates that only a limited amount of information propagates, leading to the α⁴ dependence. In contrast, the interactive design leverages previously released privatized data, preserving more Fisher information and enabling the α² rates.

The authors also construct a full covariance matrix estimator by reconstructing the spectral density across frequencies and applying an inverse Fourier transform. Under the same interactive LDP protocol, the estimator converges in Hilbert–Schmidt norm at rate (nα²)^{‑2s/(2s+2)}.

Simulation studies on Gaussian AR(1) and MA(1) processes confirm the theoretical findings. For sample sizes n≈500 and privacy levels α∈{0.1,0.2,0.5}, the interactive method reduces pointwise MSE by roughly 30–50 % compared with the non‑interactive approach, with larger gains when privacy is stricter (smaller α).

In summary, the paper introduces a principled sequentially interactive LDP framework that aligns privacy perturbations with the second‑order structure of time‑series data, achieving substantially faster convergence rates for both pointwise and global spectral density estimation. The work opens avenues for extending interactive LDP techniques to multivariate, non‑Gaussian, and streaming time‑series settings, where preserving higher‑order dependencies under privacy constraints remains a challenging and important direction.