Nonparametric spectral density estimation using interactive mechanisms under local differential privacy (2504.00919v1)

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 H\"older and Sobolev smoothness classes, we demonstrate that our estimators improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $\alpha$, since the pointwise rates depend on $n\alpha^2$ instead of $n\alpha^4$. Moreover, we show that the rate $(n\alpha^4)^{-1}$ 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.

Nonparametric Spectral Density Estimation under Local Differential Privacy

The paper addresses the challenge of estimating the spectral density for Gaussian stationary time series while ensuring local differential privacy (LDP). Traditional data privacy techniques have proven inadequate in the face of increasing data collection and analysis, prompting the emergence of differential privacy models. In particular, local differential privacy (LDP) provides strong privacy by allowing each data holder to independently create a privatized version of their information. The authors focus on nonparametric techniques suitable for stationary Gaussian time series data under LDP constraints.

Key Contributions

1. Interactive Privacy Mechanisms: The paper proposes novel interactive LDP mechanisms for three specific tasks:
   • Estimating a covariance coefficient.
   • Estimating the spectral density at a fixed frequency.
   • Estimating the entire spectral density function.

These mechanisms achieve faster convergence rates by incorporating the dependence structure manifested in time series data, specifically through a two-step mechanism: truncating the data and using the Laplace mechanism to add noise.

1. Improved Estimation Rates: The results show that interactive mechanisms provide significant improvements over non-interactive ones. For instance, the interactive estimation of a covariance coefficient achieves a convergence rate of $(n \alpha^2)^{-1}$, surpassing the best-known rate for non-interactive schemes. Likewise, the pointwise estimate of the spectral density at a fixed frequency reaches optimal convergence rates of $(n \alpha^2)^{-\frac{2s}{2s+1}}$ for Hӧlder smooth functions and $(n \alpha^2)^{-\frac{2s-1}{2s}}$ for Sobolev smooth functions, reflecting the added utility of interactive methodologies.
2. Estimation of Global Spectral Density: Utilizing mechanisms designed for vectors within $\ell_\infty$ balls, the authors construct an estimator for the global spectral density function that achieves the rate of $(n \alpha^2)^{-\frac{2s}{2s+2}}$. Despite achieving the $L_2$ norm goals, this global rate reveals an intriguing divergence from the pointwise rates.

Theoretical and Practical Implications

The implications of this research are twofold:

• Theoretical Implications: The interactive mechanisms suggest that engaging with temporal or structural dependencies can dramatically improve convergence rates in contrast to independent data settings. Additionally, the research uncovers a unique distinction between pointwise and global estimation rates, offering avenues for further exploration into the bounds of local differential privacy with dependent data.
• Practical Implications: Practically, the proposed methodologies provide a substantial benefit for privacy-preserving applications in time series analysis, such as smart meter data or other sensor-related data where respecting individual user privacy is crucial. This work presents methods for sustaining data utility while ensuring privacy, crucial in industries reliant on fine-grained, sequentially collected data.

Future Directions

The paper opens various avenues for future research, notably:

• Minimax Optimality: Investigating the possibility of minimizing the bounds identified in this paper. Determining whether these rates are indeed optimal could yield significant insights into the interaction between data structure and privacy constraints.
• Extending to Other Data Models: Applying these mechanisms beyond Gaussian processes, possibly extending to more complex or non-linear models common in time series analysis, broadening the potential application of these interactive privacy techniques.

In sum, this paper offers a rigorous mathematical foundation for balancing utility and privacy in local differential privacy settings for spectral density estimation. By advancing understanding of both theoretical boundaries and practical strategies, it contributes significantly to the field of privacy-preserving data analysis.