Minimax optimality of interactive LDP rates for spectral density estimation

Determine whether, in the problem of estimating the spectral density of a centered stationary Gaussian time series under α-local differential privacy, the convergence rates achieved by the sequentially interactive mechanisms proposed in the paper—namely the pointwise mean-squared error rates (n α^2)^{-2s/(2s+1)} for f in the Hölder class W^{s,∞}(L0, L) and (n α^2)^{-(2s−1)/(2s)} for f in the Sobolev class W^{s,2}(L), together with the global mean integrated squared error rate (n α^2)^{-2s/(2s+2)}—are minimax optimal over all α-local differentially private mechanisms (including non-interactive and sequentially interactive) and estimators.

Background

The paper introduces sequentially interactive local differential privacy mechanisms for estimating covariance coefficients, pointwise values of the spectral density, and the entire spectral density function of centered stationary Gaussian time series. Compared to existing non-interactive mechanisms, these interactive procedures achieve rates that depend on nα^2 rather than nα^4 for pointwise tasks, and provide an L2 rate for the global estimator of order (nα^2)^{-2s/(2s+2).

A notable feature of the results is that the derived nonparametric rates differ between the pointwise mean-squared error and the mean integrated squared error, which is unusual in related literature. The authors explicitly state that it remains unresolved whether these interactive-LDP rates are minimax optimal when considered against all possible local differentially private mechanisms (interactive and non-interactive).