Minimax spectral-density estimation under local differential privacy for dependent Gaussian data

Let $X_1,\ldots,X_N$ be observations from a centered stationary Gaussian process with spectral density $f$ belonging to a Sobolev-type class $\mathcal F_s$ , and let the observations be released through an $\alpha$ -locally differentially private mechanism, possibly interactive. Determine the minimax $L_2$ estimation risk $\inf_{Q,\widehat f}\sup_{f\in\mathcal F_s}\mathbb E_f\left[\lVert\widehat f-f\rVert_2^2\right]$ as a function of $N$ , $\alpha$ , and the smoothness parameters. In particular, determine whether, in the high-privacy regime, the effective sample size is $N\alpha^4$ rather than $N\alpha^2$ because of temporal dependence.

References

Primary source

arXiv

Additional references

On the privacy cost for dependent Gaussian data: spectral density estimation under local differential privacy — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to settle the privacy cost of estimating dependence in Gaussian time series, but the result has not been independently checked.

The problem asks for the optimal accuracy of spectral-density estimation from dependent Gaussian data under local privacy. The latest preprint claims a sharp dependence on the privacy level and resolves earlier rate gaps.

Known results

Kroll (2024): non-interactive methods achieved rates involving $n\alpha^{4}$ , with a corresponding non-interactive lower bound for one covariance coefficient.
The 2025 interactive preprint improved pointwise upper bounds to dependence on $n\alpha^{2}$ .
For global $L_{2}$ estimation, that preprint retained logarithmic losses and left optimality over all mechanisms open.

August 2026 claimed resolution

The latest paper claims that temporal dependence causes the stronger privacy cost, proves the $\alpha^{4}$ minimax dependence for the full problem, removes polylogarithmic losses from the upper bound, and closes the logarithmic fixed-lag autocovariance gap. These claims are currently unverified.

Current status (as of August 2026): A preprint claims the minimax problem is solved, but independent verification is absent; absent confirmation, the claim remains unverified.

Sources

No solutions have been posted yet.