Sharp interactive privacy rates for spectral density estimation

Fix $s > 1/2$, $C_0 > 1$ and $C_1 > 0$. For an even, nonnegative, $2\pi$-periodic spectral density $f$ write $$\gamma_f(h) = \int_{-\pi}^{\pi} f(\lambda) e^{ih\lambda}\,d\lambda, \qquad \rho_f(h) = \gamma_f(h)/\gamma_f(0),$$ and let $$\mathcal F(C_0, C_1, s) = \Bigl\{ f : C_0^{-1} \le \gamma_f(0) \le C_0, \ \sum_{h \in \mathbb Z \setminus \{0\}} |h|^{2s} |\rho_f(h)|^2 \le C_1^2 \Bigr\}.$$ The data $X_1, \dots, X_n$ are consecutive observations of a centered real stationary Gaussian process with spectral density $f \in \mathcal F(C_0, C_1, s)$. A sequentially interactive mechanism releases each $Z_i$ through a Markov kernel $Q_i(\cdot \mid X_i, Z_{< i})$ into a standard Borel space, with $$Q_i(B \mid x, z_{< i}) \le e^{\alpha} Q_i(B \mid x', z_{< i})$$ for every measurable output set $B$, every history $z_{< i}$ and all $x, x' \in \mathbb R$. Each release may use its own raw observation and the earlier releases, but not the earlier raw observations. Let $\mathcal Q^{\mathrm{seq}}_{n,\alpha}$ be the class of these mechanisms and $$\mathcal R^{\mathrm{seq}}_{n,\alpha} = \inf_{Q \in \mathcal Q^{\mathrm{seq}}_{n,\alpha}} \inf_{\widehat f} \sup_{f \in \mathcal F(C_0, C_1, s)} \mathbb E_{f,Q}\Bigl[\int_{-\pi}^{\pi} \bigl|\widehat f(Z_{1:n})(\lambda) - f(\lambda)\bigr|^2\,d\lambda\Bigr],$$ where $\widehat f$ ranges over measurable $L^2([-\pi, \pi])$-valued estimators; the class parameters are known to the mechanism and the estimator.

Butucea, Klockmann and Krivobokova constructed sequentially interactive mechanisms whose errors for a single autocovariance and for $f$ at a point depend on $n\alpha^2$ where those of the known non-interactive mechanism depend on $n\alpha^4$, and a global estimator with mean integrated squared error of order $(n\alpha^2)^{-2s/(2s+2)}$ up to logarithmic factors, for large $n$, over Sobolev and Hölder classes. For non-interactive mechanisms, in which each $Z_i$ depends on $X_i$ alone, Issartel and Roueff proved that the minimax risk over $\mathcal F(C_0, C_1, s)$ is of order $1 \wedge \bigl((\alpha^4 \wedge 1) n\bigr)^{-2s/(2s+1)}$.

Problem. Determine the joint dependence of $\mathcal R^{\mathrm{seq}}_{n,\alpha}$ on $n$ and $\alpha$, uniformly for $n \ge 2$ and $0 < \alpha \le 1$: give matching upper and lower bounds up to constants depending only on $C_0$, $C_1$ and $s$, including any necessary logarithmic factors and the regime in which consistent estimation is impossible.