Minimax rates for private density estimation with mixed anisotropy

Fix an integer $d \ge 2$, $R > 0$ with $R^2 > d$, $A > 0$, and vectors $\boldsymbol\beta, \boldsymbol\delta \in \mathbb{N}_{>0}^d$ such that $\beta_m/\delta_m$ does not depend on $m$, with $\delta_m < d$ for some $m$ and $\delta_k > d$ for some $k$. Let $\phi_1(t) = 1$, $\phi_{2j}(t) = \sqrt2 \cos(2\pi j t)$ and $\phi_{2j+1}(t) = \sqrt2 \sin(2\pi j t)$ for $j \ge 1$, let $\phi_{\mathbf j}(x) = \prod_{m} \phi_{j_m}(x_m)$ for $\mathbf j \in \mathbb{N}_{>0}^d$, and write $\theta_{\mathbf j}(h) = \int_{[0,1]^d} h \phi_{\mathbf j}$. For a vector $\mathbf s$ and $r > 0$ let $$W^{\mathbf s}(r) = \Bigl\{h \in L^2([0,1]^d) : \sum_{\mathbf j \in \mathbb{N}_{>0}^d} \Bigl(\sum_{m=1}^d j_m^{2 s_m}\Bigr) |\theta_{\mathbf j}(h)|^2 \le r^2\Bigr\},$$ let $\mathcal{F}_{\boldsymbol\beta,R}$ be the set of probability densities in $W^{\boldsymbol\beta}(R)$, and let $$d_{\boldsymbol\delta}(h, f) = \sup_{g \in W^{\boldsymbol\delta}(1)} \Bigl|\int_{[0,1]^d} (h - f) g\Bigr|.$$ Given independent observations $X_1, \dots, X_n$ with density $f$, a sequentially interactive $\alpha$-locally private mechanism releases each $Z_i$ through a channel $Q_i(\cdot \mid X_i, Z_1, \dots, Z_{i-1})$ with $$Q_i(B \mid x, z_{<i}) \le e^{\alpha} Q_i(B \mid x', z_{<i})$$ for every measurable $B$, all $x, x'$ and all histories $z_{<i}$; let $\mathcal{Q}_\alpha$ be the class of these mechanisms, and let $$\mathcal{R}_{n,\alpha}(\boldsymbol\beta, \boldsymbol\delta) = \inf_{Q \in \mathcal{Q}_\alpha} \inf_{\widehat f} \sup_{f \in \mathcal{F}_{\boldsymbol\beta,R}} \mathbb{E}_{f,Q}\, d_{\boldsymbol\delta}(\widehat f, f),$$ where $\widehat f$ ranges over measurable $L^2$-valued functions of the released data.

Let $\bar\beta$ and $\bar\delta$ be the harmonic means of the $\beta_m$ and of the $\delta_m$. Albert, Chevallier, Laurent and Sacko proved that, for a constant $c > 0$ depending only on $d$, $R$, $A$, $\boldsymbol\beta$ and $\boldsymbol\delta$, $$\mathcal{R}_{n,\alpha}(\boldsymbol\beta, \boldsymbol\delta) \ge c \max\bigl\{(n\alpha^2)^{-(\bar\beta + \bar\delta)/(2\bar\beta + 2d)},\ (n\alpha^2)^{-1/2}\bigr\}$$ for all $0 < \alpha \le A$ with $n\alpha^2 \ge 1$, for proportional integer smoothness vectors without the condition on the signs of $\delta_m - d$, and gave a non-interactive mechanism and projection estimator attaining this order when $\delta_m < d$ for every $m$ or $\delta_m > d$ for every $m$.

Problem. Determine $\mathcal{R}_{n,\alpha}(\boldsymbol\beta, \boldsymbol\delta)$, up to multiplicative constants independent of $n$ and $\alpha$, for $0 < \alpha \le A$ and $n\alpha^2 \ge 1$, with matching lower and upper bounds that include any necessary logarithmic factors. The parameters $d$, $R$, $\boldsymbol\beta$ and $\boldsymbol\delta$ are known and fixed.