SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) ] = 0, \quad \mathbb {E} _ {\mathbf {p} _ {z}} \bigl [ \phi_ {z, i} (X) ^ {2} \bigr ] = \frac {\gamma^ {2}}{1 - \frac {1}{2} \gamma^ {2} (1 + z _ {i})}, \mathrm {a n d} \mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) \phi_ {z, j} (X) ] = 0,
$$
so that Assumptions 1 and 2 are satisfied for  $\alpha^2 \coloneqq 2\gamma^2$ . Moreover, using, e.g., Hoeffding's lemma (cf. [9]), for  $\gamma < 1$ , the random vector  $\phi_z(X) = (\phi_{z,i}(X))_{i\in [d]}$  is  $\frac{\gamma^2}{(1 - \gamma^2)^2}$ -subgaussian. Thus, Assumption 3 holds as well, and we can invoke both parts of Theorem 2.
Let  $\| z \|_+ \coloneqq |\{i \in [d] \mid z_i = 1\}|$ , so that  $\| \mu_z \|_0 = \sum_{i=1}^d \frac{1}{2}(1 + z_i) = \| z \|_+$ . The next claim, which follows from standard bounds for binomial random variables, states that when  $Z \sim \mathrm{Rad}(\tau)^{\otimes d}$ ,  $\mu_Z$  is  $s$ -sparse with high probability.
Fact 2. Let  $Z \sim \operatorname{Rad}(\tau)^{\otimes d}$ , where  $\tau d \geq 4\log d$ . Then  $\operatorname*{Pr}\big[\| Z\|_{+} \leq 2\tau d\big] \geq 1 - \tau / 4$ .
Hence the construction satisfies  $\operatorname{Pr}_Z[\mathbf{p}_Z \in \mathcal{B}_{d,s}] \leq 1 - \tau/4$ , as required in Lemma 1.
We now choose  $\gamma = \gamma(p) \coloneqq \frac{4\gamma}{(s/2)^{1/p}} \in (0, 1/2]$ , which implies that Assumption 4 holds since
$$
\ell_ {p} (\mu (\mathbf {p} _ {z}), \mu (\mathbf {p} _ {z ^ {\prime}})) = \gamma \mathrm {d} _ {\mathrm {H a m}} (z, z ^ {\prime}) ^ {1 / p} = 4 \gamma \left(\frac {\mathrm {d} _ {\mathrm {H a m}} (z , z ^ {\prime})}{\tau d}\right) ^ {1 / p}.
$$
Therefore, we can apply Lemma 1 as well. For  $\mathcal{W}^{\mathrm{priv},\varepsilon}$ , we prove the two parts of the lower bound separately, depending on whether  $\varepsilon \leq 1$ . First, upon combining the bounds obtained by Corollary 1 and Lemma 1 (specifically, for the former, (6)), we get
$$
d \leq 1 1 2 n \alpha^ {2} (e ^ {\varepsilon} - 1) ^ {2},
$$
whereby, upon recalling that  $\alpha^2 = 2\gamma^2$ , and using the value of  $\gamma = \gamma(p)$  above, it follows that
$$
\frac {1}{3 5 8 4} \cdot \frac {d (s / 2) ^ {\frac {2}{p}}}{n (e ^ {\varepsilon} - 1) ^ {2}} \leq \gamma^ {2}.
$$
Thus,  $\mathcal{E}_p(\mathcal{B}_{d,s},\mathcal{W}^{\mathrm{priv},\varepsilon},n) = \Omega \left(\sqrt{\frac{ds^2 / p}{n\varepsilon^2}}\right)$  for  $\varepsilon \in (0,1]$ . For the second part of the bound, which dominates for  $\varepsilon > 1$ , observe that Assumption 3 holds with  $\sigma^2 \coloneqq \frac{\gamma^2}{(1 - \gamma^2)^2} \leq 2\gamma^2$ ; allowing us to apply the second part of Corollary 1, (7), which as before combined with Lemma 1 yields
$$
d \leq 2 2 4 n \sigma^ {2} \varepsilon \leq 4 4 8 n \gamma^ {2} \varepsilon ,
$$
and again from the setting of  $\gamma$  we get  $\mathcal{E}_p(\mathcal{B}_{d,s},\mathcal{W}^{\mathrm{priv},\varepsilon},n) = \Omega \left(\sqrt{\frac{ds^{2 / p}}{n\varepsilon}}\right)$ .
Similarly, for  $\mathcal{W}^{\mathrm{comm},\ell}$ , again since Assumption 3 holds with  $\sigma^2 \leq 2\gamma^2$ , upon combining the bounds obtained by Corollary 2 and Lemma 1, we get
$$
\frac {d s ^ {\frac {2}{p}}}{2 8 6 7 2 n \ell} \leq \gamma^ {2},
$$
which gives  $\mathcal{E}_p(\mathcal{B}_{d,s},\mathcal{W}^{\mathrm{comm},\ell},n) = \Omega \bigl (\sqrt{\frac{ds^{2 / p}}{n\ell}}\wedge 1\bigr)$ . Finally, note that for  $\ell \geq d$ , the lower bound follows from the minimax rate in the unconstrained setting, which can be seen to be  $\Omega \big(\sqrt{s^{2 / p}\log(2d / s) / n}\big)$  [28, 30]. This completes the proof.
This handles the case  $p \in [1, \infty)$ . For  $p = \infty$ , the lower bounds immediately follow from plugging  $p = \log s$  in the previous expressions, as discussed in Footnote 3.
We denote the mean by  $\mu$  instead of  $\theta$ , denote the estimator by  $\hat{\mu}$ , and consider the minimax error rate  $\mathcal{E}_p(\mathcal{G}_{d,s},\mathcal{W},n)$  of mean estimation for  $\mathcal{P}_{\Theta} = \mathcal{G}_{d,s}$  using  $\mathcal{W}$  under  $\ell_p$  loss.
Proof of Theorem 4. Let  $\varphi$  denote the probability density function of the standard Gaussian distribution  $\mathcal{G}(\mathbf{0},\mathbb{I})$ . Fix  $p\in [1,\infty)$ . Let  $k = d$ ,  $\mathcal{Z} = \{-1, + 1\} ^d$ , and  $\tau = \frac{s}{2d}$ ; and suppose that, for some  $\gamma \in (0,1 / 8]$ , there exists an  $(n,\gamma)$ -estimator for  $\mathcal{G}_{d,s}$  under  $\ell_p$  loss. We fix a parameter  $\gamma \coloneqq \gamma (p)\coloneqq \frac{4\gamma}{(s / 2)^{1 / p}}\in (0,1 / 2]$ , and consider the set of distributions  $\{\mathbf{p}_z\}_{z\in \mathcal{Z}}$  of all  $2^{d}$  spherical Gaussian distributions with mean  $\mu_z\coloneqq \gamma (z + \mathbf{1}_d)$ , where  $z\in \mathcal{Z}$ . Again, note that  $\| \mu_z\| _0 = \sum_{i = 1}^d\mathbb{1}\{z_i = 1\} = \| z\| _+$ , and Fact 2 applies here too. Then by the definition of Gaussian density, for  $z\in \mathcal{Z}$ ,
$$
\mathbf {p} _ {z} (x) = e ^ {- \gamma^ {2} \| \mu_ {z} \| _ {2} ^ {2} / 2} \cdot e ^ {\gamma \langle x, z + \mathbf {1} _ {d} \rangle} \cdot \varphi (x). \tag {41}
$$
Therefore, for  $z\in \mathcal{Z}$  and  $i\in [d]$ , we have
$$
\mathbf {p} _ {z ^ {\oplus i}} (x) = e ^ {- 2 \gamma x _ {i} z _ {i}} e ^ {2 \gamma^ {2} z _ {i}} \cdot \mathbf {p} _ {z} (x) = \left(1 + \phi_ {z, i} (x)\right) \cdot \mathbf {p} _ {z} (x), \tag {42}
$$
where  $\phi_{z,i}(x)\coloneqq 1 - e^{-2\gamma x_i z_i}e^{2\gamma^2 z_i}$ . By using the Gaussian moment-generating function, for  $i\neq j$
$$
\mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) ] = 0, \quad \mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) ^ {2} ] = e ^ {4 \gamma^ {2}} - 1, \mathrm {a n d} \mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) \phi_ {z, j} (X) ] = 0,
$$
so that Assumptions 1 and 2 are satisfied for  $\alpha^2 \coloneqq e^{4\gamma^2} - 1$ . By our choice of  $\gamma$  and the assumption on  $\gamma$ , one can check that Assumption 4 holds:
$$
\ell_ {p} (\mu (\mathbf {p} _ {z}), \mu (\mathbf {p} _ {z ^ {\prime}})) = 4 \gamma \left(\frac {\mathrm {d} _ {\mathrm {H a m}} (z , z ^ {\prime})}{\tau d}\right) ^ {1 / p}.
$$
Moreover, similar to the product of Bernoulli case, using Fact 2, we can show that  $\operatorname{Pr}_Z[\mathbf{p}_Z \in \mathcal{G}_{d,s}] \leq 1 - \tau / 4$ . This allows us to apply Lemma 1.
G.2.1 Privacy constraints for  $\varepsilon \in (0,1)$
For  $\mathcal{W}^{\mathrm{priv},\varepsilon}$ , upon combining the bounds obtained by Corollary 1 and Lemma 1, we get
whereby, upon noting that  $\alpha^2 = e^{4\gamma^2} - 1 \leq 8\gamma^2$  holds since  $\gamma \leq 1/2$ , and using the value of  $\gamma = \gamma(p)$  above, it follows that
$$
\gamma^ {2} \geq \frac {d (s / 2) ^ {\frac {2}{p}}}{1 4 3 3 6 \cdot n (e ^ {\varepsilon} - 1) ^ {2}}.
$$
Thus,  $\mathcal{E}_p(\mathcal{G}_{d,s},\mathcal{W}^{\mathrm{priv},\varepsilon},n) = \Omega \left(\sqrt{\frac{ds^2 / p}{n\varepsilon^2}}\wedge 1\right)$ . This establishes the lower bounds for  $\mathcal{W}^{\mathrm{priv},\varepsilon}$ . (Recall that the bound for  $p = \infty$  then follows from setting  $p = \log d$ .)
G.2.2 Communication constraints, and privacy constraints for  $\varepsilon \geq 1$
For these cases, to prove a lower bound with the desired dependence on  $\varepsilon$  or  $\ell$ , we will need to use the tighter bounds in Corollaries 1 and 2 which hold only under Assumption 3. This, however, leads to an issue: the random vector  $\phi_z(X) = (\phi_{z,i}(X))_{i\in [d]}$  is not subgaussian, due to the one-sided exponential growth, and therefore Assumption 3 does not hold.
To overcome this and still obtain a linear dependence on  $\ell$  (or  $\varepsilon$ ) (instead of the suboptimal  $2^{\ell}$  (or  $e^{\varepsilon}$ )), we will consider instead the class of "truncated" Gaussian distributions, whose corresponding  $\phi$  functions are subgaussian; and argue that these truncated distributions are close enough to the original Gaussian distributions such a lower bound in the truncated case implies one in the original Gaussian case.
In particular, we consider the following collection of truncated Gaussian distributions. For  $z \in \mathcal{Z}$ , let  $\mathbf{p}_z$  be the density function of a spherical Gaussian distribution with mean  $\mu_z$  as defined in Eq. (41). For a truncation bound  $B$ , let  $\mathbf{p}_{z,B}$  be the distribution of  $X \sim \mathbf{p}_z$  conditioned on the event that  $\| X \|_{\infty} \leq B$ . That is, we have, for  $x \in \mathbb{R}^d$ ,
$$
\mathbf {p} _ {z, B} (x) = C _ {z} \mathbf {p} _ {z} (x) \mathbb {1} \left\{\| X \| _ {\infty} \leq B \right\},
$$
where  $C_z = 1 / \operatorname*{Pr}_{X\sim \mathbf{p}_z}[\| X\|_\infty \leq B]$ . Then the following bound follows from standard Gaussian concentration bound on each dimension and a union bound over all dimensions.
Fact 3. Setting  $B \coloneqq 4\sqrt{\ln(dn)}$ , we have, for every  $z \in \mathcal{Z}$ ,  $\mathrm{d}_{\mathrm{TV}}(\mathbf{p}_{z,B}, \mathbf{p}_z) \leq \frac{1}{d^7 n^8}$ .
Let  $\mathbf{p}_{z,B}^{Y^n}$  be the distribution of the messages obtained by executing the protocol when each user gets a sample from  $\mathbf{p}_{z,B}$  and let the corresponding mixtures be denoted by  $\mathbf{p}_{+i,B}^{Y^n}$  and  $\mathbf{p}_{-i,B}^{Y^n}$ . Then we have
$$
\begin{array}{l} \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i} ^ {Y ^ {n}}, \mathbf {p} _ {- i} ^ {Y ^ {n}}\right) \leq \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right) + \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i} ^ {Y ^ {n}}, \mathbf {p} _ {+ i, B} ^ {Y ^ {n}}\right) + \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {- i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i} ^ {Y ^ {n}}\right) \\ \leq \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right) + \max  _ {z} \left\{\mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {z} ^ {Y ^ {n}}, \mathbf {p} _ {z, B} ^ {Y ^ {n}}\right) + \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {z, B} ^ {Y ^ {n}}, \mathbf {p} _ {z} ^ {Y ^ {n}}\right) \right\} \\ \leq \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right) + 2 \max  _ {z} \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {z, B} ^ {\otimes n}, \mathbf {p} _ {z} ^ {\otimes n}\right) \\ \leq \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right) + 2 n \max  _ {z} \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {z, B}, \mathbf {p} _ {z}\right) \\ \leq \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right) + \frac {2}{d ^ {7} n ^ {7}}. \\ \end{array}
$$
The third inequality follows from data processing inequality and the fourth inequality follows from subadditivity of TV distance.
Combining this with Lemma 1, for any protocol that correctly learns the Gaussian family, we must have
$$
\frac {1}{d} \sum_ {i = 1} ^ {d} \mathrm {d} _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right) \geq \frac {1}{8}. \tag {43}
$$
Next we show that the  $\phi$  functions corresponding to  $\mathbf{p}_{z,B}$  's are subgaussian and establish the corresponding upper bounds on the average information bound above. Note that
$$
\phi_ {z, i} ^ {B} (x) := \frac {\mathbf {p} _ {z \oplus i} ^ {B} (x)}{\mathbf {p} _ {z} ^ {B} (x)} - 1 = \frac {C _ {z \oplus i}}{C _ {z}} e ^ {- 2 \gamma x _ {i} z _ {i}} e ^ {2 \gamma^ {2} z _ {i}} \mathbb {1} \left\{\| x \| _ {\infty} \leq B \right\} - 1 \tag {44}
$$
By the inequality  $|ab - 1| \leq |a| \cdot |b - 1| + |a - 1|$ , we have have, for all  $z \in \mathcal{Z}$
$$
\begin{array}{l} \left| \frac {C _ {z ^ {\oplus i}}}{C _ {z}} - 1 \right| \leq \frac {1}{C _ {z}} | C _ {z ^ {\oplus i}} - 1 | + \left| \frac {1}{C _ {z}} - 1 \right| \leq \left| \frac {1}{\operatorname * {P r} _ {X \sim \mathbf {p} _ {z ^ {\oplus i}}} [ \| X \| _ {\infty} \leq B ]} - 1 \right| + \left| \operatorname * {P r} _ {X \sim \mathbf {p} _ {z}} [ \| X \| _ {\infty} \leq B ] - 1 \right| \\ \leq \frac {1 0}{d ^ {7} n ^ {7}}. \\ \end{array}
$$
Moreover, for all  $z\in \mathcal{Z}$  , for  $\gamma \leq \frac{1}{3B}$
$$
\left| e ^ {- 2 \gamma x _ {i} z _ {i}} e ^ {2 \gamma^ {2} z _ {i}} \mathbb {1} \left\{\left\| x \right\| _ {\infty} \leq B \right\} - 1 \right| \leq \left| e ^ {2 \gamma^ {2} + 2 \gamma B} - 1 \right| \leq \left| e ^ {3 \gamma B} - 1 \right| \leq 6 \gamma B. \tag {45}
$$
Hence, applying the inequality  $|ab - 1| \leq |a| \cdot |b - 1| + |a - 1|$  again on Eq. (44), we have for  $\gamma \leq \frac{1}{3B}$ ,
$$
| \phi_ {z, i} ^ {B} (x) | \leq 1 2 \gamma B + \frac {1 0}{d ^ {7} n ^ {7}}.
$$
Thus, we get that for all  $z\in \mathcal{Z},i\in [d]$ $\phi_{z,i}^{B}$  is subgaussian with proxy  $\sigma_B = 12\gamma B + \frac{10}{d^7n^7}$
Under communication constraints, applying Corollary 2, we get
$$
\left(\frac {1}{d} \sum_ {i = 1} ^ {d} d _ {\mathrm {T V}} \left(\mathbf {p} _ {+ i, B} ^ {Y ^ {n}}, \mathbf {p} _ {- i, B} ^ {Y ^ {n}}\right)\right) ^ {2} \leq \frac {1 4}{d} \sigma_ {B} ^ {2} n \ell .
$$
To conclude, we observe that by plugging our setting of  $\gamma = \gamma(p)$  in the above inequality, we must have
$$
\gamma^ {2} \geq \frac {d (s / 2) ^ {\frac {2}{p}}}{1 4 3 3 6 \cdot n \cdot B ^ {2} \ell}
$$
in order to satisfy Eq. (43), hence proving the desired lower bound. The lower bound for LDP with  $\varepsilon > 1$  follows similarly by applying Corollary 1.
G.3 Detailed results for discrete family
We derive a lower bound for  $\mathcal{E}_p(\Delta_d, \mathcal{W}, n)$ , the minimax rate for discrete density estimation, under local privacy and communication constraints.
Theorem 6. Fix  $p\in [1,\infty)$ . For  $\varepsilon >0$ , and  $\ell \geq 1$ , we have
$$
\mathcal {E} _ {p} \left(\Delta_ {d}, \mathcal {W} ^ {\text {p r i v}, \varepsilon}, n\right) \gtrsim \sqrt {\frac {d ^ {2 / p}}{n \left((e ^ {\varepsilon} - 1) ^ {2} \wedge e ^ {\varepsilon}\right)} \wedge \left(\frac {1}{n \left((e ^ {\varepsilon} - 1) ^ {2} \wedge e ^ {\varepsilon}\right)}\right) ^ {\frac {p - 1}{p}}} \wedge 1 \tag {46}
$$
$$
\mathcal {E} _ {p} \left(\Delta_ {d}, \mathcal {W} ^ {\operatorname {c o m m}, \ell}, n\right) \gtrsim \sqrt {\frac {d ^ {2 / p}}{n 2 ^ {\ell}} \wedge \left(\frac {1}{n 2 ^ {\ell}}\right) ^ {\frac {p - 1}{p}}} \wedge 1. \tag {47}
$$
In particular, for  $n\big((e^{\varepsilon} - 1)^{2}\wedge e^{\varepsilon}\big)\geq d^{2}$  and  $n(2^{\ell}\wedge d)\geq d^{2}$ , the first term of the corresponding lower bounds dominates. Before turning to the proof of this theorem, we note that Corollary 3 and Corollary 4 are direct corollaries of the theorem.
We now establish Theorem 6.
Proof of Theorem 6. Fix  $p \in [1, \infty)$ , and suppose that, for some  $\gamma \in (0, 1/16]$ , there exists an  $(n, \gamma)$ -estimator for  $\Delta_d$  under  $\ell_p$  loss. Set
$$
D := d \wedge \left\lfloor \left(\frac {1}{1 6 \gamma}\right) ^ {\frac {p}{p - 1}} \right\rfloor
$$
and assume, without loss of generality, that  $D$  is even. By definition, we then have  $\gamma \in (0,1 / (16D^{1 - 1 / p}))$  and  $D \leq d$ ; we can therefore restrict ourselves to the first  $D$  elements of the domain, embedding  $\Delta_D$  into  $\Delta_d$ , to prove our lower bound.
Let  $k = \frac{D}{2}$ ,  $\mathcal{Z} = \{-1, +1\}^{D/2}$ , and  $\tau = \frac{1}{2}$ ; and suppose that, for some  $\gamma \in (0, 1/(16D^{1-1/p})]$  there exists an  $(n, \gamma)$ -estimator for  $\Delta_D$  under  $\ell_p$  loss. (We will use the fact that  $\gamma \leq 1/(16D^{1-1/p})$  for Eq. (49) to be a valid distribution with positive mass, as we will need  $|\gamma| \leq \frac{1}{D}$ ; and to bound  $\alpha^2$  later on, as we will require  $|\gamma| \leq \frac{1}{2D}$ .) Define  $\gamma = \gamma(p)$  as
$$
\gamma (p) := \frac {4 \cdot 2 ^ {1 / p} \gamma}{D ^ {1 / p}}, \tag {48}
$$
which implies  $\gamma \in [0,1 / (2D)]$ . Consider the set of  $D$ -ary distributions  $\mathcal{P}_{\mathrm{Discrete}}^{\gamma} = \{\mathbf{p}_z\}_{z\in \mathcal{Z}}$  defined as follows. For  $z\in \mathcal{Z}$ , and  $x\in \mathcal{X} = [D]$
$$
\mathbf {p} _ {z} (x) = \left\{ \begin{array}{l l} \frac {1}{D} + \gamma z _ {i}, & \text {i f} x = 2 i, \\ \frac {1}{D} - \gamma z _ {i}, & \text {i f} x = 2 i - 1. \end{array} \right. \tag {49}
$$
For  $z\in \mathcal{Z}$  and  $i\in [D / 2]$ , we have
$$
\begin{array}{l} \mathbf {p} _ {z ^ {\oplus i}} (x) = \left(1 - \frac {2 D \gamma z _ {i}}{1 + D \gamma z _ {i}} \mathbb {1} \{x = 2 i \} + \frac {2 D \gamma z _ {i}}{1 - D \gamma z _ {i}} \mathbb {1} \{x = 2 i - 1 \}\right) \mathbf {p} _ {z} (x) \\ = \left(1 + \phi_ {z, i} (x)\right) \mathbf {p} _ {z} (x), \tag {50} \\ \end{array}
$$
$$
\phi_ {z, i} (x) := z _ {i} \cdot \frac {2 D \gamma}{1 - D ^ {2} \gamma^ {2}} ((1 + D \gamma z _ {i}) \mathbb {1} \{x = 2 i - 1 \} - (1 - D \gamma z _ {i}) \mathbb {1} \{x = 2 i \}).
$$
Once again, we can verify that for  $i \neq j$
$$
\mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) ] = 0, \quad \mathbb {E} _ {\mathbf {p} _ {z}} \big [ \phi_ {z, i} (X) ^ {2} \big ] = \frac {8 \gamma^ {2} D}{1 - \gamma^ {2} D ^ {2}}, \mathrm {a n d} \mathbb {E} _ {\mathbf {p} _ {z}} [ \phi_ {z, i} (X) \phi_ {z, j} (X) ] = 0,
$$
so that Assumptions 1 and 2 are satisfied for  $\alpha^2 \coloneqq 16\gamma^2 D$  (using that  $D\gamma \leq 1/2$  to simplify the bound).<sup>10</sup> Thus, we can invoke the first part of Theorem 2. Note that Assumption 4 holds, since  $\ell_p(\mathbf{p}_z, \mathbf{p}_{z'}) = \gamma \mathrm{d}_{\mathrm{Ham}}(z, z')^{1/p} = 4\gamma \left(\frac{\mathrm{d}_{\mathrm{Ham}}(z, z')}{\tau D}\right)^{1/p}$ . Therefore, we can apply Lemma 1 as well.
For  $\mathcal{W}^{\mathrm{priv},\varepsilon}$ , by combining the bounds obtained by Corollary 1 and Lemma 1, we get
$$
D \leq 5 6 n \alpha^ {2} \left(\left(e ^ {\varepsilon} - 1\right) ^ {2} \wedge e ^ {\varepsilon}\right),
$$
whereby, upon recalling the value of  $\alpha^2$  and using the setting of  $\gamma = \gamma(p)$  from Eq. (48), it follows that
$$
\gamma^ {2} \geq \frac {D ^ {\frac {2}{p}}}{7 1 6 8 \cdot 2 ^ {2 / p} \cdot n ((e ^ {\varepsilon} - 1) ^ {2} \wedge e ^ {\varepsilon})} \asymp \frac {d ^ {2 / p} \wedge \gamma^ {- 2 / (p - 1)}}{n ((e ^ {\varepsilon} - 1) ^ {2} \wedge e ^ {\varepsilon})}.
$$
Thus we obtain the bound Eq. (46) as claimed.
Similarly, for  $\mathcal{W}^{\mathrm{comm},\ell}$ , upon combining the bounds obtained by Corollary 2 and Lemma 1 and recalling that  $|\mathcal{Y}| = 2^{\ell}$ , we get
$$
\gamma^ {2} \geq \frac {D _ {p} ^ {\frac {2}{p}}}{7 1 6 8 \cdot 2 ^ {2 / p} \cdot n 2 ^ {\ell}},
$$
which gives  $\mathcal{E}_p(\Delta_D, \mathcal{W}^{\mathrm{comm},\ell}, n) = \Omega \left(\sqrt{\frac{d^2/p}{n2^\ell} \wedge \left(\frac{1}{n2^\ell}\right)^{\frac{p-1}{p}}}\right)$ , concluding the proof.
<sup>10</sup>It is worth noting that Assumption 3 will not hold for any useful choice of the subgaussianity parameter.
Finally, note that we could replace the quantity  $2^{\ell}$  above by  $2^{\ell} \wedge d$ , or even  $2^{\ell} \wedge D$ , as for  $2^{\ell} \geq D$  there is no additional information any player can send beyond the first  $\log_2 D$  bits, which encode their full observation. However, this small improvement would lead to more cumbersome expressions, and not make any difference for the main case of interest,  $p = 1$ .