SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
h _ {i} (X _ {i}, Y _ {i}, (Z _ {i}, \eta_ {i} ^ {2})) = \frac {\frac {1}{\sigma^ {2}} \sum_ {x \in X _ {i}} x + \frac {1}{\sigma^ {2} + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1} \sigma^ {2}} \sum_ {x \in Z _ {i}} x}{\frac {n _ {i} ^ {\star}}{\sigma^ {2}} + \frac {(m - 1) n _ {i} ^ {\star}}{\sigma^ {2} + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1} \sigma^ {2}}}
$$
The maximum risk of  $h_i$  is
$$
\begin{array}{l} \mathbb {E} \Big [ \big (h _ {i} (X _ {i}, Y _ {i}, (Z _ {i}, \eta_ {i} ^ {2})) - \mu \big) ^ {2} \Big ] = \frac {1}{\frac {n _ {i} ^ {\star}}{\sigma^ {2}} + \frac {(m - 1) n _ {i} ^ {\star}}{\sigma^ {2} + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1} \sigma^ {2}}} = \frac {1}{1 + \frac {m - 1}{1 + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1}}} \frac {\sigma^ {2}}{n _ {i} ^ {\star}} = \frac {1 + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1}}{m + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1}} \frac {\sigma^ {2}}{n _ {i} ^ {\star}} \\ <   \frac {\left(1 + \frac {1}{k _ {\epsilon}}\right) \left(1 + \frac {1}{k _ {\epsilon}} \frac {1}{m - 1}\right)}{m + \frac {1}{k _ {\epsilon}} \frac {m}{m - 1}} \frac {\sigma^ {2}}{n _ {i} ^ {\star}} = \left(1 + \frac {1}{k _ {\epsilon}}\right) \frac {\sigma^ {2}}{m n _ {i} ^ {\star}} \tag {33} \\ \end{array}
$$
Note that the RHS of (33) is the risk of the sample average deployed by  $M_{\mathrm{CDED}}$ . This means, suppose all other agents choose  $s^{\star}$ , then agent  $i$  can choose a weighted average to reduce her penalty without collecting more data.
E High dimensional mean estimation with bounded variance
In this section, we will study estimating a  $d$ -dimensional mean  $\mu(\theta) \in \mathbb{R}^d$  for distributions  $\theta$  with bounded variance. We will focus on our original setting in §2, but will outline the modifications to the formalism to accommodate the generality. For  $x \in \mathbb{R}^d$ , let  $x^{(i)}$  denote the  $i^{\text{th}}$  dimension.
Modifications to the setting in §2: First, we should change the definitions of  $\mathcal{F},\mathcal{H}$  and  $\mathcal{M}$  in equations 1 and (2) to account for the fact that the data is  $d$  dimensional. For instance, the space of functions mapping the dataset collected to the dataset submitted should be defined as  $\mathcal{F} = \{f:\bigcup_{n\geq 0}\mathbb{R}^{d\times n}\to \bigcup_{n\geq 0}\mathbb{R}^{d\times n}\}$ . Next, let  $\Theta = \{\theta ;\mathrm{supp}(\theta)\subset \mathbb{R}^d,\mathbb{E}_{x\sim \theta}\left[(x^{(i)} - \mu (\theta)^{(i)})^2\right]\leq \sigma^2,\forall i\in [d]\}$  be the class of all  $d$ -dimensional distributions where the variance along each dimension is bounded by  $\sigma^2$ . Here, the maximum variance  $\sigma^2$  is known and is public information. Note that we do not assume that the individual dimensions are independent. An agent's penalty  $p_i$  is defined similar to (3) but considers the maximum risk over  $\Theta$ , i.e
$$
p _ {i} (M, s) = \sup  _ {\theta \in \Theta} \mathbb {E} \left[ \| h _ {i} \left(X _ {i}, Y _ {i}, A _ {i}\right) - \mu (\theta) \| _ {2} ^ {2} | \theta \right] + c n _ {i}. \tag {34}
$$
Finally, the social penalty and ratio PR are as defined in (5), but with the above definition for  $p_i$ .
**Mechanism:** Our mechanism for this problem is the same as the one outlined in Algorithm 1, with the following cosmetic modifications. First, the allocation space should now be  $\mathcal{A} = \bigcup_{n\geq 0}\mathbb{R}^{d\times n}\times$ $\bigcup_{n\geq 0}\mathbb{R}^{d\times n}\times \mathbb{R}_{+}^{d}$ . The noise modulating parameter  $\alpha$  is determined by a similar equation as in (7), but with  $c$  replaced with  $c / d$ . In line 12 of Algorithm 1, we should set the size of the dataset  $Z_{i}$  to be  $\min \{|Y_{-i}|,\sigma \sqrt{d / (cm)}\}$ . Finally, the operations in lines 13 and 14 should be interpreted as  $d-$  dimensional operations that are performed elementwise. The recommended strategy  $s_i^\star = (n_i^\star ,f_i^\star ,h_i^\star)$  for agent  $i$  is as follows:
$$
n _ {i} ^ {\star} = \left\{ \begin{array}{l l} \frac {\sigma}{m} \sqrt {\frac {d}{c}} & \text {i f} m \leq 4, \\ \sigma \sqrt {\frac {d}{c m}} & \text {i f} m \geq 5 \end{array} , \quad f _ {i} ^ {\star} = \mathbf {I}, \right. \tag {35}
$$
$$
h _ {i} ^ {\star} (X _ {i}, Y _ {i}, (Z _ {i}, Z _ {i} ^ {\prime}, \eta_ {i} ^ {2})) = \frac {\frac {1}{\sigma^ {2}} \sum_ {u \in X _ {i} \cup Z _ {i}} u + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} \sum_ {u \in Z _ {i} ^ {\prime}} u}{\frac {1}{\sigma^ {2}} | X _ {i} \cup Z _ {i} ^ {\prime} | + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} | Z _ {i} ^ {\prime} |}, \qquad \mathrm {w h e r e ,} \tau_ {i} ^ {2} = \frac {2 \alpha^ {2} \sigma^ {2}}{n _ {i} ^ {\star}} \in \mathbb {R} _ {+}.
$$
Above, one difference worth highlighting is the change in the recommended estimator  $h_i^\star$ . Previously, the weighting used the  $\eta_i^2$  term returned by the mechanism, which is a function of  $Y_{i}$  and  $Z_{i}$ . This data-dependent weighting was necessary to obtain an exactly (i.e including constants) minimax optimal estimator for the corrupted dataset, which in turn was necessary to achieve an exact Nash equilibrium. However, bounding the risk when using a data-dependent weighting is challenging when the Gaussian assumption does not hold. Instead, here we use a deterministic weighting via the quantity  $\tau_i^2$ . While this is not exactly minimax optimal, we can show that its maximum risk is very close to a lower bound, which helps us obtain an approximate Nash equilibrium. It is worth pointing out that designing exactly minimax optimal estimators, even under i.i.d assumptions, is challenging for general classes of distributions [24].
The following theorem states the main properties of this mechanism.
Theorem 7. The following statements are true about the mechanism  $M_{\mathrm{C3D}}$  in Algorithm 1 with the above modifications. (i) The strategy profile  $s^{\star}$  as defined in (35) is an approximate Nash equilibrium, i.e. if all agents except  $i$  are following  $s^{\star}$ , then for any alternative strategy  $s_i$  for agent  $i$ , we have  $p_i(M_{\mathrm{C3D}}, s^{\star}) \leq p_i(M_{\mathrm{C3D}}, (s_{-i}^{\star}, s_i))(1 + 5/m)$  (ii) The mechanism is individually rational at  $s^{\star}$ . (iii) The mechanism is approximately efficient at  $s_i^{\star}$ , with  $\mathrm{PR}(M_{\mathrm{C3D}}, s^{\star}) < 2 + 10/m$ .
We see that even under this more general setting, our mechanism retains its main properties with only a slight weakening of the results. We now have approximate, instead of exact, NIC, with the benefit of deviation diminishing as there are more agents. Similarly, the bound on the efficiency is only slightly weaker than the one in Theorem 1.
When  $m \leq 4$ , the claims follow using the exact steps in §A.1. Therefore, we focus on the case  $m \geq 5$ . Moreover, some of the key steps of this proof follows along similar lines to Theorem 1, so we will provide an outline and focus on the differences.
Approximate Nash incentive compatibility. We will first prove the statement  $(i)$  of Theorem 7, which states that  $s_i^\star$ , as defined in (35), is an approximate Nash equilibrium for  $M_{\mathrm{c3D}}$ . That is, we will show that the maximum possible reduction in penalty for an agent  $i$  when deviating from  $s_i^\star$  is small, provided that all other agents are following  $s_{-i}^\star$ .
For this, we will first lower bound the penalty  $p_i$  (34) using the family of independent Gaussian distributions. Let  $\Theta_{\mathcal{N}} = \{\mathcal{N}(\mu, \sigma^2 I_d) : \mu \in \mathbb{R}^d\}$  denote the space of  $d$ -dimensional normal distributions with identity covariance matrix. For any mechanism  $M$  and strategy profile  $s \in S^m$ , we define the penalty of agent  $i$  restricted to  $\Theta_{\mathcal{N}}$  as:
$$
p _ {i} ^ {\mathcal {N}} (M, s) = \sup _ {\theta \in \Theta_ {\mathcal {N}}} \mathbb {E} \left[ \| h _ {i} (X _ {i}, Y _ {i}, A _ {i}) - \mu (\theta) \| _ {2} ^ {2} \mid \theta \right] + c n _ {i}.
$$
Since  $\Theta_{\mathcal{N}} \subset \Theta$ , it is straightforward to see that for all  $M \in \mathcal{M}$  and  $s \in S^{m}$ ,
$$
p _ {i} ^ {\mathcal {N}} (M, s) \leq p _ {i} (M, s). \tag {36}
$$
We will now use this result to lower bound the penalty of an agent for any other alternative strategy. First note that, by independence, the mean estimation problem on  $\Theta_{\mathcal{N}}$  can be viewed as  $d$  independent copies of the univariate normal mean estimation problem considered in Theorem 1 but with  $c$  replaced with  $c / d$ . Let  $\tilde{h}_i^\star$  be the weighted average that applies the estimator in (8) along each dimension. And let  $\tilde{s}_i^\star = (n_i^\star, f_i^\star, \tilde{h}_i^\star)$ . We can now lower bound the penalty of agent  $i$  when following any (alternative) strategy  $s_i \in S$ , provided that other agents are following  $s_{-i}^\star$ . We have:
$$
\begin{array}{l} p _ {i} \left(M _ {\mathrm {C 3 D}}, \left(s _ {i}, s _ {- i} ^ {\star}\right)\right) = p _ {i} \left(M _ {\mathrm {C 3 D}}, \left(s _ {i}, \left(n _ {- i} ^ {\star}, f _ {- i} ^ {\star}, h _ {- i} ^ {\star}\right)\right)\right) \\ \geq p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {c 3 d}}, \left(s _ {i}, \left(n _ {- i} ^ {\star}, f _ {- i} ^ {\star}, h _ {- i} ^ {\star}\right)\right)\right) \tag {By(36)} \\ = p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}}, \left(s _ {i}, \left(n _ {- i} ^ {\star}, f _ {- i} ^ {\star}, \tilde {h} _ {- i} ^ {\star}\right)\right)\right) \\ \end{array}
$$
(As agent  $i$ 's penalty will not be affected by other agents' estimators)
$$
\geq p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}}, \left(\left(n _ {i} ^ {\star}, f _ {i} ^ {\star}, \tilde {h} _ {i} ^ {\star}\right), \left(n _ {- i} ^ {\star}, f _ {- i} ^ {\star}, \tilde {h} _ {- i} ^ {\star}\right)\right)\right)
$$
( By adapting the analysis in §A.3.)
$$
= p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}}, \tilde {s} ^ {\star}\right) \tag {37}
$$
Above, the second step uses (36) and the third step uses the fact that other agent's estimator will not affect agent  $i$ 's penalty. The fourth step uses the fact that for estimation problems in  $\Theta_{\mathcal{N}}$ , the strategy profile  $\tilde{s}^{\star} = \{(n_i^{\star}, f_i^{\star}, \tilde{h}_i^{\star})\}_i$  is a Nash equilibrium; in §A.3, we showed this for the one-dimensional case, but this proof can be easily adapted to  $d$  dimensions since we are assuming an identity covariance matrix in  $\Theta_{\mathcal{N}}$ . Finally, by adapting the analysis in §A.5, we can obtain the following expression for agent  $i$ 's penalty  $p_i^{\mathcal{N}}(M_{\mathrm{c3D}}, \tilde{s}^{\star})$  in  $\Theta_{\mathcal{N}}$ :
$$
p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}}, \tilde {s} ^ {\star}\right) = d \sigma \sqrt {\frac {c / d}{m}} \left(\frac {\frac {1 0 \alpha^ {2}}{n _ {i} ^ {\star}} - 1}{\frac {4 \alpha^ {2}}{n _ {i} ^ {\star}} \frac {m + 1}{m} - 1} + 1\right) \tag {38}
$$
To state the approximate NIC result, we will now upper bound the penalty of the agent when following  $s_i^*$ . Using the bounded variance assumption, we have:
$$
\begin{array}{l} p _ {i} (M, s ^ {\star}) = \sup  _ {\theta \in \Theta} \mathbb {E} \left[ \right.\left\| \frac {\frac {1}{\sigma^ {2}} \sum_ {u \in X _ {i} \cup Z _ {i}} u + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} \sum_ {u \in Z _ {i} ^ {\prime}} u}{\frac {1}{\sigma^ {2}} | X _ {i} \cup Z _ {i} ^ {\prime} | + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} | Z _ {i} ^ {\prime} |} - \mu (\theta) \right\| _ {2} ^ {2} \left. \right\rvert   \theta \left. \right] + c n _ {i} ^ {\star} \\ = \sup _ {\theta \in \Theta} \sum_ {k = 1} ^ {d} \mathbb {E} \left[ \left. \left(\frac {\frac {1}{\sigma^ {2}} \sum_ {u \in X _ {i} \cup Z _ {i}} \big (u ^ {(k)} - \mu (\theta) ^ {(k)} \big) + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} \sum_ {u \in Z _ {i} ^ {\prime}} \big (u ^ {(k)} - \mu (\theta) ^ {(k)} \big)}{\frac {1}{\sigma^ {2}} | X _ {i} \cup Z _ {i} ^ {\prime} | + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} | Z _ {i} ^ {\prime} |}\right) ^ {2} \right| \theta \right] + c n _ {i} ^ {\star} \\ \end{array}
$$
Figure 2:  $E(m)$  plot. See G_em_plot.py.
$$
\begin{array}{l} = \sup  _ {\theta \in \Theta} \sum_ {k = 1} ^ {d} \frac {\frac {1}{\sigma^ {2}} \sum_ {u \in X _ {i} \cup Z _ {i}} \mathbb {E} \left[ \left(u ^ {(k)} - \mu (\theta) ^ {(k)}\right) \right] + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} \sum_ {u \in Z _ {i} ^ {\prime}} \mathbb {E} \left[ \left(u ^ {(k)} - \mu (\theta) ^ {(k)}\right) \right]}{\frac {1}{\sigma^ {2}} | X _ {i} \cup Z _ {i} ^ {\prime} | + \frac {1}{\sigma^ {2} + \tau_ {i} ^ {2}} | Z _ {i} ^ {\prime} |} + c n _ {i} ^ {\star} (39) \\ \leq \frac {d}{\frac {2 n _ {i} ^ {\star}}{\sigma^ {2}} + \frac {(m - 2) n _ {i} ^ {\star}}{\sigma^ {2} + \frac {2 \alpha^ {2} \sigma^ {2}}{n _ {i} ^ {\star}}}} + c n _ {i} ^ {\star} = \frac {\sigma^ {2}}{n _ {i} ^ {\star}} \frac {d}{2 + \frac {m - 2}{1 + \frac {2 \alpha^ {2}}{n _ {i} ^ {\star}}}} + c n _ {i} ^ {\star} = \sigma \sqrt {\frac {c d}{m}} \left(\frac {m}{2 + \frac {m - 2}{1 + \frac {2 \alpha^ {2}}{n _ {i} ^ {\star}}}} + 1\right), (40) \\ \end{array}
$$
where (39) is because: for all  $k \in [d]$ ,  $\forall x_1^{(k)}, x_2^{(k)} \in X_i \cup Z_i$ ,  $\forall z_1^{(k)}, z_2^{(k)} \in Z_i'$ ,  $x_1^{(k)} - \mu^{(k)}, x_2^{(k)} - \mu^{(k)}, z_1^{(k)} - \mu^{(k)}, z_2^{(k)} - \mu^{(k)}$  are uncorrelated pairwise. The final inequality is due to the bounded variance assumption.
Next, for brevity, let us write  $A_{m} \coloneqq \frac{\alpha}{\sqrt{n_{i}^{*}}}$  where  $\alpha$  is as defined in (7). By adapting the analysis in §A.2, we can show that
$$
A _ {m} := \frac {\alpha}{\sqrt {n _ {i} ^ {*}}} \in \left(1, 1 + \frac {C _ {m}}{m}\right), \quad \text {w h e r e ,} \quad C _ {m} = \left\{ \begin{array}{l l} 2 0, & \text {i f} m \leq 2 0 \\ 5, & \text {i f} m > 2 0 \end{array} . \right. \tag {41}
$$
By combining the results in (37), (40), and (41), we obtain the following bound:
$$
\begin{array}{l} \frac {p _ {i} (M _ {\mathrm {C 3 D}} , s ^ {\star})}{\inf _ {s _ {i}} p _ {i} (M _ {\mathrm {C 3 D}} , (s _ {i} , s _ {- i} ^ {\star}))} - 1 \leq \frac {p _ {i} (M _ {\mathrm {C 3 D}} , s ^ {\star})}{p _ {i} ^ {\mathcal {N}} (M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star})} - 1 \\ \leq \frac {\sigma \sqrt {\frac {c d}{m}} \left(\frac {m}{2 + \frac {m - 2}{1 + 2 A _ {m} ^ {2}}} + 1\right)}{d \sigma \sqrt {\frac {c / d}{m}} \left(\frac {1 0 A _ {m} ^ {2} - 1}{4 A _ {m} ^ {2} \frac {m + 1}{m} - 1} + 1\right)} - 1 = \frac {\frac {m}{2 + \frac {m - 2}{1 + 2 \frac {\alpha^ {2}}{n _ {i} ^ {\star}}}} + 1}{\frac {\frac {1 0 \alpha^ {2}}{n _ {i} ^ {\star}} - 1}{\frac {4 \alpha^ {2}}{n _ {i} ^ {\star}} \frac {m + 1}{m} - 1} + 1} - 1 \\ = \frac {4 A _ {m} ^ {2} \left((A _ {m} ^ {2} - 1) m + 1 - 4 A _ {m} ^ {2}\right) m}{(4 A _ {m} ^ {2} + m) ((7 A _ {m} ^ {2} - 1) m + 2 A _ {m} ^ {2})} =: E (m). \tag {42} \\ \end{array}
$$
Let  $E(m)$  denote the final upper bound obtained above. Next, we will prove  $E(m) < 5 / m$ . When  $m \in [5, 500]$ , this can be individually verified for each value of  $E(m)$  (see Figure 2). When  $m \geq 500$ , we have  $A_{m} \leq 1.01$  (see (41)). From this we can conclude,
$$
E (m) \leq \frac {4 \times 1 . 0 1 ^ {2} \times (2 . 0 1 \times \frac {5}{m} m - 3) m}{6 m ^ {2}} <   \frac {5}{m}. \tag {43}
$$
Combining the results in (42) and (43), we obtain the following approximate NIC result:
$$
\forall i \in [ m ], s _ {i} \in \mathcal {S}, \quad p _ {i} \left(M _ {\mathrm {C 3 D}}, s ^ {\star}\right) \leq p _ {i} \left(M _ {\mathrm {C 3 D}}, \left(s _ {i}, s _ {- i} ^ {\star}\right)\right) \left(1 + \frac {5}{m}\right).
$$
Individual rationality: This proof is very similar to the proof in §A.4. In particular, using calculations similar to (40), we can show that regardless of the choice of  $n_i$ , the agent's penalty is strictly
smaller when using the uncorrupted  $(Z_{i})$  and corrupted  $(Z_{i}^{\prime})$  datasets along with the weighted average in (35).
Approximate efficiency: To bound the penalty ratio, first note that by (38) and using the same reasoning as §A.5, we have that
$$
\frac {\sum_ {i} p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star}\right)}{\inf  _ {M \in \mathcal {M} , s \in \mathcal {S} ^ {m}} \sum_ {i} p _ {i} ^ {\mathcal {N}} (M , s)} = \frac {m p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star}\right)}{\inf  _ {M \in \mathcal {M} , s \in \mathcal {S} ^ {m}} \sum_ {i} p _ {i} ^ {\mathcal {N}} (M , s)} = \frac {m p _ {i} ^ {N} \left(M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star}\right)}{2 \sigma \sqrt {c m d}} \leq 2. \tag {44}
$$
Next, as  $\Theta_{\mathcal{N}} \subset \Theta$ , and noting that  $P(M,s) = \sum_{i} p_{i}(M,s)$  for all  $M,s$ , we can also write,
$$
\inf  _ {M \in \mathcal {M}, s \in \mathcal {S} ^ {m}} \sum_ {i} p _ {i} ^ {\mathcal {N}} (M, s) \leq \inf  _ {M \in \mathcal {M}, s \in \mathcal {S} ^ {m}} P (M, s). \tag {45}
$$
We can combine the above results to obtain the following upper bound on PR:
$$
\begin{array}{l} \Pr \left(M _ {\mathrm {C} 3 \mathrm {D}}, s ^ {\star}\right) = \frac {P \left(M _ {\mathrm {C} 3 \mathrm {D}} , s ^ {\star}\right)}{\inf  _ {M \in \mathcal {M} , s \in \mathcal {S} ^ {m}} P (M , s)} \leq \frac {m p _ {i} \left(M _ {\mathrm {C} 3 \mathrm {D}} , s ^ {\star}\right)}{\inf  _ {M \in \mathcal {M} , s \in \mathcal {S} ^ {m}} \sum_ {i} p _ {i} ^ {\mathcal {N}} (M , s)} \quad (\text {B y}) \\ = \frac {m p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star}\right)}{\inf  _ {M \in \mathcal {M} , s \in \mathcal {S} ^ {m}} \sum_ {i} p _ {i} ^ {\mathcal {N}} (M , s)} \frac {p _ {i} \left(M _ {\mathrm {C 3 D}} , s ^ {\star}\right)}{p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star}\right)} \\ \leq 2 \frac {p _ {i} \left(M _ {\mathrm {C 3 D}} , s ^ {\star}\right)}{p _ {i} ^ {\mathcal {N}} \left(M _ {\mathrm {C 3 D}} , \tilde {s} ^ {\star}\right)} (By(44)) \\ = 2 (1 + E (m))) \quad (\text {B y}) \\ <   2 + \frac {1 0}{m}. (By(43)) \\ \end{array}
$$
This establishes approximate efficiency for  $M_{\mathbb{C3D}}$  for the high dimensional setting.
F Application to Bayesian Settings
While our results study the Normal mean estimation in frequentist statistics, the main ideas can also be applied to the Bayesian setting. When the Normal mean admits a zero-mean normal prior, the major proof steps remain the same. Specifically, our current analysis constructs a sequence of Gaussian priors and takes the limit to prove the minimax optimality. In the Bayesian setting, one can simply skip the step in (22), which takes the limit w.r.t. the prior sequence. The other steps remain the same.
G Useful Results
In this section, we will state some useful results that we have used throughout this proof.
Lemma 8 (Hardy-Littlewood inequality, Lemma 1.6 in Burchard [10]). Let  $f$  and  $g$  be non-negative measurable functions that vanish at infinity. Let  $f^{*}$  and  $g^{*}$  to denote the symmetric decreasing rearrangement of  $f$  and  $g$ . If  $\int f^{*}g^{*} < \infty$ , then,
$$
\int f g \leq \int f ^ {*} g ^ {*}.
$$
Next, we will use the above result to derive a corollary that will be useful in our proofs.
Lemma 9 (A corollary of Hardy-Littlewood). Let  $f, g$  be nonnegative even functions such that,
-  $f$  is monotonically increasing on  $[0, \infty)$ .
-  $g$  is monotonically decreasing on  $[0, \infty)$ , and has a finite integral  $\int_{\mathbb{R}} g(x) dx < \infty$ .
$\forall a,\int_{\mathbb{R}}f(x - a)g(x)dx <   \infty .$
Then for all  $a$
$$
\int_ {\mathbb {R}} f (x) g (x) d x \leq \int_ {\mathbb {R}} f (x - a) g (x) d x
$$
Proof. We will break this proof into two cases. The first is when  $\sup f < \infty$  and the second is when  $\sup f = \infty$ . First consider the case  $\sup f < \infty$ . Let
$$
M := \lim  _ {x \to \infty} f (x).
$$
By using Lemma 8,  $\forall a$
$$
\int_ {\mathbb {R}} (M - f (x)) g (x) d x \geq \int_ {\mathbb {R}} (M - f (x - a)) g (x) d x.
$$
The result follows after rearrangement.
If  $\sup f = \infty$ , let  $f_{n}(x) \coloneqq \min \{f(x), n\}$ . For all  $n$  and  $a$ , by Lemma 8,
$$
\int_ {\mathbb {R}} \left(n - f _ {n} (x)\right) g (x) d x \geq \int_ {\mathbb {R}} \left(n - f _ {n} (x - a)\right) g (x) d x,
$$
$$
\int_ {\mathbb {R}} f _ {n} (x) g (x) d x \leq \int_ {\mathbb {R}} f _ {n} (x - a) g (x) d x.
$$
Note that  $|f_n(x)g(x)| \leq f(x)g(x)$ , the result follows by letting  $n \to \infty$  on both sides and using dominated convergence theorem.
Below, we provide a brief example on using Lemma 9 to calculate the Bayes risk in a normal mean estimation problem with i.i.d data. While it is not necessary to use Hardy-Littlewood for this problem, this example will illustrate how we have used it in our proofs.
Example 1. Consider the Normal mean estimation problem given samples  $X_{[n]}\sim \mathcal{N}(\mu ,\sigma^2)$ , where  $\mu$  admits a prior distribution  $\mathcal{N}(0,\ell^2)$ . The goal is to minimize the average risk:
$$
\mathbb {E} _ {\mu \sim \mathcal {N} (0, \ell^ {2})} \left[ \mathbb {E} _ {X _ {[ n ]} \sim \mathcal {N} (\mu , \sigma^ {2})} [ L (\hat {\mu} - \mu) | \mu ] \right],
$$
where the loss function,  $L(\cdot)$ , is an even function that increases on  $[0,\infty)$ . By a standard argument, one can show that the posterior distribution of  $\mu$  conditioned on  $X_{[n]}$  is Gaussian with data-dependent parameters  $\bar{\mu},\bar{\sigma}^2$ :
$$
\mu | X _ {[ n ]} \sim \mathcal {N} (\bar {\mu}, \bar {\sigma} ^ {2}).
$$
The posterior risk is:
$$
\mathbb{E}_{\mu |X_{[n]}}[L(\hat{\mu} -\mu)] = \mathbb{E}_{\mu |X_{[n]}}[L((\mu -\overline{\mu}) + (\overline{\mu} -\hat{\mu}))] = \int_{\mathbb{R}}\underbrace{L(x + (\overline{\mu} -\hat{\mu}))}_{=: f(x + (\overline{\mu} -\hat{\mu}))}\underbrace{\exp\left(-\frac{x^{2}}{2\overline{\sigma}^{2}}\right)}_{=: g(x)}dx
$$
By applying Lemma 9 with  $f$  and  $g$ , the posterior risk above is minimized when  $\hat{\mu} = \overline{\mu}$ . So is the average risk.
The next Lemma shows that convexity is preserved under expectation under certain conditions.
Lemma 10. Let  $y$  be a random variable and  $f(x,y)$  be a function s.t.
$f(x,y)$  is convex in  $x$
-  $\mathbb{E}_y[|f(x,y)|] < \infty$  for all  $x$ .
Then  $\mathbb{E}_y[f(x,y)]$  is also convex in  $x$ .
Proof. For any  $x_1, x_2$ , we have
$$
\frac {\mathbb {E} _ {y} [ f (x _ {1} , y) ] + \mathbb {E} _ {y} [ f (x _ {2} , y) ]}{2} = \mathbb {E} _ {y} \left[ \frac {f (x _ {1} , y) + f (x _ {2} , y)}{2} \right] \geq \mathbb {E} _ {y} \left[ f \left(\frac {x _ {1} + x _ {2}}{2}, y\right) \right]
$$
Lemma 11 (Centered moments of normal random variable). Let  $X \sim \mathcal{N}(\mu, \sigma^2)$  be a normal random variable and  $p \in \mathbb{Z}_+$ , then
$$
\mathbb {E} [ (X - \mu) ^ {p} ] = \left\{ \begin{array}{l l} 0 & \text {i f p i s o d d} \\ \sigma^ {p} (p - 1)!! & \text {i f p i s e v e n} \end{array} \right..
$$
G.1 Some technical results
Next, we will state some technical results that were obtained purely using algebraic manipulations and are not central to the main proof ideas. The first result states upper and lower bounds on the Gaussian complementary error function using an asymptotic expansion.
Lemma 12 (Erfc bound). For all  $x > 0$
$$
\operatorname {E r f c} (x) \leq \frac {1}{\sqrt {\pi}} \left(\frac {\exp (- x ^ {2})}{x} - \frac {\exp (- x ^ {2})}{2 x ^ {3}} + \frac {3 \exp (- x ^ {2})}{4 x ^ {5}}\right) \tag {46}
$$
$$
\operatorname {E r f c} (x) \geq \frac {1}{\sqrt {\pi}} \left(\frac {\exp (- x ^ {2})}{x} - \frac {\exp (- x ^ {2})}{2 x ^ {3}}\right) \tag {47}
$$
Proof. By integration by parts: