SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} T _ {1} = E \left[ \frac {n}{n _ {k}} \widehat {\zeta} _ {k} (V _ {i}) \left\{\frac {I (A _ {i} = a , R _ {i} = k)}{\widehat {\pi} _ {a , k} (X _ {i})} - I (R _ {i} = k) \right\} \left\{m _ {a, k} (X _ {i}) - \widehat {m} _ {a, k} (X _ {i}) \right\} \right] \\ = E \left[ \frac {n}{n _ {k}} \widehat {\zeta} _ {k} (V _ {i}) E \left\{\frac {I \left(A _ {i} = a , R _ {i} = k\right)}{\widehat {\pi} _ {a , k} (X _ {i})} - I \left(R _ {i} = k\right) \mid X _ {i} \right\} \left\{m _ {a, k} \left(X _ {i}\right) - \widehat {m} _ {a, k} \left(X _ {i}\right) \right\} \right] \\ = E \left[ \frac {n}{n _ {k}} \widehat {\zeta} _ {k} (V _ {i}) P (R _ {i} = k \mid X _ {i}) \left\{\frac {\pi_ {a , k} (X _ {i})}{\widehat {\pi} _ {a , k} (X _ {i})} - 1 \right\} \left\{m _ {a, k} (X _ {i}) - \widehat {m} _ {a, k} (X _ {i}) \right\} \right] \\ = E \left[ \frac {\widehat {\zeta} _ {k} \left(V _ {i}\right)}{\widehat {\pi} _ {a , k} \left(X _ {i}\right)} \left\{\pi_ {a, k} \left(X _ {i}\right) - \widehat {\pi} _ {a, k} \left(X _ {i}\right) \right\} \left\{m _ {a, k} \left(X _ {i}\right) - \widehat {m} _ {a, k} \left(X _ {i}\right) \right\} \right] \tag {46} \\ \end{array}
$$
Given the boundedness of  $\widehat{\zeta}_k$  and  $\widehat{\pi}_{a,k}$ , it can be concluded that  $T_{1} = O_{p}(\| \widehat{\pi}_{a,k} - \pi_{a,k}\| \| \widehat{m}_{a,k} - m_{a,k}\|)$ . Now let us consider  $T_{2}$ . Conditioning on  $V_{i}$ , we have the following expression:
$$
\begin{array}{l} T _ {2} = E \left[ \left\{\frac {n}{n _ {T}} I (R _ {i} = T) - \frac {n}{n _ {k}} I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \right\} \left\{\widehat {\tau} _ {a, k} (V _ {i}) - \tau_ {a, T} (V _ {i}) \right\} \right] \\ = E \left[ E \left\{\frac {n}{n _ {T}} I (R _ {i} = T) - \frac {n}{n _ {k}} I (R _ {i} = k) \widehat {\zeta} _ {k} (V _ {i}) \mid V _ {i} \right\} \left\{\widehat {\tau} _ {a, k} (V _ {i}) - \tau_ {a, T} (V _ {i}) \right\} \right] \\ = E \left[ \left\{\frac {n}{n _ {T}} P \left(R _ {i} = T \mid V _ {i}\right) - \frac {n}{n _ {k}} P \left(R _ {i} = k \mid V _ {i}\right) \widehat {\zeta} _ {k} \left(V _ {i}\right) \right\} \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \tau_ {a, T} \left(V _ {i}\right) \right\} \right] \\ = E \left[ \left\{\frac {1}{P \left(R _ {i} = T\right)} P \left(R _ {i} = T \mid V _ {i}\right) - \frac {1}{P \left(R _ {i} = k\right)} P \left(R _ {i} = k \mid V _ {i}\right) \widehat {\zeta} _ {k} \left(V _ {i}\right) \right\} \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \tau_ {a, T} \left(V _ {i}\right) \right\} \right] \\ = E \left[ \frac {P \left(R _ {i} = k \mid V _ {i}\right)}{P \left(R _ {i} = k\right)} \left\{\zeta_ {k} \left(V _ {i}\right) - \widehat {\zeta} _ {k} \left(V _ {i}\right) \right\} \left\{\widehat {\tau} _ {a, k} \left(V _ {i}\right) - \tau_ {a, T} \left(V _ {i}\right) \right\} \right]. \tag {47} \\ \end{array}
$$
Therefore,  $T_{2} = O_{p}(\| \widehat{\zeta}_{k} - \zeta_{k}\| \| \widehat{\tau}_{a,k} - \tau_{a,k}\|)$
Combining all results from the above three steps yields
$$
\left\| \widehat {\mu} _ {a, k} - \mu_ {a, T} \right\| = O _ {p} \left(n ^ {- 1 / 2} + \left\| \widehat {\pi} _ {a, k} - \pi_ {a, k} \right\| \left\| \widehat {m} _ {a, k} - m _ {a, k} \right\| + \left\| \widehat {\zeta} _ {k} - \zeta_ {k} \right\| \left\| \widehat {\tau} _ {a, k} - \tau_ {a, k} \right\|\right). \tag {48}
$$
$$
\left\| \widehat {m} _ {a, k} - m _ {a, k} \right\| \left\| \widehat {\pi} _ {a, k} - \pi_ {a, k} \right\| = o _ {p} (1 / \sqrt {n}), \quad \left\| \widehat {\zeta} _ {k} - \zeta_ {k} \right\| \left\| \widehat {\tau} _ {a, k} - \tau_ {a, k} \right\| = o _ {p} (1 / \sqrt {n}), \tag {49}
$$
then the conditional bias,  $E\left(\widehat{\mu}_{a,k} - \mu_{a,T}\right) = o_p(1 / \sqrt{n})$  since  $O_{p}(o_{p}(1 / \sqrt{n})) = o_{p}(1 / \sqrt{n})$ .
Consequently, by Slutsky's theorem,  $\sqrt{n} (\widehat{\mu}_{a,k} - \mu_{a,T})\stackrel {d}{\to}\mathcal{N}(0,\sigma^2)$ . Here,  $\sigma^2$  represents the semiparametric efficiency bound, which has been derived and defined in (43).
E.5 Proof of Theorem 3
Proof. The proof presented herein closely follows the methodology outlined in [5]; we start by establishing the consistency and asymptotic normality of the global estimator, assuming a fixed  $\eta_{k}$ . We then invoke Lemma 4 and 5 in [5], which state that the proposed adaptive estimation for  $\eta_{k}$  as shown in (9) allows for (i) the recovery of the optimal  $\bar{\eta}_{k}$  by the estimator  $\hat{\eta}_k$ , and (ii) the uncertainty introduced by  $\widehat{\eta}_k$  is negligible when estimating  $\Delta_T$ . Essentially, we require Assumptions (A1)-(A7) and the assumptions stated in Theorem 2 to hold. The additional regularity conditions we require are the variance and covariance of the influence functions to be properly bounded. Note that we can invoke Lemma 4 in [5] to prove the negligible uncertainty of our proposed estimator because both the optimization problem posed in [5] and the optimization problem in our formulation aim to minimize the asymptotic variance of the global federated estimator while controlling for possible estimation bias due to the introduction of the penalty term. Specifically, to show the equivalence between S.9 in [5] and (9) in our formulation, we first see that the penalty terms are equivalent under assumptions in Theorem 3. We then establish the equivalence of the leading term by comparing the asymptotic variance of the global federated estimator derived in (55) and the S.9 in [5].
Lemma E.3 demonstrates the consistency of the site-specific estimators given that the source site  $k$  satisfies the conditions outlined in Theorem 2. We denote the set of source sites that fulfill the conditions in Theorem 2 as  $S^{*}$ , and consider a fixed  $\eta_{k}$  such that  $\eta_{k} = 0$  for  $k \notin S^{*}$ , then
$$
\widehat {\mu} _ {a, G} \left(\eta_ {k}\right) = \widehat {\mu} _ {a, T} + \sum_ {k \in \mathcal {K}} \eta_ {k} \left\{\widehat {\mu} _ {a, k} - \widehat {\mu} _ {a, T} \right\} \tag {50}
$$
is consistent for  $\mu_{a,T}$  since  $\widehat{\mu}_{a,k}$  for  $k\in S^{*}$  are consistent estimators for  $\mu_{a,T}$ . Hence, we can establish that  $\widehat{\Delta}_G(\eta_k) = \widehat{\mu}_{1,G}(\eta_k) - \widehat{\mu}_{0,G}(\eta_k)$  consistently estimates  $\Delta_T = \mu_{1,T} - \mu_{0,T}$ . Moving forward, we proceed to examine the asymptotic normality of the global estimator by utilizing the influence functions for the site-specific estimators. First, we rewrite the global estimator with the fixed  $\eta_{k}$  as
$$
\widehat {\Delta} _ {G} \left(\eta_ {k}\right) = \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) \left(\widehat {\mu} _ {1, T} - \widehat {\mu} _ {0, T}\right) + \sum_ {k \in \mathcal {S}} \eta_ {k} \left(\widehat {\mu} _ {1, k} - \widehat {\mu} _ {0, k}\right) = \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) \widehat {\Delta} _ {T} + \sum_ {k \in \mathcal {S}} \eta_ {k} \widehat {\Delta} _ {k} \tag {51}
$$
In Appendix E.3, the influence functions for the source site and target site estimators have been derived. To facilitate representation, we decompose the influence functions for the source site estimators into two parts, each defined on the target sample and source sample, respectively,
$$
\begin{array}{l} \xi_ {a, k} ^ {(1)} (Z _ {i}) = \frac {1}{P (R _ {i} = k)} \left[ \frac {I (A _ {i} = a , R _ {i} = k)}{\pi_ {a , k} (X _ {i})} \zeta_ {k} (V _ {i}) \{Y _ {i} - m _ {a, k} (X _ {i}) \} \right] \\ + \frac {1}{P (R _ {i} = k)} \bigg [ I (R _ {i} = k) \zeta_ {k} (V _ {i}) \{m _ {a, k} (X _ {i}) - \tau_ {a, k} (V _ {i}) \} \bigg ] \\ \end{array}
$$
$$
\xi_ {a, k} ^ {(2)} \left(Z _ {i}\right) = \frac {1}{P \left(R _ {i} = T\right)} \left[ I \left(R _ {i} = T\right) \tau_ {a, k} \left(V _ {i}\right) \right] - \mu_ {a, T}. \tag {52}
$$
Here, it should be noted that the expectations of where  $\xi_{a,k}^{(1)}(Z_i)$  and  $\xi_{a,k}^{(2)}(Z_i)$  are both equal to zero. Furthermore, for the target site estimator, we define:
$$
\xi_ {a, T} ^ {(2)} \left(Z _ {i}\right) = \frac {1}{P \left(R _ {i} = T\right)} \left[ \frac {I \left(A _ {i} = a , R _ {i} = T\right)}{\pi_ {a , T} \left(V _ {i}\right)} \left\{Y _ {i} - m _ {a, T} \left(V _ {i}\right) \right\} + I \left(R _ {i} = T\right) m _ {a, T} \left(V _ {i}\right) \right] - \mu_ {a, T} \tag {53}
$$
and the expectation of  $\xi_{a,T}^{(2)}(Z_i)$  is also zero. We denote the influence function for  $\widehat{\Delta}_k$  as  $\xi_{1,k}^{(1)}(Z_i) - \xi_{0,k}^{(1)}(Z_i) + \xi_{1,k}^{(2)}(Z_i) - \xi_{0,k}^{(2)}(Z_i)$ . Similarly, the influence function for  $\widehat{\Delta}_T$  is given by  $\xi_{1,T}^{(2)}(Z_i) -$
$\xi_{0,T}^{(2)}(Z_i)$ . Thus, the influence function for the global estimator evaluated with target and source samples can be expressed as:
$$
\begin{array}{l} \widehat {\Delta} _ {G} (\eta_ {k}) - \Delta_ {T} = \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) \left(\widehat {\Delta} _ {T} - \Delta_ {T}\right) + \sum_ {k \in \mathcal {S}} \eta_ {k} \left(\widehat {\Delta} _ {k} - \Delta_ {T}\right) \\ = \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) \frac {1}{n _ {T}} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \left\{\xi_ {1, T} ^ {(2)} (Z _ {i}) - \xi_ {0, T} ^ {(2)} (Z _ {i}) \right\} \\ + \left(\sum_ {k \in \mathcal {S}} \eta_ {k}\right) \frac {1}{n _ {T}} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \left\{\xi_ {1, k} ^ {(2)} (Z _ {i}) - \xi_ {0, k} ^ {(2)} (Z _ {i}) \right\} \\ + \sum_ {k \in \mathcal {S}} \eta_ {k} \frac {1}{n _ {k}} \sum_ {i = 1} ^ {N} I (R _ {i} = k) \left\{\xi_ {1, k} ^ {(1)} (Z _ {i}) - \xi_ {0, k} ^ {(1)} (Z _ {i}) \right\} \\ = \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) \frac {\left\{\xi_ {1 , T} ^ {(2)} (Z _ {i}) - \xi_ {0 , T} ^ {(2)} (Z _ {i}) \right\}}{P (R _ {i} = T)} \\ + \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \left(\sum_ {k \in \mathcal {S}} \eta_ {k}\right) \frac {\left\{\xi_ {1 , k} ^ {(2)} (Z _ {i}) - \xi_ {0 , k} ^ {(2)} (Z _ {i}) \right\}}{P (R _ {i} = T)} \\ + \frac {1}{N} \sum_ {k \in S} \sum_ {i = 1} ^ {N} I \left(R _ {i} = k\right) \eta_ {k} \frac {\left\{\xi_ {1 , k} ^ {(1)} \left(Z _ {i}\right) - \xi_ {0 , k} ^ {(1)} \left(Z _ {i}\right) \right\}}{P \left(R _ {i} = k\right)} \tag {54} \\ \end{array}
$$
The asymptotic variance for  $\widehat{\Delta}_G(\eta_k)$  equals the variance of the influence function (54). Its derivation is similar to that of 43. Let us denote this asymptotic variance as  $\mathcal{V}(\eta_k)$ . Under the assumption of i.i.d. units within each site, we have:
$$
\begin{array}{l} \mathcal {V} (\eta_ {k}) = \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) ^ {2} \frac {\operatorname {V a r} \left\{\xi_ {1 , T} ^ {(2)} (Z _ {i}) - \xi_ {0 , T} ^ {(2)} (Z _ {i}) \mid R _ {i} = T \right\}}{P (R _ {i} = T)} \\ + \left(\sum_ {k \in \mathcal {S}} \eta_ {k}\right) ^ {2} \frac {\operatorname {V a r} \left\{\xi_ {1 , k} ^ {(2)} \left(Z _ {i}\right) - \xi_ {0 , k} ^ {(2)} \left(Z _ {i}\right) \mid R _ {i} = T \right\}}{P \left(R _ {i} = T\right)} \\ + 2 \left(1 - \sum_ {k \in \mathcal {S}} \eta_ {k}\right) \left(\sum_ {k \in \mathcal {S}} \eta_ {k}\right) \frac {\operatorname {C o v} \left\{\xi_ {1 , T} ^ {(2)} (Z _ {i}) - \xi_ {0 , T} ^ {(2)} (Z _ {i}) , \xi_ {1 , k} ^ {(2)} (Z _ {i}) - \xi_ {0 , k} ^ {(2)} (Z _ {i}) \mid R _ {i} = T \right\}}{P (R _ {i} = T)} \\ + \sum_ {k \in S} \eta_ {k} ^ {2} \frac {\operatorname {V a r} \left\{\xi_ {1 , k} ^ {(1)} \left(Z _ {i}\right) - \xi_ {0 , k} ^ {(1)} \left(Z _ {i}\right) \mid R _ {i} = k \right\}}{P \left(R _ {i} = k\right)} \tag {55} \\ \end{array}
$$
Under the boundedness conditions of the variance and covariance of the influence functions, this variance is finite. Consequently, we can express the asymptotic distribution of  $\sqrt{N}\left(\widehat{\Delta} G(\eta_k) - \Delta T\right)$  as:
$$
\sqrt {N} \left(\widehat {\Delta} _ {G} \left(\eta_ {k}\right) - \Delta_ {T}\right) \xrightarrow {d} \mathcal {N} (0, \mathcal {V} \left(\eta_ {k}\right)) \tag {56}
$$
We further define the optimal adaptive weight  $\bar{\eta}_k$  as follows:
$$
\bar {\eta} _ {k} = \arg \min  _ {\eta_ {k} = 0 \forall k \notin \mathcal {S} ^ {*}} \mathcal {V} (\eta_ {k}) \tag {57}
$$
By leveraging Lemmas 4 and 5 from [5], we can recover the optimal  $\bar{\eta}_k$  with negligible uncertainty for estimating  $\Delta_T$  if we estimate  $\widehat{\eta}_{k,L_1}$  using (9). The consistency of  $\widehat{\mathcal{V}} (\widehat{\eta}_{k,L_1})$  follows when we can effectively approximate  $\mathcal{V}(\bar{\eta}_k)$  with  $\widehat{\mathcal{V}} (\widehat{\eta}_{k,L_1})$ . Thus,
$$
\sqrt {N / \widehat {\mathcal {V}} (\widehat {\eta} _ {k , L _ {1}})} \left(\widehat {\Delta} _ {G} \left(\widehat {\eta} _ {k, L _ {1}}\right) - \Delta_ {T}\right) \xrightarrow {d} \mathcal {N} (0, 1) \tag {58}
$$
We now proceed to analyze the efficiency gain resulting from the federation process. The estimator that relies solely on the target data is denoted as  $\widehat{\Delta}_T = \widehat{\Delta}_G(\eta_0)$ , where  $\eta_0$  assigns all weights to the target and none to the source. In contrast, the estimator that leverages the proposed adaptive ensemble approach is denoted as  $\widehat{\Delta}_G(\widehat{\eta}_{k,L_1}) = (1 - \sum_{k\in S}\widehat{\eta}_{k,L_1})\widehat{\Delta}_T + \sum_{k\in S}\widehat{\eta}_{k,L_1}\widehat{\Delta}_k$ . Here,  $\widehat{\eta}_{k,L_1}$  can recover the optimal weight  $\bar{\eta}_k$  that is associated with the minimum asymptotic variance. Consequently, the variance of  $\widehat{\Delta}_G(\widehat{\eta}_{k,L_1})$  is no larger than that of the estimator relying solely on the target data by definition.
To establish that the asymptotic variance of  $\widehat{\Delta}_G(\widehat{\eta}_{k,L_1})$  is strictly smaller than that of the estimator based solely on the target data  $\widehat{\Delta}_T$ , we adapt Proposition 1 in [5] with a modified informative source condition. Specifically, for each source site  $s \in S^*$ , we define  $\widehat{\Delta}_G(\eta_s)$  a global estimator where  $\eta_s$  is the optimal ensemble weight if we only consider target site and this source site  $s$ . Then the modified informative source condition is  $\mathrm{Cov}\left\{\sqrt{N}\widehat{\Delta}_T, \sqrt{N}\left(\widehat{\Delta}_G(\eta_s) - \widehat{\Delta}_T\right)\right\}$ , where  $\widehat{\Delta}_G(\eta_s) - \widehat{\Delta}_T$  can be expressed as
$$
\begin{array}{l} \widehat {\Delta} _ {G} (\eta_ {s}) - \widehat {\Delta} _ {T} = \widehat {\Delta} _ {G} (\eta_ {s}) - \Delta_ {T} - \left(\widehat {\Delta} _ {T} - \Delta_ {T}\right) \\ = \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = T) (1 - \eta_ {s}) \frac {\left\{\xi_ {1 , T} ^ {(2)} (Z _ {i}) - \xi_ {0 , T} ^ {(2)} (Z _ {i}) \right\}}{P (R _ {i} = T)} \\ + \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \eta_ {s} \frac {\left\{\xi_ {1 , s} ^ {(2)} (Z _ {i}) - \xi_ {0 , s} ^ {(2)} (Z _ {i}) \right\}}{P (R _ {i} = T)} \\ + \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = s) \eta_ {s} \frac {\left\{\xi_ {1 , s} ^ {(1)} (Z _ {i}) - \xi_ {0 , s} ^ {(1)} (Z _ {i}) \right\}}{P (R _ {i} = s)} \\ - \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \frac {\left\{\xi_ {1 , T} ^ {(2)} (Z _ {i}) - \xi_ {0 , T} ^ {(2)} (Z _ {i}) \right\}}{P (R _ {i} = T)} \\ = \frac {1}{N} \sum_ {i = 1} ^ {N} I (R _ {i} = T) \eta_ {s} \frac {\left\{\xi_ {1 , s} ^ {(2)} (Z _ {i}) - \xi_ {0 , s} ^ {(2)} (Z _ {i}) - \xi_ {1 , T} ^ {(2)} (Z _ {i}) + \xi_ {0 , T} ^ {(2)} (Z _ {i}) \right\}}{P (R _ {i} = T)} \\ + \frac {1}{N} \sum_ {i = 1} ^ {N} I \left(R _ {i} = s\right) \eta_ {s} \frac {\left\{\xi_ {1 , s} ^ {(1)} \left(Z _ {i}\right) - \xi_ {0 , s} ^ {(1)} \left(Z _ {i}\right) \right\}}{P \left(R _ {i} = s\right)}. \tag {59} \\ \end{array}
$$
In summary, if there exist a source site  $s$  with consistent estimator of  $\Delta_T$  and further satisfy  $\left|\operatorname{Cov}\left\{\sqrt{N}\widehat{\Delta}_T,\sqrt{N}\left(\widehat{\Delta}_G(\eta_s) - \widehat{\Delta}_T\right)\right\} \right| \geq \varepsilon$  where  $\varepsilon$  denotes a positive constant, the variance of  $\widehat{\Delta}_G$  is strictly smaller than  $\widehat{\Delta}_T$ .
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