$\hat{e}=\tilde{e}$ and $B_{n}$ and $\beta$ are as in (99). Fix $M>-\infty$
and an arbitrary sequence $a_{n}\uparrow\infty$. For each $j>M$ with
$2^{j}a_{n}\alpha_{n}\leqslant r$, define the “shell”
$S_{j}=\\{e\in\mathcal{F}_{n}:2^{j-1}a_{n}\alpha_{n}<\|e-\tilde{e}\|_{2,P_{n}}\leqslant
2^{j}a_{n}\alpha_{n}\\}$. It follows that for each such $j$, whenever $e\in
S_{j}$ we have $r_{0}\geqslant r\geqslant\|e-\tilde{e}\|_{2,P_{n}}\geqslant
2^{j-1}a_{n}\alpha_{n}$, and so by (76), we have

$(\phi_{n}(e)-\phi_{n}(\tilde{e}))\bm{1}(C_{n})\leqslant-
c_{0}\|e-\tilde{e}\|_{2,P_{n}}^{2}\bm{1}(C_{n})\leqslant-
c_{0}2^{2j-2}a_{n}^{2}\alpha_{n}^{2}\bm{1}(C_{n})$

for all $e\in S_{j}$. Hence using the definition of $\hat{e}$

$\displaystyle\bm{1}(\hat{e}\in S_{j})\bm{1}(C_{n})$
$\displaystyle\leqslant\bm{1}\biggl{(}\,\sup_{e\in
S_{j}}\hat{\phi}_{n}(e)-\hat{\phi}_{n}(\tilde{e})\geqslant
0\biggr{)}\bm{1}(C_{n})$
$\displaystyle\leqslant\bm{1}\biggl{(}\,\sup_{e:\|e-\tilde{e}\|_{2,P_{n}}\leqslant
2^{j}a_{n}\alpha_{n}}(\hat{\phi}_{n}(e)-\hat{\phi}_{n}(\tilde{e}))-(\phi_{n}(e)-\phi_{n}(\tilde{e}))\geqslant
c_{0}2^{2j-2}a_{n}^{2}\alpha_{n}^{2}\biggr{)}\bm{1}(C_{n})$
$\displaystyle\leqslant\bm{1}\bigl{(}2^{j}a_{n}\alpha_{n}\|\hat{\eta}-\eta\|_{2,P_{n}}+\|\hat{\eta}-\eta\|_{2,P_{n}}^{2}\geqslant
c_{0}C_{0}^{-1}2^{2j-2}a_{n}^{2}\alpha_{n}^{2}\bigr{)}\bm{1}(C_{n})$
$\displaystyle\leqslant\frac{2^{j}a_{n}\alpha_{n}\|\hat{\eta}-\eta\|_{2,P_{n}}+\|\hat{\eta}-\eta\|_{2,P_{n}}^{2}}{c_{0}C_{0}^{-1}2^{2j-2}a_{n}^{2}\alpha_{n}^{2}}\bm{1}(C_{n})$
$\displaystyle=\frac{C_{0}}{c_{0}}\biggl{(}\frac{\|\hat{\eta}-\eta\|_{2,P_{n}}}{2^{j-2}a_{n}\alpha_{n}}+\frac{\|\hat{\eta}-\eta\|_{2,P_{n}}^{2}}{2^{2j-2}a_{n}^{2}\alpha_{n}^{2}}\biggr{)}\bm{1}(C_{n})$

where the last inequality follows from Lemma C.4. Then

$\displaystyle\bm{1}\left(\frac{r}{2}\geqslant\|\hat{e}-\tilde{e}\|_{2,P_{n}}>2^{M}a_{n}\alpha_{n}\right)\bm{1}(C_{n})$
$\displaystyle\leqslant\sum_{j>M,2^{j}a_{n}\alpha_{n}\leqslant
r}\bm{1}(\hat{e}\in S_{j})\bm{1}(C_{n})$
$\displaystyle\leqslant\frac{C_{0}}{c_{0}}\sum_{j=M+1}^{\infty}\left(\frac{\|\hat{\eta}-\eta\|_{2,P_{n}}}{2^{j-2}a_{n}\alpha_{n}}+\frac{\|\hat{\eta}-\eta\|_{2,P_{n}}^{2}}{2^{2j-2}a_{n}^{2}\alpha_{n}^{2}}\right)$
$\displaystyle=\frac{C_{0}}{c_{0}}\left(\frac{\|\hat{\eta}-\eta\|_{2,P_{n}}}{2^{M-2}a_{n}\alpha_{n}}+\frac{4}{3}\frac{\|\hat{\eta}-\eta\|_{2,P_{n}}^{2}}{2^{2M}a_{n}^{2}\alpha_{n}^{2}}\right).$

Since $\|\hat{\eta}-\eta\|_{2,P}=o_{p}(a_{n}\alpha_{n})$, we conclude

$\textnormal{Pr}(r/2\geqslant\|\hat{e}-\tilde{e}\|_{2,P_{n}}>2^{M}a_{n}\alpha_{n},C_{n})=o(1)$

for each $M$. Now, by step 3 we know that
$\|\hat{e}-e^{*}\|_{2,P_{n}}=o_{p}(1)$. Applying step 3 again but with
$\hat{\eta}=\eta$ shows
$\|\tilde{e}-e^{*}\|_{2,P_{n}}=O_{p}(n^{-1/4})=o_{p}(1)$. Thus

$\|\hat{e}-\tilde{e}\|_{2,P_{n}}\leqslant\|\hat{e}-e^{*}\|_{2,P_{n}}+\|e^{*}-\tilde{e}\|_{2,P_{n}}=o_{p}(1)$

so that $\textnormal{Pr}(\|\hat{e}-\tilde{e}\|_{2,P_{n}}>r/2)=o(1)$. Since
$\textnormal{Pr}(C_{n})\rightarrow 1$ as $n\rightarrow\infty$, we can conclude
that

$\textnormal{Pr}(\|\hat{e}-\tilde{e}\|_{2,P_{n}}>2^{M}a_{n}\alpha_{n})=o(1).$

With $a_{n}\uparrow\infty$ arbitrary, by the preceding display and Lemma B.2
we have $\|\hat{e}-\tilde{e}\|_{2,P_{n}}=O_{p}(\alpha_{n})$ and

$\|\hat{e}-e^{*}\|_{2,P_{n}}\leqslant\|\hat{e}-\tilde{e}\|_{2,P_{n}}+\|\tilde{e}-e^{*}\|_{2,P_{n}}=O_{p}(n^{-1/4})+O_{p}(\alpha_{n})$

as desired.

It remains to show the same convergence rate holds out of sample, i.e.,
$\|\hat{e}-e^{*}\|_{2,P}=O_{p}(n^{-1/4})+O_{p}(\alpha_{n})$. We do this by
showing that
$\big{|}\|\hat{e}-e^{*}\|_{2,P_{n}}-\|\hat{e}-e^{*}\|_{2,P}\big{|}=O_{p}(n^{-1/4})$.
With $\mathcal{E}_{2}^{-}$ as defined in Lemma B.5 in terms of the collection
$\mathcal{E}$ of Assumption 5.1, we know that

$\displaystyle\big{|}\|\hat{e}-e^{*}\|_{2,P_{n}}^{2}-\|\hat{e}-e^{*}\|_{2,P}^{2}\big{|}$
$\displaystyle=\Big{|}\frac{1}{n}\sum_{i=1}^{n}[(\hat{e}(X_{i})-e^{*}(X_{i}))^{2}-\int_{\mathcal{X}}(\hat{e}(x)-e^{*}(x))^{2}\,\mathrm{d}P(x)]\Big{|}$
$\displaystyle\leqslant\sup_{e\in\mathcal{E}_{2}^{-}}|(P_{n}-P)e|.$

Furthermore, by Lemma B.3 and Lemma B.5 we know that for some $K_{0}<\infty$
we have

$\displaystyle\sup_{e\in\mathcal{E}_{2}^{-}}|(P_{n}-P)f|$
$\displaystyle\leqslant
K_{0}n^{-1/2}\int_{0}^{1}\sqrt{\log\mathcal{N}(\epsilon,\mathcal{E}_{2}^{-},L^{2}(P_{n}))}\,\mathrm{d}\epsilon$
$\displaystyle\leqslant
K_{0}\sqrt{2}n^{-1/2}\int_{0}^{1}\sqrt{\log\mathcal{N}(\epsilon/4,\mathcal{E},L^{2}(P_{n}))}\,\mathrm{d}\epsilon$
$\displaystyle=4K_{0}\sqrt{2}n^{-1/2}\int_{0}^{1/4}\sqrt{\log\mathcal{N}(\delta,\mathcal{E},L^{2}(P_{n}))}\,\mathrm{d}\delta.$

Then by (30) we conclude
$\sup_{e\in\mathcal{E}_{2}^{-}}|(P_{n}-P)e|=O_{p}(n^{-1/2})$. Since

$|a-b|\leqslant
a+b\implies\sqrt{|a-b|}\leqslant\sqrt{a+b}\implies|a-b|\leqslant\sqrt{a+b}\sqrt{|a-b|}=\sqrt{|a^{2}-b^{2}|}$

for any $a,b\geqslant 0$, we have

$\big{|}\|\hat{e}-e^{*}\|_{2,P_{n}}-\|\hat{e}-e^{*}\|_{2,P}\big{|}=O_{p}(n^{-1/4})$

as desired. ∎

###  C.10 Proof of Theorem 5.1

Here we prove convergence of Algorithm 1, our concave maximization procedure
for designing an optimal CSBAE. We begin by proving we can design for
$\hat{\theta}_{\textnormal{AIPW}}$, as stated in the first numbered condition
of Theorem 5.1. The proof proceeds by showing that the objective defining
$e_{t,\textnormal{AIPW}}^{*}(\cdot)$ in (35) can be written in a form so that
Assumption 5.1 and (34) are satisfied, the latter with $\alpha_{N}=N^{-1/4}$.
Then we conclude by applying Lemma 5.2.

Many of our expressions will include the cumulative sum of batch frequencies
$\sum_{u=1}^{t}\kappa_{u}$. We use $\kappa_{1:t}$ to denote this quantity
below. We similarly abbreviate $\sum_{u=1}^{t}N_{u}$ to $N_{1:t}$.

With $V_{0:t,\textnormal{AIPW}}$ scalar, the information function
$\Psi=\Psi(\cdot)$ is simply an increasing scalar-valued function by
Assumption 5.2, and hence we have

$\displaystyle
e_{t,\textnormal{AIPW}}^{*}(\cdot)=\operatorname*{arg\,max}_{e_{t}(\cdot)\in\mathcal{F}_{*,t}}(V_{0:t,\textnormal{AIPW}})^{-1}$
$\displaystyle=\operatorname*{arg\,min}_{e_{t}(\cdot)\in\mathcal{F}_{*,t}}V_{0:t,\textnormal{AIPW}}$
$\displaystyle=\operatorname*{arg\,max}_{e_{t}(\cdot)\in\mathcal{F}_{*,t}}h(V_{0:t,\textnormal{AIPW}})$
$\displaystyle=\operatorname*{arg\,max}_{e_{t}(\cdot)\in\mathcal{F}_{t,*}}\mathbb{E}_{P^{X}}\left[\frac{2C}{\gamma_{0}}-\frac{v_{0}(1,X)}{e_{0}^{(t)}(X)}-\frac{v_{0}(0,X)}{1-e_{0}^{(t)}(X)}\right]$

where in the final equality we dropped the additive term
$\mathbb{E}[(\tau_{0}(X)-\theta_{0})^{2}]$ which is independent of
$e_{t}(\cdot)$, and defined $h(x)=2C/\gamma_{0}-x$ for

$\gamma_{0}:=\frac{\epsilon_{1}}{2}\frac{1}{\kappa_{1:t}}\min(\kappa_{1},\kappa_{t})>0.$

Evidently $h(\cdot)$ is decreasing. We can now define the information matrix

$\mathcal{I}=\mathcal{I}(e_{t},\eta_{0})=\mathbb{E}_{P^{X}}\left[\frac{2C}{\gamma_{0}}-\frac{v_{0}(1,X)}{e_{0}^{(t)}(X)}-\frac{v_{0}(0,X)}{1-e_{0}^{(t)}(X)}\right]$

which is of the form (28) with

$f(e,w)=\frac{2C}{\gamma_{0}}-\frac{w_{1}}{1-w_{3}-w_{4}e}-\frac{w_{2}}{w_{3}+w_{4}e}$

and

$\eta(x)=\biggl{(}v_{0}(0,x),\,v_{0}(1,x),\,\frac{1}{\kappa_{1:t}}\sum_{u=1}^{t-1}\kappa_{u}e_{u}(x),\,\frac{1}{\kappa_{1:t}}\kappa_{t}\biggr{)}.$

Let $\mathcal{W}=[c,C]^{2}\times\mathcal{W}_{+}$ where
$\mathcal{W}_{+}=\\{(x,y)\in{}^{2}\mid
x\geqslant\gamma_{0},y\geqslant\gamma_{0},x+y\leqslant 1-\gamma_{0}\\}$. By
(3) and the assumptions of the Theorem (specifically the uniform bounds on the
variance functions and the assumption that $\epsilon_{1}\leqslant
e_{1}(x)=\hat{e}_{1}^{(k)}(x)\leqslant 1-\epsilon_{1}$ for all
$x\in\mathcal{X}$ and folds $k=1,\ldots,K$), we can verify that
$\eta(x)\in\mathcal{W}$ and $\hat{\eta}^{(k)}(x)\in\mathcal{W}$ for all
$x\in\mathcal{X}$, folds $k=1,\ldots,K$, and sufficiently large $N$, where

$\hat{\eta}^{(k)}(x)=\biggl{(}\hat{v}^{(k)}(0,x),\,\hat{v}^{(k)}(1,x),\,\frac{1}{N_{1:t}}\sum_{u=1}^{t-1}N_{u}\hat{e}_{u}^{(k)}(x),\,\frac{N_{t}}{N_{1:t}}\biggr{)}.$

Also we have

$1-\gamma_{0}\geqslant w_{3}+w_{4}e\geqslant\gamma_{0}\quad\text{and}\quad
f(e,w)\geqslant 0,\quad\forall(e,w)\in[0,1]\times\mathcal{W}.$

Evidently $\mathcal{W}$ is closed and bounded, hence compact. With
$w_{3}+w_{4}e$ linear in $e$, there exists $\delta<0$ such that for all
$(e,w)$ in a neighborhood containing $(\delta,1-\delta)\times\mathcal{W}$,
$w_{3}+w_{4}e$ is uniformly bounded away from 0 and then $f(e,w)$ evidently
has continuous second partial derivatives on this neighborhood. Finally, we
compute

$-f^{\prime\prime}(e,w)=\frac{2w_{2}w_{4}^{2}}{(w_{3}+w_{4}e)^{3}}+\frac{2w_{1}w_{4}^{2}}{(1-w_{3}-w_{4}e)^{3}}\geqslant\frac{2c\gamma_{0}^{2}}{(1-\gamma_{0})^{3}}>0,\quad\forall(e,w)\in[0,1]\times\mathcal{W}$

which shows that all conditions of Assumption 5.1 have been satisfied.
Equation (34) holds with $\alpha_{N}=N^{-1/4}$ by (46) and (17), completing
the proof of the first numbered condition of Theorem 5.1, pertaining to design
for $\hat{\theta}_{\textnormal{AIPW}}$.

It remains to show the second numbered condition holds. As above, the proof
proceeds by showing the objective for
$\hat{e}_{t,\textnormal{EPL}}^{*}(\cdot)$ in (35) can be written in a form so
that Assumption 5.1 and (34) are satisfied, the latter with
$\alpha_{N}=N^{-1/4}$. To that end, we take
$\mathcal{I}=\mathcal{I}(e_{t},\eta_{0})=V_{0:t,\textnormal{EPL}}^{-1}$ which
takes the form (28) with

$f(e,w)=f(e,w_{1},w_{2},w_{3},w_{4},w_{5})=\frac{(w_{3}+w_{4}e)(1-w_{3}-w_{4}e)}{w_{1}(w_{3}+w_{4}e)+w_{2}(1-w_{3}-w_{4}e)}w_{5}w_{5}^{\top}$

and

$\eta(x)=\biggl{(}v_{0}(0,x),\,v_{0}(1,x),\,\frac{1}{\kappa_{1:t}}\sum_{u=1}^{t-1}\kappa_{u}e_{u}(x),\,\frac{\kappa_{t}}{\kappa_{1:t}},\,\psi(x)\biggr{)}.$

Let
$\mathcal{W}=[c,C]^{2}\times\mathcal{W}_{+}\times[-C,C]^{p}\subseteq{}^{4+p}$,
where once again $\mathcal{W}_{+}=\\{(x,y)\in{}^{2}\mid
x\geqslant\gamma_{0},y\geqslant\gamma_{0},x+y\leqslant 1-\gamma_{0}\\}$ for

$\gamma_{0}:=\frac{\epsilon_{1}}{2\kappa_{1:t}}\min(\kappa_{1},\kappa_{t})>0.$

Evidently $\mathcal{W}$ is closed and bounded, hence compact. Then the
assumptions of the Theorem (specifically the uniform bounds on the variance
functions and the assumption that $\epsilon_{1}\leqslant
e_{1}(x)=\hat{e}_{1}^{(k)}(x)\leqslant 1-\epsilon_{1}$ for all
$x\in\mathcal{X}$ and folds $k=1,\ldots,K$) ensure that for each fold
$k=1,\ldots,K$, if we take

$\hat{\eta}^{(k)}(x)=\biggl{(}\hat{v}^{(k)}(0,x),\,\hat{v}^{(k)}(1,x),\,\frac{1}{N_{1:t}}\sum_{u=1}^{t-1}N_{u}\hat{e}_{u}^{(k)}(x),\,\frac{N_{t}}{N_{1:t}},\,\psi(x)\biggr{)}$

then $\eta(x)\in\mathcal{W}$ and $\hat{\eta}^{(k)}(x)\in\mathcal{W}$ for all
$x\in\mathcal{X}$ whenever $N$ is sufficiently large. We note that for each
$e\in[0,1]$,

$f(e,w)=\frac{(w_{3}+w_{4}e)(1-w_{3}-w_{4}e)}{w_{1}(w_{3}+w_{4}e)+w_{2}(1-w_{3}-w_{4}e)}w_{5}w_{5}^{\top}$

is positive semidefinite because the lead constant above is nonnegative.
Furthermore, note that the denominator
$w_{1}(w_{3}+w_{4}e)+w_{2}(1-w_{3}-w_{4}e)$ is bounded below by $c$ for any
$(e,w)\in[0,1]\times\mathcal{W}$ and continuous on
$(e,w_{1},w_{2},w_{3},w_{4})\in{}^{5}$. This denominator is linear in $e$ (for
fixed $w$) and so additionally there exists $\delta<0$ such that on some open
neighborhood containing $(\delta,1-\delta)\times\mathcal{W}$, this denominator
is strictly positive. Therefore $f(e,w)$ has two continuous partial
derivatives with respect to $e$. Finally, we compute

$-f^{\prime\prime}(e,w)=\frac{2w_{4}^{2}(w_{3}+w_{4}e)(1-w_{3}-w_{4}e)}{(w_{1}(w_{3}+w_{4}e)+w_{2}(1-w_{3}-w_{4}e))^{3}}w_{5}w_{5}^{\top}.$

This is positive semidefinite since as above, $1-\gamma_{0}\geqslant
w_{3}+w_{4}e\geqslant\gamma_{0}$ for all $(e,w)\in[0,1]\times\mathcal{W}$, so

$\frac{2w_{4}^{2}(w_{3}+w_{4}e)(1-w_{3}-w_{4}e)}{(w_{1}(w_{3}+w_{4}e)+w_{2}(1-w_{3}-w_{4}e))^{3}}>0$

on $[0,1]\times\mathcal{W}$. Furthermore, as all diagonal entries of
$w_{5}w_{5}^{\top}$ are nonnegative, the inclusion of an intercept in
$\psi(x)$ ensures that

$\displaystyle\inf_{(e,w)\in[0,1]\times\mathcal{W}}\textnormal{tr}(-f^{\prime\prime}(e,w))$
$\displaystyle\geqslant\inf_{(e,w)\in[0,1]\times\mathcal{W}}\,\frac{2w_{4}^{2}(w_{3}+w_{4}e)(1-w_{3}-w_{4}e)}{[w_{1}(w_{3}+w_{4}e)+w_{2}(1-w_{3}-w_{4}e)]^{3}}$
$\displaystyle\geqslant\frac{2\gamma_{0}^{3}(1-\gamma_{0})}{C^{3}}.$

As before, equation (34) holds with $\alpha_{N}=N^{-1/4}$ by (46) and (17),
enabling us to apply Lemma 5.2 and completing the proof of the Theorem.

##  Appendix D Additional simulations

We present results from some additional numerical simulations in the framework
of Section 6.

###  D.1 Unequal budget constraints

Tables 3 and 4 reproduce Tables 1 and 2 using simulations with budget
constraints $m_{L,2}=m_{H,2}=0.4$. The results are qualitatively similar to
those in the main text. One notable difference is that there seem to be some
additional gains to pooling in ATE estimation, both asymptotically and in
finite samples. For example, in the homoskedastic DGPs, we see about a 5%
asymptotic efficiency gain from pooling in Table 3 when either $d=1$ or
$d=10$, which translates well to finite sample gains, particularly for the
$d=10$ DGP. By contrast there is no asymptotic gain for the homoskedastic DGPs
in Table 1.

Table 3: Same as Table 1, but for simulations with $m_{L,2}=m_{H,2}=0.4$
DGP | Estimator | Design | Sim. rel. eff. (90% CI) | Asymp. rel. eff.
---|---|---|---|---
$d=1$, Homoskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.051 (1.015, 1.088) | 1.050
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 1.034 (0.995, 1.073) | 1.050
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.050 (1.024, 1.077) | 1.050
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 0.993 (0.977, 1.008) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.916 (0.871, 0.962) | 0.919
$d=1$, Heteroskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.024 (0.978, 1.071) | 1.073
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 1.099 (1.046, 1.154) | 1.067
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.043 (1.019, 1.067) | 1.034
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 1.032 (1.003, 1.063) | 1.012
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 1.016 (0.961, 1.071) | 0.993
$d=10$, Homoskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.084 (1.031, 1.139) | 1.050
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 1.081 (1.030, 1.133) | 1.029
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.128 (1.096, 1.161) | 1.050
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 0.994 (0.970, 1.018) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.520 (0.478, 0.565) | 0.448
$d=10$, Heteroskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.051 (0.999, 1.107) | 1.064
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 1.001 (0.952, 1.051) | 1.045
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.062 (1.031, 1.094) | 1.035
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 0.981 (0.948,1.015) | 1.012
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.626 (0.582, 0.671) | 0.611
Table 4: Same as Table 2, but for simulations with $m_{L,2}=m_{H,2}=0.4$
DGP | Estimator | Design | Sim. rel. eff. (90% CI) | Asymp. rel. eff.
---|---|---|---|---
$d=1$, Homoskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.118 (1.076, 1.162) | 1.092
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 1.084 (1.038, 1.131) | 1.051
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.099 (1.072, 1.126) | 1.050
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 0.995 (0.976, 1.015) | 1.006
$d=1$, Heteroskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.139 (1.035, 1.249) | 1.090
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 1.050 (0.905, 1.188) | 1.013
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.074 (0.980, 1.172) | 1.034
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 1.028 (0.970, 1.090) | 1.017
$d=10$, Homoskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.234 (1.203, 1.266) | 1.056
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 1.204 (1.172, 1.237) | 0.994
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.262 (1.234, 1.292) | 1.050
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 0.997 (0.982, 1.013) | 1.002
$d=10$, Heteroskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.104 (1.065, 1.144) | 1.069
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 1.055 (1.016, 1.094) | 1.043
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.078 (1.042, 1.116) | 1.035
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 0.993 (0.967,1.019) | 1.013

###  D.2 Perfect nuisance estimation

To better isolate the performance effects of our specific choices of nuisance
estimation methods in the numerical study of Section 6, in Tables 5 and 6 we
reproduce Tables 1 and 2, respectively, but assume all nuisance functions are
known exactly at both the design and estimation stages. For ATE estimation
(Table 5), the flexible designs are aware of a perfectly constant variance
function in the homoskedastic DGPs, which induces them to always learn the
(optimal) simple RCT in every simulation. However, for the binned designs in
the homoskedastic DGPs and both the binned and flexible designs in the
heteroskedastic DGPs, there is some cross-simulation variability in the
propensity learned. This stems from variation in the parts of the variance
function being sampled due to variation in the covariates across simulations.
Consequently, the simulated finite sample efficiency gain ends up being
somewhat lower than the asymptotic gain. This suggests that the finite sample
efficiency gains from pooling observed in Table 1 are due to improved use of
nuisance function estimates by the pooled estimator
$\hat{\theta}_{\textnormal{AIPW}}$. One reason we might expect this is that
the pooled estimator uses nuisance estimates from observations pooled across
both batches of the experiment, while each component of the linearly
aggregated estimator only uses nuisance estimates from a single batch.

Table 5: Same as Table 1, but for oracle simulations that assume knowledge of
the true nuisance functions
DGP | Estimator | Design | Sim. rel. eff. (90% CI) | Asymp. rel. eff.
---|---|---|---|---
$d=1$, Homoskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 0.978 (0.960, 0.997) | 0.999
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.828 (0.794, 0.863) | 0.875
$d=1$, Heteroskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.005 (0.960, 1.051) | 1.051
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 0.977 (0.935, 1.020) | 1.044
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 1.008 (0.977, 1.039) | 1.026
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.905 (0.859, 0.952) | 0.976
$d=10$, Homoskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 0.968 (0.923, 1.013) | 0.965
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 1.000 (1.000,1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.402 (0.368, 0.439) | 0.432
$d=10$, Heteroskedastic | $\hat{\theta}_{\textnormal{AIPW}}$ | Flexible | 1.023 (0.986, 1.060) | 1.043
$\hat{\theta}_{\textnormal{AIPW}}$ | Binned | 1.010 (0.967, 1.054) | 1.019
$\hat{\theta}_{\textnormal{AIPW}}$ | Simple RCT | 1.000 (1.000, 1.000) | 1.000
$\hat{\theta}_{\textnormal{AIPW}}^{(\mathrm{LA})}$ | Flexible | 1.020 (0.990,1.051) | 1.026
$\hat{\theta}_{\textnormal{AIPW}}^{(\text{bin})}$ | Binned | 0.618 (0.575, 0.663) | 0.612
Table 6: Same as Table 2, but for oracle simulations that assume knowledge of the true nuisance functions  DGP | Estimator | Design | Sim. rel. eff. (90% CI) | Asymp. rel. eff.
---|---|---|---|---
$d=1$, Homoskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.129 (1.080, 1.179) | 1.100
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 1.026 (0.984, 1.069) | 1.021
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.009 (1.003, 1.016) | 1.000
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 1.073 (1.041, 1.107) | 1.056
$d=1$, Heteroskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.126 (1.070, 1.184) | 1.128
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 0.966 (0.916, 1.017) | 0.969
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.008 (1.001, 1.015) | 1.000
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 1.082 (1.043, 1.123) | 1.079
$d=10$, Homoskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.079 (1.068, 1.090) | 1.025
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 0.997 (0.979, 1.015) | 0.966
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.047 (1.041, 1.053) | 1.000
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 1.021 (1.013, 1.028) | 1.019
$d=10$, Heteroskedastic | $\hat{\theta}_{\textnormal{EPL}}$ | Flexible | 1.132 (1.115, 1.148) | 1.075
$\hat{\theta}_{\textnormal{EPL}}$ | Binned | 1.082 (1.058, 1.106) | 1.054
$\hat{\theta}_{\textnormal{EPL}}$ | Simple RCT | 1.059 (1.052, 1.066) | 1.000
$\hat{\theta}_{\textnormal{EPL}}^{(\mathrm{LA})}$ | Flexible | 1.055 (1.041,1.068) | 1.062

In Table 6, however, we still see some finite sample efficiency gains from
pooling, though the effect is not as large as in Table 2 in the main text. We
attribute this to the fact that for estimating $\theta_{0,\textnormal{PL}}$,
pooling allows the asymptotic variance to be approached more quickly as a
function of the total sample size $N$. Such an effect does not show up in ATE
estimation with AIPW, since for $\hat{\theta}_{\textnormal{AIPW}}^{*}$ and
$\hat{\theta}_{\textnormal{EPL}}^{*}$ the oracle estimators of Section 3, we
can see that
$N\textnormal{Var}(\hat{\theta}_{\textnormal{AIPW}}^{*})=V_{0,\textnormal{AIPW}}$
exactly for all $N$ while
$N\textnormal{Var}(\hat{\theta}_{\textnormal{EPL}}^{*})$ only approaches
$V_{0,\textnormal{EPL}}$ asymptotically as $N\rightarrow\infty$. So letting
$A_{N}^{*}$ be the (finite sample) AMSE of
$\hat{\theta}_{\textnormal{EPL}}^{*}$ computed on a sample of size $N$, we’d
expect $NA_{N}^{*}>2NA_{2N}^{*}$. Then averaging two independent copies of
$\hat{\theta}_{\textnormal{EPL}}^{*}$ on $N$ observations yields an estimator
with AMSE $A_{N}^{*}/2$, while pooling would yield an estimator with AMSE
$A_{2N^{*}}$.

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