A Neural Mean Embedding Approach for Back-door and Front-door Adjustment
Authors: Liyuan Xu
Arthur Gretton
Venue: ICLR 2023 poster (2023)

Abstract: We consider the estimation of average and counterfactual treatment effects, under two settings:  back-door adjustment and front-door adjustment. The goal in both cases is to recover the treatment effect without having an access to a hidden confounder. This objective is attained by first estimating the conditional mean of the desired outcome variable given relevant covariates (the ``first stage" regression), and then taking the (conditional) expectation of this function as a ``second stage" procedure.  
We propose to compute these conditional expectations directly using a regression function to the learned input features of the first stage, thus avoiding the need for sampling or density estimation. All functions and features (and in particular, the output features in the second stage) are neural networks learned adaptively from data, with the sole requirement that the final layer of the first stage should be linear. The proposed method is shown to converge to the true causal parameter, and outperforms the recent state-of-the-art methods on challenging causal benchmarks, including settings involving high-dimensional image data. 

A NEURAL MEAN EMBEDDING APPROACH FOR BACK- DOOR AND FRONT-DOOR ADJUSTMENT
Liyuan Xu
Arthur Gretton
Gatsby Unit
Gatsby Unit
1 1iyuan. jo. 1 119ulcl..... a@ucl.ac.uk
arthur. -eccurrgreete grettoontttottteaaoooooo com
ABSTRACT
We consider the estimation of average and counterfactual treatment effects, under two settings: back-door adjustment and front-door adjustment. The goal in both cases is to recover the treatment effect without having an access to a hidden con- founder. This objective is attained by first estimating the conditional mean of the desired outcome variable given relevant covariates (the "first stage" regression), and then taking the (conditional) expectation of this function as a "second stage" procedure. We propose to compute these conditional expectations directly using regression function to the learned input features of the first stage, thus avoid- ing the need for sampling or density estimation. All functions and features (and in particular, the output features in the second stage) are neural networks learned adaptively from data, with the sole requirement that the final layer of the first stage should be linear. The proposed method is shown to converge the true causal pa- rameter, and outperforms the recent state-of-the- methods on challenging causal benchmarks, including settings involving nigh-dimensional image data.
INTRODUCTION
The goal of causal inference from observational data is to predict the effect of our actions, or treat- ments, on the outcome without performing interventions. Questions of interest can include what is life expectancy? or counterfactual questions, such given the observed
the effect of smoking health outcome for a smoker. how long would they have lived had they quit smoking? Answering these questions becomes challenging when confounder exists, which affects both treatment and the outcome, and causes bias the estimation. Causal estimation requires us to correct for this confounding bias.
A popular assumption in causal inference is the no unmeasured confounder requirement, which means that we observe all the confounders that cause the bias in the estimation. Although a number of causal inference methods are proposed under this assumption (Hill 2011 Shalit et al., 2017 Shi al. 2019 Schwab et al. 2020). rarely holds in practice. In the smoking example, the confounder can be one's genetic characteristics or social status, which are difficult to measure for both technical and ethical reasons.
To address this issue, Pearl 1995) proposed back-door adjustment and front-door adjustment, which recover the causal effect the presence of hidden confounders using a back variable or front- door variable, respectively. The back-door variable is covariate that blocks all causal effects directed from the confounder to the treatment. In health care, patients may have underlying pre- dispositions to illness due to genetic social factors (hidden). which cause measurable symptoms. The symptoms can be used as the bock variable if the treatment is chosen based on these.
By contrast, front-door variable blocks the path from treatment to outcome. In perhaps the best- known example, the amount of tar in a smoker's 's lungs serves as a front-door variable, since it is increased by smoking, shortens life expectancy, and has direct link to underlying (hidden) socio- logical traits. Pearl ( (1995) showed that causal quantities can be obtained by taking the (conditional) expectation of the conditional average outcome.
While|Pearl (1995) only considered the discrete case, this framework was extended to the continuous case Singh et al. 22020, using two-stage regression (a review of this and other recent approaches

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for the continuous case is given in Section 5). In the first stage, the approach regresses from the relevant covariates to the outcome of interest, expressing the function as a linear combination of non-linear feature maps. Then, in the second stage, the causal parameters are estimated by learning the (conditional) expectation of the non-linear feature map used in the first stage. Unlike competing methods (Colangelo & Lee, 2020 Kennedy et al., 2017), two-stage regression avoids fitting proba- bility densities, which is challenging in high-dimensional settings (Wasserman) 2006 Section 6.5).
Singh et al. (2020)'s 's method is shown to converge the true causal parameters and exhibits better empirical performance than competing methods.
One limitation of the methods in [Sinhh al. (2020) that they use fixed pre-specified feature maps from reproducing kernel Hilbert spaces, which have limited expressive capacity when data are complex (images, text, audio). To overcome this, we propose to employ neural mean embedding approach to learning tpsk-specific adaptive feature dictionaries. At high level, we first employ a neural network with a linear final layer in the first stage. For the second stage, we learn the (conditional) mean of the stage I features in the penultimate layer, again with a neural net. The approach develops the technique of of(Xu et al. (2021a|b) and enables the model to capture complex causal relationships for nigh-dimensional covariates and treatments. Neural network feature means are also used to represent (conditional) probabilities in other machine learning settings, such as representation learning (Zaheer et al.) (017) and approximate Bayesian inference (Xu et al. 2022). We derive the consistency of the method based on the Rademacher complexity, a result of which is of independent interest and may be relevant in establishing consistency for broader categories of neural mean embedding approaches, including Xu et al. (2021a b). We empirically show that the proposed method performs better than other state-of-the-art neural causal inference methods, including those using kernel feature dictionaries.
This paper is structured as follows. In Section we introduce the causal parameters we are inter- ested in and give a detailed description of the proposed method in Section 3 The theoretical analysis
presented in Section 4 followed by a review of related work Section 5 We demonstrate the empirical performance of the proposed method in Section door adjustment problem with a binary treatment, and a challenging back-door and front-door setting where the treatment consists of high-dimensional image data.
[ covering two settings: a classical back-
2 PROBLEM SETTING
In this section, we introduce the causal parameters and methods to estimate these causal methods, namely a back-door adjustment and front-door adjustment. Throughout the paper, we denote random variable in a capital letter (c.g. A), the realization of this random variable in lowercase (e.g. a), and the set where random variable takes values in calligraphic letter (e.g. A). We assume data is generated from a distribution P.
Causal Parameters We introduce the target causal parameters using the potential outcome frame- work (Rubin) 2005). Let the treatment and the observed outcome be A E A and Y E J R). We denote the potential outcome given treatment a as ((a) E V. Here, we assume no inference, which means that we observe = ((a) when = a. We denote the hidden confounder as U E U and assume conditional exchangeability Va E A, (a)) II A/U, which means that the potential out- comes are not affected by the treatment assignment. A typical causal graph is shown in Figure Ia We may additionally consider the observable confounder O E O, which is discussed in Appendix
A first goal of causal inference is to estimate the Average Treatment Effect (ATE 0 (TTE(A) [y(w),, which is the average potential outcome of A a. We also consider Average Treatment Effect on the Treated (ATT) OATT(a; a') E [ A = a'), which is the expected potential out-
come of A = a for those who received the treatment A = a'. Given no inference and conditional exchangeability assumptions, these causal parameters
be written in the following form.
Proposition 1 (Rosenbaum Rubin 1983 Robins 1986). Given unobserved confounder U. which satisfies no inference and conditional exchangeability, we have
0 (ATE(a) = Ey [E [Y|A = a, U]] OATT(a; a') = Eu E = U]|A= o'].
If we observable additional confounder O we may also consider conditional average treatment effect (CATE): the average potential outcome for the sub-population of O o, which is discussed
2

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X
A
A
M
(a) General causal graph
(b) Back-doot adjustment
(c) Front-door adjustment
Figure I: Causal graphs we consider. The dotted circle means the unobservable variable.
in Appendix C Note that since the confounder U is not observed, cannot recover these causal parameters only from (A,
Back-door Adjustment back-door adjustment, we assume the access to the back variable X X, which blocks all causal paths from unobserved confounder U to treatment A. See Figure for a typical causal graph. Given the back-door variable, causal parameters can be written only from observable variables (A, Y, X) as follows.
Proposition 2 (Pearl 1995 Theorem 1). Given the back-door variable X, have
OATE 0) Ex (a(a, X)), Óxrr(a; a') = Ex [o(a, X) A = a'),
where g(a, x) E [YIA 0, X m|.
By comparing Proposition to Proposition
that causal parameters can be learned by treating the back-door variable X as the only "confounder", despite the presence of the additional hidden confounder U. Hence, we may apply any method based the "no unobservable confounder" assumption to back-door adjustment.
Front-door Adjustment Another adjustment for causal estimation is front-door adjustment, which uses the causal mechanism to determine the causal effect. Assume observe the front-door variable M M,, which blocks all causal paths from treatment A outcome Y, as in Figure Ic Then, recover the causal parameters follows.
Proposition 3 (Pear 1995 Theorem 2). Given the front-door variable M, we have = a]], @xrr(a; EM ig(a', M)]A= a), where g(a, m) E IYIA a, M m] and A' A is random variable that follows the same distribution as treatment
ATTE a) = E Ex
Unlike the case of the back-door adjustment, we cannot naively apply methods based on the "no un- measured confounder" assumption here, since Proposition 3 takes different form to Proposition I
ALGORITHMS
In this section, we present our proposed methods. We first present the case with back-door adjust- ment and then move to front-door adjustment. The algorithm is summarized in Appendix A
Back-door adjustment The algorithm consists of two stages; In the first stage, we learn the con- ditional expectation g E [ a, z] with specific form. We then compute the causal parameter by estimating the expectation of the input features to g.
The conditional expectation g(a,x) is learned by regressing XA Y. Here, we consider specific model g(a, .²) = w (a) & xx (z)), where Φ A + Rd. Øx X + Rd are feature trainable weight vector, and denotes
maps represented by neural networks, w E Rd4d2 tensor product < b = vec(ab") This tensor form used for g(a,z)) explicitly separates out the treatment of the features X and of A; the event that is much higher dimension than A,
In the binary treatment case A (0, 1}. the ATE is typically defined the expectation of the difference of potential outcome E[y - y (0)) However, we define ATE as the expectation of potential outcome E (o)], which is primary target of interest in continuous treatment case, also known as dose response curve. The applies to the ATT well.

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then concatenating both single input tends to downplay the information in A. In addition, can take advantage of linearity and focus on estimating the relevant (conditional) expectation as discussed later.
Given data {(a, Vr, -;) ~ P size of n, the feature maps A Øx and the weight w can be trained by minimizing the following empirical loss:
i (w, DA, xx) = nh Σ (m=-w w (a)((a)) @ (x;)))))
We may add any regularization term to this loss, such as weight decay A//w|/. Let the minimizer of the loss L be in, Φx = arg min 2 and the learned regression function be g(a, = i (aa(a) @ (x())) Then, by substituting 9 for g in Proposition @NTE(A) = T (aa(a) &0 E (xx(x)). OxTT(a; d') = i (da(a) oF
2
we have
(x) A A= ad)
This is the advantage of assuming the specific form of g(a,z) w ( & xx(x)) From linearity, recover the causal parameters by estimating [[)x E[ox (X) a'}. Such (conditional) expectations of features are called (conditional) mean embedding, and thus, we name method "neural (conditional) mean embedding".
We can estimate the marginal expectation [X)x
simple empirical average [x)x ~
1 n En= dx (zz). The conditional mean embedding E Elxx (X) A a'] requires more care, how- (2021a). in which we train another ever: it can be learned by a technique proposed in Xu et al. regression function from treatment A to the back-door feature Φx (X). Specifically, we estimate Elox(X)]A = dd) by føx (a'), where the regression function f f0x A + Rd be given by
føx argmin 2 (f:
(x:
(
3-A-5
Here, ||- denotes the Euclidean norm. The loss 23 may include the additional regularization term such weight decay term for parameters in f. We have
@TTE(a) i (
Exx(1.)) @ArT(a;a')= i (fx(a)
as the final estimator for the back-door adjustment. The estimator for the ATE ÔATE is reduced to اا ((,,,;),, This coincides with other neural network causal methods (Shalit et al.) 2017 Chernozhukov et al., 2022b), which do not assume g(a, z) w ( (a) @ (x)). As we have seen, however, this tensor product formulation essential for estimating ATT by back-door adjustment. It will also be necessary for the front-door adjustment, as we will see next.
the average the predictions ÔATE
Front-door adjustment We can obtain the estimator for front-door adjustment by following the almost same procedure as the back-door adjustment. Given data {(a;, yh, m)) we again fit the regression model ğ(a, m) ó (δala) @ (m)) by minimizing
(w S ΦM)
Σ (us - w ((p)(aa)) @ (mm()))))
where ΦM M + Rd is feature map represented as the neural network. From Proposi-
tion for Tx (a) = E (M) A A a. we have @ATE(a) w (B (A(A)) X fax (a)
and 0 A (arr(a;0)) (a:aa) ~ Tt (ox(a') 8 f (a)) Again, we estimate feature embedding by empirical average for (A)) or solving another regression problem for p& (a). The final estimator for front-door adjustment given as
@xTe(a) - Σ da(a.) @ fan((a)) Oxrr(a;a')= = (da(a') @ for (a)) T=
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where fØr is given by minimizing loss L- 1 E (l f(a:)1? (with additional regularization term).
THEORETICAL ANALYSIS
In this section, prove the consistency of the proposed method. We focus the back-door adjustment case, since the consistency of front-door adjustment can be derived identically. The proposed method consists of successive regression problems. In the first stage, we learn the conditional expectation 9, and then in the second stage, we estimate the feature embeddings. First, we show each stage's 's consistency, then present the overall convergence rate the causal parameter.
Consistency for the first stage: In this section, consider the hypothesis space of g Rd (x) E IRd (a) S 1, max
Hg {w ( @ @x(x)) u E RRddd Φ (a)
||w|| R, max aEA
|1 S 11.
DE
Here, we denote & -norm and infinity norm of vector b E Rd as ||b|| |6ف and |161 = maXield] b, Note that from inequality (a) @ xx(x)) I a | (x)
and Hölder's s inequality, we can show that h(a, E [-R, R] for all E Hg. First, we discuss the richness of this hypothesis space by the following theorem.
Theorem 1. Let A, X C Id be compact. Given sufficiently large R, R,,1,, d2, for any continuous function f A x X + I and constant e >> 0, there exists h E Hg which satisfies supa, If(a, M(a,x) S.
The proof uses the modified version of universal approximation theorem (Cybenko net, which will be given in Appendix [ [..I] Theorem[] tells function f with an arbitrary accuracy, which suggests the richness of our function class. Given this hypothesis space, the following lemma bounds the deviation of estimated conditional expectation and the true one.
1989) for neural
tells that we can approximate any continuous
Lemma 1. Given data S = {a., Yis yi. Fi>' let minimizer of loss LN be 9 = arg min 27. If the true conditional expectation g is in the hypothesis space g E Hy. w.p. at least 1 26, have
lgg - Pll 1 + 8R² (log 2/8)/2m.
where (Hg is empirical Rademacher complexity of Hy given data S.
The proof is given in Appendix B.3
Here, we present the empirical Rademacher complexity when apply feed-forwart neural network for features.
Lemma 2.
The empirical Rudemacher complexity Rg(H)) scales as Rs(H)) S O(C4) /Vn) for some constani C if we use specific L-layer neural net for feutures A Φx.
See Lemma 7 in Appendix for for the detailed expression of the upper bound. Note that this may be of independent interest since the similar hypothesis class considered in Xu et al. (2021a/b), (). and no explicit upper bound is provided on the empirical Rademacher complexity in that work.
Consistency for the second stage: Next, we consider the second stage of regression. In back-door adjustment, we estimate the feature embedding E (X)) and the conditional feature embedding E[σx(X)IA a'). We first state the consistency of the estimation of marginal expectation, which can be shown by Hoeffding's 's inequality.
Lemma 3. Given data {ar) and feature map Φxx, w.p. at least - 5, we have
E [xx(x)] - n
xx
2 log (2gg)))))))))
S
n
For conditional feature embedding xlxx(()) a'], we solve the regression problem føx argmin, L (f: 2x), the consistency of which is stated follows.
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Lemma 4. Let hypothesis space H be
Hz {a A- (fi(a)...... (a)(()) (a)) [-1,, fi,.... ......... faz E Hy)).
where Hy is some hypothesis space of functions of f X - [-1,1]. Let the true function be fox (a) = =[(xx(X)14 = a), and we assume fox Hg. Let fax arg min (f: (x) given data S {(a 3.)}. Then, we have
V (16 + 8v w.p. at least - 28, where (I)())] = max; (IAill(C)) and Rs(H)) is the empirical Rademacher complexity Hy given data S.
|1 (A) - fax (A)) S
The proof is identical to Lemma]] We use neural network hypothesis class for Hy whose empirical Rademacher complexity is bounded by O(1/ /n) as discussed in Proposition in Appendix B.3
Consistency of the causal estimator Finally, we show that if these two estimators converge uni- formly, we can recover the true causal parameters. To derive the consistency of the causal parameter, we put the following assumption on hypothesis spaces in order to guarantee that convergence in f-- norm leads to uniform convergence.
Assumption 1, For functions h1, h2 E Hgg there exists constant e > 0 and B that
(a, h2(a,, _ 11) (A, X) 1) P(A.X)- a
sup
Intuitively, this ensures that we have non-zero probability observing all elements in A X. We can see that Assumption I is satisfied with B and (a, Axx P(A = a, X = x) when treatment and back-door variables are discrete. A similar intuition holds for the continuous case; in Appendix B.2 we show that Assumption | holds when with 3 2d when A, X' are d-dimensional intervals the density function of P(A, X) is bounded away from zero and all functions in Hy are Lipschitz continuous.
Theorem 2. Under conditions in Lemmas to 3 and Assumption 7 w.p. at least 1 - 48, we have
sup (a) (a(((a)--AAAaa @ATE(a)] O(n de). aEA
If we furthermore assume that for all f. f.
(A) f(A)ll((A)))) $
sup [[f(a)= - - S aEA
max (E
then, w.p. at least 46, we have su (0ATT((; - ÓArT((; (a; d'll S O(n do + 434)
The proof is given in Appendix B.3 This rate is slow compared to the existing work (Singh et al.) 2020), which can be as fast as (n (nn-1/4). However, (Siggh et al. (2020) assumes that the correct regression function g is in a certain reproducing kernel Hilbert space (RKHS), which a stronger assumption than ours, which only assumes a Lipschitz hypothesis space. Deriving the matching minimax rates under the Lipschitz assumption remains topic for future work.
RELATED WORK
Meanwhile learning approaches to the back-door adjustment have been extensively explored in re- cent work, including tree models (Hill 2011) et al., 2019)), kernel models (Singh et al.) 2020) and neural networks (Shi et al. 2019 Chernozhukoy et al., 2022b (Shalit al. [[[])) most liter- ature considers binary treatment cases, and few methods can be applied to continuous treatments.
Schwab et al. (200 proposed to discretize the continuous treatments and Kennedy et al. (2017);
Colangelo & Lee (2020) conducted density estimation of P(X) and (XX These simple to implement but sufter from the curse of dimensionality (Wasserman] 2006 Section 6.5).
Recently, the automatic debiased machine learner (Auto-DML) approach (Chernozhukov et al. 2022a) has gained increasing attention, and can handle continuous treatments in the back-door ad- justment. Consider functional that maps g to causal parameter EE [m[g, (A,))). For

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the ATE case, have m(g. (A, g(a, since ( E [g(a, We may estimate both g and the Riesz representer a that satisfies E [m (A.X)) = E (9 by the least-square regression to get the causal estimator. Although Auto- DML can learn a complex causal relationship with neural network model (Chernozhukov et al. 2022b), it requires a considerable amount of computation when the treatment is continuous, since have learn different Riesz representer for each treatment a. Furthermore, as discussed in Appendix B.4] the error bound on can grow exponentially with respect the dimension of the probability space, which may harm performance in sigh-dimensional settings.
X))
Singh et (2020) proposed a feature embedding approach, in which feature maps are specified as the fixed feature maps in a reproducing kernel Hilbert space (RKHS). Although this strategy can be applied to number of different causal parameters, the flexibility of the model limited since uses pre-specified features. Our main contribution is to generalize this feature embedding approach to adaptive features which enables us to capture more complex causal relationships. Similar tech- niques used in the additional causal inference settings, such as deep feature instrumental variable method (Xu et al. 2021a) or deep proxy causal learning (Xu et al. 2021b).
By contrast with the back-door case, there is little literature that discusses non-linear front-door adjustment. The idea was originally introduced for the discrete treatment setting (Pearl 1995) and was later discussed using the linear causal model (Pearl 2009). To the best of our knowledge, |Singh et al. (2020) is the only work that considers the nonlinear front-door adjustment, where fixed kernel feature dictionaries are used. We generalize this approach using adaptive neural feature dictionaries and obtain promising performance.
EXPERIMENTS
In this section, we evaluate the performance of the proposed method based on two scenarios. One considers the back-door adjustment methods with binary treatment based on IHDP dataset (Gross) 1993 and ACIC dataset (Shimoni al., 2018). Another tests the performance on high-dimensional treatment based on dSprite image dataset (Matthey et al.) (017). We first describe the training procedure we apply for our proposed method, and then report the results of each benchmark. The
details of hyperparameters used in the experiment are summarized in Appendix D
6.1 TRAINING PROCEDURE
During the training, we use the learning procedure proposed by Xu et al. (2021a). Let us consider the first stage regression in a back-door adjustment, in which we consider the following loss L with weight decay regularization
1 E (w, SA. (A.
n >>00 (y) ( @ (x(x)))) + A||w||/
To minimize Dt with respect to (w, A., ), we can use the closed form solution of weight w. If fix features A,, Φx, the minimizer of w be written
1
n
u(... xx) =
XI
n Σ
y: n. R.),
n2
where (a,x
(l(a)= & Φz(a). Then, we optimize the features as Φ. Φx
arg min Sn dx Lx (m((A, Φx), A Φx) using Adam (Kingma & Ba 2015). We empirically found that this stabilizes the learning and improves the performance of the proposed method.
6.2 BINARY TREATMENT SCENARIO
In this section, report the performans two classical causal datasets: IHDP dataset and ACIC dataset. The IHDP dataset widely used to evaluate the performance of the estimators for the ATE (Shi et al.) 2019; Chernozhukov et al. 2022b: Athey al. 2019). This is semi-synthetic dataset based on the Infant Health and Development Program (IHDP) (Gross) 1993). Following existing work, generate 1000 sets of 747 observations of outcomes and binary treatments based
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Table 1: Mean and standard error of the ATE prediction error.
IHDP
ACIC
DragonNet
0.146 1 0.010
0.241 + 0.123
RieszNet(Direod
0.123 0.004
0.334 0.133
(I)s)
0.122 + 0.037
52.73 + 40.71
RieszNet((RR)
0.110 + 0.003
1.071 + 0.555
RKHS Embedding
0.166 + 0.003
1.785 + 1.398
NN Embedding (Proposed)
0.117 0.002
0.231 + 0.112
on the 25-dimensional observable confounder in the original data.The ACIC dataset introduced in (Shi et al. 2019), which is based on linked birth and infant death data (LBIDD) (Mathews & MacDorman 2006). This is considered a more challenging benchmark dataset than IHDP since contains data points with extreme propensity scores (i.e. P(A 1 can be very close to 0 or 1). We select 101 datasets following Shi et al. (2019) and remove outliers in each dataset using the procedure described in Appendix
We compare our method to competing causal methods, DragonNet (Shi et al. 2019), RieszNet (Chernozhukov al. 2022b), and RKHS Embedding (Singh al. 2020). DragonNet is neural causal inference method specially designed for the binary treatment, which applies the targeted regularization (van der Laan & Rubin 2006) ATE estimation. RieszNet implements Auto-DMi with a neural network, which learns the conditional expectation and Riesz representer a jointly while sharing the intermediate features. Given estimated 9, it proposes three ways calculate the causal parameter;
Direct E [m (A,X))] IPW; E [Yá(A,X)) DR E (g.
where functional m maps to the causal parameter (See Section 5 for the example of functional m). We report the performance of each estimator in RieszNet. RKHS Embedding employs the feature embedding approach with fixed kernel feature dictionaries.
The results are summarized in Table Although RieszNet(IPW) estimator performs promisingly in IHDP, the performance degenerates for the ACIC dataset, which suggests RieszNet(IPW) is prone to extreme propensity scores. This not surprising, since the true Riesz representer in this case (A, P[A=1|X] P(A=o|X] which can be very large if P(A 1|X) becomes close to 0 or 1. This also harms the performance of RieszNet(DRR. We see that the proposed method outper- forms all competing methods besides RieszNet(DR) in the IHDP dataset, for which the performance comparable (0.117 0.002 v.s. 0.110 0.003).
A
6.3 HIGH-DIMENSIONAL TREATMENT SCENARIO
To test the performance of our method of causal inference in a more complex setting, we used dSprite data (Matthey al. 2017), which is also used the benchmark for other gigheiiensional causal inference methods (Xu et al. 202 b). The dSprite dataset consists of images that are 64 x 64 = 4096-dimensional, described by five latent parameters (shape, scale, rotation. inn, posX and posY). Throughout this paper, we fix (shape, scale, rotation) and use posX E [0, 1] and posY E [0. 1] as the latent parameters. Based on this dataset, we propose two experiments; one ATE estimation based on the back-door adjustment, and the other is ATT estimation based on front-door adjustment.
our back-door adjustment experiment, we consider the case where the image is the treatment. Let sample hidden confounder U (0, 1), and consider the back- door as (X1,X2) = (U cos + E1, sin / 10++5 52) where E1, ~ (O0 0.09), 0 (n, 2n). We define treatment as the image, where the parameters are set as posx- 1 We add Gaussian noise (O, 0.01) each pixel of images. The outcome is given follows. 3
Back-door Adjustment
3 posy = X2+15
(AA 100
64
Y
+ 4(U - 0.5) +
h(A)
Σ
Apil
64
64
where Asy) denotes the value of the pixel at (i.j) and ey is the noise variable sampled from ey (O, 0.25). Each dataset consists of 5000 samples of (Y, A, X1, and consider the problem of estimating (ATE(a) = (A)/100, We compare the proposed method to RieszNet and RKHS Embedding, since DragonNet is designed for binary treatments and is not applicable here. We
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MSE
Algorithm
MSE
RKHS
3
Proposed
ReiszNet
ReiszNet ReiszNet Proposed
100
1000
5000
Emb.
(DR)
(IPW)
Data Size
Algorithm
Figure ATT experiment based dSprite data
Figure 2: ATE experiment based on dSprite data
generate 10 datasets and the average of squared eπror (#ATE(a) - (ATE(a))² at 9 test points is reported in Figure
We can see that the proposed method performs best in the setting, which shows the power of the method for complex high- bigh-dimensional inputs. The RKHS Embedding method suffers from the limited flexibility of the model for the case of complex high-dimensional treatment, and performs worse than all neural methods besides RieszNet(IPW), This suggests that it is difficult to estimate Riesz representer in high-dimensional scenario, which is also suggested by the exponential growth of the error bound to the dimension as discussed in Appendix B.4 We conjecture this also harms the performance of RieszNet(Direct) and RieszNet(DR), since the models for conditional expectation i and Riesz representer & share the intermediate features the network and are jointly trained in RieszNet.
Frontdoor Adjustment We use dSprite dataset to consider front-door adjustment. Again, we sample hidden confounder U1. U2 Unif(- 1.5, 1.5), and we set the image to be the treatment, where the parameters are set as posX +15 posY 1. We add Gaussian noise (00, 0.01) 3
3 to each pixel of the images. We use M = (A) + EM as the front-door variable M, where EM (O, 0.04). The outcome is given as follows,
M
Y
+ 5(U + U2) + +ey, Ey ~ NO
100
We consider the problem of estimating 0xTT(a; a; a') and obtain the average squared error on 121 points of a while fixing a' to the image of posX 0.6, posY 0.6. We compare against RKHS Embedding, where the result is given Figure 3 Note that RieszNet has not been developed for this setting. Again, the RKHS Embedding method suffers from the limited flexibility of the model, whereas our proposed model successfully captures the complex causal relationships.
7 CONCLUSION
We have proposed novel method for back-door and front-door adjustment, based the neural mean embedding. We established consistency of the proposed method based on Rademacher complexity argument, which contains a new analysis of the hypothesis space with the tensor product features. Our empirical evaluation shows that the proposed method outperforms existing estimators, especially when high-dimensional image observations are involved.
As future work, it would be promising to apply similar adaptive feature embedding approach to other causal parameters, such as marginal average effect V.0ATE (a) (Imbens & Newey 2009). Furthermore, it would be interesting to consider sequential treatments, as in dynamic treatment effect estimation, in which the treatment may depend on the past covariates, treatments and outcomes. Recently, kernel feature embedding approach (Singh et al.) 2021) has been developed estimate the dynamic treatment effect, and we expect that applying the neural mean embedding would benefit the performance.

---

ACKNOWLEDGEMENT
This work was supported by the Gatsby Charitable Foundation.
REFERENCES
Susan Athey. Julie Tibshirani, and Stefan Wager. Generalized random forests. The Annals of Statis- fics, 47(2): 148 11788 2019.
Victor Chernozhukov, Whitney K. Newey, Vietor Quintas-Martinez, and Vasilis Syrgkanis. Auto- matic debiased machine learning via neural nets for generalized linear regression, 2021.
Victor Chernozhukov, Whitney Newey, and Rahul Singh. Automatic debiased machine learning of causal and structural effects. Econometrica, 2022a.
Victor Chernozhukov, Whitney K. Newey, Victor Mhintas-Martines and Vasilis Syrgkanis. Riesznet and forestriesz: Automatic debiased machine learning with neural nets and random forests. In ICML 2022, 2022b.
Kyle Colangelo and Ying-Ying Lee. Double debiased machine learning nonparametric inference with continuous treatments. arXiv:2004.030366 2020.
G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems (MCSS), 2(4):303-314, 1989.
R. Gross. Infant health and development program (IHDP): Enhancing the outcomes low birth weight, premature infants in the united states, 1985-1988, 1993.
Jennifer L. Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217-240, 2011.
Guido W. Imbens and Whitney K. Newey. Identification and estimation of triangular simultancous
equations models without additivity. Econometrica, 7765):148-11512 2009.
Edward H Kennedy, Zongming Ma. Matthew D McHugh, and Dylan S Small. Nonparametric meth- ods for doubly robust estimation of continuous treatment effects. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(4): 1229, 2017.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015.
Tj Mathews and Marian MacDorman. Infant mortality statistics from the 2008 period linked birth/infant death data set. National vital statistics reports from the Centers for Disease Control and Prevention, National Center for Health Statistics, National Vital Statistics System, 54:1-29, 06 2006. doi: 00.1037/655995200--000
Loic Matthey, Irina Higgins, Demis Hassabis, and Alexander Lerehner. dSprites: Dis- entanglement testing sprites dataset, 2017. URL https://githus: com/ deepmind/
dsprites-dataset/
Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. In International Conference Learning Representations, 2018.
Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of Machine Learning. MIT Press, 2012.
Behnam Neyshabur, Ryota Tomioka, and Nathan Srebro. Norm-based capacity control in neural networks. Proceedings The 28th Conference on Learning Theory, volume 40 Proceedings of Machine Learning Research, pp. 1376-1401, 2015.
Judea Pearl. Causal diagrams for empirical research. Biometrika, $2(4):669-688 1995.
Judea Pearl. Causality. Cambridge University Press, 2 edition, 2009.
10

---

James Robins. A new approach causal inference in mortality studies with sustained exposure period application to control of the healthy worker survivor effect. Mathematical Modelling, 1511199--112,, 1986.
Paul Rosenbaum and Donald B. Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):4 1983.
Donald B. Rubin. Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American Statistical Association, 100(469):3223332) 2005.
Patrick Schwab, Lorenz Linhardt, Stefan Bauer, Joachim M. Buhmann, and Walter Karlen. Learning counterfactual representations for estimating individual dose-response curves. In AAAI, 2020.
Uri Shalit, Fredrik D. Johansson, and David Sontag. Estimating individual treatment effect: general- ization bounds and algorithms. In Proceedings of the 34th International Conference on Machine Learning. PMLR, 2017.
Claudia Shi, David Blei, and Victor Veitch. Adapting neural networks for the estimation of treatment effects. In Advances in Neural Information Processing Systems, volume 32, 2019.
Y. Shimoni, C. Yanover, E. Karavani, and Y. Goldschmnidt. Benchmarking Framework for Performance-Evaluatio of Causal Inference Analysis. ArXiv preprint 180.00000 2018.
Rahul Singh, Liyuan Xu, and Arthur Gretton. Kernel methods for causal functions: Dose, heteroge- neous, and incremental response curves, 2020.
Rahul Singh, Liyuan Xu, and Arthur Gretton. Kernel methods for multistage causal inference: Mediation analysis and dynamic treatment effects, 2021.
Mark J. van der Laan and Daniel Rubin. Targeted maximum likelihood learning. The International
Journal of Biostatistics, 2(1), 2006.
Larry Wasserman. All of Nonparametric Statistics (Springer Texts in Statistics). Springer-Verlat Verlag, 2006. ISBN 0387251455
Liyuan Xu, Yutian Chen, Siddarth Srinivasan, Nando de Freitas, Arnaud Doucet, and Arthur Gret- ton. Learning deep features in instrumental variable regression. In International Conference on Learning Representations, 2021a
Liyuan Xu, Heishiro Kanagawa, and Arthur Gretton. Deep proxy causal learning and its application to confounded bandit policy evaluation. In Advances in Neural Information Processing Systems, volume 34, pp. 26264 26275, 2021b.
Liyuan Xu, Yutian Chen, Arnaud Doucet, and Arthur Gretton. Importance weighted kernel Bayes' rule. In Proceedings of the 39th International Conference on Machine Learning, 2022.
Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poezos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in Neural Information Processing Systems, vol- ume 30, 2017.

---

A
ALGORITHM SUMMARY
Here, we provide the summary of algorithm.
Algorithm 1: Back-door Adjustment
Data: Back-door adjustment data {at, Mb. [44
Learn weights and features
w A.A. Øx arg min 2T,
E = EE= (y: w - @ (x(x))))?
2 Learn conditional embedding
fex = arg min -:A-R
2 (f: Φx),
f(a,)]
Compute causal parameters as
θATE(a) w (da(a) & 1 -
OATT(a;a')=u - w (δi(a) @ fsx((')
Algorithm 2: Front-door Adjustment
Data: Front-door adjustment data {a,, yks m,;
Learn weights and features
w AAA ØM arg min L,M,
CM + Ei - w (os(a))) @ (m())))
2 Learn conditional embedding
fon arg min AA-
LM @M), (M a E (o f(a))1
Compute causal parameters as
θATR(U) w0 1 Ei @a(a)) @ (a)
@xrr(a:a') T (δi(a') f fzm.(0) (a)
B TECHNICAL DETAILS
B.1 UNIVERSAL APPROXIMATION THEORY
In this section, we provide the proof of Theorem Recall our hypothesis space is
Hg = (w" T (a) & (x)) w E Rddd (a) Rd 6x (I) E R2
|| R, max
@a(a))l I max [1xx(x)]l> < 1]}
aeA
Let consider the feature be
(a = [a[ a + 01) (s a + +aa, )
xx(x) = io(t] r 31),..,, (t,, + Baa)1T
where is the sigmoid function and 8,,4 E RD, a, B, E R are parameters. By considering the case of dj = dy and setting "non-diagonal" elements of w to zero, we can see that
41
g(a, x) Σ wooos a a:)a(
member of of Hg. Next, present the following lemma.
Lemma 5. Let be a finite, signed regular Borel measures on A x X. If satisfies the followings:
Vs, E R Vo, E TR, { assT + a)a(t x + B)dp(a, + == 0.
(3)
12

---

then, given any continuous function f : A A - R and there is finite sum
g(2) wio(s a a aa)((t 8t).
which satisfies
max o,
- x))
The proof is identical to Theorem in Cybenko (189).. Now, all we have to prove is that the Sigmoid function satisfies (3. This can be shown by the similar discussion in the Lemma I in (Cybenko 1989).
Proof of Theorem Assume that
V8, E RR,, ER, f JAX (o =0
Then, for all 7,0 E R. we have
f AxX
lim lim A 400 22
a a) +0)
lim lim (Ai( a0)) Jaxx
+yd((a,
-10000 A2
S (alo)Ex(()du(a..)) AxX
where
aa < 0) (s 0) (x(x) (s=a+a=0)
((x++<<0)
Ea(a)
(1x++>>)) (f=++B=0)
((7)
(5)
We used the Lesbegue Bounded Convergence Theorem the second equation. From definition, we have
0 = / JAXX (=(a)xx(=))u(a,z))
x II (7))((0)
+ +H + ((6))(HHH xIXaa) + H(H x H
where
n {a Als a a==
IT. fa X11 x + 3 0}
НA 8.0 = {a E Als T a a a a 00
H = ta E x11 T I x + 0}.
Hence for all s, 3, we have
p(II x (1 = I(IIAA x = (H x xI = μ( s.a
x x = O.
Based on this, we show μ 0. Fix 8, and consider functional F(h) defined as
F(h) f
Axx hiss a,t x))d((a
where h is bounded measurable function h(u,v) [u, u] [0, + R, where min T x. zEX
T max s a, u min s a, T maxt" .EAA
aEAA
xEx
Let indicator function [(,.] (d.a) (u. u) defined
(u, = f
(u (b,e), e
E (d,e))
otherwise
Then, we have
F ((p.o0))(eoo)) 1 (1I1 U H (II) U -.,-)))) 0.
Since
lbbc)xxdde) (1,oc)x (1,20) (1,00)x((e,00) + [1,,0)x(e,20)
13

---

have ((()))))((s)) 0 for all b, d, R. For linearlity, we have
Σ n (e 0.
Note that Σ 1/ ex) uniformly converges to any bounded measurable function h [ū. u] x [8,e] + R. Hence, F(h) = O. In particular, h(u, = cos(u + v), sin(u + v) are bounded measurable functions, and thus,
f JAXX expli(s' T a + L x))dp(a,
cos(s a a t σ) + isnn(s T a t x))
Axx
F(cos(u + u)) + (sinn( + U)) 0.
Thus, the Fourier transform of is 0 and so μ must be zero as well. From Lemma 5 we see Theorem I
B.2 IMPLICATION OF ASSUMPTION
In this section, discuss the implication of Assumption I especially when the back-door and treatment variables are continuous. First, we show the upper bound of the sup norm of Lipschitz function.
Lemma 6. Let Z Z be the probability variable following P(Z) and Z [0, 11 Then, for all L-Lipschitz function bounded in h(2) [-R, Rl,, we have
[M<2]| S (2)
max ZE2
(2R AR+22VlL
P(Z)
the density function (2) is bounded away from zero ((z) 2 22>
Proof. Since 2 is compact, there exists x" such that
||<=*|| = max |*(=)].
2
Let M = |/((2)) and we consider the following rectangle
M
M
2 E Vi [d] max
0, zma
S 2h S min
11 +
2R 2VdL
2R 2
where 2/g denotes the i-th element of Then, from Lipschitz continuity, for all B, we have
[x(2)| 2 [n(=")
L -
2112
M L
Σ |2
2/g 12
M
2
2 M L
2R 2 2ddT.
M
2
2 M L
2
M/2
2/dl
Now, consider the volume of B. Since
R
R
5
<
2R 2/al 2R 2/dL
2R
2'
the events 0 2 2/1
and 1 < 2/1 + 2R+2VdL
M
do not occur simultancously. Therefore, we
2R++2VL
have
min
1, 1,
M
max (0,
M
M
2R + 2/
2R + 2/0L)
2R+2200
14

---

and
/ fM)))(((=)dd
[1bllp(2) =
/ [h(=)] f())d=
M
M²
2R + 2V/L
Since M = max [h(2),, we have
max #62
[h(2)l $
(2R 2v
2Vd
| P(Z)
By this, we can give the Assumption follows for the interval probability space.
Corollary 1. IfA [0, 1jda, X' [0, 1 1xxx and all function h E Hg are L- Lipschitz continuous, we have
h2 N2
max 2xxEAxA
1h (a, x)) hz(a, a²1 S Clih1
h22lp((.X))
dxtox where C' C== Tイ+ (4R + 4VdA L) 11x
Note that the assumption on hypothesis space easy to satisfy since all neural network Lipchitz function if we use the ReLU activation and regularize the operator norm of the weight in each layer.
B.3 CONSISTENCY RESULTS
Proof of Lemma
We use the following Rademacher bound to prove the consistency (Mohri
al. 2012)
Proposition 4. (Mhrri al. 2012 Theorem 11.3) Let be a measurable space and H be a family of functions mapping from X to y C [-R, R). Given fixed dataset S = (1)1,21), (y2, 22)..... (yn,
(., xn)) E (X y)), the empirical Rademacher complexity given by
1
Rg (7) = E.
sup n
Eorh(zi)) MEH
where = (01..... σn), with independent random variables taking values in {-1, +1} with equal probability. Then, for any 8 O, with probability at least 8 over the draw of an i.dd sample S of size each of following holds for all h E H::
log 2/5
E [V - M(X)) <
(U) (xx))² + 4R²
2n
log 2/5
n U h(x:)) < E (Y M(X))² &RRAs + 4R²
2n
Given Proposition we can prove the consistency of conditional expectation.
Proof of Lemmal followings.
From Proposition and 0.g Hg. for the probability least 1 25, we have
log 2/5
E
[Y ((A,)))
< Σ (y) g(a;, (()1, SRRis (Hly) + 4R V
2n
1
log 2/5
(1 g(a,, x.))² < E (Y x() - SRRRs(H.)) 4R2
2n
1=1
15

---

From the minimality of ğ arg min LT have
llg 2/6
E [ - ((A,-9(A,A))) x)) E ((A + 16RR)s(Hg) + 8R² V
2n
log 2/5
+ B [[((A - ((A, x))* < 16R (H₂) + 8R² V
2n
Taking the square root of both sides completes the proof.
Empirical Rademacher Complexity of Hg We discuss the empirical Rademacher complexity of Hy when we use feed-forward neural network for features A Φx here. The discussion is based on "peeling" argument proposed Neyshabur al. (2015).
Proposition 5 ((Neyshabur al. 2015). Theorem 1). Let hypothesis space of layer neural net be
HNN that
[Tww"I S )
HNN S:RD - f(() = w( (
snj.
where a is ReLU function and W E DDH w(L) E RIIH W ..... w(L-1) E -B xH are weights. The norm || llp.g matrix Lpp.- norm supzz0 [Wall// ||/1p/ Then, for any L, 2 any IS PS co. and any set {s1...., 8nh, the empirical Rademacher complexity is bounded as
2(L-1)
1 (12291() #1,) Rs(HHNN) < V
(minfv", (100,,,)l(2)))) max ||silp
for p" = 11(1111/1) and [x)+ = max
Given this, we can bound the empirical Rademacher complexity of Hg when each coordinate of features truncated member of HNN-
Lemma 7. Let A,X C IDD and define hypothesis set HNNFeat. (d) that
HNNFeat. (d) { RD (ss)) (f(f
.f1...., HNN}
where ō is ramp function 5(x) min(1, max(0,x))) Consider H, that
Hy tw- (ds(a) & (x))) w IRdida da(a)= E Rdd ,x (x) Rd ||/|| 11 S R, Φ HNNFeat. (di), 6x E HNNFest. (d2)) Given data set S ((a1, ... (an, kn)}, we have
1 Rs S 6R (,2211+11) 11p
2(6-1)
(min(p",4 (log(2)))) (max l|aillo- + max (10zlll=))
Note that we have
max aEA
(a)all)loo < 1, max [x)())
#EX
since we apply in the features. The proof is given as follows.
Proof. Let define the following hypothesis spaces.
HNN (ā flf HNN).
z2 ((( HNN).
Then, from the definition, we have
d2
Hlg t
Wights (a)
may!
Vi,j hig Alao
16

---

Since the maximum of linear function of the constraint ||||| I R is achieved for the values satisfying |oo| = R. we have
d
de
d1
d2
(HH) = Rs
wiyhes (a,
R, Vi.j his H&N
de
Rs
Wiyh (a)
Weyl
R, Vi, j his HlNN
|ayl = 1, his E HPSN })
< RRSs
Wishis (a, x))
i=1
Let HNN Has be the function space defined
HNN HNN (a, xz) x)) hr, h2 Hix)-
Since HNN contains the zero function, the final hypothesis space is the subset the convex hull of HNN
N because
d1
d2 wg) = Σ ())(h)(a,))-0)-
ΣΣ
Σ
| - (a,
20
Therefore, have
RzHH))) S HF -H2N)= S s
Now, we can bound Rs (H(Rrr)
Rs (
Ris ((2 (Fi(()
fe(s)) ([i(a)+
(fi(a))² ((((a)2-((((0)))) J..J2 E (nwot) HHNN)
1 Ms (((h(o)+
+ fa())) F..
+ 1 As (((A(a))) Fr E (lNx))
2
Rs (((((()))) fe E (WNN))
+ 2
< 2g (fr((0)+ + fa(x)) fi, E HHNN)) Rs. (PNNN) + Rsx (HNN)
R (HNN) + 32ss (HNN)).
where SA = {a:} and Sx {z Here, we used Talagrand's contraction lemma (Mohri et al. 2012 Lemma 5.1 in the inequality. Again, from Talagrand's s contraction lemma, we have
Rs (HNN) I Rs, (HNN), Rsx (HNN) S Rsx (HNN).
since ē is an 1-Lipchitz function.
Combining them, we have
As(Hy) S 6R( (NNN)) Rsx (HNN)).
This and Proposition 5 completes the proof.
Now, we derive the final theorem to show the consistency of the method.
Proof of Theorem 2 From the triangular inequality, we have
(()re(a) (ATE(a))
le - E [o(a,X))) |0IIE(O) - E [](a,
17

---

For the first term of r.h.s, we have
(ATT(()) - [0(a, x)]] Blg(a, X) g(a, X)]l
E (lg X) g(a, X)I]
sup aEA.
l0(a, x) 2()
For the second term, we have
0 - / E2 l0(a, 10(axx)]
(x(xx) - (a) 2 E [xx(x))
T ΦA(a) &
1=1
(xx(1))) (a(())aa)))))))))))()) 60 (
< |
Φ (a)
8
|
Σ
xx (2x) E [xx(x)]
n2
6x (2;) - [xx(X)]
R
71
Therefore, we have
n xx(x;) - E [xx(x)
(0ATE(a) - θATE(A)] S sup lg(a, x)- g(a, x)1 + R
0,2
Using Lemmas
and and Assumption
we have T
1/28
sup (g(a, - g(a,,)) 1 16HH)s(HL)) + +&R a,æ
log 2/6
2n
E (x)(x)) - - Σ - (x(())
log (2gg22//))
with probability at least - 46. Combining them and applying Lemma completes the proof for ATE bound. For ATT, we can derive the followings with the same discussion
10xTT(a; d') - (ATTT(as a')l < sup lg(a, x) J(a, x)1 + sup R
fox (a') - TE (X)
0,
a'EAA
Using Lemma 4 and the assumption made in Theorem 2 we have
1/288
Fox (a') E (xx(A a s 1H6)s + 8//
log (og(222/8)
2n
If we use neural network hypothesis space Hy considered in Proposition 5 we can see that the ATT bound holds.
B.4 LIMITATION OF SMOOTHNESS ASSUMPTION ON RIESZ REPRESENTER
In Chernozhukov et al. (2022b), we consider a functional m such that the causal param-
eter 0 can be written as 0 E [m[g, (A, X))], where g is the conditional expectation g(a, x) = E |Y|A= X æ). Then, Riesz Representer a, which satisfies E [m[g. (4, [a X), exists as long as
E [(m² (a, (A, S M |all
for all a E Ha and a smoothness parameter M. When we consider ATE (xTe(a), the corresponding functional m would be
m(a, (), = X).
Chernozhukov al. (2021 Theorem 1) shows that the deviation of estimated the Riesz Representer à and the true one ag scales as linear to the smoothness parameter M.
||á -ooolffA & O,Mo, + nn -1/2),
18

---

A
M
(a) General causal graph
(h) Back-door adjustment
(c) Front-door adjustment
Figure 4: Causal graph with observable confounder. The bidirectional arrows mean that allow both directions or even common ancestor variable.
where 8n is the critical radius that scales
log n
6n 00
when we consider fully connected neural networks. Now, we show that the smoothness parameter M can have an exponential dependency on the dimension of the space, even for simple a. Consider A = [-1, 1dd and some compact space X. We assume the uniform distribution for PX) Consider following å
ā(a, x) = max 1 -
Σ
2/4
where d[j denotes i-th element of a, and here we consider ā that does not depend on z. Say, we are interested in estimating (ATE(a) of a [0, (0, 00T, for which
E [I) X)) E [(ā x))
Now consider B that
Vi 1 a S 11 2
B E A
Then, since a(a, x) = 0 for all a & B, we have
/
||á ((A))
(o(A,))
(((A,)))
We use the assumption that P(A, X) is the uniform distribution to have the last equality. Hence, if Ha. the smoothness parameter must have the exponential dependency
M 2 2d
C OBSERVABLE CONFOUNDER
In this section, we consider the case where we have the additional observable confounder, the causal graph of which is given in Figure 4
Given the causal graph in Figure 4 ATE and ATT are defined as follows.
aATE (a) Ev, [E [Y]U, a]), OxTE(a; a') Eu,o E (YU,O,A= al]-
Furthermore, we can consider another causal parameter called conditional average trealment effect (CATE), which is conditional average of the potential outcome given O o;
#CATE (a; o) E (a)) O
Given exchangeability and no inference assumption, we have
(CATE(a; Eu |Y U, o, A=
19

---

These causal parameters can be recovered if the back-door the front-door variable is provided as follows.
First, we present the Proposition stating these causal parameters can be recovered if are given the back-door variable X.
Back-door adjustments:
Proposition 6 (Pearl 1995). Civen the back-door adjustment X in Figure 4b @xTe(a) = Ex,o lg(a, X,),
we have
OxTT(a; a²) Ex,o (a,, O, { = [d],,
OCTTE a;; o) Ex (9(0, o, X) = ol
where g(a, 0, = E a, 0 = x = x].
Now, we present the deep adaptive feature embedding approach to this. We first learn conditional expectation 9 as ~(a, 0, = ib (Q @ Φo(0) @ Φx (x)), where
A. do dx (z) arg min
T Σ(.
[ w (oa(as) @o(a)) @ (x(x:))))
(4)
given data (1,, a. o x;). Here, w is the weight and Φ Φo, Øx are the feature maps. From Proposition 6
we have
@TTE(A) = T (òx(a) @ Ex.o (oo(0) dx (xx (x))
AATT = T (a^(a) & Exoo (o(0) ox (X) a' d))
OCATE(a;O) T (a) @ do(o) & - Therefore, by estimating the feature embeddings, have
( )x(ex)) θATE(A) = = (ax(a) & Σ (0o(:)) @
θATT(a;a') = u (0^(aa) @ foose) (a)), ÓCNTE(a; i (δ.(a) @ 2ol0) @ fox (0)
where foosdx fox (0) are learned from
foogex arg min f n
[ @ )x(xx) f(a))
føx arg min
((x) f(o.)11².
Front-door adjustment: fied follows.
Given the front-door variable M, these causal parameters can be identi-
Proposition 7 (Pearl 1995). Given the front-door variable M in Figure 4c we have
@aTE(a) = E [Eo [Mmio.A=a [g(A', O, M)]]] 0xrr(a; d') = Eo 1EMi0.A-alo(d O,]]
(CATE(a; = EA, [EM ( M)]
where g(a, m) = = E O = o. = m] and A' follows the identical distribution as A.
For front-door adjustment, we learn conditional expectation 9 as ğ(a, 0, x)) = u (6 dm (m)). where
w. L.. So (m)) arg min
(y: w T(bA(as) 8 2ol0s)) @ Ox 22s (m:)))
20

---

Then, from Proposition 7 have
(TT = EE (A(4) & Bo (o(()) @ a O (M)
( ~ w T (ás(a') @ Eo (0) @ E s (M []]
(CATE();0) = T E (A)( @ (0(() @ EM 2 (M))
The conditional expectation Exi00=o,Eaa (M( is estimated as Eaii0=o,Aza [M) fou ()) a). where
1
f&n = arg min
Σ
(l - ffonm))-f(oo,,n (o),,
nt i=1
Then, by replacing the marginal expectation with the empirical average, have
(
((# )a(a))
θATE(A) - v
Σ
(dolos) @ San (05)
i=1
(ōald)) 8
Σ (ōo(o)) @ Saw (os, (1) ())
( = :
-
@OATE(a; = T H )x(دa) (
@ (0) @ f (0,
D EXPERIMENT DETAILS
Here, we describe the network architecture and hyper-parameters of all experiments. Unless other- wise specified, we used Adam with learning rate = 0.001, 81 = 0.9, 82 = 0.999 and æ 10 For RKHS Embedding, we used Gaussian kernel for continuous variable where the bandwidth is determined by the median trick.
D.1 BINARY TREATMENT SCENARIO
In this scenario, all treatments are binary A E f0, 1}. In RKHS Embedding and Neural Embedding, used the feature & given
(1 = [], (0) [0, 1JT
both IHDP setting and ACIC setting. This is equivalent to learn two models
E Y x, 0j = w/ (x E [Y|X= x, A 1] = w xx
with shared nonlinear feature Φ (X).
IHDP Dataset We used the 1000 data used in (Chernozhukov et al. 2022b), which is publicly available at Github page of the paper. The network structure for back-door feature (X) is shown in Table 2 Note that is much smaller network than Dragonnet or Riesznet, but increasing network size did not affect the result much
Table 2: Network structures of Neural Embedding for IHDP dataset. For the input layer, we provide the input variable. For the fully-connected layers (FC), we provide the input and output dimensions.
Back-door feature Φx (X)
Layer
Configuration Input(X)
2 FC(25, 200), ReLU
3
FC(200, 200), ReLU
21

---

ACIC Dataset We used the 101 data used in (Shi et al.
2019). which satisfies overlap assumption.
(i.e. Not all data points has the extreme propensity score P(A 1[X).) We noticed that some data contains outliers and we only consider the data points with the outcome Y is in the range of
E (Q SIQR, Qa + 5IQR
where Qs(Y) 25%, 75%-quantile value of outcome, respectively, and IQR Qa(Y) Q)
We run Dragonnet and RieszNet estimators with the same network architecture IHDP dataset. The network structure for back-door feature Φx (X) is shown in Table 3 Note that the same structure is used in Dragonnet and Riesznet predict conditional expectation E [Y]X,
Table Network structures of Neural Embedding for ACIC dataset. For the input layer, we provide the input variable. For the conne-connected layers (FC), we provide the input and output dimensions.
Back-door feature Φx (X)
Layer
Configuration
Input(X)
2
FC(177, 200), ELU
3
FC(200, 200), ELU
4
FC(200, 200), ELU
FC(200, 100), ELU
D.2 HIGH-DIIMNNIONAL TREATMENT SCENARIO
Here, we generate all dataset by ourselves from original dSprite dataset (Matthey et al. 2017).
Back-door ATE estimation The network features for the proposed method is summarized in Ta-
ble 4 The network structures for RieszNet is given Table S. Note that they share the similar feature extractor for images.
Table 4: Network structures of the neural embedding method in dSprite back-door adjustment periment. For the input layer, we provide the input variable. For the fully- layers (FC), we provide the input and output dimensions. SN denotes Spectral Normalization (Miyato et al. 2018)
Treatment Feature (A)
Back-door feature Φx (X)
Layer
Configuration
Layer
Configuration
Input(A) FC(4096, 1024), SN, ReLU
Input(X X) FC(2, 36), ReLU
FC(1024, 512), SN, ReLU, BN FC(512, 128), SN, ReLU
3
FC(36, 5), ReLU
FC(128, 32), SN, BN, Tanh
Front-door ATT estimation Here, we used the same network architecture as in the back-door adjustment summarized in Table
22

---

Table 5: Network structures RieszNet in dSprite back-door adjustment experiment. For the fully- connected layers (FC), we provide the input and output dimensions. SN denotes Spectral Normal-
ization Miyato et al. 2018
Common Feature a(A, X)
Regressor
Layer
Configuration
Layer
Configuration
Input( A, X) FC(4098, 1024), SN. ReLU
Input(A, X)
2
2
Common Feature Φ(A, X)
3
FC(1024, 512), SN, ReLU, BN
FC(32, 32), ReLU FC(32, 32), ReLU
4
FC(512, 128), SN, ReLU
5
FC(128, 32), SN, BN, Tanh
5
FC(32, 1)
Riesz representer learning
Layer
Configuration
Input( A, X)
2
Common Feature of (4, X)
3
FC(32, 1)
23


Reviews:
[{"review_id": "c9KTQh0Eat", "reviewer": "Reviewer_Cbhn", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work studies front-door and back-door adjustment using tensor product of learned kernels, generalizing Singh et al (2021) which employed fixed-form kernels.  The authors establish consistency for the estimators, and demonstrate improved performance on IHDP and a synthetic high-dimensional benchmark.\n\n---\n\nPost-rebuttal update: Thank you for your response. I'm raising my score given the new experiments.", "summary_of_the_review": "Pros:\n\n+ Proposed algorithm is a natural and intuitive extension\n\nCons: \n\n- Novelty is lacking\n- Neither theoretical nor practical benefits were clearly demonstrated", "strength_and_weaknesses": "The proposed method combines the tensor product formulation, which simplifies the estimation of ATT, and kernel learning, which leads to improved flexibility.\n\nMy main concerns are the relative lack of novelty, and the significance of the contribution: \n* Broadly speaking, the idea of using learned kernels in kernelized estimators for causal effect have been well-studied (Xu et al, 2021a; 2021b).  For the specific problems of back-door and front-door adjustment in this work, the generalization of the corresponding kernel estimator (Singh et al, 2021) is straightforward, and does not appear to introduce new challenges.\n* The lack of novelty will not be an issue if the benefits of the proposed method are more clearly demonstrated, but I'm not really sure if this is the case here: on the theoretical side, the authors established consistency results with worse rates than Singh et al (2021); thus, no provable benefits over fixed-form kernels were demonstrated.  On the empirical side, the empirical improvements are mainly demonstrated on the dSprite dataset, which is highly synthetic, and does not appear to relate to real-world problems well.\n\nIn aggregate, I feel there should be more demonstration of the contributions from either side, before there is a compelling case for acceptance.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "QQqogRENce8", "reviewer": "Reviewer_jNzo", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper targets continuous intervention and counterfactual problem which is usually handled by back-door and front-door adjustment in causal inference. In particular, the authors aim to improve two-stage regression limited by fixed pre-specified feature maps. To improve, the authors propose employing two neural networks to extract adaptive features and estimate the mean embedding, respectively. The loss function is the regression loss with regularization. More importantly, the authors provide with theoretical analysis of the consistency of the proposed method using Rademacher complexity, which bounds the estimated ATE, assuming all the functions in the hypothesis space are Lipschitz continuous. Lastly, the authors demonstrate the proposed method on two popular causal data sets to support their claim. \n\n", "summary_of_the_review": "The paper proposes to train two neural networks for estimating treatment effect, an important question in causal inference. The two neural networks are trained based on regression objectives, which can be easily reproduced. The authors then provide with the theoretic analysis to bound the estimated treatment effect for intervention and counterfactual setting. The experimental results look promising and intriguing. ", "strength_and_weaknesses": "Strength:\n1) The authors focus on the important problem, treatment effect, in the causal inference community. They leverage the approximate power from the neural networks to focus on back-door adjustment and front-door adjustment. \n2) The idea is straightforward and intuitive. The theoretic analysis provides insights into the proposed method. \n3) The experiment results are promising. \n\nWeaknesses:\n1) It seems that it assumes the best approximation function exists in the hypothesis function space. However, what if that is not the case?\n2) It is a bit unclear to me why the function form of g(a,x) is designed in the way stated in the paper. Maybe more explanation will be better. \n\nQuestion:\n1) Could you please explain a bit more about why you design the L2 loss function in Equation 2?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "NTrCVATXDy2", "reviewer": "Reviewer_BgB4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "A method is presented for estimating causal treatment effects from observational data using neural networks for the sake of modeling flexible nonlinear relationships in the relevant conditional expectation functions. Techniques for both backdoor and frontdoor adjustments based on the directed acyclic graph causal inference theory of Pearl are presented.  The conditional expectation of outcome Y given treatment A and backdoor control X is assumed to be a linear combination of a separable tensor product of individual neural network transformations of A and X. The method has better scaling properties than competing neural-network-based causal inference methods when treatment effects are continuous and/or the problem space is high dimensional. Theoretical results based on Rademacher complexity convergence theory demonstrating the consistency of the methodology is presented. Positive experimental results on semi-synthetic health policy data and synthetic benchmark image data often used for causal inference research on images are presented.", "summary_of_the_review": "A method is presented for estimating causal effects from observational data using neural nets to model conditional expectations. The abstract promises no assumptions or requirements for the functions to be learned aside from linearity in one output layer, but in fact the method assumes a separability between neural net transformations of the treatment and the backdoor variable. Experimental results are resonably strong but the method does not always win and one of the experiments chooses a very simple relationship between hidden confounder and backdoor variable which does not seem to take full advantage of the upside of the flexible neural net methodology.\n\n***Update post-rebuttal ***\n\nIn light of the newly added approximation theorem and the new dSprite experiment with a more complex nonlinear control variable relationship, I have raised my score to a 6.", "strength_and_weaknesses": "The methodology is novel to the best of my knowledge.\n\nFor the most part, I think the paper is very well written, although there are a few spots where I think the paper makes claims which are a bit too overbroad (I will elaborate shortly). \n\nThe experimental results look reasonably strong but I have a few concerns there. One concern is that ReiszNet(DR) does beat the proposed method on the health policy dataset, to a degree that may not be large but does seem to be reasonably statistically significant. I commend the authors for having the honesty to report this result, but it does weaken the case for neural mean embedding at least slightly. For the dSprite experiment, I thought the relationship between the backdoor control X and the hidden confounder U was disappointingly simple  (X1, X2 are just U1, U2 + noise). I would have thought/hoped that the authors would want to demonstrate the merits of their neural network-based technique by demonstrating the ability to capture a more complex relationship between the hidden confounder and the backdoor control.\n\nThe main overstated claim I see is that the abstract claims \"Alll functions and features (and in particular, the output features in the second stage) are neural networks learned adaptively from data, with the sole requirement that the final layer of the first stage\nshould be linear\"...but in fact, that is not the sole requirement. At the bottom of page 3, an assumption is made that g(A,X) can be represented as a linear combination of a separable function of individual neural net transformations of A and X.  I say \"separable\" here because that would seem to be an appropriate term, given that g has to be a linear combination of a tensor product of neural_net_1(A) times neural_net_2(X).  So A and X cannot interact arbitrarily in a nonlinear way. On page, 4, the authors refer to the advantage of this specific functional form and I don't doubt that there are advantages, but there is also the disadvantage of the possibility that some g(A,X)'s may not be implementable by the architecture, regardless of the number of hidden units used in either of the neural nets (unless I have misunderstood something). So the final layer of the first stage being linear is not the \"sole requirement\". I feel like the implications of this separability choice needed to be discussed/justified more. In my opinion, that would be a better use of space than the Rademacher complexity consistency/convergence stuff, which strikes me as perfectly OK but unsurprising...in general, we know that a neural net with a given number of hidden units has finite Rademacher complexity and you will converge on the correct hypothesis in the infinite data limit ...so this section seems unsurprising.\n\nThe other clarity concern I have is more minor but I would still like to see it cleaned up. There are a few spots where the writing could be taken to suggest that you can always solve the problem of having an hidden unobserved confounder with a backdoor adjustment. In fact, you have to be lucky enough to have an observed backdoor variable which closes the backdoor path between treatment and outcome. The abstract says \"the goal in both cases is to recover the treatment effect without having an access to a hidden confounder\".  The introduction says that \"In health care, patients may have underlying predispositions to illness due to genetic or social factors (hidden), from which measurable symptoms will arise (back-door variable)\".  I don't think \"will arise\" is appropriate here...you might be lucky enough to have access to measurable symptoms which close the backdoor paths from treatment to outcome via the hidden factors, but you also might not have access to any such measurable symptoms. Not a big problem, but some readers might get the impression that you can always deal with hidden confounders this way, and that is not true- you might not have any observed backdoor variable. \n\nI am open to persuasion during the rebuttal phase, however.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "F-IrA0qJgr", "reviewer": "Reviewer_VeMx", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a two-stage regression method to estimate the average treatment effects under back-door and front-door adjustment. Both stages use neural network models to learn the relevant functions. They conducted several experiments to illustrate the performance of their proposed method, using several state-of-the-art methods as baselines.", "summary_of_the_review": " I think the proposed method is flexible to handle multiple treatments or continuous treatments. But there are some major issues with the proposed method which I outlined in the Strength and Weakness section, e.g., it does not take into account the treatment assignment bias in the observational data. I also have doubts in the reliability of the experimental results, e.g., whether they did a fair comparison with other methods.", "strength_and_weaknesses": "Strength:\n\n1. This paper has a thorough literature review.\n\n2. I think the proposed method is flexible to handle multiple treatments or continuous treatments. That is an advantage compared to other methods in the literature. But on the other hand, I also have doubts / questions on the effectiveness of the method, as outlined below.\n\nWeakness:\n\n1. On a high level, in either back-door or front-door adjustment, how do you get the adjustment variables, X or M in the paper? Some intuitions might help the reader understand better about the problem. Also, in the experiments, you took X or M to be a noisy version of U. If the noise is small, does it imply that we have too good an adjustment variable?\n\n2. Even though the paper claims that their method was developed in a front-door or back-door adjustment scenario, the truth is that, the authors simply assumed the existence of an adjustment variable (X or M), and developed a method that applies to the adjustment variables. In this sense, it is no different than other methodologies which make the assumption that all confounders can be observed. This is also why the proposed method can be directly compared to, e.g., dragonnet, which assumes observable confounders. \n\n3. One big problem I can see with the proposed method is that, it does not take into account the problems with doing causal inference on observational data, namely, the distribution of X in different treatment groups could be very different. That is why various methods have been developed in the literature to account for this treatment assignment bias, e.g., doing distributional matching in Tarnet (Shalit, U., Johansson, F.D. and Sontag, D., 2017, July. Estimating individual treatment effect: generalization bounds and algorithms. In International Conference on Machine Learning (pp. 3076-3085). PMLR.) \n\n4. The other disadvantage I can see with this two-stage regression method lies in that, the estimation of the regression models in the two stages is somehow independent. In other words, the 1st stage model cannot be informed by the 2nd stage. This is the shortcoming of a two-stage model compared to a joint model.\n\n5. I am also curious that, when you have discrete treatments, does it pose any challenge for you to estimate the \\phi_A function? How do you estimate \\phi_A in this case?\n\n6. I think you need a better way to illustrate your algorithm, either graphically or a step-by-step procedure.\n\n7. On Page 4, bottom line, left equation, is it a typo to use f_{\\phi(M)}(a)? Should you use \\phi_M(m)?\n\n8. In the experimental section, when you compared with other methods such as dragonnet, did you ensure that you used the same neural network architecture for a fair comparison? I know that the original paper of dragonnet used a very simple 2-layer neural net to estimate the conditional outcome. I am wondering if you used a more complex neural network in your method which could explain the better performance of yours. Also, dragonnet can be extended to multiple treatments case, which can also serve as a baseline in your multi-treatment experiment. You might also want to include tarnet as a baseline in the binary treatment experiment.\n\n9. In terms of the metrics, why did you use MAE in Table 1, but MSE in Figs. 2 and 3? Also in the IHDP dataset, you did not have a front-door or back-door adjustment, right?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster