Paper: Adversarial Learning of Decomposed Representations for Treatment Effect Estimation (ICLR-2024)

Reviewer comment (Theory):
Q1, `To deal with the issue, the common practice is to introduce pre-treatment covariates such that {Y (t)|x} =d {Y |t, x} (ignorability assumption).' Is it correct? If there are only pre-treatment covariates, it implies that there are no descendants of both $T$ and $Y$ in the set of covariates. How can we ensure $ignorability$ hold?

Author response:
i) To our understanding, we believe that *"introducing all the pre-treatment covariates"* is a safe choice, but may not be an efficient one.  
Actually, the quoted sentence was referenced from [1]. We have also mentioned this in the paper (the 1st line of page 2).  
The full discussion can be accessed from https://ftp.cs.ucla.edu/pub/stat_ser/r345-sim.pdf.  

ii) *If there are only pre-treatment covariates, it implies that there are no descendants of both $T$ and $Y$ in the set of covariates. How can we ensure ignorability hold?*\  
The **ignorabity** assumption means that $\boldsymbol{X}$ blocked all the \textit{\bf{back-door paths}} from $T$ to $Y$ in the form $T\leftarrow \cdots Y$.  
Therefore, the descendants of $T$ and $Y$ are not necessary to block such paths.  
 

References:\  
[1] DB Rubin. Author’s reply (to judea pearl’s and arvid sjölander’s letters to the editor). Statistics in  
Medicine, 28:1420–1423, 2009.

Reviewer comment (Theory):
Q2, Eq.(2): `E[Y(t)|x]=E[Y(t)|x,T =t]=E[Y|x,T =t]', Can we really transform the potential outcome prediction problem into a supervised learning problem?

Author response:
Sure, under the ignorability assumption, $\mathbb{E}[Y(t)|x]=\mathbb{E}[Y|x, t]$, which is the common practice in literature of treatment effect estimation.  
However, we care about the potential outcome for all the units, instead of the ones that receive treatment $T_i=t$.  
That is, we need to estimate the counterfactual outcome $Y_i(t)$ for the units with $T_i\neq t$  
This is why the inference of potential outcomes (and also the CATE) is more challenging than the common supervised learning problem.

Reviewer comment (Theory):
Q3. For the causal DAG in Fig. 1 (b),  does ADR also apply when $X_1$ is an unobserved variable.

Author response:
Thanks for the question.  
In this case, since $X_1$ is not a confounder, ADR can be applied when $X_1$ is unobserved.

Reviewer comment (Theory):
*Th 3.1* (variance lower bound). The statement seems incorrect or has typo(s). For CATE, the bound should depend on the value of x, but your eq of V takes expectation on X. Moreover, for consistent estimators, V should depend on n, and V → 0 as n → inf, but your V is a constant wrt n. Anyway, I don’t see this result has a strong relationship to the method (or else I will give a lower score), you could remove this result if you cannot fix it.

Author response:
- Q1 [questions on the Thm 3.1]

Thanks for the interest in Thm 3.1. It seems like there might be some understanding due to this result.
	
i) *Why the variance bound is constant w.r.t. $n$.*

Please note that Thm 3.1  is about the \textit{asymptotic variance bound}, instead of the \textit{variance} of the estimation. 
	The asymptotic variance refers to the variance of the limit distribution, instead of the limit of the variance.
	Please allow us explain this with a simple example. 
	Suppose that $\{X_i\stackrel{i.i.d.}{\sim}N(\mu, \sigma^2)\}$ and $\hat{\mu}=\frac{\sum_i X_i}{n}$ is the estimation of $\mu$, then $\sqrt{n}(\hat{\mu}-{\mu})\stackrel{d}{\rightarrow} N(0, \sigma^2)$.
	In this case, the asymptotic variance of $\hat{\mu}$ is $\sigma^2$, and the variance of $\hat{\mu}$ is $\sigma^2/n$. Please kindly refer to [1] for the formal definition.
	
ii) *Why the variance bound is not relevant to $x$*. 

Suppose that $\tau_0$ is the underling value, and $\widehat{\tau}(x)$ is the estimation. Then the variance of $\tau(x)$ refers to $\mathbb{E}[\tau_0-\widehat{\tau}(X)]^2$, a real number. The conditional variance  $\mathbb{E}[\tau_x-\widehat{\tau}(X)]^2$ is a function of $X$.

References:\
[1] Van der Vaart A W. Asymptotic statistics[M]. Cambridge university press, 2000.

Reviewer comment (Theory):
*Def 3.1* deviates from standard notions in the literature and also has practical limitations. For example, in your Fig 1a, if there is an X4 that is a parent of both X1 and X2, this is usually understood as a confounder and can improve the precision of estimation, but is excluded from your approach. In your Fig 1b, which is actually the “M-bias” case (see model 7 [here](https://ftp.cs.ucla.edu/pub/stat_ser/r493.pdf)), X1 and X3 both satisfy the backdoor criteria and could be understood as confounders. Here, besides X3, both X1 and X2, which are excluded from your approach, could possibly improve the precision (though X2 alone is a bad control).

Author response:
**On Def 3.1**

Thanks for recommending the literature and We have read it through.

i) I could not agree that $X_4$ is similar to the setting of Model 8 and 13 (*possibly good for precision*). Because in our case, $p(y|x_2, x_3, do(t)) = p(y|x_2, x_3, x_4, do(t))$, while in Model 8, $p(y|do(x), z)\neq p(y|do(x))$. 
Although explanation by graph is intuitive and inspiring, I suppose justifying by the interventional distribution is also a plausible and clear way. 
Besides, as [1] mentioned in the ``Model 8", "*controlling for $Z$ reduces the variation of the outcomes $Y$ and helps to improve the precision of the ACE estimate in finite sample*". I find this this motivating because in this case controlling for $Z$ is not necessary (in terms of bias), but benefits the precision in finite sample. 

ii) I agree that all the subsets except from $\{X_2\}$ are admissible sets to block the back-door paths. Meanwhile, I also think it sufficies to include $X_3$ in modeling the ITE.

Reviewer comment (Theory):
I suggest being clear that the definition in *Def 3.1* is nonstandard, discussing and comparing it to usual notions (possibly in the Appendix). In particular, you should mention there are possible variables in your IVs that are good to add as controls.

Author response:
- Q2 [questions on Def 3.1]

The terms "Instrument variables" and "Adjustment Variables" were borrowed from the notations from DeR-CFR [2].
We admit that such names may run at a risk of deviating from the standard notions. 

i)  *``in your Fig 1a, if there is an $X_4$ that is a parent of both $X_1$ and $X_2$, this is usually understood as a confounder and can improve the precision of estimation, but is excluded from your approach."*

In this case, $P(Y|X_2, X_3, do(t))=P(Y|X_2, X_3, X_4, do(t))$, which means that adding $X_4$ is spurious after including $\{X_2, X_3\}$.
More generally, we may prove that $p(Y|do(t), \boldsymbol{X})=p(y|do(t), \boldsymbol{C}, \boldsymbol{A})$. We have added this result in Prop 3.2.

ii) *``your Fig 1b is actually the “M-bias” case (see model 7 here). X1 and X3 both satisfy the backdoor criteria and could be understood as confounders. Here, besides X3, both X1 and X2, which are excluded from your approach, could possibly improve the precision (though X2 alone is a bad control)."*

In the example of Fig 1b, there is no unblocked back-door path(s) from $T$ to $Y$. Therefore, we could not agree that $X_1$ and $X_3$ could be called as confounders. However, both $X_1$ and $X_3$ are admissible set in [3] that satisfies $p(y|x_1, do(t))=p(y|x_1, t)$ and  $p(y|x_1, do(t))=p(y|x_1, t)$.


References:\
[2] Anpeng Wu, Junkun Yuan, Kun Kuang, Bo Li, Runze Wu, Qiang Zhu, Yueting Zhuang, and Fei
Wu. Learning decomposed representations for treatment effect estimation. IEEE Transactions on Knowledge and Data Engineering, 35(5):4989–5001, 2022.\
[3] Pearl J, Paz A. Confounding equivalence in causal inference[J]. Journal of Causal Inference, 2014, 2(1): 75-93.

Reviewer comment (Theory):
Prop 3.2 seems correct but the learning approach is not sufficient. Taking (i), I agree that independence means larger L_A than dependence, but there can be many different functions A that give the independence. Worse, some A could take a confounder but “cleverly” throw away the dependence on T. Similar comments apply to (ii). Could your theory rule out these concerns?

Author response:
- Q3 [questions on Prop 3.2]

The learning approach is a direct implementation of Prop 3.2.
The $\hat{h}\_{A\rightarrow T}(\cdot)$ in Prop 3.2 corresponds to the ancillary predictor $h_{A\rightarrow T}(\cdot)$.
Prop 3.2 claims when $\mathcal{L}_A$ is maximized when $A\perp T$, so we maximize $\mathcal{L}_A$ in the learning process.
Could you please be more specific on the question *``Worse, some A could take a confounder but “cleverly” through away the dependence on T. Similar comments apply to (ii). "*.

Reviewer comment (Theory):
The ADR algorithm does not precisely enforce the required independence or even the approach in Prop 3.2, because L_A, L_C, L_I contain both prediction and adversarial terms, so the ADR is a trade-off but not a direct implementation of the theory. Moreover, training the losses together with hyper-parameters adds yet another layer of trade-off.

Author response:
**On Prop 3.2**

Thanks for the futher explanation. If $A(C)\perp T$, it means that the information of $C$ may not be catched by $A(\cdot)$. Perhaps this can be appreciated by the results in Figure 3, we investigate the $\overline{W}_I$, $\overline{W}_C$, $\overline{W}_A$ to justify the owenership of the *learned* decomposed representations.
By saying that "a direct implementation of Prop 3.2", we intend to express that the loss function and the iterative learning process is designed according to Prop 3.2. 
For the ADR algorithm, Prop 3.2 plays the same role as Prop 1 (global optimality) for GAN.

Reviewer comment (Writing):
In "Algorithm 1," the text should be edited to remove the reference to the specific optimizer Adam. Since optimizers change over time, and the contribution is presumably more general, other optimizers would likely work as well in principle.

Author response:
As suggested, we haved revised Algorithm 1 remove the reference to  the specific optimizer Adamin the updated version. Thanks for the kind suggestion.

Reviewer comment (Writing):
It is unclear whether the authors intend investigators to examine the inferred decomposition of X, or whether the motivation is mainly or exclusively efficient CATE estimation.

Author response:
Thanks for the question. 
We admit that the original purpose is to facilitate a more efficient CATE estimation.
However, we believe that such decomposition is beneficial for further analyses on the covariates, especially in the case where interpretation is important.

Reviewer comment (Writing):
The aim is not clear. The disentanglement itself seems to be the aim, and it is not clear how it contributes to the accuracy of the CATE (see Question 1).

Author response:
- [Q1. purpose of the decomposition]

We agree that it is important to answer the question for implementing such decomposition for the ITE estimation.
It is absolutely correct that the purpose is to `` limit the confounder variables to necessary ones to alleviate the estimation variance due to extreme weights". 
However, it should be noted that, even we do not adopt the inverse probability weighting method, the estimation variance also increase with the aggravation of data imbalance as Thm 3.1 suggests.
We may have a further understanding by reviewing the development of the literature.


Firstly, the balanced representation leaning algorithms [1, 2] were proposed to reduce the data imbalance caused by selection bias. 
Except from the empirical evidence, [2] explained the benefit from the view of generalization bound.

Secondly, [3] pointed out that leaning the balanced representation alone may overlook the necessary confounding information and hence should learn the decomposed representation. However, the loss terms of DR algorthm in [3] was not complete and hence was not able to learn the decomposed representations. [4] proposed the DeR algorithm to realize the decomposition learning under the situation of binary treatment and response.
To deal with the case for continuous treatment and adapt to both categorical and continuous response, we analyzed the theoretical property of the decomposed representations and propose the ADR algorithm.




References:\
[1] Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual
inference. In International Conference on Machine Learning, pp. 3020–3029. PMLR, 2016.\
[2] Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: generalization bounds and algorithms. In International Conference on Machine Learning, pp. 3076–3085.
PMLR, 2017.\
[3] Negar Hassanpour and Russell Greiner. Learning disentangled representations for counterfactual
regression. In International Conference on Learning Representations, 2020\
[4] Anpeng Wu, Junkun Yuan, Kun Kuang, Bo Li, Runze Wu, Qiang Zhu, Yueting Zhuang, and Fei
Wu. Learning decomposed representations for treatment effect estimation. IEEE Transactions on
Knowledge and Data Engineering, 35(5):4989–5001, 2022.

Reviewer comment (Theory):
The adversarial joint objective is not in a convex-concave formulation, which means there is no guarantee of convergence. Intuitively, it seems very unstable.

Author response:
- [Q3. the convergence guarantee]
Thanks for the insightful question.

i) ``*Are there any existing studies of such a formulation that maximizes the loss function such as the MSE?*"  

A similar formulation is that in GAN [5], the generator is updated by maximizing the cross-entropy loss (the discriminator us undated by minimizing the cross-entropy loss). In this case, the optimal predictor has an explicit solution and such that the convergence can be readily proved (see Sec 4.2 in [5])
	
ii) ``*While maximizing the MSE by the adversary is easily accomplished by making the predictions infinity, it seems to be difficult to predict it accurately.*"

Please kindly note that in maximizing the loss $\\mathcal{L}\_A^h:=\sum_i l(h\_{A\\rightarrow T}(A(\\boldsymbol{x}_i)), t_i)$, 
the $h\_{A\rightarrow T}(\cdot)$ is fixed and supposed to be (close to) the optimal predictor given $A(\cdot)$. 
	As is suggested by the proof of Prop 3.2, 
	$\mathcal{L}\_A^h\approx Var[\mathbb{E}[T|A(X)]]=Var(T)-\mathbb{E}[Var[T|A(X)]]$ that is less equal than $Var(T)$ and is maximized when $A(X)\perp T$.
	Practically, we restrict the values of representation networks by the $\mathcal{L}_O$ in eq (7) such that $(\sum \overline{W}[k]-1)^2$ is close to zero.
	
iii) ``*It may be helpful to show realistic convergence using a learning curve.*" 

Thanks for the suggestion. We have checked the tensorboard results saved in the training process. For example, in the case of Synthetic Dataset (continuous treatment). It can be seen that $\mathcal{L}_A^h$ has an oscillation at the very beginning and then starts to increase and end at a steady level after 2.5k steps.
	We have added the relevant results in the Supplementary.
	Unfortunately, we have not yet come up with a way to theoretically prove the convergence for continuous treatment.
	We suppose that the result would be similar to Proposition 2 in GAN [5].
	
	
References:\
[5] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair,
Aaron Courville, and Y. Bengio. Generative adversarial nets. In Neural Information Processing
Systems, 2014.

Reviewer comment (Novelty):
Are there any existing studies of such a formulation that *maximizes* the loss function such as the MSE?

Author response:
- [Q3. the convergence guarantee]
Thanks for the insightful question.

i) ``*Are there any existing studies of such a formulation that maximizes the loss function such as the MSE?*"  

A similar formulation is that in GAN [5], the generator is updated by maximizing the cross-entropy loss (the discriminator us undated by minimizing the cross-entropy loss). In this case, the optimal predictor has an explicit solution and such that the convergence can be readily proved (see Sec 4.2 in [5])
	
ii) ``*While maximizing the MSE by the adversary is easily accomplished by making the predictions infinity, it seems to be difficult to predict it accurately.*"

Please kindly note that in maximizing the loss $\\mathcal{L}\_A^h:=\sum_i l(h\_{A\\rightarrow T}(A(\\boldsymbol{x}_i)), t_i)$, 
the $h\_{A\rightarrow T}(\cdot)$ is fixed and supposed to be (close to) the optimal predictor given $A(\cdot)$. 
	As is suggested by the proof of Prop 3.2, 
	$\mathcal{L}\_A^h\approx Var[\mathbb{E}[T|A(X)]]=Var(T)-\mathbb{E}[Var[T|A(X)]]$ that is less equal than $Var(T)$ and is maximized when $A(X)\perp T$.
	Practically, we restrict the values of representation networks by the $\mathcal{L}_O$ in eq (7) such that $(\sum \overline{W}[k]-1)^2$ is close to zero.
	
iii) ``*It may be helpful to show realistic convergence using a learning curve.*" 

Thanks for the suggestion. We have checked the tensorboard results saved in the training process. For example, in the case of Synthetic Dataset (continuous treatment). It can be seen that $\mathcal{L}_A^h$ has an oscillation at the very beginning and then starts to increase and end at a steady level after 2.5k steps.
	We have added the relevant results in the Supplementary.
	Unfortunately, we have not yet come up with a way to theoretically prove the convergence for continuous treatment.
	We suppose that the result would be similar to Proposition 2 in GAN [5].
	
	
References:\
[5] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair,
Aaron Courville, and Y. Bengio. Generative adversarial nets. In Neural Information Processing
Systems, 2014.

Reviewer comment (Theory):
While maximizing the MSE by the adversary is easily accomplished by making the predictions infinity, it seems to be difficult to predict it accurately.

Author response:
- [Q3. the convergence guarantee]
Thanks for the insightful question.

i) ``*Are there any existing studies of such a formulation that maximizes the loss function such as the MSE?*"  

A similar formulation is that in GAN [5], the generator is updated by maximizing the cross-entropy loss (the discriminator us undated by minimizing the cross-entropy loss). In this case, the optimal predictor has an explicit solution and such that the convergence can be readily proved (see Sec 4.2 in [5])
	
ii) ``*While maximizing the MSE by the adversary is easily accomplished by making the predictions infinity, it seems to be difficult to predict it accurately.*"

Please kindly note that in maximizing the loss $\\mathcal{L}\_A^h:=\sum_i l(h\_{A\\rightarrow T}(A(\\boldsymbol{x}_i)), t_i)$, 
the $h\_{A\rightarrow T}(\cdot)$ is fixed and supposed to be (close to) the optimal predictor given $A(\cdot)$. 
	As is suggested by the proof of Prop 3.2, 
	$\mathcal{L}\_A^h\approx Var[\mathbb{E}[T|A(X)]]=Var(T)-\mathbb{E}[Var[T|A(X)]]$ that is less equal than $Var(T)$ and is maximized when $A(X)\perp T$.
	Practically, we restrict the values of representation networks by the $\mathcal{L}_O$ in eq (7) such that $(\sum \overline{W}[k]-1)^2$ is close to zero.
	
iii) ``*It may be helpful to show realistic convergence using a learning curve.*" 

Thanks for the suggestion. We have checked the tensorboard results saved in the training process. For example, in the case of Synthetic Dataset (continuous treatment). It can be seen that $\mathcal{L}_A^h$ has an oscillation at the very beginning and then starts to increase and end at a steady level after 2.5k steps.
	We have added the relevant results in the Supplementary.
	Unfortunately, we have not yet come up with a way to theoretically prove the convergence for continuous treatment.
	We suppose that the result would be similar to Proposition 2 in GAN [5].
	
	
References:\
[5] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair,
Aaron Courville, and Y. Bengio. Generative adversarial nets. In Neural Information Processing
Systems, 2014.

Reviewer comment (Evaluation):
It may be helpful to show realistic convergence using a learning curve.

Author response:
- [Q3. the convergence guarantee]
Thanks for the insightful question.

i) ``*Are there any existing studies of such a formulation that maximizes the loss function such as the MSE?*"  

A similar formulation is that in GAN [5], the generator is updated by maximizing the cross-entropy loss (the discriminator us undated by minimizing the cross-entropy loss). In this case, the optimal predictor has an explicit solution and such that the convergence can be readily proved (see Sec 4.2 in [5])
	
ii) ``*While maximizing the MSE by the adversary is easily accomplished by making the predictions infinity, it seems to be difficult to predict it accurately.*"

Please kindly note that in maximizing the loss $\\mathcal{L}\_A^h:=\sum_i l(h\_{A\\rightarrow T}(A(\\boldsymbol{x}_i)), t_i)$, 
the $h\_{A\rightarrow T}(\cdot)$ is fixed and supposed to be (close to) the optimal predictor given $A(\cdot)$. 
	As is suggested by the proof of Prop 3.2, 
	$\mathcal{L}\_A^h\approx Var[\mathbb{E}[T|A(X)]]=Var(T)-\mathbb{E}[Var[T|A(X)]]$ that is less equal than $Var(T)$ and is maximized when $A(X)\perp T$.
	Practically, we restrict the values of representation networks by the $\mathcal{L}_O$ in eq (7) such that $(\sum \overline{W}[k]-1)^2$ is close to zero.
	
iii) ``*It may be helpful to show realistic convergence using a learning curve.*" 

Thanks for the suggestion. We have checked the tensorboard results saved in the training process. For example, in the case of Synthetic Dataset (continuous treatment). It can be seen that $\mathcal{L}_A^h$ has an oscillation at the very beginning and then starts to increase and end at a steady level after 2.5k steps.
	We have added the relevant results in the Supplementary.
	Unfortunately, we have not yet come up with a way to theoretically prove the convergence for continuous treatment.
	We suppose that the result would be similar to Proposition 2 in GAN [5].
	
	
References:\
[5] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair,
Aaron Courville, and Y. Bengio. Generative adversarial nets. In Neural Information Processing
Systems, 2014.

Reviewer comment (Novelty):
What is the purpose of the decomposition? The original purpose was to combine weighting only w.r.t. the confounders in DR-CFR, in my understanding. Confounder variables should be limited to necessary ones to alleviate the estimation variance due to extreme weights. The proposed method does not use weighting and thus I am confused about its aim.

Author response:
- [Q1. purpose of the decomposition]

We agree that it is important to answer the question for implementing such decomposition for the ITE estimation.
It is absolutely correct that the purpose is to `` limit the confounder variables to necessary ones to alleviate the estimation variance due to extreme weights". 
However, it should be noted that, even we do not adopt the inverse probability weighting method, the estimation variance also increase with the aggravation of data imbalance as Thm 3.1 suggests.
We may have a further understanding by reviewing the development of the literature.


Firstly, the balanced representation leaning algorithms [1, 2] were proposed to reduce the data imbalance caused by selection bias. 
Except from the empirical evidence, [2] explained the benefit from the view of generalization bound.

Secondly, [3] pointed out that leaning the balanced representation alone may overlook the necessary confounding information and hence should learn the decomposed representation. However, the loss terms of DR algorthm in [3] was not complete and hence was not able to learn the decomposed representations. [4] proposed the DeR algorithm to realize the decomposition learning under the situation of binary treatment and response.
To deal with the case for continuous treatment and adapt to both categorical and continuous response, we analyzed the theoretical property of the decomposed representations and propose the ADR algorithm.




References:\
[1] Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual
inference. In International Conference on Machine Learning, pp. 3020–3029. PMLR, 2016.\
[2] Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: generalization bounds and algorithms. In International Conference on Machine Learning, pp. 3076–3085.
PMLR, 2017.\
[3] Negar Hassanpour and Russell Greiner. Learning disentangled representations for counterfactual
regression. In International Conference on Learning Representations, 2020\
[4] Anpeng Wu, Junkun Yuan, Kun Kuang, Bo Li, Runze Wu, Qiang Zhu, Yueting Zhuang, and Fei
Wu. Learning decomposed representations for treatment effect estimation. IEEE Transactions on
Knowledge and Data Engineering, 35(5):4989–5001, 2022.

Reviewer comment (Theory):
Why \( L_A \) does not include the accuracy of \( f_{C\cup A\cup I \to Y} \)? A(x) is input to \( f_{C\cup A\cup I \to Y} \), but the gradient for the connection is stopped. Does not this have any negative impact on the whole design of the optimization procedure?

Author response:
- [Q2. the design of $\mathcal{L}_A$]

We have added the loss of $f\_{A\cup T\rightarrow Y}(\cdot)$ in $\mathcal{L}\_A$ to learn the representation that are predictive to $Y$.
The loss were based on the result in Thm 3.2.
We minimize the loss of $f\_{A\cup T\rightarrow Y}(\cdot)$ to realize $\boldsymbol{A}\not\perp Y$ and
maximize the loss of $h\_{A\rightarrow T}(\cdot)$ to realize $\boldsymbol{A}\perp T$.

Reviewer comment (Theory):
Soundness: the method is considered as a heuristic alternate update and not formalized as a joint objective; thus the optimization can be unstable (maybe it does not converge).

Author response:
- [Q3. the convergence guarantee]
Thanks for the insightful question.

i) ``*Are there any existing studies of such a formulation that maximizes the loss function such as the MSE?*"  

A similar formulation is that in GAN [5], the generator is updated by maximizing the cross-entropy loss (the discriminator us undated by minimizing the cross-entropy loss). In this case, the optimal predictor has an explicit solution and such that the convergence can be readily proved (see Sec 4.2 in [5])
	
ii) ``*While maximizing the MSE by the adversary is easily accomplished by making the predictions infinity, it seems to be difficult to predict it accurately.*"

Please kindly note that in maximizing the loss $\\mathcal{L}\_A^h:=\sum_i l(h\_{A\\rightarrow T}(A(\\boldsymbol{x}_i)), t_i)$, 
the $h\_{A\rightarrow T}(\cdot)$ is fixed and supposed to be (close to) the optimal predictor given $A(\cdot)$. 
	As is suggested by the proof of Prop 3.2, 
	$\mathcal{L}\_A^h\approx Var[\mathbb{E}[T|A(X)]]=Var(T)-\mathbb{E}[Var[T|A(X)]]$ that is less equal than $Var(T)$ and is maximized when $A(X)\perp T$.
	Practically, we restrict the values of representation networks by the $\mathcal{L}_O$ in eq (7) such that $(\sum \overline{W}[k]-1)^2$ is close to zero.
	
iii) ``*It may be helpful to show realistic convergence using a learning curve.*" 

Thanks for the suggestion. We have checked the tensorboard results saved in the training process. For example, in the case of Synthetic Dataset (continuous treatment). It can be seen that $\mathcal{L}_A^h$ has an oscillation at the very beginning and then starts to increase and end at a steady level after 2.5k steps.
	We have added the relevant results in the Supplementary.
	Unfortunately, we have not yet come up with a way to theoretically prove the convergence for continuous treatment.
	We suppose that the result would be similar to Proposition 2 in GAN [5].
	
	
References:\
[5] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair,
Aaron Courville, and Y. Bengio. Generative adversarial nets. In Neural Information Processing
Systems, 2014.

Reviewer comment (Theory):
Error in derivations. I spotted two issues. First, Theorem 3.2 claims that $\mathbf{C}$ is a valid set $\mathbf{X}’$ in the definition of the instrumental variables. On the other hand, by looking at the example in Fig. 1 (b), $\mathbf{C} = \varnothing$, but $X_2 \notindependent Y \mid T$. Second, there seems to be an erroneous statement in the proof of Prop. 3.2, that the equality in the expectation $\mathbb{E} (T \mid A(X) ) = \mathbb{E}(T)$ implies the independence, $T \independent A(X)$, which is not true, if $T$ is continuous. Specifically, there could be inequalities wrt. to higher moments. Those two issues are further very important for the correct implementation of the ADR.

Author response:
- [Q1] Thanks a lot for pointing out the questionable part in our work.

i) As for the claim that $\boldsymbol{I}\perp Y|T\cup \boldsymbol{C}$ in Thm 3.2, we admit the definition for $\boldsymbol{I}$ (instrumental variables) was inappropriate.
We should rule out the "collider" variables in the definition of $\boldsymbol{I}$.
Under this updated definition, $\boldsymbol{I}=\{X_1\}$ for Figure 1(b). We have revised the paper accordingly, and also corrected the corresponding proof for Thm 3.2 in the Supplementary.

ii) We apologize that the careless statement in the proof of Prop 3.2 caused your confusion.
We have changed \textit{"when $\mathbb{E}(T|A(X))=\mathbb{E}(T)$, i.e., $T\perp A(X)$"} as \textit{"when $T\perp A(X)$, we have $\mathbb{E}(T|A(X))=\mathbb{E}(T)$"}.
It should be noted that Prop 3.2 claims that $\mathcal{L}_A$ is maximized when $A\perp T$, which is in align with the corrected statement.

Reviewer comment (Reproducibility):
Implementation and tuning. Some details are missing on the implementation of the baselines, e.g., the dimensionalities of the representations. Also, the authors did not provide any details on how to choose the dimensionalities of the decomposed representations in their method, which is a very important issue in practice, e.g., for the IHDP benchmark. Therefore, it is impossible to say, whether the empirical evaluation was fair.

Author response:
- [Q3] Implementation Details

We adopted the same representation dimension (256) and the same structured MLPs for the predictors on top of the representations in the experiments.
For a fair comparison, the values of such parameters were also in align with the compared literature [2]. The implementation details were added in the Supplementary (see \verb|Supplementary/code|).
Thanks for pointing out this problem, we have  
added the relevant information in Section 5.

References: \
[2] Anpeng Wu, Junkun Yuan, Kun Kuang, Bo Li, Runze Wu, Qiang Zhu, Yueting Zhuang, and Fei
Wu. Learning decomposed representations for treatment effect estimation. IEEE Transactions on
Knowledge and Data Engineering, 35(5):4989–5001, 2022.