Paper: Conditional Instrumental Variable Regression with Representation Learning for Causal Inference (ICLR-2024)

Reviewer comment (Novelty):
My main concern is the similarity of this work to Wu.2022 in ICML. Even though the contribution of this work is introducing CIV to handle the restrictive conditions of IV, your methods are similar to Wu et.al.

Author response:
Thanks for your question. While our CBRL.CIV method shares a similar idea as CBIV in Wu et al. 2022 [1] in terms of using representation learning in the context of IV-based causal effect estimation, our contribution under the CIV framework is distinct in several key aspects.

(1) Different problem settings as shown in Fig. 1(c) (for CBIV) and Fig. 1(d) (for  our method CBRL.CIV) respectively. Our extends CBIV by providing a more flexible framework that can handle weaker, more realistic conditions. Specifically, CBRL.CIV is designed to work under the setting where $S$ is unconfounded with $W$ and $Y$ conditioning on a set of measured covariates, while CBIV requires the absence of confounding bias between $S$ and $Y$.  

(2) The technical contribution of CBRL.CIV lies in its innovative application of confounding balancing to address the confounding bias between $S$ and $W$, especially under the condition of limited sample size. This advancement enables CBRL.CIV to extract a balanced representation $\mathbf{Z}$ for addressing the confounding bias between $S$ and $W$, as well as the confounding bias between $W$ and $Y$. Note that CBIV only mitigates the confounding bias between $W$ and $Y$, but not deal with the confounding bias between $S$ and $W$. 

(3) These improvements showcase the practicality and adaptability of CBRL.CIV across a wider range of scenarios in representation learning, where the CBIV method may incur significant estimation bias due to the confounding bias between $S$ and $W$.
We have revised the following descriptions to make the difference between CBRL.CIV and CBIV clearly in our updated version. 

>Current solutions [1,2] consider a standard IV and use a regression model conditioning on $\mathbf{C}$, and they are not applicable to a CIV case since they do not mitigate the confounding bias between $S$ and $W$. 

and 

>CBIV [2] combines confounding balancing with IV regression to remove the confounding bias between $W$ and $Y$ caused by the unobserved confounders and the observed confounders. Different from these methods, our CBLR.CIV addresses the confounding bias between $S$ and $W$, as well as the confounding bias between $W$ and $Y$. Therefore, CBLR.CIV has a wider application than a method based on a standard IV and it also deals with non-linear cases.

[1] Jason S. Hartford, Greg Lewis, Kevin Leyton-Brown, and Matt Taddy. Deep IV: A flexible approach for counterfactual prediction[C]. In Proceedings of the 34th International Conference on Machine Learning, pp. 1414–1423, 2017. 

[2]Anpeng Wu, Kun Kuang, Bo Li, and Fei Wu. Instrumental variable regression with confounder balancing[C]. In International Conference on Machine Learning, pp. 24056–24075. PMLR, 2022.

Reviewer comment (Novelty):
Can you state the difference, or the unique technical contribution of CIV for representation learning?

Author response:
Thanks for your question. While our CBRL.CIV method shares a similar idea as CBIV in Wu et al. 2022 [1] in terms of using representation learning in the context of IV-based causal effect estimation, our contribution under the CIV framework is distinct in several key aspects.

(1) Different problem settings as shown in Fig. 1(c) (for CBIV) and Fig. 1(d) (for  our method CBRL.CIV) respectively. Our extends CBIV by providing a more flexible framework that can handle weaker, more realistic conditions. Specifically, CBRL.CIV is designed to work under the setting where $S$ is unconfounded with $W$ and $Y$ conditioning on a set of measured covariates, while CBIV requires the absence of confounding bias between $S$ and $Y$.  

(2) The technical contribution of CBRL.CIV lies in its innovative application of confounding balancing to address the confounding bias between $S$ and $W$, especially under the condition of limited sample size. This advancement enables CBRL.CIV to extract a balanced representation $\mathbf{Z}$ for addressing the confounding bias between $S$ and $W$, as well as the confounding bias between $W$ and $Y$. Note that CBIV only mitigates the confounding bias between $W$ and $Y$, but not deal with the confounding bias between $S$ and $W$. 

(3) These improvements showcase the practicality and adaptability of CBRL.CIV across a wider range of scenarios in representation learning, where the CBIV method may incur significant estimation bias due to the confounding bias between $S$ and $W$.
We have revised the following descriptions to make the difference between CBRL.CIV and CBIV clearly in our updated version. 

>Current solutions [1,2] consider a standard IV and use a regression model conditioning on $\mathbf{C}$, and they are not applicable to a CIV case since they do not mitigate the confounding bias between $S$ and $W$. 

and 

>CBIV [2] combines confounding balancing with IV regression to remove the confounding bias between $W$ and $Y$ caused by the unobserved confounders and the observed confounders. Different from these methods, our CBLR.CIV addresses the confounding bias between $S$ and $W$, as well as the confounding bias between $W$ and $Y$. Therefore, CBLR.CIV has a wider application than a method based on a standard IV and it also deals with non-linear cases.

[1] Jason S. Hartford, Greg Lewis, Kevin Leyton-Brown, and Matt Taddy. Deep IV: A flexible approach for counterfactual prediction[C]. In Proceedings of the 34th International Conference on Machine Learning, pp. 1414–1423, 2017. 

[2]Anpeng Wu, Kun Kuang, Bo Li, and Fei Wu. Instrumental variable regression with confounder balancing[C]. In International Conference on Machine Learning, pp. 24056–24075. PMLR, 2022.

Reviewer comment (Novelty):
Using confounder balancing to migrate the gap between treated and controlled groups are very commo. I wonder the specific beneficial point on the combination of CIV and confounder balancing.

Author response:
Confounding balance is commonly employed to reduce bias in the estimation of causal effects by making the treated and control groups comparable when there are confounders between treatment $W$ and outcome $Y$ and all the confounders are observed. In this work, however, we consider a more complicated problem as depicted in Fig. 1(d), where there exist latent confounders $\mathbf{U}$ affecting $W$ and $Y$ and the distribution of the observed confounders $\mathbf{C}$ is imbalanced not only across the different groups as specified by the different values of $W$ (i.e. the treated and control groups), but also the different groups corresponding to the different values of the CIV $S$. In this case, the usual confounding balancing (on $\mathbf{C}$) cannot be used as some confounders ($\mathbf{U}$) are unobserved, and hence IV is used. Here, as explained in the main text, the standard IV approach is inadequate, thus we use CIV (to address the confounding bias between $W$ and $Y$), and to address the imbalanced distribution of $\mathbf{C}$ across the different groups specified by the different values of CIV $S$ (i.e. the subsequent issue arisen with the use of CIV), we adopt the idea of confounding balancing. 

To the best of our knowledge, CBRL.CIV is the first work that considers the imbalance issues in the applications of CIVs when there are finite samples.

We have added some discussion on the specific beneficial points in our updated version. 

>To enhance the practical use of the IV approach, specifically to address the double imbalance problem caused by both observed and unobserved confounders in non-linear cases when using a CIV, we propose CBRL.CIV, a novel \underline{C}onfounding \underline{B}alancing \underline{R}epresentation \underline{L}earning method for non-linear \underline{CIV} regression. We further demonstrate its theoretical soundness and practical usefulness.  By integrating CIV with confounding balance, CBRL.CIV emerges as a comprehensive solution that adeptly navigates the intricacies of confounder effects in non-linear models, offering a significant advancement in causal inference.

Reviewer comment (Writing):
When the authors or the publication are included in the sentence, the citation should not be in parenthesis, e.g., “it is well-known that the causal effect of the treatment on the outcome is non-identifiable Pearl (2009); Shpitser & Pearl (2006).” The citation should be in parenthesis using \citep{}.

Author response:
Thanks. We have corrected them accordingly.

Reviewer comment (Theory):
In Eq. (12), there is a missing square, and the same missing happens in the pseudocode of Algorithm 1.

Author response:
Thank you for pointing out these typos. We have corrected them in our updated version.