Conditional Instrumental Variable Regression with Representation Learning for Causal Inference
Authors: Debo Cheng
Ziqi Xu
Jiuyong Li
Lin Liu
Jixue Liu
Thuc Duy Le
Venue: ICLR 2024 poster (2024)
Keywords: Causal Inference
Variational AutoEncoder
Instrumental Variable
Primary area: causal reasoning
TL;DR: We propose a novel confounding balance representation learning method for non-linear CIV regression in average causal effect estimation from observational data with latent confounders.

Abstract: This paper studies the challenging problem of estimating causal effects from observational data, in the presence of unobserved confounders. The two-stage least square (TSLS) method and its variants with a standard instrumental variable (IV) are commonly used to eliminate confounding bias, including the bias caused by unobserved confounders, but they rely on the linearity assumption. Besides, the strict condition of unconfounded instruments posed on a standard IV is too strong to be practical. To address these challenging and practical problems of the standard IV method (linearity assumption and the strict condition), in this paper, we use a conditional IV (CIV) to relax the unconfounded instrument condition of standard IV and propose a non-linear \underline{CIV} regression with \underline{C}onfounding \underline{B}alancing \underline{R}epresentation \underline{L}earning,  CBRL.CIV, for jointly eliminating the confounding bias from unobserved confounders and balancing the observed confounders, without the linearity assumption. We theoretically demonstrate the soundness of CBRL.CIV. Extensive experiments on synthetic and two real-world datasets show the competitive performance of CBRL.CIV against state-of-the-art IV-based estimators and superiority in dealing with the non-linear situation.

Published as a conference paper at ICLR 2024
CONDITIONAL INSTRUMENTAL VARIABLE REGRES- SION WITH REPRESENTATION LEARNING FOR CAUSAL INFERENCE
Debo Cheng 1; Ziqi Xu 12:, ,Jiyyong , Lin Liu 1, Jixue Liu 1 & Thuc Duy Le University of South Australia Data61, CSIRO (firstname lastn.mme a
edu. ' ziqi xu@data61 csiro. au
ABSTRACT
This paper studies the challenging problem of estimating causal effects from ob- servational data, in the presence of unobserved confounders. The two-stage least square (TSLS) method and its variants with a standard instrumental variable (IV) are commonly used to eliminate confounding bias, including the bias caused by unobserved confounders, but they rely on the linearity assumption. Besides, the strict condition unconfounded instruments posed on standard is too strong to be practical. To address these challenging and practical problems of the stan- dard IV method (linearity assumption and the strict condition), in this paper, we conditional IV (CIV) to relax the unconfounded instrument condition of standard IV and propose non-linear CIV regression with Confounding Balancing Representation Learning. CBRL.CIV, for jointly eliminating the confounding bias from unobserved confounders and balancing the observed confounders, with- out the linearity assumption. We theoretically demonstrate the soundness of CBRL.CIV. Extensive experiments on synthetic and two real-world datasets show the competitive performance of CBRL.CIV against state-of- -fate-of-the-art IV-based esti- mators and superiority in dealing with the non-linear situation.
INTRODUCTION
Causal inference is a fundamental problem in various fields such as economics, epidemiology, and medicine (Hernán& & Robins 2010 Imbens & Rubin 2013). Estimating treatment (i.e., causal) ef- fects of given treatment on an outcome of interest from observational data plays an important role in causal inference. Nevertheless, estimating causal effects from observational data suffers from the confounding bias caused by confounders] and unobserved confounders make causal effect estima- tion more challenging. When there is unobserved confounder affecting both the treatment and outcome variables. as illustrated by the causal DAG in Fig. (a), the endogeneity problem") (An- tonakis et al., 2010 Cheng et (024) arises, and in this situation, it is well-known that the causal effect of the treatment on the outcome is non-identifiable (Pearl 2009 Shpitser & Pearl 2006).
Instrumental variable (IV) is a powerful approach addressing the endogeneity problem and iden- tifying causal effects in the presence of unobserved confounders. but under strong assumptions (An- derson & Rubin 1949. Hernán Robins 2006). Typically, the IV approach requires a valid standard IV (denoted as 5) that satisfies the following conditions: (i) S affects the treatment W (a.k.a. the relevance condition), (ii) the causal effect of on the outcome Y is only through W (a.k.a. the exclusion condition), and (iii) S and Y are not confounded by measured or unmeasured variables (a.k.. the unconfounded instrument condition). For instance, the variable S in the causal DAG of Fig. 1 (b) is standard IV that satisfies the three conditions. Under the linearity assumption, the two-
"These authors contributed equally to this work.
'A confounder is common cause that affects two variables simultancously.
2 A directed acyclic graph (DAG) a graph that contains only directed edges and no cycles.
'hhe endogeneity comes from the presence of unobserved confounders, which leads to causal ambiguities
the causal relations between the treatment and the outcome (Antonakis al. 2010 Greenland 2003

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W
(a)
(b)
(c)
(d)
Figure I: An illustration presents four scenarios: (a) unobserved confounders, (b) standard IV, (c) the imbalance issue with standard IV, and (d) the imbalance issue with CIV. Variables include treatment W. outcome Y, the set of observed confounders C, the set of unobserved confounders U, and IV S.. Directed edges indicate causal relationships and dotted lines represent statistical associations. In (a), the causal effect of W on Y is non-identifiable due to U affecting both W and Y; In (b), the causal effect of W on Y can be estimated by using the IV S; In (c), the dependence of U and C affecting the potential for confounding and distribution imbalance in causal effect estimation, addressed in Wu et al. (2022); In (d), S is a conditional IV (CIV). In this case, C induces bias across the different groups specified by the different values of S, potentially biasing the estimated causal effect of W on Y. This paper addresses this problem.
stage least squares (TSLS) method is often employed to estimate the causal effects in the presence of unobserved confounders using a given standard IV (Angrist & Imbens 1995 Imbens 2014).
In many real-world scenarios, non-linear variable relationships combined with the effect of unob- served confounders make estimating causal effects from observational data more challenging. Re- cently, some methods were proposed to address the challenging case with non-linear relationships when standard IV is used, such as DeepIV (Hartford et al. (2017). kemnelVV regression (Singh et al. 2019). double machine learning IV (DMLIV) method (Syrgkanis et al. 2019). etc. These methods are built on two-stage regression models. Using Fig. (b) as an example, the first stage of such a two-stage approach is to fit non-linear regression model for the conditional distribution of the treatment W conditioning on the standard IV S and the set of measured pretreatment variables C, i.c., the estimated treatment W is obtained based on P(W S, C). The second stage is to fit a non-linear regression model on Y conditioning on C and W obtained in the first stage.
However, due the spurious associations between U and C (as illustrated in Fig. (c)), the distri- butions of C in the treated and control groups may be imbalanced (Shalit et al. 2017). which can result in biased estimations in the non-linear outcome regression model (Abadie 2003 especially when the outcome regression model is misspecified. To address the imbalance issue in the outcome regression model, Wu Wu al. (2022) proposed confounder balaneing IV regression (CBIV) algorithm, and demonstrated the effectiveness of the confounding balancing representation learning approach. However, CBIV ignores the confounding bias caused by the imbalance distribution of C across different groups specified by the different values of S in the first stage.
To address the imbalance problems and the practical issues of the standard IV approach (linearity assumption and the strict condition), consider conditional IV (CIV) in this work. CIV relaxes the third condition of the standard IV, and allows a variable to be a valid IV by conditioning on a set of measured variables (Brito & Pearl 2002 van der Zander al., 2015). In real-wolrd applications, a CIV is less restrictive than a standard IV and is easier to find. For example, we aim to estimate the effect of a treatment (W) on the recovery from a disease (Y). W and are affected by unobserved confounders U, such as the socioeconomic status of patients. The variable of clinical practice guide- lines (S) for treatment decision W is CIV with the conditioning set (C), which includes factors such as age, health condition, and an individual's propensity to seek medical assistance. This exam- ple can be represented by the causal DAG Fig. (d). In contrast, a standard IV affecting only W but not Y via other paths is difficult to find in the real-world.
two-stage regression is commonly used to obtain an estimation of the causal effect by an IV or a CIV approach. There are two risks of biases in the estimation using a two-stage solution in Fig. (d). Firstly, the distribution imbalance of between the treated and control groups leads to a bias in the outcome regression model. Secondly, unobserved confounders could cause a distribution imbalance of C between the different groups specified by different values of S as C leading to a bias in the treatment regression model. Current solutions (Hartford et al., 2017; Wu et al.. 2022) consider standard IV and use a regression model conditioning on C, and they are not applicable to a CIV case since they do not mitigate the confounding bias between S and W.
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To enhance the practical use of the IV approach, specifically to address the double imbalance prob- lem caused by both observed and unobserved confounders non-linear cases when using a CIV, propose CBRL.CIV, a novel Confounding Balancing Representation Learning method for non- linear CIV regression. We further demonstrate its theoretical soundness and practical usefulness. By integrating CIV with confounding halance, CBRL,CIV emerges comprehensive solution that adeptly navigates the intricacies of confounder effects in non-linear models, offering signifi- cant advancement in causal inference. We summarise the main contributions of work as follows.
We study the problem of causal effect estimation from data with unobserved confounders in a non-linear setting and design a novel and practical CIV method to solve the problem. To the best of our knowledge, this is the first work that considers the imbalance issues in the applications of CIVs when there are finite samples.
We propose a novel representation learning method, CBRL.CIV for balancing the distri- butions and removing the confounding bias simultaneously. We theoretically show the soundness of the proposed CBRL.CIV method. Specifically, CBRL.CIV utilises two bal- ance terms to enhance generalisation across different groups specified by different values S in the treatment network, and different groups specified by different values of W in the outcome regression network.
Extensive experiments are conducted on synthetic and real-world datasets and the results demonstrate the effectiveness of CBRL.CIV in average causal effect estimation.
PRELIMINARY
We introduce some important notations and definitions used in the paper. Let W (1,0) denote a binary treatment variable where W = represents treated and W = 0 untreated, Y the outcome variable of interest, and U set of unobserved confounders. Let X denote the set of pretreatment variables, namely, all variables in X are measured before W is measured. Finally, we assume that the complete data are independent and identically distributed realizations of (W, Y, U, X), and the observed data is O = Y.Y.
We aim at estimating the average causal effect of W on Y from O with the presence of unobserved confounders U. In the observational data O for each individual i, the potential outcomes are de- noted as Y, = 1) and Y,(W = 0) corresponding to W (Imbens & Rubin 2015 Rosenbaum & Rubin 1983 Note that only one of the potential outcomes can be observed for given individual, which is known as the fundamental problem in causal inference (Imbens & Rubin 2015).
Furthermore, we use causal DAG G (V, E) to represent the underlying mechanism of the studied system, where V represents the set of nodes (variables) of the DAG, and E C V x V is the set of directed edges (Pearl 2009 Spirtes et al. 2000). For any X, V and * j. the directed edge X; + X; in the causal DAG G indicates that X; is a direct cause (parent) of Xj. and Xj is direct effect (child) of X,. A path from X; to X, is directed path or causal path when all edges along it are directed towards Xy. Additionally, on causal path from X; to Xy. X; is an ancestor of Xy. and conversely, X, descendant of X,. Using a causal DAG, CIV (Definition 7.4.1 on Page 248 (Pearl 2009h) can be defined as:
Definition (Conditional IV), Let G (V, E) be a causal DAG with V (W.), variable S E X is conditional IV w.r.t. W +Y if there exists set of measured variables C C X such that (i) S aL W C, (ii) S La Y C in Gw, and (ii)) AC E C, C is not a descendant of Y, where Gw is manipulated DAG of G by deleting the direct edge W + Y from G, and and and IL are d-connection and d-separation (Pearl 2009), respectively.
In the above definition, C is known as the set of variables that instrumentalise S. the conditioning set of S. Note that a CIV is a more general IV than the standard IV and when is an empty set in Definition [ S is standard IV. More definitions are provided in Appendix A.
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3 THE PROPOSED CBRL.CIV METHOD
3.1 PROBLEM SETTING & ASSUMPTIONS
We assume that a binary CTV S E X is known and further assume that the rest variables in X, denoted as C, are pre-IV variables, i.e., they are measured before S is measured. Our problem is illustrated in Fig. (d), typical setting where CIV is used, and in the figure, S is a CIV that instrumentalised by C (i.e., C is the conditioning set of S). C affects both S and W. and C affects both W and Y. C has two roles here: observed confounders C between W and Y and conditioning set of CIV w.r.t. W and Y. As discussed previously, when CTV used, in addition to the imbalanced distributions of C between W and Y as discussed in the previous work (Wu jet al,, 2022), C is imbalanced among the groups specified by the different values of S. That is, we face a double distribution imbalance problem, one between S and W, and one between W and Y. Therefore the existing solution addressing the distribution imbalance of between W and Y such CBIV (Wu al. 2022) cannot be applied because the confounding bias resulted from the imbalance of among the groups specified by the different values S remain unresolved.
To address the double imbalance issue caused in using CIV approach, especially in non-linear scenarios, we have designed a novel three-stage method based on balanced representation learning techniques (Shalit al. 2017 Wu et al. 2022).
The following assumptions are essential when using CIV method for causal effect estimation in data. Assumption 1. (IV positivity) is almost surely that P(S C)
The IV positivity assumption is necessary for non con-paraettric identification (Greenland 2000
Assumption 2. (Independent compliance type (Cui & Tchetgen Tchetgen 2021| Hernán & Robins 2006g) It almost surely that ob = P(W = =, -
= -,
where (C, = P(W = 1 = 1, 1, - P(W = =1|S=
This assumption means that there is no additive interaction between S on U in system for the treatment probability conditioning on C and U. It also demonstrates that U is independent of an individual's 's compliance type (Imbens & Rubin 2015). In a binary S. there are four compliance types for an individual: Compliers, Always-takers, Never-takers and Defiers (Angrist & Imbens, 1995
Imbens& & Rubin 2015). The details of compliance types are provided in Appendix A.
3.2 CBRL.CIV
Our proposed CBRL.CIV method tackles the double distribution imbalance issue by learning a confounding balancing representation of C, denoted Z using representation network vho(-). i.e., Z = wo(() (C) to achieve confounding balance in counterfactual inference and remove the con- founding bias from unobserved confounders without lincarity assumption. CBRL.CIV consists of three components, CIV regression for balancing the distributions of C between S and W, treat- ment regression for balancing the distribution between W and Y. and outcome regression using the learned balancing representation Z.
CIV Regression for Confounding Balance. To balance the distributions of C between different values of the CIV S, we need to ensure that S and the representation Z are independent conditioning on C. It is challenging to directly learn a representation Z such that S We propose to achieve this in two steps. In Step one, we employ a logistic regression network sp (c:) with j as parameter of the network for an individual to predict the conditional probability distribution of the CIV S, P(S C). and we sample Š from P(S | C), i..., ~ P(S | C). For the
from the data.
CIV regression model, we have the following loss function:
1 n D ss (og(((((.))))))))))...)))))())))
Ls
s;) (o)(()))(())))))))
(1)
where n is the number of samples in the data. Step two learns Z such that Š JL Z. For achieving this goal, we train the representation network Z tha (C) with the learnable parameter 0 and minimise the discrepancy of the distributions of Z for different values of Š by using IPM(.), which is the
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integral probability metric for measuring the discrepancy of distributions. The above requirement for confounding balance is formally stated as the following objective:
D(S, (o((C))) = (PMM(1e (C) - 0 c. c.)
(2)
where (V)(C))( = c.))=.. | E 1) is the distribution of the representation Z (i.e., the (C()
In this way, we obtain S L. Z, which implies S JL. Z C because S is obtianed by conditioning Wasserstein distance as the metric for evaluating the discrepancies for CFR, CBIV, and our method CBRL.CIV, enabling fair comparison in our experiments because Wasserstein distance is used in the implementations of CFR and CBIV.
in the sub-group S
C and Z is also obtained by conditioning C. We utilise the
Treatment Regression with Confounding Balance. To address the distribution imbalance of C between W and Y and the effect unobserved confounders U, we follow the previous non-linear IV-based methods, such as DeepIV (Hartford et al., 2017) and CBIV (Wu al. 2022). to build treatment regression network by using the estimated S obtained in the first stage and C to regress W for obtaining an estimated W that is not affected by U. Specifically, we use a logistic regression net- work (2, 0,) with the learnable parameter v to calculate the conditional probability distribution of the treatment W, i.e., W P(W| C), and optimise the following loss function:
Lw 1 Σ (ws log((eu (8;; c:))
w1:(( log (8,, c:))))
(3)
After obtaining W, achieve W Z for alleviating the effect of the distribution imbalance of C between W and Y. we minimise the discrepancy of the distributions for different values of W, and the objective function can be formed as:
D(W, the (C)) = IPM( (C) = 0 &; c.) (C) = 1 6;, (.)) & c.)
(4)
where (C Pc,
(0. 1) represents the distribution of the representation = = (C) the sub-group W = w. The confounding balance term is a constraint term that is included the learning process to ensure that vo(()) and W are independent, i.c., W TT Z.
Outcome Regression. In the final stage, we propose to regress the outcome using Z (i.e., tho (C)) and the estimated W obtained in the treatment regression stage. Furthermore, regression networks for counterfactual inference, as done in Hartford et al.] (2017). Both networks, denoted as fa² (vbo C)) and f (vbo (C)), have leamable parameters 7 and 70 respectively.
two separate
The estimated treatment W P(W š, C) and the balancing representation vho (C) are used to regress Y in the outcome regression network. The objective function for the two separate regression networks is defined as follows:
Ly
1
Σ 1=1
Vi
Σ (0,10,1)
(1bo (C)) 8;; c;)
(5)
where y the value of observed outcome, and P(w; &6, c,) is obtained from the treatment regres- sion network (3;, c;) in the second stage.
Optimisation. We employ three-stage optimisation for our proposed CBRL.CIV method. Firstly, we optimise the CIV regression network (,((c) for minimising the loss function Ls in Eq.[ Then, we optimise the treatment regression network (,, c,) for minimising the loss func- tion Tw in Eq. 3 Finally, we simultaneously optimise the two confounding balance representation learning networks and the outcome regression network by adding the two confounding balance terms as type of regularisation to the objective function of the outcome regression network, as follows:
min Ly + aD(W, vi (C)) + 3D(S, (o(C)
(6)
where a and are tuning parameters.
After we obtain vho( C) and parameters 700 and ,1, the average causal effect (ACE) of W on Y can be calculated by
AČE BB] (V)) (v (C))
The pseudo-code of the proposed CBRL.CIV method is provided in Appendix B.I.
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3.3 THEORETICAL ANALYSIS
The deep learning based IV estimator, DeepIV (Hartford et al. 2017), provides the counterfactual prediction function:
S(W, = (W, EE
EU C]
(8)
where U is only conditioning on C, but not W. E|U C| is non-zero, but stays constant with the change of W 1 to W = O. With valid standard IV S, the counterfactual prediction function
f(W, C) can be solved over the observed confounders:
Y S. - (WW, ()C
where dP(W S, C) is the conditional treatment distribution. The relationship in Eq. defines
an inverse problem for the function f by using two observed functions E|Y S. C and P(W
C) (Hartford al., 2017 Newey & Powell, 2003). Most IV methods (Chen & Pouzo 2012 Singh et al. 2019) employ a two-stage process for solving this problem: estimating P(W S, C) for the true P(W S, C), and then approximating / (W, C) given the estimated P(W | C).
We have the following theorem to identify (solve) the counterfactual prediction in Eq. by using a CIV and deep neural networks Bennett al. (2019); LeCun et al. (2015) in our problem setting.
Theorem 1. Given the observational data O = (C,S, W,W.WYY Y') generated from a causal DAG G = (CUUU(S, W, Y7,E) E) where E the set of directed edges as shown Fig. [d), suppose that is CIV and C is a set of pre-IV variables. Under Assumptions I and 2, if the learned representation Z = vho ()) of C is independent of estimated $ and W, then f(W, C) can be identified by using the ( S, C), where P(W] S, C) is the
$ and the representation Z as: E
estimated conditional treatment distribution by using S and C.
The proof is given in Appendix B.2. Theorem ensures the soundness of the proposed CBRL.CV method by solving an inverse problem for counterfactual outcomes prediction.
4 EXPERIMENTS
The goal of the experiments is to assess the performance of the proposed CBRL.CIV method in ACE estimation from data in the presence of unobserved confounders. Thus, we conduct extensive experiments on synthetic datasets since they allow the true ACE to be known. To show the potential of the proposed CBRL.CIV method in real-world applications, we also evaluate the performance of CBRI.CIVV on two real-world datasets that have been widely used in the evaluation of IV-based estimators (Abadic 2003; Card 1993] Cheng et al. 2024). Due to the page limitation, we report the ablation study, and the effect of and 3 CBRL. CIV in Appendix C.3 and C.4, respectively.
4.1 EXPERIMENTAL SETUP
Baselines. We assess the performance of CBRL.CIV by comparing it with eight state-of-the-an art causal effect estimators for causal effect estimation from observational data. These estimators can be categorised into three groups: (I) most closely related work: CBIV, the confounding balancing IV regression Wu et al. 2022); (II) Other IV methods using machine leaming/repreesntation learn- ing, including (1) the DeepIV (Hartford et al. 2017) (a deep neural network based IV estimator), (2) OrthIV [Sjolander & Martinussen (2019) (an orthogonal regularisation machine learning based IV estimator), (3) DMLIV (Syrgkanis et al. 2019) (double machine learning based IV estimator), (4) DRIV (Syrgkanis et al., 2019) (doubly robust learning based IV estimator). (5) SDRIV (Cher- nozhukov et al.) (0188) (sparse linear doubly DRIV), and (6) ForestDRIV (Syrgkanis et al., 2019) (random forest DRIV); (III) Other estimator, not based, but a popular representation learning method which considers confounding balance, i.c., CounterFactual Regression by using deep repre- sentation learning (CFR) (Shalit et al. 2017).
Implementation. We have implemented CBRL.CIV using Python programming language. We used lulti-layer perceptrons with ReLU activation function and BatchNorm to implement the lo- gistic regression networks for CIV regression and treatment regression. For optimising the CIV

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Table 1: The table summarises the estimation error (Tea over 30 synthetic datasets with different settings. The smallest estimation error in each setting is highlighted. Note that CBRL.CIV obtains the smallest estimation error in terms of mean and STD.
Within Sample
Out-of-Sample
Methods Syn-4-4
Syn-8-8
Syn-12-12
Syn-4-4
Syn-8-8
Syn-12-12
DeepIV
1.58.1.111
1.49
1.45:10117
1.58.10.77
1.45.10.18
1.45.10.22
OrthIV
0.69:10:43
1.86.10.15
1.72:10.08
3.86
2.14:0.29
1.86:10.11
DMLIV
0.69::0.5
1.8440.14
1.73+0008
1.06=1.66
2.05=0.32
1886:0.12
DRIV
1.96#t0.51
1.14
0.640.35
2.03++0.34
1.42=0.39
0.92=t0,43
SDRIV
1911:0.36
0.72:10.5
0.46:t0.35
2.13=:0.39
1.09=0.44
0.52:t0.49
ForestDRIV
187++033
0.72+00.4
0.46
1.99+0044
0.94+0.6
0.75++0.52
CFR
0.6+00.38
0.58+00.3
0.51+0.22
0.6+0.388
0.58+0031
0.51++0.25
CBIV 0.48+0027
0.4+0.11
0.41+0.11
0.48+0.22
0.4+0.11
0.41+0.114
CBRL.CIV
0.14:10.05
0.14100.3
0.12.10.03
0.14
0.14100.3
0.12:10.03
and treatment regression networks, we used stochastic gradient descent (SGD) (Duchi et al. 201 Furthermore, used the ADAM method (Kingma & Ba 2014) to optimise the outcome network jointing the confounding balance modules. To prevent overfitting, we employed an le-reuularisation term to regularise the objective function in Eq. (6). The code of CBRL.CIV, network parameters, and parameter tuning are provided the supplementary materials. The implementations and parameter settings of compared estimators are provided in Appendix C.I.
Evaluation Metries. For the experiments on synthetic and two real-world datasets, we use the estimated ACE error, (AĆE - ACE) to evaluate the performance of estimators ACE esti-
mation from data with unobserved confounders.
4.2 SIMULATION STUDY
We follow the data generation process in (Hassanpour & Greiner 2020 Wu et al. 2022 to generate
synthetic datasets for the evaluation. Note that the underlying causal DAG Fig. I (d) is used to generate all synthetic datasets for evaluation of the performance of CBRL.CIV is relative to the double imbalance issues causal effect estimation from data with unobserved confounders. The details of the synthetic data generation process and more results are provided in Appendix C.2.
We assess the performance of CBRL.CIV under different settings of the number of observed and unobserved confounders, and we denote the settings as 'Syn-p-q', i.e., setting/dataset with ob- served confounders and / unobserved confounders. In our experiments, we sample 10k individuals with three settings, Syn-4-4, Syn-8-8, and Syn-12-12. We report the mean and the standard devia- tion (STD) of the estimation error of all methods in Table Note that Within-Sample" denotes the estimation error calculated over the training samples, and Out--of-Sampp implies the estimation error calculated over the testing samples.
From the estimation errors of all estimators in Table I have four conclusions: (1) The pro- posed CBRL.CIV method obtains the smallest estimation error on all synthetic datasets because of CBRL.CIV balances the distribution of C between W and Y the outcome regression stage, as well as the distribution of C between Š and W in the treatment regression. (2) CBIV has the second smallest estimation error but is worse than CBRL.CIV since it ignores the distribution imbalance of between $ and W in the treatment regression. (3) The estimators in the second group have a large estimation error since these methods ignore the confounding bias in both stages. (4) The confounding balance-based estimator, CFR has the third smallest estimation error, but CFR does not deal with the affect of U.
We also evaluate the impact of sample size on the performance of CBRL.CIV uses synthetic datasets with range of sample sizes: 2k, 4k, 6k, 8k, 10k, and 20k. The estimation errors of all estimators on Out-of-Sample are visualised with boxplots in Fig. 3 From Fig. 3 we have the following ob- servations: (1) CBRL.CIV has the lowest error and smallest variance of error in different sample sizes. (2) As the sample size increases, the variance of the estimation error of CBRL.CIV decreases slightly. (3) CBRL.CIV has the narrowest upper and lower bounds compared all other methods.
In summary, CBRL.CIV achieves the best performance among all estimators in ACE estimation from synthetic data in the presence of unobserved confounders.
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DeepIV
OrthTV
DMLIV
DRIV
SDRIV
ForestDRIV
CFR
CBIV
CBRL.CIV
6.00
4.00
3.00
2.00
1.00
0.00
2k
4k 6k 8k 10k
20k
Figure 2: The experimental results of all estimators on synthetic datasets with unobserved con- founders non-linear scenarios. Our CBRL.CIV achieves the lowest error ACE estimation. And as the sample increases, CBRL.CIV has slight decrease in variance slightly.
Table The estimation errors (mean=STDD over the out of samples on two real-world datasets with different dimensions of p and q.
Methods
IHDP-2-2
J-DP-6-6
Twins-2-2
Twins-4-4
Twins-6-6
DeepIV
2.642:10.407
2.222
2.471:0.419
0.088
0.081=0.070
0.081
OrthlV
14.678::64.451 3702233.843
11.71117.026
0.465=10.779
0.533:10.336
0.675:0.882
DMLIV
1..
8.945=1.0029
4.805=22.229
3.167:18.197
1.695=1.839
0,412:10.269
DRIV
3.865+1.5001
7.163++9.311
4.875++3.889
0.588+00.334
0.602+0.853
0.589++0.421
SDRIV
17.73
4.009+6.502
6.373+7.868
0.799+0.585
0.591+0.403
1.773++4.778
ForestDRIV
16,25244-2.996
12.197433.402
5.283+55.484
0.763++0.647
0.357+0.307
0.673+-0.544
CFR
2.570:11.064 1.75410.501
2.917-10.419
0.037110.008
0.03910.021 0.040:10.026
CBIV
2.312:10.944 1.34910.452
2.86.10.490
0.033100001
0.03410.026
0.03510.0116
CBRL.CIV
0.25110.082
0.454.10.068
0.108.10.063
0.030 0.027-1.018
0.030.L0.024
4.3
EXPERIMENTS ON TWO WORLD DATASETS
We also conduct experiments on two real-world datasets, IHDP (iill 2011) and Twins (Shalit et al,, 2017), to evaluate the performance of CBRL.CIV method and demonstrate its potential in real- world applications. The two datasets are widely used in the evaluation data-driven causal effect estimators (Cheng et al. 2024 Guo al. 2020).
The Infant Health and Development Program (IHDP) dataset consists of 747 infants (139 treated infants and 608 control infants). A total of p + q variables, serving as confounders, selected from the original covariates that are ancestors of, or direct causes of Y. in which variables are observed confounders C, and q variables are unobserved confounders U (where p and q take values of 2, 4, and 6 respectively). Then we have the CIV assignment probability: P(S C) (Hexp(-(EE)) C+ u.)) S Bern(P(S C)), and the treatment assignment probability:
IHDP:
P(W () Itexp(-(S-))), CALTCHLT U.) Bern(P(W [S, C)), where Bern(.) is the Bernoulli distribution.
Twins: The dataset is collected from the recorded twin births and deaths in the USA between 1989-1999 1991 (Almond et al. 2005). Similar to Wu et al. Ww et al. (2022), we use 5,271 records from same-sex twins with weights less than 2,000g from the original data. Following the same process with IHDP, we choose p + variables from the original covariates as confounders and generate the CIV S and the treatment W.
The experiments on the two real-world datasets also conducted over 30 replications with the 63/27/10 splitting of train/validation/tes samples. Following the work (Wu et al.) 2022). in each replication, we shuffle the data and split the data into train/validationntest samples randomly.
Results. Table 2 shows the results where IHDP-p-q or Twins-p-g, e.g. IHDP-2-4 denotes the use of dataset with 2 observed confounders and 4 unobserved confounders. From Table 2 have
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the following observations: (1) CBRL.CIV achieves the lowest estimation errors in all experimental settings of both datasets. (2) CBIV achieves the second-best performance on both datasets. (3) CFR and DeepIV outperform the traditional IV methods.
In a word, our CBRL.CIV method obtains the best performance on both synthetic and real-world datasets compared to the state-of-the-an IV methods and confounding balance based methods.
5 RELATED WORKS
The IV approach is useful way to eliminate the confounding bias due unobserved confounders in causal effect estimation (Anderson & Rubin 1949] Cheng et al. 2023 Wang et al.) 2022). Two- Stage Least Squares (TSLS) is one of the most commonly used IV-based methods for causal effect estimation from data in the presence of unobserved confounders through linear regression (Angrist
& Imbens 1995; Imbens 2014). In non-linear setting, some methods use non-linear regression to regress the treatment conditioning on the IV and the other observed confounders in the first stage, and then regress the outcome variable by using the resampled treatment and the observed confounders (Newey & Powell| 2003). Recently, machine learning based IV methods have been developed for reducing estimation bias, such as Kernel IV (Singh et al. 2019), OrthI (Sjolander & Martinussen 2019), and Deep Features IV Regress (Xu et al. 2021). Furthermore, DeepIV Hartford ct al. (2017) and OneSIV (Lin et al. 2019) fit the treatment assignment based on the IV and the observed confounders, and then resample the treatment based on the fitted treatment assignment mechanism. CBIV (Wu et al. 2022) 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 wider application than method based on standard IV and it also deals with non-linear cases.
Another line of related works is confounding balance with representation learning (Hassanpour & Greiner 2020 Shalit et al.) 2017). They consider confounding balancing as domain adaption problem (Daumé III 2007) under the unconfoundedness assumption (Imbens & Rubin 2015). In contrast, our CBLR. does not rely on the unconfoundedness, and it reduces the confounding bias due to both unobserved and observed confounders by using representation learning techniques.
CONCLUSION
In this paper, we propose a novel method called Confounding Balancing Representation Learning for non-linear CIV regression (CBRL,CIV) for causal effect estimation from data with unobserved confounders in non-linear scenarios. CBRL.CIV addresses the imbalance problems of observed con- founders and the confounding bias of unobserved confounders by using balancing representation learning network to obtain a balancing representation of the observed confounders and removing the confounding bias caused by unobserved confounders simultaneously in non-linear system. Exten- sive experiments conducted on synthetic and two real-world datasets have demonstrated the effec- tiveness of CBRL.CIV in average causal effect estimation from data with unobserved confounders. The proposed CBRL.CIV method makes significant contributions to causal effect estimation by considering the distribution imbalance of observed confounders between the CIV and the treatment, and the imbalance distributions of observed confounders between the treatment and the outcome simultaneously. which is the first work address the challenging problem causal inference. The proposed CBRL.CIV method has potential applications in various fields, including healthcare, so- cial sciences, and economics, where causal effect estimation is essential for preision-makinge The CBRL.CIV method relies some specific assumptions, such as the existence of valid CIV and the assumption on independent compliance type (Assumption 2). If these assumptions fail, the method may produce incorrect causal effects. It is vital to verify these conditions and conduct robustness checks and sensitivity analyses to ensure the validity of the causal estimates by CBRL.CIV when the domain experts not sure whether the conditions satisfied.

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ACKNOWLEDGMENTS
We wish acknowledge the support from the Australian Research Council Discovery Project 230101122. Thuc Duy Le is supported by DECRA (DE200100200).
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A APPENDIX
This is the appendix to the paper 'Conditional Instrumental Variable Regression with Representa- tion Learning for Causal Inference'. We provide additional definitions of causal DAG, the pseudo- code of CBRL.CIV, the proof of Theorem I, the details of synthetic and semi-synthetic real-world datasets, and more experimental results.
A PRELIMINARY
The definitions of Markov property and faithfulness are usually required in causal graphical mod- elling.
Definition 2 (Markov property Pearl (2009). Given a DAG G = (X, E) and the joint probability distribution of X. (P(X)), G satisfies the Markov property for vX, X. X, is probabilistically independent of all of its non- den-deseensntss given the set of parent nodes of X4.
Definition (Faithfulness Spirtes et al. (2000)). DAG G = (X. E) is faithful to a joint distribution P(X) over the of variables X is and only if every independence present in P(X) is entailed by G and satisfies the Markov property. A joint distribution P(X) over the set of variables X. is faithful to the DAG G if and only if the DAG G is faithful to the joint distribution P(X).
When the faithfulness assumption is satisfied between joint distribution P(X) and DAG of set of variables X, the dependencyyindependencyy relations among the variables can be read from the DAG Pearl (2009): Spirtes al.) (2000). In a DAG, d-separation is a well-known graphical criterion used to read off the conditional independence between variables entailed in the DAG when the Markov property and faithfulness are satisfied Pearl (2009); Spirtes et al. (2000).
Definition 4 (d (d-separation Pearl (200)). path π in a DAG G (X, E) said to be d-separated (or blocked) by a set of nodes M if and only if (i) π contains chain X; + Xx + X, or fork X, & Xx - Xx such that the middle node X2 is in M, or (ii) π contains a collider X₃ such that X is not in M and no descendant of X is in M. A set M is said to d-separate X; from X (X; 11 X, |M) if and only if M blocks every path between X; Xj. Otherwise, they are said to be d-connected by M, denoted as X, JL X, M.
The instrumental variable (IV) approach is powerful tool for estimating causal effects from ob- servational data in the presence of unobserved confounders and noncompliance with the assigned treatment W Angrist & Imbens (1995); [mbens] (2014); Imbens & Rubin (2015). The main strata
are based on the instrument assignment to follow the treatment W (or not). Compliers always comply with W: W(S
1) 1 and W(S 0) 0; Always-takers always follow W:
W(S = 1) = W(S = 0) = 1; Never-takers never follow W: W(S = 1) = W(S = 0) = 0;
Defters always against W: W(S 1) and W(S 0) 1.
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B THE PROPOSED CBRL.CIV METHOD
In this section, the pseudo-code of CBRL,CIV, the hyperparameters, the tuning parameters setting, and the proof of Theorem are provided.
B. PSEUDO-CODE OF CBRL.CIV METHOD
The proposed three networks be optimized sequentially. The loss functions of the three networks are formulated as follows.
m
+ (1-)))))++()))())0(((())))))
(10)
Lw 1
(w (1- - (11)
mi Ly + aLCBB + BLCBS
yi
Σ
frre (vo (v)(C)))(w) 1&, (i,c.))
(12)
(.E(0.1)
DDWW, (b( + (D($$,
where a and are tuning parameters. LCBw and LoBs are the confounding balance constraints in Eq.s (2) and (4), respectively. The pseudo-code of CBRL.CIV is listed in Algorithm 1.
Network Structures & Tuning Parameters. For the CIV and treatment regression networks, we use multi-layer perceptrons with ReLU activation functions for the logistic regression networks ipp and pv respectively. We optimize the loss functions Lg and Lw using SGD with a learning rate of 0.05. For the outcome regression network, the loss function Ly + aL + BLCBS- For the two tuning parameters a and 8, we select them from the range (0.0001, 0.001, 0.01, 0.01, 1.1,
use ADAM with learning rate of 0,0005 to optimize
B.2 THRORETICAL ANALYSIS
Theorem 2. Given the observational data O = (C,S,W,Y) generated from a causal DAG G (CUU U (S. W, Y), E) where is the set of directed edges as shown in Fig. 1 (d) in the main text, suppose that S is CIV and C is set of pre-CIV CIV variables. Under Assumptions and 2, if the learned representation Z vh C) of C is independent of the estimated S and W, then f(W, C) can be identified by using Š and Z by solving the inverse problem using $ and Z as: Eyy S,C f(W, Z)dP(W C) where P(W) (,C) the estimated conditional treatment distribution by using Š and C.
Proof. We prove that the estimated $, W. and the learned representation Z by CBRL.CIV can be used to identify f(W, C) from the observational data O. We first note that $ and W, as well as S and Y are confounded by C respectively in O and the CBRL.CIV method uses the previously described three-stage strategy to adjust for the confounding bias due to C and U.
In the CIV regression stage, aim to obtain S IL Z to eliminate the confounding bias between S and W. For this, we first perform logistic regression using deep neural networks for predicting the conditional probability distribution of the CIV S conditioning on C, namely P(S C). Next, $ is sampled from P(S C) assuming P(S C) P(S C, Z) since then the confounding balance phase, the learned representation Z and the estimated š are forced to be independent, i.c., Z when the optimal solution of Eq. (6) in the main text is obtained. Therefore, the causal relationship between S and W is not confounded by Z.
In the treatment regression stage, we aim to obtain W TL C. So we conduct logistic regression using deep neural networks for predicting the conditional probability distribution of the treatment W
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Algorithm CBRL.CIV: CIV Regression with Confounding Balancing Representation Learning
Observational dataset O = (W, Y, X) with the treatment W, the outcome Y, the CIV 8 and the observed variables C; the maximum number of iterations t. Output: Yu- = foo (Po(C)) and y1 far (C))
Input:
CTV Regression:
for ite=l to do
{c, + (c,) + P(s; c;)
Ls = * Ei _ (s))
st) (o0())(())))
update μ SGD(Ls)
end for
Treatment Regression:
for ite=1 to i do
{c.)i=1 + (c;)
(5:, 1==1 + pu(3;, + P(wy sú, ci)
Lw + === (px ci)) + (1 w;)(1 (8)
update v + SGD(LW)
end for
Outcome Regression:
for ite=1 to do
(c.) + Pa(c,) + P(s: c.)
(6, 1-1 + pu((i, c.) P(w, c;)
( D(S, (C))
(Vo(c,), ú D.
D(W,)o(C))
1 - (y: -E (We 6/ ci)) + aD(W, the (C)(C) + +3D( (C))
Ly - aLCBW 3LoBs
update 0, 0,70,7 & ADAM(LY + aLoBW + BLCBB)
end for
conditioning C and S, namely P(W S C). Next, W is sampled from P(W S, C,C) assuming P(W C) P(W S, C, since then through the confounding balance phase, the learned representation Z and the estimated W are independent, i.c.. W when the optimal solution of Eq. (6) in the main S, C. U) as it almost surely that @(C) = o(C, U) as stated Assumption Thus, the causal relationship between W and Y is not confounded by Z and U.
is obtained. Furthermore, P(W S, C) P(W S,C) P(W
As the defined inverse problem of the standard IV in (q.(9) in the main text, in outcome gression stage, W P(W C)C) and the (C) are used to predict the counterfactual out- comes. Thus, E[ S.C = /(W, Z)d(WW S, C) is the inverse problem S(W, Z) given E| S, C) and P(W S)C..
C EXPERIMENTS
C.1 EXPERIMENTAL SETUP
Implementation sources for comparsion methods. The implementations for DeepIV, OrthIV, DMLIV, DRIV, SDRIV, and ForestDRIV methods are sourced from EconML (Battocchi et al., 2019). The CBIV implementation is avaiable on the authors' GitHub page at https:::ss /tttps://qithutt com/ anpwu/ CB- IV Similarly, the CFR implementation can be found on the authors' GitHub repository at https: //gittttb//////// com/ c wwinicalmm/cfnnee
Parameters setting. To ensure a fair comparison, we use the common setting for the tuning pa- rameters done in Shalit et al. (2017); Wu al. (2022). such learning rate and number of
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DeepiV
OrthiV
DMLIV
DRIV
SDRIV
ForestDRRVV
CFR
CRIV
CBRL.CIV
3.00
2,50
2.00
1.50
20k
Figure 3: The experimental results of all estimators within-sample of synthetic datasets with unobserved confounders in non-linear scenarios. The CBRL.CIV obtained the lowest estimation error against all comparisons.
epochs. Furthermore, we set the same parameters, such as the dimension of learned representation for CFR, CBIV, and our CBRL.CIV.
C.2 SIMULATION STUDY
We generate the synthetic datasets following the works Hassanpour & Greiner (2020); Wu et al. (2022) for the evaluation. Especially, the set pre-IV variables C and the set unobserved vari- ables U drive from C1 C2, ... Cp. U1, U2 ... U, N(0, Ep q). where p and g the dimen- sions of and U, respectively, Ep + ppt 0.95 *0.05 denotes that all values excluding those the main diagonal are 0.05 in the covariance matrix, and lyta is p + q degree all-ones matrix. The CIV S, treatment variable W, and outcome variable Y are generated as follows:
1+ eap( e C+E U.) P(SIC)
(13)
S (P(e)(P(SS)))
P(W (,
1 cxpl -( exp(-
E/ C.+ C. EI C U.))
(14)
W ~ Bern(P(WW |)C))
W P J(W
W
Σ 9
C
Σ
U?
p Σ G
U.)
(15)
p+
where (e C)) and (eW C)) are the true CIV and treatment assignment, respectively. In order avoid bias caused by the data generation process, repeated the data generation process 30 times (i.e., 30 are generated datasets for each setting) for all experiments. We randomly split dataset into training (70%) and testing (30%) samples for all simulations. The generation process is widely used in the evaluation of data-driven causal effect estimation meth-
ods Hassanpour & Greiner 2020): Wu al. (2022); Cheng et al. (2022).
Results.
The estimation errors all estimators on within-sample are visualised with boxplots in Based on Fig. 3 have the same conclusion drawn in the main text.
Fig. 3
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Table The table summarises the estimation error of CBRL.CIV with the original S (mean STD) over 30 synthetic datasets.
Syn-4-4
Syn-8-8
Syn-12-12
0.48 + 0.27
0.40 + 0.19
0.41 + 0.14
Table 4: The table summarises the estimation enror (mean + STD) over 30 synthetic datasets with different settings and sample sizes. The compared methods have CBRL,CIV, CBRL.CIV with Equa- tion 2 unused (CBRL), and CBRL.CIV with both Equations 2 & 4 unused (CBRL).
Syn-4-4
Syn-8-8
Syn-12-12
CBRL.CIV
0.19 1.93
0.13 1.95
0.09 1.97
2k
CBRL
0,26 1.77
0.45 # 1.70
0.54 + 1.72
CBRL
0.32 1.63
0.61 1.68
0.76 1.68
CBRL.CIV
0.18 0.07
0.14 = 0.03
0.13 = 0.03
4k
CBRL
0.43 0.29
0.35 0.20
0.39 0.11
CBRL
0.56 0.43
0,44 # 0.32
0.61 # 0.24
CBRL.CIV
0.14 0.05
0.14 = 0.03
0.12 + 0.03
6k
CBRL
0.48 # 0.27
0,40 = 0.19
0.41 = 0.14
CBRL
0.60 0.38
0.58 + 0.31
0.51 0.25
CBRL.CIV
0.16 0.05
0.03
0.11 0.03
8k
CBRL
0.56 0.33
0.47 0.13
0.38 0.12
CBRL
0.68 0.40
0.25
0.47 = 0.25
CBRL.CIV
0.18 # 0.05
0.14
0.11 # 0.04
10k
CBRL
0.59 0.25
0.10
0.46 0.14
CBRL
0.74 0.34
0.53 0.24
0.60 = 0.20
CBRL.CIV
0.15 0.06
0.04
0.11 0.03
20k
CBRL
0.57 + 0.20
0.49 0.14
0.48 : 0.08
CBRL
0.60 0.24
0.61 # 0.28
0.58 0.21
C.3 ABLATION STUDY
We have tried using Eq. 2 (i.e., longer using loss 1) the synthetic datasets used in Section 4.2. The experimental results on datasets with out-of-samples are reported in Table
Results. From Table 3 we know that CBRL.CIV without loss has a larger estimation error than CBRL.CIV with loss i.e., S Z is not enough for mitigating the confounding bias between S and W.
To show the necessity of Eq. (and Eq. 4), conduct ablation study and report the experimental results in Table The synthetic dataset generation process is the same as that Section 4.2.
Results. From Table 3 CBRL.CIV has the smallest estimation error on all experimental settings than CBRL (CBRL.CIV without CIV regression, i.e. Eq. 2 is not used) and CBRL (CBRL.CIV with both Eqs. 2 & unused). The results indicate that the CIV regression for confounding balance necessary (and that the treatment regression with confounding balance is also needed).
C.4 THE EFFECTS OF a AND 3
We have conducted the experiment to examine the effects of and B the performance of CBRL.CIV. The results are summarised in Table Note that we only adjust the values of a and 3, and Eqs. and 3 were still worked in Eq.
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Table 5: The table summarises the estimation eπror (mean + STD) over 30 synthetic datasets with different values for a, B. The sample size is 6k with the setting Syn-4-4.
Wathin-Sample
Out-of-Sample
a, a,
0.1594 + 0.0371
0.1597 0.037
a.000.01
0.1553 # 0.0418
0.1556 = 0.0411
a.3
0.1406 # 0.0533
0.1412 + 0.0525
8=
0.1877 # 0.0691
0.1878 = 0.0679
a, 0.2198 0.0511
0.2198 = 0.0511
Table The estimation errors (mean=STD) over the within-sampi on two real-worke datasets with different dimensions of p and
Methods
IHDP-2-2
IHDD-444
IHDP-6-6
Twins-2-2
Twins-4-4
Twins-6-6
DeepIV
2.692:10.407
2.900-10.200
2.516.10.39
0.091110.045
0.068.10.048
0.082:10.041
OrthIV
0.773:10.407 2.13511.210
3.899:10.735
0.256:10.158
0.434:10.079
0.352-10.113
DMLIV
1.134:10881
5.010112.113
4.239
0.244:10.156
0.425:100799
0.343-1.1118
DRIV
3.872=11.17 3.956=t0.740
4.483+2.550
0.285++02228
0.246::0.205
0.478=0.329
SDRIV
8,212=1,106 2.932:t4.171
3.733
0.347::0.22
0.286=0.193
0.572=0.414
ForesDDRIV
10.176:19.89) 6.333-77838
4.669=4.17071
0.377:10.883
0.255=:.228 0.490=0.355
CFR
1.568 1.757+00.59
2.917++0.28
0.037+00.008
0.039+0.0021 0.040=000227
CBIV
2.397 1.344+00443
2.836+-0.486
0.033++0.011
0.03440.026 0.035+0.016
CBRLCCV
0.081
0.066
0.120+00078
0.030-H0.019 0.027++0018 0.031-0.024
Results. From Table
we observe that when and are set 0.1, the minimum estimation error
is achieved. Furthermore, even when and & are both set to O, CBRL.CIV still exhibits minimal estimation error.
C.5
TWO REAL- WORLD DATASETS
IHDP. IHDP is the Infant Health and Development Program (IHDP) dataset collected from ran- domised controlled experiment that investigated the effect home visits by health professionals future cognitive test scores Hill (2011). There 24 pretreatment variables and 747 infants, including 139 treated (having home visits by health professionals) and 608 controls.
Twins. Twins is benchmark dataset frequently used in causal inference literature. It focuses on twin births and deaths in the USA between 1989 and 1991 Almond et al. (2005). Each twin pair in the dataset includes 40 pretreatment variables that provide information about the parents, pregnancy, and birth of the twins. Both the treated group (W=1, representing the heavier twin) and the control group (W=0, representing the lighter twin) are observed. The true outcome of interest the mortality rate after one year for each twin.
Results. Table shows the estimation errors over the within sammle-sammle on two real-world datasets
with different dimensions of p and g. where p is and q is. From Table 6 we have the same conclu- sions as described the main text.
17


Reviews:
[{"review_id": "AG8WIV6HQ3", "reviewer": "Reviewer_MNhf", "summary": "This paper introduces a method aimed at addressing the challenge of conditional instrumental variables, which refer to variables that may not be valid unconditional instruments, but satisfy the instrumental variable assumptions when conditioned on an appropriate adjustment set. The paper argues that the primary issue associated with conditional instrumental variables lies in the distributional imbalance observed among the groups defined by the values within the conditioning set. To address this, the paper proposes a methodology that uses regularization similar to the approach employed by Shalit et al. [2017] to minimize these distributional shifts.", "questions": "How were the baselines tuned?\n\nWhy are they so much higher variance than the proposed method?\n\nIf you consider a setting with a standard IV, do the baselines then outperform your method?\n\nHow did you select instruments for IHDP and Twins?\n\nWhat is a plausible source of unobserved confounding for IHDP and Twins?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper studies balancing in the conditional instrumental variable setting, and they use an approach reminiscent of Shalit et al [2017] to penalize changes in the conditional distribution of the treatment and instrument are stable across conditions. \n\nI think that it is under appreciated that conditional IVs are possible (although they are implicit in most standard IV assumptions), so I'm glad this topic is getting attention. \n\nI reviewed an earlier version of this paper and most of my critiques from then have now been fixed (with the exception of the experiments below).", "weaknesses": "The experiments remain the weakest part of this paper. The result of the proposed method are very impressive relative to the baselines, but it is not at all clear how fair the comparison was. The synthetic data is a novel benchmark, so I would expect that the baselines would all underperform without some hyper parameter tuning, but there is no mention of this (the authors do tune the hyper parameters of the proposed method).\n\nMore importantly, both IHDP and Twins are datasets that are used in the unconfounded setting, so it's not clear that they even have an instrument (particularly not a conditional instrument) or unobserved confounding. \n\nCode is supplied but only for the proposed method (not the baselines), and the data generating process is only supplied", "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": "", "extra_scores": {}}, {"review_id": "8mVsSSKfu0", "reviewer": "Reviewer_9EAs", "summary": "This paper presents a novel approach designed to tackle the complexities associated with conditional instrumental variables (CIVs). These are specific variables that, while not qualifying as unconditional instruments, fulfill the IV assumptions when conditioned on a suitable adjustment set.  The paper identifies that the predominant challenge with these CIVs emerges from distributional imbalances among groups delineated by the values in the conditioning set. To mitigate this, the paper proposes a framework that leverages regularization techniques to minimize these distributional discrepancies.", "questions": "Sww weakness", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1.\tThe paper is good by its deep exploration of the challenges associated with estimating causal effects in the presence of non-linear relationships and unobserved confounders. It specifically addresses the imbalance issues inherent in applications of CIVs in non-linear settings with finite samples. This is a significant contribution, as CIVs are more practical and less restrictive than traditional IVs in real-world scenarios.\n2.\tThe paper proposes a novel Confounding Balancing Representation Learning (CBRL.CIV) method for non-linear CIV regression in causal effect estimation. Notably, CBRL.CIV efficiently mitigates the confounding biases between $S$ and $W$, as well as between $W$ and $Y$.  \n3.\tThe paper delves into the theoretical underpinnings of CBRL.CIV, highlighting its robust capability to generalize across varied groups. This ensures a harmonious balance across the CIV, treatment, and outcome regression networks.\n4.\tThrough experiments on both synthetic and real-world datasets, the effectiveness of CBRL.CIV has been validated in estimating average causal effects. Such empirical evidence is crucial for the adoption and trustworthiness of the proposed method in real-world applications.", "weaknesses": "Causal inference fundamentally depends on certain assumptions and the proposed CBRL.CIV method is no exception. If these prerequisites aren't satisfied in specific scenarios, the efficacy of the CBRL.CIV method might be compromised. Thus, a discussion on the potential negative impacts of using CBRL.CIV under such conditions would be beneficial.\n\n**Presentation**:\n1.\tFigure 2 in the main content and Figure 3 in the appendix should use the same legend. \n2.\t 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{}.\n3.\tIn Eq. (12), there is a missing square, and the same missing happens in the pseudocode of Algorithm 1.", "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": "", "extra_scores": {}}, {"review_id": "GAfmTERIao", "reviewer": "Reviewer_A6cp", "summary": "This paper tackles the complex task of estimating causal effects in observational data while considering unobserved confounders. It introduces a novel approach called CBRL.CIV, which leverages a conditional instrumental variable (CIV) to relax the stringent requirements of standard instrumental variables (IV), such as linearity assumptions and strict unconfounded instrument conditions. Through theoretical analysis and extensive experiments on synthetic and real-world datasets, the paper demonstrates the effectiveness of CBRL.CIV in eliminating confounding bias, particularly in non-linear scenarios, making it a promising alternative to existing IV-based estimators.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Using CIV to relax the restrictive conditions of IV methods is practical for most realistic scenes. \n2. The overall presentation and writing structure is clear and well motivated.\n3. Extensive experimental results are convicing.", "weaknesses": "1. 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.", "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": "", "extra_scores": {}}, {"review_id": "nIXOxtHUtX", "reviewer": "Reviewer_kNXT", "summary": "This paper addresses the challenge of estimating causal effects from observational data while dealing with unobserved confounders. The paper introduces a conditional IV (CIV) approach called CBRL.CIV, which relaxes the strict conditions and allows for non-linearity. Theoretical analysis supports the soundness of CBRL.CIV, and extensive experiments demonstrate its competitive performance in handling non-linear scenarios compared to state-of-the-art IV-based estimators.", "questions": "See Weaknesses", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. CIV offers a more powerful and flexible approach to handle causal inference with unmeasured confounders.\n2. The experimental results are extensive and convincing.", "weaknesses": "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, you methods are similar to Wu et.al. Hence, can you state the difference, or the unique technical contribution of CIV for representation learning?", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)