Multi-Treatment Effect Estimation with Proxy: Contrastive Learning and Rank Weighting
Authors: Minqin Zhu
Anpeng Wu
Ruoxuan Xiong
Kun Kuang
Venue: Submitted to ICLR 2023 (2023)

Abstract: We study the treatment effect estimation problem for continuous and multi-dimensional treatments, in the setting with unobserved confounders, but high-dimension proxy variables for unobserved confounders are available. Existing methods either directly adjust the relationship between observed covariates and treatments or recover the hidden confounders by probabilistic models. However, they either rely on a correctly specified treatment assignment model or require strong prior of the unobserved confounder distribution. To relax these requirements, we propose a Contrastive Regularizer (CR) to learn the proxy representation that  contains all the relevant information in unobserved confounders. Based on the CR, we propose a novel ranked weighting method (Rw) to de-bias the treatment assignment. Combining Cr and Rw, we propose a neural network framework named CRNet to estimate the effects of multiple continuous treatments under unobserved confounders, evaluated by the Average Dose-Response Function. Empirically, we demonstrate that CRNet achieves state-of-the-art performance on both synthetic and semi-synthetic datasets.

Under review as a conference paper at ICLR 2023
MULTI-TREATMENT EFFECT ESTIMATION WITH PROXY: CONTRASTIVE LEARNING AND RANK WEIGHTING
Anonymous authors Paper under double-blind review
ABSTRACT
We study the treatment effect estimation problem for continuous and multi- dimensional treatments, in the setting with unobserved confounders, but high- dimension proxy variables for unobserved confounders are available. Existing methods either directly adjust the relationship between observed covariates and treatments recover the hidden confounders by probabilistic models. However, they either rely on correctly specified treatment assignment model or require strong prior of the unobserved confounder distribution. To relax these require- ments, we propose Contrastive regularizer (Cr) to learn the proxy representation that contains all the relevant information in unobserved confounders. Based the Cr, we propose a novel Rank weighting method (Rw) to de-bias the treatment as- signment. Combining
and Rw, we propose a neural network framework named CRNet to estimate the effects of multiple continuous treatments under unobserved confounders, evaluated by the Average Dose-Respe Response Function. Empirically. we demonstrate that CRNet achieves state-of-the-art performance on both synthetic and mnnisyytthetic datasets.
INTRODUCTION
Causal inference is widely applied for explanatory analysis and decision mak-
Observed Treotment
ing. e.g.. Precision Medicine (Raita
Observed
al. 2021), Advertisement (Lada et
Procy Repeceention
FIN
Observed Covariate (Proxy)
2019 Education (Johansoon et al.
2016 and Digital Economy (Nazarov
Unmessured Cunfoundees
2020 With accessible observation data, many existing algorithms aceu- rately estimate the effect of binary treat- ment by adjusting the confounders (i.c., the common causes of treatments and outcomes) which rely on unconfounded-
Goi Canaa Graph
(b) Procy Regresemaion
Figure 1: (a) Causal Structure of Raw Data, i.e., Y T T U: (b) Target Relationship from proxy representa- tion, i.e., (TT) LU| E(X).
ness assumption that all confounders observed. However, continuous and multi-dimensional treatments and unmeasured confounders are common in practice. For instance, practitioners seek develop precise medicine by studying the response of multiple drug dosages (i.e., treatment) on patient health state (i.e., outcome) (Shi et al. 2020). Besides, due to technique and manipulation issues, some key variables, associated with treatments and outcomes, like patient's immunity maybe missing in the historical data, which are referred to as unmeasured confounders. To detect and adjust unmeasured confounders, practitioners would record some proxy variables (noised unob- served confounders, c.g., antibodies) which don't have direct effect treatments and outcome of interest but has spurious association through shared common confounders (Fig. aa)).
In continuous treatments setting, under unconfoundedness assumption, recent works discretize the continuous treatment into multi-valued treatment (Hill 2011 Wager Athey 2018) to traditional models, develop generalize balancing methods for continuous scenario (Hirano & Imbens] 2004 Vegetabile et al., [022 Huling et al. (021). Among them, state-of-te tare--ff-he-ant that works (Wu & Fukumizu,
Schwab al. 2020 Nie et al.
2021 learn low-dimensional representation for raw data and

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balance it using minimizing mutual information, which discard the imbalance part of raw data and lose most information for predictive task in practice. In fact, the technique implements a trade-off decreasing the estimator variance at the price of increasing the bias. Furthermore, with unobserved confounders, we control the proxy rather than unobserved variables, the effect estimation will induce additional bias, referred recovery bias. To deal with this bias, instead of balancing rep- resentations and discarding information to block the relationship between observed covariates and treatments, we propose a novel Contrastive regularizer (Cr) to learn proxy representation for cap- turing all the relevant information in unobserved confounders with contrastive learning (He et al.) 2020 Chen et al., 2020; Grill et al., 2020) which regularize representation space by positive and negative pairs. In Cr, we define the positive pair is the pair of treatments and proxies from the same sample, and the negative pair is the pair of treatment from one sample and proxies from different samples. And with an ideally representation for confounders, we would adopt balancing methods to climinate confounding bias, such as generalized propensity score (Hirano & Imbens 2004).
However, one limitation is that the covariate balancing methods rely on the correct specified mod- els. we don't 't have any prior for the models of propensity score, i.e., the conditional distribution of treatment conditioning for -igh-dimensional data and continuous treatment. Besides, balancing methods still suffer from extreme values problem. Although recent methods (Fong et al. 2018) Vegetabile al., [202 Hul- ing et al. 2021) propose to clip the score value or optimize balancing weights directly, they still fail in complex data, especially, under multi-continuous treatment setting. So a balancing method that have no extreme values and adapted to unobserved confounders is urgently needed. Therefore, to control for bias from treatment assignment, we propose to rank the weights obtained from inverse propensity score for more effective balancing weighting. Based on the proxy representation learned above, we sort the propensity score based weights in descending order and record their rank (the order in sorted data) as rank weights (Rw), which is an effective and robust weights for treatment effect estimation, theoretically.
the covariates, the effect estimation would still be biased, especially
Combining Contrastive regularizer (Cr) and Rank weighting (Rw) methods, we propose neural network framework CRNet to alleviate the outcome approximate bias in estimating the Average Pose-Response Function (ADRF). CRNet can accurately estimate the effects of multiple continuous treatments with high-dimension proxy variables. Empirically, we demonstrate that CRNet achieves state-of of-the-art performance on both synthetic and semi-synthetic datasets.
RELATED WORK
Causal effect identification with proxy methods Proxy (Guo et al. 2020) assumes that the unob- served confounders can be recovered from the observed covariates. CEVAE (Louizos et al. 2017). intact-VAE (Wu & Fukumizu 2021 recover unobserved confounders with VAE (Kingma et al. 2019) constraint. Negative controls Lipsitch et af.) 2010) assume that there exist two negative con- trol variables: one is related to treatments and confounders, and another is related to outcomes and confounders. DFPV (Xu et al. I introduces neural networks to model the bridge function (Miao) let al., 2018) for estimating the causal effect. The setting of this paper is similar to the proxy. But our method need no data distribution prior and outperforms others in performance.
Estimation methods for continuous treatments For estimating the continuous treatment effect, a branch of methods include spline (Imai & Van Dyk 2004). kernel methods (Flores et al.) 2012).
ensemble methods (Hill) 2011; Wager & 2018)] representation-buseee based methods (Schwab et al.)
2020 Nie et al., 2021 Bica et al. 2020) model the relationship between treatments and outcomes. There is also a branch of methods (Hirano & Imbens] 2004: Imai & Van Dyk 2004; Robins et al.) | Huling et al. 2021] aim at balancing the covariates shifts. Few previous works take into account of unobserved variables with continuous treatment assignment bias. In this paper, we propose the contrastive regularizer to gain the balancing methods with the presence of unobserved confounders. Also, we propose new rank weighting method which have no extreme values and is not much sensitive to model misspecified. Combining Cr and Rw, we design framework CRNet to estimate continuous treatment with proxy.
2000 Vegetabile et al. |021 Arbour et al.] 2021 1
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3 PRELIMINARIES
Notation For self-consistency, we use uppercase for random variables (e.g., A) and lowercase for their realization (e.g.. a). We suggest bold the character A vector, otherwise scalar. Given a variable A", superscript represents the dimension of A, and subscript denotes the i-th sample of A. DA refers to the total number of dimensions of A. NA means the total number of samples of A. Besides, a real valued sample {X T, U, YJ} E RP+g+++] denotes a random sample with observed covariates X; RP, P..rratments T, IRS, unobserved confounders U; E R" and outcome Y; E R. And (X,; Ti, U, Y:);= refers to a set of (X,, T, U;, Yi} with n samples. E is denoted as expectation and P represents density distribution function. A calligraphic letter denoted as hypothesis space.
3.1 PROBLEM SETUP
As Fig. Ta) shown, this paper, we focus on the treatment effect estimation problem for Continuous and multi-dimensional Treatments, in the setting with unobserved confounders, but high-dimension Proxy variables for unobserved confounders are available (Briefly, CTP problem). Specifically, we formalize proxy as:
Definition (Proxy) The observed covariate and the noise view of unobserved confounder. For- mally, X = f" (1) where the noise item 61 - (T, U} and fr means the true function.
Without loss of generality, we focus on estimating the Average Dose-Response Function (ADRF) curve in this paper. We denote ADRF* as the true ADRF and define that
Definition (ADRF) The potential outcome of continuous treatments over the population: ADRF ET = t)] 16 (U, do(t))),
where do(t) means the do operation on treatment that do(t) 1 (U}.
3.2 MOTIVATION
To analyse the complex CTP problem, simplify the ADRF estimation considering additive regression model 6(X,tt t) given the observed t with no sample selection bias following Imai et al. (2008)
<(x(tt) ~ ((U,|t) + ((crs
And the the estimated ADRE of ((x,tt) can be formulated as
ADRF B[o(X,|t)),
(3)
where t means the observed treatment that t (U).
We set ADRF as baseline and define the estimation error as
A = Elo" (U, do(t)) - x" (U]t) + Elø" (U]t) -
[t]]
(4)
Given the Eq.
4 (the detailed derivation process is in the Appendix), denote the first error term
Ar Ei " do())) o" (U]t)] as the bias from treatment assignment, the second term Ay (U]t)-
as the bias from outcome approximate and the third term A., = =-E(h(er,tt)
as the bias from the recovery of U. Thus, we decompose the estimation error A into
A= A. + AT + Ay
(5)
Then we the divide ADRF estimation with multi-continuous treatments and proxy problem into three component: 1. Reduce the bias of recovery eиrоr Ae, ADRF estimation. 2. Reduce the bias Ar from treatment assignment of on U. Reduce the bias Ay from the outcome approximation.
3.3 ASSUMPTIONS
Throughout this paper, assume the two common assumptions Assumption Stable Unit Treat- ment Value Assumption, SUTVA and Assumption 2 Overlap/Posittivity assumption (Imbens & Rubin 2015) satisfied. Moreover, we assume the following assumptions.
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Assumption (Latent unconfoundedness) The potential outcome is independent of treatment signment given the unobserved confounders. Formally, Y (t) T
Assumption (Proxy assumption) The proxy is independent of treatment and outcome given un- observed confoudner. Formally, X 1 (T,YHU
To eliminate the recovery enror De,- following Louizos al. (0077), we consider it as self-supervised representation learning problem: Recovering latent representation U from Ptt, X, y); which means estimating tt, j y)),, and we formulate this problem P(ylt, = Ju P(ylt, X, U)P(U]X, t)dU Ju (tlt,, (U), t)dU. (For the discussion of proxy identi- fication, see Appendix). Then we make assumptions that
5ssumption 5
(Recoverability) The density P(U, t, 3) ofe latent confounders U can be approx- imately recovered solely from the observations {X, t. y3.
Assumption (Proxy representation) With proxies X, there exists some representations E(X(t) (briefly, E(X)) such that E(X) P(U), which means Y(t) U E(X) for polential outcome with the specific treatments
4 ESTIMATION
4.1
CONTRASTIVE REGULARIZER
Based on Assumption 6, the latent representation will be obtained when approximating the repre- sentation E(X). Existing methods (Louizos al., 2017 Bica al. 2020) address this problem by VAE (Kingma et al. 2019). GAN (Goodfellow et al. 2020 etc. They all rely on strong prior of the density form of U. In this paper, inspired by 55. propose novel contrastive learning model to preserve U and eliminate A., from data without explicit distribution prior.
Contrastive Learning There exists two functions f E F and E G. which encode X representa- tions f(X, e)) and (TT,e2)). satisfying E1), 9(T,, e2)) i + j. .), is a function that measures the similarity between representations.
>> ((T(,.)) where
Contrastive learning approximates the latent representations by constructing contrastive samples (similar and dissimilar instances), by which similar instances are closer the projection space, while dissimilar instances further away in the projection space maximize the lower bound of the mutual information.
Under CTP setting, even if from U still can not be completely eliminated based on traditional representation algorithms. There- fore, we longer rely on the representation balancing algorithm to cut off the relationship between Xand T (even we do, we cannot guarantee accurate estimation). Instead, we propose to strengthen the association between X and T using contrastive learning (Jaiswal al. (0200 model proxy representation E(X) with neural network E-)) to represent the information from U.
control the observed proxies X, the spurious association derived
The essential part for the contrastive approach is the contrastive pairs for modeling representations. Inspired by Arbour et al. (202T): Li et al. (2020), we construct contrastive pairs with no need of dis- cretizing the treatments causal inference: X and TT in original sample as positive pairs {(X,, T.) and X and T in permuted sample (shuffle X and T of data to obtain the permuted data) negative pairs (X,. T.}. Then, as Fig. 2 shown, we set E(X(t) = (t))) and adopt the NLL (Nega- tive Log-Like (iggLikelihood)) loss (Chen et al. 2020 to design novel contrastive loss to model PUU X):
((((t)))).((t))
for (c)(f()), = log
(6)
1==1 ((s((()))
where (((()) (t)) denotes the cosine similarity [F())lla(()) In contrastive aspect, representa-
(g))
tions { are queries and representations {f(X) are keys. For query g(T;), the positive key
f(X)) of the sample and the cosine similarity s( (X.)). value in the numerator in Eq.
(6) is high. In contrast, the representation s(X;) is the negative key of (TT;) where + j and the
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Pr
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Contrastive less Sample Pairs:
Input Sample
Inpet Sample N
Figure 2:
Contrastive regularizer. The covariates X are transformed to f(X) via MLPs F. In practice (Chen et al., 2020). the representation f(X) is not directly constrained by contrastive loss. f(X) transforms to Px through projection head Px. The treatments T are operated in similar way. PT and px are constrained by lcr the context represent px and pr-
For the sake of brevity, we use f(X) and g(T) in
cosine similarity 9 value the denominator of Eq. (6) low. Constrained by lor capture the proxy representations gf(X), from the data ().
In this section, we propose to strengthen association between treatments and covariates to recover unmeasured common causes U using for, which has two responsibilities: (1) strengthen the association between and T in the same sample, (2) constrain the representation space using X and T in the permuted samples. Benefiting from contrastive learning, these two responsibilities complete each other. With contrastive learning constraints, learned representation E(X) refuse the information of c and maintain the information of U, which means eliminate the error item A., in (5. Next, we consider the error term Ar in Eq. 5).
4.2
RANK WEIGHTING
4.2.1
MULTIPLE TREATMENT SCORE WEIGHT
Estimating ADRF given proxy representation, existing methods usually apply the balaneing methods D to approximate the density P[T for balancing score weights. That is, adopting the inverse of
the approximated iP(T)U) the sample weights: W, P(T, However, U) is sensitive correct specified and nearly can not be estimated accurately, especially for siggh-imensional U and multi multi-diimensional continuous T.
To approximate the U) under CTP setting, we adopt mixture density network (MDN, which uses neural network to learn the Gaussian mixture model, (Bishop 1994h) to model ((TTUU) I2 è (t9 U). As the Fig. shown, we apply MDN to approximate PTT and obtain the sample weight W; as
W3 P(T
PT Σ an (T, Mkk Σ
'Given that both matching and stratification methods can be considered special cases of weighted meth- ods, and that the first two methods require discretization of T when it is continuous values, this paper focuses weighting methods.

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The loss function of Rw is:
K Nn (T, pkx Σ
LRw
log
(8)
1=1 k=
where K' is the number of sub-Gaussian models ((-) in the Gaussian mixture model, (HR. Σ&) is the mean vector and covariance matrix of the kth sub- Gaussan model, and ax is the probability that the observation belongs to the kth sub-Gaussian model.
a1 14 of
E(X)
(DN(...)
(tl))
+
Rw
akk Hk of
Figure 3: Rank weighting. The proxy representation ( are transformed to Gaussian mixture distribution with K Gaussian submodels N(aa, M4x Σ via MLPs MDN. Then we infer the esti- mated density Ut) and sort descending order to get the rank weight Rw.
4.2.2 RANK WEIGHT
However, the sample weight W from UT still suffers from extreme values problem. Although recent methods (Fong et al. 2018 Vegetabile et al., [2021] Huling et al. (202]) propose to clip the score value or optimize balance weights directly, they still face dilemma when the data gets more complex, especially, under multi-continuous treatment setting. So a balancing method that have no extreme values and adapted to unobserved confounders urgently needed. Therefore, based on the proxy representation learned above, the core contribution of this paper is the rank weights for more effective balancing weighting.
Motivated by the balancing score problem, we normalize the IPW weights obtained from Eq. {W.)== E [0, 1] and sort them descending order. We record their rank (the order in sorted data) R, N and the difference between adjacent W stride ó E R. We define the IPW as E(R,5) U() It's clear that when sample size n is limited, the large ó causes extreme values
and when + co. 8 0, define the form of rank weight as {(R) and propose
Proposition 1 There exists some rank weight É(R) that when Z = e is the normalizing constant of É(R).
ELR)
P( where
The Proposition 1 shows that the causal effect estimation with rank weight approximate to the un- biased estimation of causal effect. We detail the definition and proposition in Appendix. When is limited, to eliminate the stride, we set it to constant ê L, and obtain the rank weight
{(Rs) = R.
(9)
The Eq. (9) shows that Rw method is adapted Cr and can be applied to data of any dimension because it only depends on the rank information of weights. when - xo, the rank weight ap- proximates to FTEjUy- With limited data samples, the rank weight don't rely on specified models and address extreme values problem.
Fig. 3 shows the process rank weighting: After training MDN Ek N (T Pk: of). Then we sort [P( in descending order and get the rank weight Rw, ((R,). Then the AT of Eq can be eliminated by weighted regression with rank weight.
we inference P(T =
Note that the rank weight is only adapted MDN, can he applied any balancing weights.

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4.3 CRNET
Combining the Contrastive regularizer and Rank weight, we propose a neural network framework CRNet estimate ADRF under CTP setting eliminate Av. As Fig. shown, the overall CRNet
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Figure 4: CRNet. For training procedure, the proxy representation EXX t) (f(X),gg [t)} con- strained by contrastive loss (X,(X, t) are concatenated and input MLPs and MDN D to obtain the estimated outcome Y and the rank weights Rw. The final objective to minimize the weighted loss in Eq. 10 For inference procedure, the estimated ADRF is obtained by ((f(X).g(t))
architecture contains three components: (1) contrastive regularizer, which contains two MLPs heads that encode proxy and the observed treatments into representations ( (2) A sample weight learner named tank weighting. This module optimizes the rank weights on &Rw in (8. using the representations {f(X). (T)) (3) A base MLPs encoder H that concatenates f(X) and g (T)) and transforms them the estimated Y to approximate the observed Y by the weighted regression loss &final (W, X, T, Y).
Combining (X,(X, T) and RRw (WV, the final loss is defined as:
Efinal
Row * (Y; - Y.)² + * Le (X, 3 & R (W),
(10)
where Rw; is the rank weight optimized by & (W). and 3 are the hyperparameters of Cr and Rw, respectively.
EXPERIMENTS
To evaluate the performance of CRNet for CTP problem, we compare statistical methods and deep-based methods as baselines in ten simulation data and four semi-synthesis data from IHDP & News. All experiments are implemented using PyTorch (Paszke et al. 2019) on Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz.
Baselines We compare our model with following baselines: For statistical methods, we use (1) CausalForest (Wager & Athey 2018). random forest algorithm for causal inference. (2)
Bart (Hill M001,/Chimme
et al. 2010), Bayesian Additive Regression Trees for causal infer-
for continuous treat- ments. CBGPS (Fong et al. [2018). a generalized covariate balancing propensity score (Imai & Ratkovic 2014) for continuous treatments. (5)EB, a continuous treatment version of entropy bal- ancing method (Hainmueller) 2012). (6)DCOWS (Huling et al. 2021), balancing method based on the distance covariance (Székely Rizzo 2009 For representation based methods, we apply (7) NN, neural network with fully MLPs. (8) MDN (Bishop) 1994), mixture density network modelling the density. (9) DRNet (Schwab et 2020). a multi-head deep model stratified
(3) GPS (Hirano& & Imbens] 22044, generalized propensity
according to T, we use a modified version (Nie et al. 2021) for estimating ADRE (10) VCNet.
(Nie et al. 202T), deep model which considers T as a varying coefficient. (11) CEVAE (Louizos)
ct al. 2017 VAE-based model to constrain the representation of covariates.
We evaluate the performance of CRNet in ten simulation data and four semi-synthesis data. For simulation experiments, design 10 simulation datasets and named five of them Datax Dr Dx (e.g.. Datax 5 means simulation with treatment, 5 covariates and no unob- served confounders). We name the other of them Dalay Dr Dx with unobserved confounders.
Datasets

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We also conduct 4 semi-synthetic experiments on real-world datasets: IHDE and News IHDP contains 747 observations on 25 covariates. Following Schwab et al. (2020). we sample 5000 samples with 2870 covariates from News. The semi-symhetic experiment IHDP is named as IHDP Dr. The other 3 semi-synthetic experiments on News are named as News Dr. For detailed descriptions of datasets, models, and hyperparameters, Appendix.
Metrics For all experiments, we perform 30 replications (E = 30) to report the average mean squared error (MSE) and the standard deviations (SD) of the average dose-response function esti- mation. For correlation measurement, we adopt distance correlation (dCor) to evaluate the quality of dCov(X.Y)
the proxy representation EX dCoo((X,Y) Va Varr(X)) with Var(X) dCoo (X,,
dCov X, T) = = E Ej= Bis, where A,,j = a,j a, - -ä a,j = |X Xyl1 a, プド ai. J.j 2ド=1 Bij a.
/ Eij=1 aij And the form of Big is similar.
Experimental results on simulation datasets To assess the performance of CRNet, conduct simulation experiments increasing the dimensions of treatments and proxies. As Table T shown, all methods except GPS perform well in the low-dimensional Datay_ 115. All baselines fails when treatments are multiple in Datay: 22200 and Datar 5.200. Increasing the dimension of treatments Dalay 2__200 and Dalay 5 5_200, we found that CEVAE, which performs well in low dimensions, fails in convergence, which has also been demonstrated in Rissanen & Marttinen (2021). The per- formance of CRNet outperforms others across different settings. We further verify the effectiveness of Cr module in CRNet below.
Table 1: Results (MSE+SD) on simulation Datar -D
E=30
Datay_1.5
Datay_1.50
Datar_1.200
Datar .2.200
Datap_5.200
Causal Forest
3.962 + 0.9519
12.22 + 15.530
76.56 + 79.527
82.30 + 82.395
96.89 + 89.262
Bart
0.90022
9.415 11.109
129.9 83.055
127.8 54.624
154.3 :: 66.021
GPS
42.25 3.0993
65.67 53.597
1030. + 119.94
1110. 140.00
1112. + 140.51
NN
1.914 0.5765
2.542 2,8093
22.78 11.503
26.71 9.8489
22.87 8.8573
DRNet
1.982 0.6883
3.422 + 1.6802
17.24 9.7990
21.77
23.95 9.2497
VCNet
1.457 + 1.4142
3.502 +.502
12.95 17.753
27.22 17.365
25.28 + 15.617
CEVAE
0.944 + 1318
3.308 + 0.2950
7.309 + 0.6142
21.44 1.2480
25.04 + 1.9198
CRNet
0.865 = 0.2212
1.360 :: 1.5138
6.651 = 9.4864
11.79 = 10.047
14.79 = 15.678
Experimental results on the effectiveness of Cr block In the setting with multiple continuous treatments and proxies, we propose the Cr to model E(X) to hold onto the information from unob- served confounders. Practitioners
representation-basee approaches to map proxies into low- dimension representation space which will lose information predictive of the predicted treatment variable. As shown in Fig. 5 the correlation between E(X) and T from conventional methods still weak. To retain the information predictive, Cr regularize the proxy representation E(X) by contrastive learning, the correlation from CRNet and CRNet(fft [ is strong. In the experiment of Fig. (b), we first train a U-toTT prediction network to obtain the representation T(U) to represent the relationship between T and U. X X_T(U) denotes the dCor of X and T(U). DRNet refers the dCor between T(U) and representation f(X). Others are operated similarly. It demonstrates that Cr successfully regularized the representation E(X) between X and T and other methods not.
Experimental results on the effectiveness of Rw block Our downstream block for estimation is Rw. It reliable iff the proxy representation is accurately measured. To evaluate the performance of Rw block, we conduct 5 simulation experiments with no unobserved confounders. As shown in Table all experiments use the same backbone NN. In all experiments, GPS and MDN which direct model the density of IT( induce excessive bias in ADRF estimation. The direct rank weight without Cr performs well in all experiments. And with Cr, our rank weighting method outperforms other weighting methods. It demonstrates that our CRNet is state-of-the-art even with no unobserved confounders.
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" T refers the dCor hetween the proxies X and treatments T.
(CRNe((t) means the correlation hetween fXX and gT)).
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0.49
0.44
X
X T(T)
DRNet
DRNet
0.39
VCNet
0.34
VCNet
CEVAB
d
CEVAB
0.29
CRNet
CRNet
CRNem())
0.24
0.19
0.14
1000
2000
3000
5000
1000
2000
4000
5000
training, samples
training samples
(a) Correlation between X and T
(b) Correlation between and T(U)
Figure 5: Correlation of treatments, unobserved confounders, and covariates. both figures, the abscissa represents the sample size, and the ordinate represents the value of dCor.
Table 2: Results SM) on simulation Datax Dr
E=30
Datax 1.5
Datex 1 50
Datax 1 1,200
Datax 2,200
Dotax 5. 6,200
NN
0,317 +: 0.2181
2.012 + 2.3314
5.440 + 5.6123
5.606 + 7.3323
12.75 + 20.344
+GPS
1.055 + 1.2795
13.31 + 25.384
36.58 = 62. 489
28.20 + 33.284
47.64 + 70.285
+CBGPS
0.323 1. 0.2011
1.353 1 1.8173
6.196 1 6.9755
5.472 1 6.4689
8.019 1: 5.2724
+EB
0.552 +: 0.9203
2,933 1: 4.0231
6.398 1 5,6084
4.217 1 5,9709
8,210 1: 3.0670
+CCOWS
0.308 = 0.2841
1.680 = 1.6652
6.048 : 5.3401
4.727 + 6.2233
9.490 = 7.9573
+MDN
0.589 1: 1.6048
13.11 1: 45.473
99.26 1: 246.86
12.03 1: 18.670
18.44 1: 19.405
+Rw
0.268 1 0.2762
1.171 1 0.8079
3.436 1 4,0289
3.935 1 4.1624
7,500 1. 3.9411
+Cr+Rw
0.105 1 O. 0.0308
1.170 = 1.0302
3.043 1 4.9246
1.915 + 5.1747
5.654 + 2.8299
Experimental results on real-world datasets We further verify the performance of CRNet in real-world datasets IHDP & News. As shown in Table the traditional methods Bart and Causal Forest cannot estimate the treatment effect accurately and suffer from the high-dimensional proxy imbalanced between different treatments. CRNet obtain high-quality representation F(X) and retain predictive information of the predicted treatments in representation using contrastive regular- izer, but other deep-based methods fails to capture the rich information between high-dimensional covariates and treatments. Therefore, CRNet shows robust performance and achieves the state-of- the-art in real-world experiments.
Table 3: Results (MSE+SD) on semi-simulation Real-Date Saal-Dat__Drr
E=30
IHDP_1
News_1
News_2
News_5
Causal Forest
0.576 + 0.5651
24.60 4.7884
20.08 1: 4.0395
21.23 1:: 4.7864
Bart
0.514 + 0.3728
6.200 : 12.018
11.83 + 3.2380
22.25 + 17.902
GPS
3.198 + 15.148
1.603 + 0.2932
2.293 + 0.7984
6.085 = 1.1495
NN
0.774 +: 1.5413
0.958 + 0.1148
2,435 1 0.2597
7.435 + 1.0865
DRNel
1,485 +: 1.4620
3.412 1.3222
6.486 +: 4.2931
10.18 + 2.8848
VCNet
0.626 + 0.7126
8,563 9.4974
7.578 + 9.2410
10.56 1 6.4256
CEVAE
1.935 + 0.7765
1.142 0.0762
3.289 + 0.1860
9.660 + 0.4348
CRNet
0.351 + 0.2122
0.867 0.2193
2.115 + 0.6143
5.081 + 0.7098
CONCLUSION
For CTP problem, we formulate the estimation error into three terms from recovery of unobserved confounder, treatment assignment and approximation of outcome. We propose the contrastive reg- ularizer to constrain the proxy representation in representation space for the bias from recovery of unobserved confounder. Based Cr, we propose rank weighting method to eliminate the extreme values problem and alleviate the sensitivity problem to model misspecified in treatment assignment model. Combining Cr and Rw, we elaborate CRNet adapted CTP problem reduce the out- come approximation bias. CRNet achieves the state-of-the-art performance in estimating ADRF of both synthetic and syntheic data.

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Under review as a conference paper at ICLR 2023
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A
APPENDIX
A.1
IDENTIFICATION
Given latent unconfoundedness and proxy assumption, we know that the causal effect is not iden- tified conditioning on X. Because 61 Y UX then (t) y T.X. We also show this problem in ADRF adjustment formula. The true ADRF is identified as
ADRF" = (t)l = Eu (t)(t)
Ull Eu E (t) = Uj = Eu Eu E
T
When proxy exists, ADRF is identified as
ADRF= = [() Ex (t) X|]] = Ex(B(((() |T T
x
Ex B TT x,]
* EU|EYY = Uj] = ADRE..
(12)
clear that the adjustment formula of ADRF from proxy is different from that of true ADRF because Yt) / T,X, it will induce the recovery error A., in the ADRF estimation phase if the unmeasured confounder U is not correct specified. The performance of using proxy to estimate ADRF depends the degree of recovery of U.
A.2 PROOF OF EQUATION (4)
We set ADRF as baseline and define the estimation error as
A ADRF" ADRF (T, t)] E(o(X,tt)
El do(t)) E[g(U,[t) + h(er/tt)]
(U,, - [[[((,,(t) B()(
(13)
(U, do(())] E[(*((Utt) + -U -Bh(ctt)
EU, (U,)o(t)) 6" (U])t + - E/h(er,tt)
-
A.3 ANALYSIS FOR RANK WEIGHIT
Based on the definition rank, we define the corresponding index function of rank I(Ra) = And record the stride
ÖR, =
W, WI(R.11
0 <<D R < n R 0
(14)
W
as the difference between two adjacent weights of the sorted data. Then can build sequence model of W as:
W = WI(n.-1-
8R.
< R <
(15)
8n.
R
12

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Under review as a conference paper at ICLR 2023
Supposing extreme value example that the maximum weight is much larger than others, this because there are many unmeasured weights between the largest weight and the second largest weight we obtained. Combining Eq.(15) and the continuity of the probability density, it is clear that the excessive stride causes the extreme value of the weights. Moreover, the sample weights with large stride will also cause the sensitivity misspecified because slight misspecifed of the large stride induce significant bias, Then is just using the rank information enough to make a covariate balance? To answer this question, we formulate the model in Eq.415) as:
W (R, (R)) E(R,6):
(16)
where W is the sample weight, the rank, is the indez function of and ó is stride. We reduce the second line to third line because I is deterministic function of R.
Based on Eq. 14 we notice that in {Wik=1 E [0, 1], if n + co, then stride - 0 because WR, WR.-1 0. It similar to the unnormalized density. Therefore, we formulate IPW via Gibbs
(( where Z fe ((n,)))) We set + 0o, then EURD
sampling: P[TUU z
U() So we propose that
2
e Proposition There exists some rank weight É(R) that when n + 0o, 2 7 2(JUy) where
2 = z=fe ((()) is the normalizing constant of E(R).
The proposition and Eq. show that when co, using rank weight {(R) is enough to make a covariate balance because the weights Rw approximate to the IPW of density F(T]U). It means when oc, the causal effect estimation with Rw approximates the unbiased estimation of causal effect (for unbiased estimation with IPW of P), see Imbens (2000). And when the data size is limited, rank function É(R) can effectively avoid the extreme value problem and alleviate the sensitivity to the model misspecified. So we direct eliminate della by setting å = _ to obtain É(R) Rs R4. The operation can be considered as enforce the distribution of delta is Uniform, which biased. But as n + oc, the obtained É(Rt) z R& is approximating to the true IPW.
A.4 EXPERIMENTAL DETAILS
The rules for defining symbols in this section are the same in the main body. Please note that when the superscript is specified as AP=2, means the dimension of A is 2, and when it is not specified (e.g., A2), it means the power of A is 2.
A.4.1 DETAILS ON DATASETS
The dataset split and dimension information corresponding to the data name are expressed in Table
Table 4: Dataset description
Nwvinn
Dr
Dx
Du
Dr
Data: 1.5
1800/300
5
Datar 1.3
1800/300
50
50
Dotay 200
1800/300
200
Dotay 200
1800/300
200
200
Datay_1_20
1800/300
5
200
200
Datax 1
1800/300
0
Datax 1 .55
1800/300
50
0
Datax -1 11.00 200
1800/300
200
0
Datax 200
1800/300
Datax 200
1800/300
200
IMDP_1
672/75
25
25
News. 1
3150/1 350
2870
20
News 2
3150/1 350
2
2870
20
News_5
3150/1350
5
2870
20
13

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Under review as a conference paper at ICLR 2023
Synthetic datasets construct synthetic datasets following EB \Vegetbblee et al. 2021). For all (0,1).
simulation datasets, the true covariates UP 1.-200 are constructed as: 1 200
For Datax -15, Datax Datax -1 datasets, T is constructed as:
pm3
T= = 0.5W 0... 1.5 *
XP
pml
And Y is constructed as:
yp=1
+ e pm1
+ X==2 + XP=3
Dp=i
2000
p=5
2.3 * XP=
X 5
4.0 *
Σ X + T( T=== (3 cos
Σ T
p= 151
p=3
For Datay-1.5 Datay_1113 _1500, Datay _atoy_11200 _12200 datasets, T is constructed as:
p=3
p=Du
T==1 = 035W 1) 0.5 0.5) + 1.5 *
UP + 0.5 * Σ U
p=1
1-1
And y==1 is constructed as:
yp=1_
Up1i
+.1 * U + 2.2 UP=3 + 2.3 UP=
e Epsl
DU
U I(Du 5)
I(Du == 200) x
50
p=151
p=200 200
100
Du + * Σ U + 4.0 *
Σ U p-151
p=6
where Du is the dimension of U and is the indicator function.
For 2.200 and Datay 55.000 datasets, 1==1 is constructed as:
p=1
D=Du
T==1 = 03, 1) + 065 0.5) + 1.5 *
UP + 0.4
Σ UP
p= 151
T 1), 1.5 5
100+p
T==3..5 = O.5. + Σ U
10100
And y is constructed as:
y==1_
1
Up=1
2.1 UP=2 +22 U + 2.3 U
p- TF
Du
+ UP=G
+ +l(Du > 5)
ll = 200) x2
50
+ 100
p=151
p= 200
* Σ UP +4.0 * Σ U
DU +
p=6
p=151
p=5
+ I(DT == 5) (0.1
T +
The observed covariates x==1--200 of Datay_155. Datay Datag Datay .2.200 and Datar 5.200 are formulated
x==1--5 U linespace (0, P
Nx)
10
X P00 UP=5--200
where linespace(O, fo Nx) means samples Nx data from [0, l.
14

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Under review as a conference paper at ICLR 2023
IHDP
The generation process of T and Y are formulated as:
T 21-=1
5Um=1 + 0.1max ( 1)
+ +.(00,0255,
2.1 min
minnUn Up=3.5.6,1 1)
Y (3T
p=15
Σ U2 +0(00,).5)
1.2 T
U are standardized to (0, 1) and T are normalized to [0, 1].
The observed covariates X 25.2 are formulated
X p-1 = 0.2 (U==
U==2 X== = sin (
X cos (UP X 0.5erp( U
X5 U== x5=5 U
News
The generation process of T and Y are formulated as:
TP=...6 sin sin(UU))e))a) (T"")UutnnU"")) + 1): yF=1 (U" + + M(U0=3LL-1000 (0.1
For News c 2, for News_5 4. are standardized N 1) and T are normalized to [0, 1].
0.2(UP)
I[moo[(p 1),5) 0}
sin (U))
I[ [mmo((pp 1).5) = 1}
fx (UP)= = (os( +
I(mod((p 1).5) = 2}
0.1 + abb("")) (bs(
I(mod((p - 1). 3)
.Imoo((p 1),5) =4)
Given /x, the observed covariates X===1..2570 are formulated as
x -=I1-20 = fx (U)),
TP=21..2870 U
A.4.2 DETAILS ON MODELS
We construct CRNet with depth (256, 128, 128, 128, 128) hidden units. consists of 5 FCs with {32, 64, 64, 32, 32} hidden units. The MDN module consists of 3 FCs with (20, 20, 20}.
As Fig.
shown, consists of 5 FCs with
NN consists of FCs with (32, 32, 32, 1} hidden units. We implement GPS, Bart, CF and DRNet following Schwab et al. (2020). We improve on DRNet and implement VCNet following Nie et al.) (202I). We implement CEVAE following Louizos al. (2017). We normalize simulation data to [0,1] for the conditional density estimator in DRNet and VCNet.
A.4.3 DETAILS ON HYPERPARAMETERS
For GPS Bart and CF, we use the default hyperparameters as Schwab et al. (2020). For all representation--asee models, fixed the random seed and search for the best performance with SGD or Adam. We also adjust the learning rate with (0.1, 0.01, 0.001, 0.0001, 0.00001}. For DRNet and VCNet, [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.9] and with {100, 1,01, 0.01, 0.001 For VC- Tet+TR, we adjust the hyperparameters TR with {0.1. 0.01, 0.001}. Besides, for CRNet, we adjust hyperparameters a with (100, 10, 1, 0,1. 0.01, 0.001) and / 1 consistently.
adjust the hyperparameters knots with {00.33, 0.66), (0.2, 0.4, 0.6, 0.8),
with 1,013 and the learning rate of
15


Reviews:
[{"review_id": "eYonEYo9CL", "reviewer": "Reviewer_S7Ex", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes the use of contrastive learning to deconfound the observed proxy variable of the confounder and treatment for unbiased continuous treatment effect estimation. In addition, some weighting methods are incorporated for score balancing. The proposed method has been evaluated using synthetic and sem-synthetic datasets. It shows that the use of contrastive learning approach can achieve the treatment effect estimation without the need to model the confounders expicitly.", "summary_of_the_review": "The paper addressed an interesting and important problem. The perspective of using contrastive learning to solve the unbiased treatment effect estimation is an interesting one and carries some novelty. The performance comparison with more recent variants of CEVAE is not included. It is hard to evaluate the significance of this work.", "strength_and_weaknesses": "Strengths:\n+ Unbiased continuous treatment effect estimation is an interesting and important problem.\n+ The idea of using contrastive learning for learning the representation of observed proxy variables and the treatments provides an interesting perspective to address the problem as compared to the existing VAE and GAN approaches.\n\nWeaknesses:\n- The key contribution is the use of the Cr regularizer. The novelty related to the weighting for the score balancing is considered limited.\n- Ablation study is missing. So, it is not clear about the relative importance of Cr and Rw.\n- Only CEVAE was compared in the experiment while there are other more recent VAE models proposed for the same application (e.g., intact-VAE). Whether the proposed CRNet can perform the more recent extensions of CEVAE is unclear.\n- The proposed approach may not be useful for applications where the cofounders are intended to be modeled explicitly.\n- The evaluation is only carried out using synthetic and semi-synthetic data, and no real data is used.\n- There are quite many typos and presentation issues in the paper as listed in the section of the review.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "bUmdLy9HKn", "reviewer": "Reviewer_kSiR", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposed a new model for causal proxy learning for average dose-response function estimation of multidimensional treatments.  This is done by based on combining contrastive regularizer to learn the proxy representation, and ranked weighting method to de-bias the treatment assignment mechanism estimation. The authors claims that the proposed method can avoid the deficiencies of existing methods that rely on correctly specified treatment assignment models and/or require strong prior of the unobserved confounder distribution. ", "summary_of_the_review": "Causal proxy learning is generally a very interesting topic, and this paper proposes a new solution to it. However, the motivation for the design choices of the paper is quite unclear; and the theoretical justification is not very convincing.  ", "strength_and_weaknesses": "**Strength**\n\n- The proposed method is new to the best of my knowledge. Under the proxy assumption, the paper proposes to combine contrastive learning of the proxies and the rank weight estimations of the propensity scores to estimate treatment effects.\n- The paper has relatively extensive set of baselines and plenty of tasks in the experimental analysis part. I also appreciate that the authors have detailed all synthetic functions in the appendix for reproducibility. \n\n**Weakness**\n- Missing important references and lacks clear motivation.  The problem setting of this paper is very closely related to (Yixin, 2021), which should definitely be mentioned and compared. I found it hard to be convinced by the author's argument that \"existing methods rely on correctly specified treatment assignment models and/or require strong prior of the unobserved confounder distribution\", since the identification results in (Wang, 2021) does not rely on any of these. Given those, the paper lacks key motivations for the Cr + Rw approach.\n\n- The claim that \"our method need no data distribution prior and outperforms others in performance\" is not very convincing. To some degree, the choice of similarity functions in the contrastive learning part already implicitly assumes certain prior preferences. Even assuming it's true that this method requires absolutely no prior, this comes at a cost of: 1), one needs to specify the similarity function as well; 2), the identifiability of both treatment effects and latent confounders become unclear. \n\n- Although the proposed method is new, I don't immediately see the key novelty of the paper. The contrastive learning is not a new method and has been used for learning latent representations for a long time; the rank weighting might have certain degree of novelty, however I don't see how this relates to the proxy causal learning setting. It looks more like a standalone estimator on this own and needs more theoretical analysis and experiment evaluation for justification.  \n- Lacks rigorous theoretical analysis. For example, the authors claim that \" learned representation E(X) refuse the information of $\\epsilon_1$ and maintain the information of $U$, which means we eliminate the error item...\", which is too heuristic and needs to be formally derived. Also, the paper lacks identifiability analysis for $U$ (or treatment effects). Without those analysis, Eq(11) that the authors proposes hardly qualifies as an \"identification\"; and simply dodging this issue by hand-waving \" The performance of using proxy to estimate ADRF depends on the degree of recovery of $U$\" is clearly unconvincing. \n\n**Minor issues**\n\n- In related works, the paper cited a bunch of methodologies and claims that the proposed method \"outperforms others in performance\". Unfortunately, many of those mentioned in this part are not used in experiments at all. The authors need to be careful on what their claims and avoid over-selling.\n\n- The writing of the paper is not very consistent. For some parts it is very clear and convincing, but for some other parts it is very confusing and I found it hard to understand what the authors are trying to do. I finally managed to understand most of it, but at the cost of reading it multiple times and making my own guess on a number of undefined terms and notations. \n\n\n**Reference**\n\nWang, Yixin, and David Blei. \"A proxy variable view of shared confounding.\" International Conference on Machine Learning. PMLR, 2021.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "HGMUyDA2UH", "reviewer": "Reviewer_X9JX", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper tackles the problem of estimating the effect of continuous and multi-dimensional treatments in the setting with unobserved confounders. The paper assumes access to high dimensional proxy variables for the unobserved confounders and proposes a Contrastive regulariser which learns a proxy representation for the unobserved confounders. The proposed method, CRNet, uses the Contrastive regulariser, together with a Rank weighting methods which de-biases the treatment assignment, to effectively estimate the effects of multiple continuous treatments. ", "summary_of_the_review": "Overall, I believe that the method developed in this paper, CRNet aims to address an important and understudied problem in the causal inference literature and achieves good performance in practice. However, I believe the authors should include more details regarding how the learnt representations recover the hidden confounders, as this is a crucial point for the paper. Moreover, the paper makes many explicit and implicit assumptions, whose implications are not discussed. ", "strength_and_weaknesses": "Strengths:\n-  The paper addresses an important and understudied problem in the causal inference literature, which is the estimation of treatment effects for continuous and multi-dimensional treatments in the challenging setting of unobserved confounders\n- The use of contrastive learning to learn the proxy regularisation + the rank weighting to address the selection bias is interesting and aims to overcome the known problems of balanced representations which may result in too strong regularisation and discard information about the confounders\n- CRNet is thoroughly evaluated on many datasets and demonstrates performance improvements over the benchmarks. \n\n\nWeaknesses:\n- The authors simplify the Average Dose-Response Function estimation problem through the additive regression model in equation (2). I think the paper would benefit from a more detailed discussion regarding how realistic this simplification is and in which cases it holds.\n- The paper makes many assumptions; while I believe that this is needed to be able to handle the setting of unobserved confounders, the papers should include a more detailed discussion about the implications of these assumptions and how realistic they are in practice. \n- It it unclear how learning the representation f(X) through contrastive learning helps with recovering the hidden confounders. In particular, why does strengthening the association between treatments T and covariates X helps to recover the unmeasured common cause U? I believe the authors should describe this in more detail, and if possible, develop theory to support this claim which I believe is crucial for the paper. \n- To approximate the treatment score $\\mathbb{P}(\\mathbf{T} \\mid \\mathbf{U})$ the paper assumes it follows a Gaussian mixture model and uses a mixture density network to learn it. I believe that this assumption about the distribution of the treatments given the unobserved confounders should be more explicitly acknowledged and again discussed in which setting it is valid. \n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "A537dyC7j5Y", "reviewer": "Reviewer_2J3F", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Estimating treatment effect in the presence of unobserved confounders is a challenging but omnipresent problem. The authors assume availability of a proxy for unobserved confounders and attempt to estimate it to then infer weights that can be used for unbiased treatment effect estimation. The main contribution of the paper is the proposal of \"rank weights\" for this purpose.", "summary_of_the_review": "My recommendation is to reject. I do believe that there are promising ideas and I am open to revise my score but at this moment there are many imprecisions that have to be corrected. ", "strength_and_weaknesses": "**Strengths**\n- The paper proposes new ideas to leverage the proxy model and show that the proposed approach outperforms in several example.\n- The experiments consider many different baselines.\n\n**Weaknesses**\n- The paper has serious flaws in presentation that hinder the comprehension of the method, assumptions, and novelty. For instance, in the preliminaries section, a potential outcome is never defined, $\\phi$ is not defined, the notation $\\phi(\\cdot | t)$ is not defined, Def. 2 and the notation do(t) independent of U are highly non standard and should be explained. Given this I don't understand the motivation section of the paper.\n- Assumptions 5 and 6 should be discussed in more detail. It is not clear how plausible they are. Especially, assumption 6 seems to directly contradict the causal graph in Fig. 1.\n- To my knowledge none of the semi-synthetic datasets incorporate a proxy structure in their data generating mechanisms. Are these changed to fit the setting in this paper. Do you loose performance when the correct thing to do is simply condition on observed covariates, i.e. if there are no unobserved confounders?\n\n**Suggestions** \n- One reference on ITE with continuous treatment could be added: Bellot, Alexis, Anish Dhir, and Giulia Prando. \"Generalization bounds and algorithms for estimating conditional average treatment effect of dosage.\" arXiv preprint arXiv:2205.14692 (2022).", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject