Leveraging a Simulator for Learning Causal Representations from Post-Treatment Covariates for CATE
Authors: Lokesh Nagalapatti
Pranava Singhal
Avishek Ghosh
Sunita Sarawagi
Venue: Accepted by TMLR ()

Abstract: Treatment effect estimation involves assessing the impact of different treatments on individual outcomes. Current methods estimate Conditional Average Treatment Effect (CATE) using observational datasets where covariates are collected before treatment assignment and outcomes are observed afterward, under assumptions like positivity and unconfoundedness. In this paper, we address a scenario where both covariates and outcomes are gathered after treatment. We show that post-treatment covariates render CATE unidentifiable, and recovering CATE requires learning treatment-independent causal representations. Prior work shows that such representations can be learned through contrastive learning if counterfactual supervision is available in observational data. However, since counterfactuals are rare, other works have explored using simulators that offer synthetic counterfactual supervision. Our goal in this paper is to systematically analyze the role of simulators in estimating CATE. We analyze the CATE error of several baselines and highlight their limitations. We then establish a generalization bound that characterizes the CATE error from jointly training on real and simulated distributions, as a function of the real-simulator mismatch. Finally, we introduce SimPONet, a novel method whose loss function is inspired from our generalization bound. We further show how SimPONet adjusts the simulator’s influence on the learning objective based on the simulator’s relevance to the CATE task. We experiment with various DGPs, by systematically varying the real-simulator distribution gap to evaluate SimPONet’s efficacy against state-of-the-art CATE baselines.

Published in Transactions on Machine Learning Research (01/2025)
Leveraging Simulator for Learning Causal Representations from Post- Treatmrettn Covariates for CATE
Lokesh Nagalapatti
nlokeshiisc@gmaal.oom com
Deportment of Computer Science and Engineering
IIT Bombay
Pranava Singhal"
pranaou.piinghhll@mmai.ccm
Department of Computer Science
Stanford University
Avishek Ghosh
avishek Siashekkgghohh@iilbe
Department of Computer Science and Engineering IIT Bombay
Sunita Sarawagi
sunita @iitb.oc oc.
Deportment of Computer Science and Engineering
11T Bombay
Reviewed on OpenReview: https: // openreveew....... net/ net/forum? id= FM3tgW33tz
Abstract
Treatment effect estimaatio involves assessing the impact of different treatments on individual outcomes. Current methods estimate Conditional Average Treatment Effect (CATE) using observational datasets where covariates are collected before treatment. assignment and out- comes are observed afterward, under assumptions like positivity and unconfoundedness. In this paper, we address a scenario where both covariates and outcomes are gathered after treatment. We show that post-treatenn treatment covariates render CATE umidentifiable, and recovering CATE requires learning treatment-indepennde representations can be learned through contrastive learning if counterfactual supervision is avail- able in observational data. However, since counterfactuals are rare, other works have explored using simulators that offer synthetic counterfactual supervision. Our goal in this paper to systematically analyze the role of simulators in estimating CATE. We analyze the CATE error of several baselines and highlight their limitations. We then establish generalization bound that characterizes the CATE error from jointly training on real and simulated distributions, as function of the real-simulator mismatch. Finally, introduce SimPONet, novel method whose loss function is inspired from our ceneralization bound. We further show how SimPONet adjusts the simulator's 's influence on the learning objective based on the simulator's relevance to the CATE task. We experiment with various DGPs, by systematically varying the real-sinulator distribution gap to evaluate SimPONet's efficacy against satte-of-theatt CATE baselines.
causal representations. Prior work shows that such
1 Introduction
In Conditional Average Treatment Effect (CATE) task, the goal is to estimate the difference in outcome Y, individual Z is subject different treatments T. The gold standard to estimating such effects is Randomized Control Trials, which are often expensive, and with the easy availability of observational data, there is extensive interest in harnessing them for deriving these estimates. The first step in estimating treatment effects from obser- vational datascts is to determine the set covariates that, when conditioned upon, make the treatment effects
Lokesh and Pranava contributed equally. This work was conducted by Pranava during their at IIT Bombay

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identifiable. Prior works (Knnzel et al. 2019 Nie & Wager 2021
Curth & van der Schaar 2023 Shalit et al.
2017: Nie et al. 2021 Shalit et al. 2016 Chauhan et al. 2023 Curth & van der Schaar 2021: Zhang et al. 2020;
Shi et al. 2019 Stuart 2010). assumne that such covariates are observed, and gathered prior to treatment, with outcomes Y observed after the treatment is given. However, collecting such datasets is challenging as it requires tracking the same individuals over two distinet time points. As a result, readily available observational datasets can sometimes contain both covariates X and the outcomes Y recorded together post-treatment. X, Y are observed for individuals characterized by latent representation Z. For example, in economics, policymakers im- plement different taxation policies T aimed at improving an outcomes Y like Gross Domestic Expenditure of the individuals Z. To identify effective policies, they may rely on domain expertise or conduct small-scale RCTs where collecting pre-treatmest covariates is feasible. However, the true effectiveness of a policy is only revealed through large-scale testing on the population after its implementation. Collecting such datasets lypically involves obtain- ing post-treatment covariates X and their associated outcomes Y together (Ashenfelter 1978
Angrist 1995
Even in other domains such as voluntary hcalthcare surveys, only post-treatment data about patients might be accessible. In medical imaging, an image laken under a specific inst.rument setting (Irealment) may be evaluated to determine whether switching to a different setting would improve subsequent diagnosis (outcome).
We present our setup in the top panel of Fig. marked RealDGP (Data Generating Process for the real distribution), where the latent variables Z causally produce the observed treatment T, outcome Y, and covariates X. We begin this paper by presenting an impossibility result in this context.
Real DGP
Py (x
N(ur()),
6(X - (Z)
T
Lemma 1. The Conditional Averuge Treatment Effect (CATE) T on Y, given X, is not identifiable using i. samples of the observed variables from the true data- eatt-generaatig process depicted in the top panel of Fig. 7
Y(T)
X(T)
9
7 Pz
Z
f= 9t
of
proof. Since X is a collider, conditioning on it opens the backdoor path T X + Z - Y. Furthermore, as Z is latent, this backdoor path remains open, making CATE unidentifiable from X, T, and Y alone (Pearl 2015).
96
X0
(1)
Y0
Y$(1)
8(X
(2))
(2
The main takeaway from the lemma is that certain additional assump- tions are unavoidable for achieving identifiability. The lemma further emphasizes that the key to identifiability lies in extracting treatment- independent causal representations from the post-treatment X that affect Y. One such assumption that allows for the recovery of causal representations is counterfactual supervision in real data. Prior work (Von Kügelgen et al. 2021 demonstrates
Simulator DGP
Figure 1: The Data Generating process for Real and Simulator.
that, under such an assumption, contrastive losses can be applied to pairs of covariates that diller by treatment. to extract Z from X and T. While some works (Nagalapatti et al., 2022 Bachman et al.) 2019) assume direct access to such counterfactual supervision in real data, others rcly simulators that generate high-quality synthetic counterfactuals (Von Kügelgen et al. 2021 Zimmermann et al. 2021). However, these are strong assumptions since counterfactuale are rarely available in real-world scenarios, and while simulators are more feasible, assessing their quality or relevance to the downstream task during training is challenging. Therefore, our goal in this paper is to leverage simulators only to the extent they remain relevant to the CATE task. We conduct a theoretical analysis to derive generalization bounds that show how CATE error worsens as the mismatch between real and simulated distributions incrcases. These insights motivate our proposed algorithm, SimPONet, which uses simulated data to apply regularizers inspired by our generalization bound. SimPONel's aim is to enhance CATE estimates beyond what is achievable with observational data alone. Through experiments, we systematically vary the distributional gap between real and synthetic data across various DGPs, demonstrating that SimPONet consistently outperforms multiple baselines in estimating CATE.
We address Treatment Effect Estimation with tretttreatment covariates a non- identifiable challenge- - by leveraging simulator that offers synthetic counterfactual supervision. 2) We assess the CATE errors for three basclines that can be trained on real /smmmlatt data, and highlight their limitations. 3) Next, we consider a joint training framework, and derive a generalization bound that characterizes the
Contributions: 1)
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CATE error as function of real-simulator distributional mismatch. 4) This analysis motivates method, SimPONel, novel algorithm that uses simulated samples to improve CATE estimates beyond what can be achieved from observational data alone. 5) To our knowledge, this is the first systematic study on the role of simulators in CATE estimation. 6) Experiments across various DGPs confirm SimPONet's effectiveness.
2
Related Work
2.1
CATE with Tre-mmeatent Covariates
The primary challenge addressed here is handling confounding that arises out of biased treatment assignment in observational datasets. The main ideas explored include: estimating pseudo-outoomes for missing treatments in the training dataset and then using these to train effect predictors (Gao & Han
2020 Curth & van der Schaar
2021
Nie & Wager 2021: Kennedy 2020 Yoon et al. 2018 Zhang et al.
2020;
Nagalapatti et al. 2024a) )
adding targeted regularizers to ensure consistent ITE estimates (Shi et al.
2019 Nie al.) 2021| Zhang et al.
2022;;
learning balanced representation of covariates across treatment groups (Shalit et al. 2016 2017 Yao
ct al.
2018 Chauhan et al.
2018hChauhan
2023
Wang ct al.
2024 Wu ct al.)
2023
matching to ncar-by covariates (tturrt
2010
Rosenbaum & Rubin
1983
lacus el al. 201 Schwab el al. 2018
Kallus
2020
Nagalapalli el al.
2024b)
and weighing losses to mitigate confounding Hassanpour & Greiner
2019a b
Jung et al. 2020
Ozery- Flato
ctt al.
2018).
2.2
CATE with Post- Treatment Covariates
This is our setting, and is more challenging because falls into the third rung (counterfactual) of Pearl's causal ladder (Pearl & Press 2000). Please refer (Pearl 2015 to for formal proof. In cconomics, post-treatment
variables in trials are known to exacerbale esaled causal effects (Coppock 2019 HOMOLA et al. 2024
King 2010). Post-treatment variables have been used to estimate selection bias Z(T in observational
data Barcinboim & Pearl 2012 Bareinboim & Tian 2015 Correa et al. 2018). A closcly related work is (Huang
et al. 2023) that leverages post-treatment variables for estimating treatment effects but differs from us since they assume: (1) covariates X causally affect Y, and (2) an entangled version of X,Z is observed; they simply focus on disentangling Z through representation learning.
2.3 Rerl- Applications of using Simulators for Estimating CATE
We provide two examples from medicine and electrochemistry to show how simulators aid CATE estimation in practice:
Medicine, Simulators play a crucial role in pharmacology, particularly for assessing drug efficacies. For instance, the SimBiology toolbox in MATLAB is commonly used to predict the effects of SGLT2 inhibitors (T) on type-2 diabetes (Y) while considering post-treatment covariates (X) such as plasma glucose levels, gut glucose levels, urinary glucose excretion, and liver insulin levels. SimBiology enables modeling these effects using differential and algebraic equations that are often calibrated on target populations to minimize the real-simulator mismatch. Despite not perfectly replicating reality, such simulators are invaluable for carly-stage clinical trial decisions and have demonstrated utility in modeling short-term treatment effects (Dalla Man el al. 2007
Electrochemistry. Another application involves recommending optimal electrode materials to maximize battery capacity (Y). By observing Y under various electrode materials (T) and post-treatment variables like charge/ charge/discharge /discharge rate, internal resistance, and temperature distribution (X), the Ansys Battery Cell and Electrode Simulator provides clectrochemical simulations. This tool has been used by Volkswagen Motorsport for comprehensive multiphysics simulations to design and validate battery models. Such simulators are highly relevant for practical prriisonnmmkingg in industries.
These examples illustrate the practical relevance of simulators across different fields. While simulators cannot fully replace real data randomized controlled trials (RCTs), they offer valuable insights that can reduce the number of RCTs needed for optimal treatment identification. Our paper aims to characterize the CATE
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error when using imperfect simulators in conjunction with real observational data. Additionally, SimPONet maximizes the utility of simulators by leveraging the highly correlated simulator's 's treatment effects with real-world effects, without relying on the exact correlation of individual potential outcomes.
3 Problem Formulation
We use random variables X.T.Y to denote post-treatment covariates, binary treatments, and outcomes respectively. The observational dataset has n samples: Dur = ((),, where t, E TT= (0.1} denotes trealment, 2; E C R denotes covariales observed after 4 is applied, and yi the resulting outcome. We use the Neyman- Rubin potential outcomes framework to denote (t) as the potential outcome and covariate for unit under treatment t. The main challenge is the absence of counterfactuals in Dura: i.c., for cach unit we observe covariates and outcomes under only one treatment ty-
We use the random variable Z E Z C F= to denote the causal representations of covariates X. Z generates X via trppeiii covariate generating functions gr 2 + X' for t E [0,1]. We assume that 9t is diffeomorphic (Locatello et al. 2019a by Von Küggeggen et al. 2021 ) i.e., it is smooth, invertible, and has a smooth inverse. The assumption that Sr is diffeomorphic is required to establish the theoretical identifiability of r. A non-invertible transformation risks losing information when mapping Z,T to to X. If lost information includes features necessary for identifiability, then 7 becomes unidentifiable from the observed X,T,Y. Diffeomorphism ensures that all factors involved in generating X are preserved within X so that there exists inverse functions ft f + 2, Vt E {0,1} that could recover the causal representations Z back. In Sec. 5.6 we experiment with non-invertible transformat tionsss and find that both the baselines and SimPONel maintain robust, performance, even when these assumptions are violated in practice.
A sample obtained from the real DGP as follows: (1) 24 ~ Pz, (2) t. ~ P(T (3),, x, (2, = 6(x 9t, (z:)), where 5 denotes the dirac-delta distribution, (4) yi P(Y (2)) is sampled from Gaussian with mean M4 (2:) and constant variance of. Here, M4 2 + Vt generates responses for individuals with latent representations under treatment t. We express the factual observed outcome for i as x(t4))=+, (fe. (z,)), and the missing counterfactual (CF) outcome under 1-t, as (1 = -1-1, (z:)).
(1-t,)
Our Goal is Conditional Average Treatment Effect (CATE) estimation which quantifies the difference in outcomes due to a change in treatment. Given a test unit (t)) ; its CATE is given by T, (T))) 0) As argued earlier, estimating T from post-treatment data involves the sub goal of learning causal representations of observed covariates X using a function fe=X + Z. We use T: Z- to express the treatment. effect using the latent 2j as
Since fo inverts X to give Z, the same effect can also be expressed for (1))4z) using tx as Tx (1),4)) /n Mn (ft, (o[ (zr;)) where tx X' X XT++ V. Notice that Tx (0.t) (+.))=rof(())) When estimating Tx, the factual outcome is easy, all we need to do is fit a regression model on the observation data. The main challenge lies in estimating the counterfactual outcome under treatment. 1-4..
Theorem 1 in (Locatello et al. 2019a) presents an impossibility result stating that ft which maps covariates X to their treatment-indeeendder causal ropresentations Z is not identifiable solely using Dim- The main hurdle is that multiple DGPs can yield the same marginal distribution P(X,T), making it impossible to isolate the true DGP. However, prior work has shown how to learn Se with counterfactuals, requiring that Dem includes both covariates (t+) and X;(1+t). Theorem 4.4 of (Von Kügelgen et al.) 2021) shows that such counterfactual supervision allows for recovery of Z up to a diffeomorphic transformation h using contrastive learning. Proposition 2 in (Zimmermann et al. 202T) further shows that h is, in particular, a rotation in an n, dimensional nort-normalized hypersphere. While some prior works assume direct access to counterfactual supervision in real data (Nagalapatti et al. 2022 Bachman et al., 2019), others rely on high-quality synthetic counterfactuals from simulators (Xie) 2018 Kaur et al.) 2021). In contrast, our approach seeks to leverage simulators only to the extent that they improve the downstream CATE task. We next formally define the simulator's s data generating process.
Simulator DGP The simulator generates paired instances giving rise to a counterfactual dataset Dayn = (0) (1) (0). (1)} generated using the DGP as shown in the lower panel in Figure The simulated instances are obtained as follows: (1) 24 Pz; i.e., Z is sampled from the same distribution as real, (2)
pest-treatment covariates az P(X5) 7 21. T = t) 6(X' (z:)) under both treatments t = (0.1).
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0 2 Yt are diffeomorphic functions, and (3) corresponding outcomes (0), yf (1) sampled from P(Y*| =1) (x))))))))) where : : Vt. Note that 2; remains hidden even in Dayn- We use "S" in the superscript to indicate a simulator component. Now we describe some metrics that assess the distance between real and simulator DGP.
Definition 1 (de)c((es{{() We assess the distance between the real and synthetic causal representation extractors fi and ff using the following expected distance: (f)e(fe.ff)) P(xj) [1l (2)
Definition 2 [ We assess the distance between the real and simulator CATE functions on the Pz distribution as: d (T,T ) 12 (T7(3) Under composition with a diffeomorphic function h, we write da ( =
Assumptions for Identifying CATE Tx- We summarise the assumptions that are needed on the real dataset Dum and simulated counterfactual dataset Dayn to identify the CATE function tx: (A1) Positivity: P(T = =|Z = 2) O, Vt , Z. (A2) Diffeomorphic Covariate Generation: Covariates in both real and synthetic distributions
obtained through diffeomorphic transformations of Z under any treatment. T. (A3) Identifiability Tf given Z: The causal factors Z that generate X form sufficient adjustment set, blocking backdoor paths between T and Y, thus making 7 identifiable from Z. Note that A2 and A3 logether ensure that X contains information about all the relevant latent factors that affect the outcome Y and is weaker notion of the commonly used seconfoundednes assumption.
CATE Error (ECATE). Given test dataset Dist = { (0). (1)) with each zy rendered under Li we compute the empirical error incurred in estimating CATE using mean squared error as ΣCATE = I E EjEDm (T - where Ty = (1) yy (0) is the true effect and ig is the predicted effect for the instance
The CATE error in general be decomposed across treatment T as
ÊCATE [(T==t) ECATE where ECATE= S tET
JEEX
Definition 3. Let define factual error Ep and counterfactual error E samples with observed treatment and missing treatment 1 1 as follows:
E = /
and Etp== #EE
NEX
Lemma 2. The CATE error is related to the factual and counterfactual error as: SCATE +2E'p (Proof in Appendiz 1.5.1
4 Learning Causal Representations for CATE
Our task involves learning four functions: Fc that extracts the causal representations from X() and cee that estimates the outcomes (t) for te {0,1). With access to counterfactual simulated data Daya and observational real data Dum: one can come up with the following approaches for estimating CATE: 1) SimOnly, which only uses Dayn: and 2) RealOnly which only uses Dum to estimate μ4, (3) Real,,imm,, which uses Daym to estimate fa and subsequently, Dtm to estimate Pr. We now discuss the training approach for each of these methods, delve into their shortcomings, and then present our proposed method SimPONet.
To illustrate the shortcomings, we consider a test instance x" generated under treatment T=1 (without loss of generality) and derive the CATE error expression for it in the population setting, as [Drmll + 00 and |Day oc.
4.1 The SimOnly Estimator
SimOnly solely uses Daya: It leverages the counterfactual supervision provided by the simulator and identifies the simulator's DGP as follows:
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(Step 1) Estimate the synthetic causal representation extractor f from covariate pairs (z);0))=; using contrastive learning Von Kügelgen et al. (2021):
Doyal
(2(1)) log Σ 2 (xp(sim((((s))))))ppp
( argmin U8 (
where ((f)))(((((())
(1)
where sim(.,0 is cosine similarity, (1)(1) (1 (1-1)) denotes a positive pair with the same underlying latent z;- A negative pair (a (f), (t)) has different (z),2)) Contrastive learning increases similarity of representations of positive pairs (1) while pushing apart the negative pairs (2)(t),))z:(t)),
Lemma 3. As DDyall + 00, contrastive training with paired counterfactual covariales as shown in Eq. recovers f = f. where h is a diffeomorphic transformation. Moreover, when the latent space 2 CC(n=-11) (unit-norm hepeersphere in ), h is a rolation transform by Extended Mazur- Ulam Theorem as shown (Zmmerrmnn
et al. 2021) (Proposilion 2).
RRefe Appendur A.5.2 for more details.]
The main insight from the above lemma is that, given counterfactual supervision, it possible recover causal representations Z from post-treatment covariates X up to rotation h, making CATE identifiable in the simulated distribution, as we demonstrate below.
(Step 2) Estimate (5
(z) with supervision on difference of
a8
S
argmin
Σ (00))
(2)
x0 EDya
The SimOnly method uses these estimates as-is on real data, i.e 7= and f We.. We analyze below the error incurred with such estimates on real data.
CATE error: In the population setting, since Jä =o we see that the optimization problem in Eq 2 yields S - as its solution. Thus, for an instance x" from the real distribution under treatment 1, the true CATE ((fi((()))) (x:)). The CATE error using SimOnly becomes:
This shows that for SimOnly to provide accurate CATE estimates, the simulator must perfectly align with the real world; ie., كي = and ft= fl for all However, designing such simulators is highly challenging in practice, making this method unsuitable for CATE.
4.2 The RealOnly Estimator
RealOnly solely uses real observational data Dun Since Dun lacks counterfactual covariates, this model cannot apply contrastive training and therefore cannot explicitly supervise the recovery of causal representations. Instead, it focuses on regressing the factual outcomes y:(44) from post-treatment covariates (x:4)). In terms of the four learning parameters, its learning objective is:
Dual
argmin Po.PP s=1
(u (u--De De, (F.,
However, since Se are not individually supervised, we might well collapse them into composition of = fri yielding uE, (æ,)=y, and thereby CATE Fx = of (z) -ff (æ).
RealOnly is consistent in estimating the factual outcomes, because as Duual + co, we have p. = argming: Ee-P(a))) - (x)) = of and therefore, the factual error Ef = O. However, RealOnly

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incurs significant error when estimating the counterfactual outcome, which in turn contributes to the CATE error, as shown below.
CATE error: The true CATE for x" obtained using treatment 1 can be written as ((f(==))==(((((x)) po(fi(²*)) Then, the CATE error for RealOnly is computed as:
(x*) - (a
In the population setting, using the consistency of factual estimates, the CATE error reduces to [a=)]] This error is zero when f1(=")= Thus, for RealOnly to provide accurate CATE estimates, the treatment must not affect the post-treatment covariates, Le., = Vz in which case their inverse equal fo= fi. However, this assumption is often unrealistie. For instance, in pharmacology, different drugs typically induce distinct effects on patient covariates, limiting the applicability of this model in such settings.
Remark: The post-treatment covariates X can be viewed as a special case of the pre-treatment covariates Z when the covariate- searriat-generating functions 90=91 91. In such cases, our proposed SimPONet algorithm offers no distinct advantage, and existing CATE methods designed for pre-treatment covariates suffice and should be used instead.
4.3 The Simy Estimator
Unlike SimOnly, which uses Dayn to learn both ft and p., this approach leverages Dayn solely to learn the representation extractor Je. Specifically, it assumes that f. - -/ as obtained from Eq. Thereafter, it learns the te parameters by applying factual loss on Dim to estimate
Po = argmin -u
(1o-k1) } Dem
We call this method Sim since learns the outcome parameters μ from real samples while learning representation extractor ft from the simulator.
condition under which the Sim,,,Sim, model achieves zero zero ia wsen f. f for cach treatment I. This requires that the simulator aligns with real-world covariates, specifically æ; = 9===)())==)² x This limitation arises because the model learns the representation extractor solely from Dayn, without making adjustments for real covariates.
CATE crror:
In summary, we described three possible CATE estimators and showed that each method would provide accurate CATE estimates under certain strong assumptions about the real and simulator DGPs. Given that none of these assumptions would hold in practice, we now turn exploring a joint training framework that learns simultaneously from both real and simulated samples.
4.4 The SimPONet Estimator
We first conduct a theoretical analysis to derive generalization bound that characterizes the CATE error as function of the mismatch between the real and simulator distributions. This analysis forms the basis for our proposed method, SimPONet, whose loss function is inspired by the bound.
Lemma 4, Assume T is K,-Lipschitz, and f and 78 are estimates from the simulator DGP obtained from the oplimization in Eq. Then, the CATE error on the estimotes Je and F admits the following bound:
(F.
where d2(e) are distance functions in Sec. and Ep is the factual loss. (Proof in Appendiz A.5.2
The expressions in blue are constants that capture the discrepancy between real and simulated distributions and cannot be minimized. In contrast, the remaining terms can be minimized by training on Dum and Dayn- As Duaal approachees infinity, the factual error E can be made to approach zero, while the other minimizable distance terms act as regularizers. The term da 77 can assist in regularizing the outcome parameters P4,
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whereas daje (f. can aid in regularizing the parameters of the causal representation extractor functions fa. This analysis leads to our proposed approach SimPONel whose overall loss is as follows:
min
E(v-- Me, (2il))
(F
2 +X₂ Σ 11 - (x)))
Σ Σ
(T. T--- (of
(1)())
ft) Dern
Dewn
DoyntC(0.1)
Factual Loss on Dern
a(se fi) regularizer
5 regularizer on Daya
(3)
where (1) and Xr 0 loss weights. 7(0)= ( denotes the estimated CATE.
SimPONet relaxes the strict equality fi used by Simy, and instead uses fi as a regularizer, while ensuring that T4 accurately predicts the factual outcomes for instances in Dim- Tt also imposes the 7 loss on simulated instances to leverage any potential similarity between the true treatment effect, т, and the simulated treatment effect, -5. Furthermore, the 7 loss is essential to prevent degenerate solutions that would cause SimPONel to collapse to the S.im, estimator. This is because applying regularization solely on Se can drive the regularizer (lf{ f. (z)]1 to zero, leading to f ft, while still minimizing the factual error E by updating Ti accordingly. Consequently, SimPONet would collapse into the Real,Simm estimator, making the S5 loss critical in avoiding such degeneracies.
Adjusting Loss Weights. SimPONet adjusts the loss weight Xy for learning fc by comparing the factual errors of the RealOnly model, which trains on X, with those of the Real,.Simy model, trained using simulated causal representations f. (z)) If RealOnly consistently outperforms Rcal,,Simy in factual crror, we infer that the simulated ed representations aions may not generalize well the real distribution, prompling SimPONel to reduce Xy. By default, xy is set 1; however, if Simm exhibits a notably higher factual error, SimPONet lowers Xy to 10-4.
In rast, luning X, requires 7 supervision on real data, which is unavailable. Prior work (Curth & van der Schaar 2021 Nagalapatti et al. 2024b Künzel et al. 2019) argue that while outcome functions μe can be complex, the difference function T - 1o is often simpler. For instance, if we consider n Pr + (with <> 0), we can make the factual outcomes to diverge arbitrarily while their corresponding 7 and 7 remain equal. Therefore, while comparing the factual errors between SimOnly and RealOnly models set X is appealing, it maybe a poor choice in practice. So, SimPONet always sets A to its default
We present the SimPONet's pseudocode in Appendix A.4
5
Experiments
We conduct experiments that designed to address the following research questions:
RQ1 How do different methods compare with varying discrepancies between real and simulator in settings with closed form estimates; i.e., without errors due to finite-sample training?
RQ2 How does SimPONet compare to other SOTA basclines that assume pro-treatment covariates?
RQ3 What are the contributions of individual loss terms in SimPONet?
RQ4 How does SimPONel fare against the baselines when μa exhibits complex non-linear behavior?
RQ5 How do CATE methods perform when trained directly (a) post-trealment. covariates X, or (b) pre-treatment Z, or (c) simulated causal representations fs from Eq.
5.1 Neural Architecture and Hyperparameters
For the Lincar experiments RQL, we omit shared layers Fig. 5 and sct fo and fi as n, x ny matrices, and Po and fi as n2 x1 vectors. For the real-world experiments in RQ2, with both ferpyy as non con-linear functions, set Pig and ii as 2-layer MLPs with hidden layers of 100 and 50 neurons. The shared layers have 2 hidden
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layers with 50 and ny neurons, respectively. We impose the d(ft.ft) loss for SimPONet on outputs of the shared layers. For experiments in RQ4, we omit the shared layers while sotting ft as lincar layer. But since p is non-linear, we use an MLP with one hidden layer of 50 neurons and ReLU activations for each Tt.-
Hyperparameters: We implemented all baselines and SimPONet using jax within CATENets @Curth et al, 2021), a standard library for benchmarking state-of-the-art CATE estimation methods. To ensure consistency, we used the same MLP architecture, learning rates, optimizers, and other hyperparameters as the default. settings in CATENets for baseline approaches. The unique hyperparameters for SimPONet are the loss weights Xy and Xy As described Sec. 4.4 PiNPONet tunes ' by comparing the factual errors of the RealOnly and Real models, while X is always set to CATENets applies carly stopping based on factual error in the validation datasel, common practice in CATE training. To ensure a fair comparison, we maintained consistent. training and validation splits across all methods.
For CATE experiments, standard deviation is sometimes mislcading to comment on the statistical significance of empirical results as noted in (Curth et al. 2021). So, for all experiments, we conduct a one-sided paired t-test with SimPONet as the baseline and enclose p-values in brackets to indicate statistical significance. Lower p-values favor SimPONet.
Assessing Statistical Significance:
5.2 RQ1: Assessing Baselines Under Settings Without Finite Training Sample Errors
To address RQI, we consider a setting where both the real and simulator DGPs as shown in Fig. I are linear. In particular, we generate the training datasets Dern and Dayn as follows: (1) Latent variables = RR= are sampled from distribution Pz. (2) Real and simulated covariates for treatment t are computed as g((z) R, and gf =2)= where R. and S, are invertible matrices. (3) Outcomes are generated as px(2)= =2 w, and p (2)== wy where Wy and wy are vectors in RT= We consider two datasets for RQ1: (1) Synthetie-Gaussian and (2) Real World-HHDP, differing in how Z is obtained. In setting (1), 2 E 10 is sampled from a standard Gaussian ((011), while for (2), Z is taken from the real-world IHDP dataset (Appendix A.7) as-is.
Now, to systematically control the real-simulator mismatch, we need means to vary the following distances: d( R R) di R, " ", and d(r, We achieve this follows: (1) Initialize Ro ww ~ (0., (2) To inject a distance 7R E (0, 0.5) between R and RT set R = (1 - 7R) + (R, (3) Set We use 0.4 in all experiments, (4) Similarly, inject Yns gap between R, and
wi + S, (5) For treatment effect parameters w, = =1 We in the real DGP, we sample its simulator counterpart with a gap 7 as = -1->)>11, +,, and set wi$ accordingly.
In linear settings, the optimization problems for the CATE estimators SimOnly, RealOnly, and Real,,Simy admit closed-form solutions. show the closed-form solutions in Table 7 in the Appendix, and a detailed derivation in Appendix A.6
For SimPONet, a closed-form solution is not possible, so we solve it to a local optimum using alternating minimization over ūe and Je, with each alternating update in closed-form. We show the SimPONet's update equations in Appendix(A.6.] In summary, the setting of RQ1 allows study of the impact of varying discrepancies between the real and simulator distributions without approximation errors due to finite training samples.
We show the results comparing SimPONet with the three baselines in Table | where we observe: (a) Across both synthetic and real-world settings, SimPONel achieves either the best or second-best, performance. The CATE error for SimPONet remains controlled primarily due to its ability to bound errors in the counterfactual distribution. (b) In contrast, the RealOnly and Real,,Sim, y models perform well only under certain restrictive conditions favorable to them, providing zero error on the fact distribution but very high counterfactual error, leading to poor CATE estimates.
5.3
RQ2: Comparing SimPONet with State-of state-of-the-art art CATE Baselines
We conduct experiments using semi-synthetic observational datasets commonly used to assess efficacy of treatment effect estimation methods in the literature: the Infant Health Development Program (IHDP) and
3 *Each cutry in the table reports the CATE Error alougside its corresponding p-value CATE Error (p-value). The p-value denotes the statistical significance of the hypothesis that SimPONet outperforms the baseline. We one-sided paired t-test for this assessment. Smaller p-vuuu indicate stronger evidence in favor of Sim PONet. We show the best performing method in green, and the second best method yellow across all our tables.
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Table 1: RQI: In a linear DGP setring, we vary the gaps using YR for fu..fi fi) in the first column, Yrs for d(ft,fe) and To for ). "low" refers to 0.1 that simulates small distance, while "high" refers to 0.4. We run all experiments with five different seeds and report g-values of comparing the mean performance in bracket.
Gamesian
Real World-IHDP
d(Jo.fi)
d(oof))
d(r==9)
SimOnly
RealOnly
SonPOONet
SimOnly
RealOnly
Rosi,,Simn
SimeONee
0.00
high
ligh
2.82 (0.27)
0.00 (1.00)
15.75 (0.01)
2.58 (0.00)
3.57 (0.11)
0.00 (1.00)
48.76 (0.05)
3.20 (0.00)
low
low
low
0.63 (0.00)
2.47 (0.02)
1.19 (0.01)
0.54 (0.00)
1.00 (0.44)
3.43 (0.02)
2.78 (0.00)
(0.00)
low
low
high
1.57 (0.16)
2.47 (0.08)
1.19 (0.83)
1.39 (0.00)
1.62 (0.26)
3.43 (0.04)
2.73 (0.02)
1.49 (0.00)
low
high
low
2.14 (0.22)
2.17 (0.00)
15.75 (0.01)
1.85 (0.00)
3.67 (0.31)
3.43 (0.48)
48.76 (0.05)
3.37 (0.00)
low
high
high
2.82 (0.26)
2.47 (0.56)
15.75 (0.01)
2.57 (0.00)
3.57 (0.11)
3.43 (0.39)
48.76 (0.05)
3.19 (0.00)
high
low
low
0.63 (0.00)
13.86 (0.02)
1.19 (0.01)
0.54 (0.00)
1.00 (0.47)
47.78 (0.06)
2.73 (0.00)
0.98 (0.00)
high
low
high
1.57 (0.16)
13.86 (0.03)
1.19 (0.83)
1.39 (0.00)
1.62 (0.27)
47.78 (0.06)
2.78 (0.02)
1.50 (0.00)
high
high
low
2.14 (0.21)
13.86 (0.03)
15.75 (0.01)
1.85 (0.00)
3.67 (0.31)
47.78 (0.06)
48.76 (0.05)
3.38 (0.00)
high
high
high
2.82 (0.26)
13.96 (0.01)
15.75 (0.01)
2.57 (0.00)
3.57 (0.11)
17.78 (0.06)
48.76 (0.05)
3.19 (0.00)
the Atlantic Causal Inference Conference (ACTC) datasets. These datasets contain real-world pre-treatment covariates (Z). Please refer Appendix A.7 for more details on these datasets.
To align these datasets with our study, apply RealNVP Normalizing Flows (Dinh et al. 2016) to transform pre-treatment covariates Z into post-treatment X. Flows are non-linear deep neural networks that ensure invertibility of the covariate generating functions 9 We consider randomly initialized flows with two coupling layers. We used the flows 9o.91 on real data, and two other distinct flows go to obtain covariates in synthetic data from Z. We borrow the real outcomes as-is from the ground truth dataset. However, we synthesize simulator outcomes with a gap of 7/- as follows: (1) sample w RR (0,,1) (2) set 7 (3) ((=))=(z)+) (a(()) -7.-2" 2 "T=")""") where (()) is the standard deviation of CATE labels in the real dataset. Scaling by a (T) ensured comparability between 7 and S Thus, when 7r 00 T= when 7-=1, 1, disparate from 7
We evaluated SimPONet against various basclines from the well-known CATENets (Cuthh et al.) 2021), a benchmarking library for CATE estimation. Since the baseline methods are not designed to extract the causal representations, we provided them with representations extracted by simulated causal representation extractor f." (x) as input for fair comparison. Running these baselines with post-treatment covariates X directly input yiclded much poorer results shown in Fig. 3 We also compared SimPONel with SimOnly, RealOnly, and Real, Simy baselines that we developed in our theoretical analysis. We present the results in Table 2 for 7 = 0.1, and defer the results for
Comparing Fartual Emers Validation Data
AeaiOny
p=0.08
p=0.47
p= 0.19
0.000
HDP
ACC226
Dataset
Figure 2: Factual errors with v-vauues shown above bars. For IHDP, RealOnly consistently outperforms Real,,Simy.
larger 77 to Appendix 5.7 We make the following key observations:
(a) IHDP:
This dataset contains 25 pre-treatment covariates, 19 of which are binary. Contrastive training struggled to capture these binary features, causing Real,,Simy, which uses f (x) as input, to consistently underperform RealOnly, which directly uses a. As shown in Fig 2 the p-value zero for the THDP dataset, but significantly larger for the others. As a result, we set the weight of the regularizer d(fi.f() controlled by Xy, to le-4 for IHDP, while keeping Ay at its default value of for ACIC. Overall, SimPONel achieved the best. performance.
(b) ACIC-2:
This dataset is unique in that the true CATE, is constant across all individuals in the =0 0 for real samples. As a result, our approach to syuthesizing the simulated CATE, -5, given by (+)==(<>)) + (a(T) w yields Soolddss(33)
observational data, implying that its standard deviation a
for all a. This leads to perfect alignment between the synthetic and true CATE, causing SimOnly to outperform all other methods on this dataset. Although SimPONet could have improved by assigning a higher weight the S regularizer, tuning this weight would typically require supervision T, which we avoid.
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Table 2: RQ2: Comparison of SOet with several pre-treatment baselines and post-treatment proposals. p-values for paired t-tests against SimPONet are in brackets. Lower p-values indicate statistical significance. SimPONet outperforms others overall, while SimOnly performs best ACIC-2 since T= 7
Method
IHDP
ACIC-2
ACIC-7
ACIC-26
ACIC-AII
RNet
(Nie Wager
2021
1.54 (0.00)
3.30 (0.00)
5.91 (0.04)
6.06 (0.18)
5.78 (0.03)
XNet
Kiinzel et al.
2019
1.0 (0.00)
0.43 (0.15)
5.49 (0.17)
5.1 (0.38)
4.45 (0.41)
DRNet
Schwab et
2020
0.96 (0.00)
0.24 (0.59)
5.53 (0.15)
5.08 (0.39)
4.45 (0.40)
CFRNet
Shalit et
2017
0.96 (0.00)
0.36 (0.26)
5.55 (0.15)
5.09 (0.38)
4.67 (0.32)
FlexTENet (Curth & van der Schaar
2021
0.96 (0.00)
0.32 (0.32)
5.46 (0.19)
5.04 (0.40)
4.85 (0.37)
DragonNet
[Shi al. 2019
0.96 (0.00)
0.29 (0.41)
5.57 (0.14)
5.09 (0.38)
4.60 (0.40)
IPW (Robins et 1994
0.96 (0.00)
0.36 (0.24)
5.56 (0.15)
5.09 (0.38)
4.45 (0.40)
k-NN Stuart 2010
0.96 (0.00)
0.33 (0.33)
5.48 (0.18)
5.13 (0.37)
4.44 (0.40)
Match
SScwwbb et al.
2018
0.98 (0.00)
0.56 (0.11)
5.75 (0.08)
5.13 (0.37)
4.56 (0.28)
StableCFR
Wu et al. 2023
1.01 (0.00)
1.09 (0.03)
5.56 (0.15)
5.08 (0.43)
4.82 (0.14)
ESCFR
Wang ct al. 2024
0.96 (0.00)
0.27 (0.47)
5.55 (0.15)
5.79 (0.21)
4.73 (0.18)
PairNet
Nagalapatti et al. 2024b
0.97 (0.00)
0.12 (0.85)
5.46 (0.23)
5.05 (0.37)
4.44 (0.41)
SimOuly
0.94 (0.00)
0.00 (0.98)
6.65 (0.00)
6.60 (0.12)
6.45 (0.02)
RealOuly
0.83 (0.13)
11.23 (0.01)
14.81 (0.05)
8.18 (0.01)
9.82 (0.00)
Seml,
0.96 (0.00)
0.17 (0.76)
5.57 (0.14)
5.09 (0.38)
4.32 (0.38)
SimPONet
0.79 (0.00)
0.26 (0.00)
5.04 (0.00)
4.67 (0.00)
4.36 (0.00)
Table 3: RQ3: Impact of regularizers. Here, -d(fe.F() ) represents our loss 3 with =0 and - means A 0. A
negative value implies SimPONet with all regularizers outperforms the ablation where regularizer is disabled.
THDP Linear fc, Linear 14
GP - Linear ft, Non-Linear
d(fo=/i)
d(h.f?))
d(r,r")
JsF...))
S
-s(
0..00
high
high
+1.29 (1.00)
-1.07 (0.18)
-0.55 (0.31)
-0.62 (0.26)
high
low
low
-0.64 (0.04)
+0.01 (0.51)
-0.29 (0.24)
-0.04 (0.47)
high
low
high
-0.40 (0.11)
+0.00 (0.50)
-0.42 (0.10)
-0.19 (0.34)
high
high
low
+1.74 (0.99)
-0.03 (0.48)
-0.66 (0.27)
-0.71 (0.25)
high
high
high
+ +1.30 (1.00)
-0.03 (0.45)
-0.72 (0.21)
-0.97 (0.18)
(c) ACIC-7 and ACIC-26: The CATENets baselines significantly outperform the RealOnly model, because of high-quality causal representations extracted by F.S. This demonstrated by Real,,Sim, outperforming RealOnly on factual error (see Fig. 2 In ACIC-7 and ACIC-26, SimPONet achieves the best results by leveraging the closeness between F and FlexTENet (Curth & van der Schaar 2021), which shares parameters between ño and jA1, and Net (Nagalapatti et al. 2024b), which applies losses on pairs of close-by by samples are strong contenders to SimPONet.
The last column in the table presents results across all 77 seeds of the ACIC dataset. SimPONet achieves the lowest mean CATE error overall. However, for certain seeds, SimPONet's CATE error remains comparable to baselines such as PairNet, k-NN, as indicated by them having p-values close to 0.4. The RealOnly model consistently performs worse across all seeds due to the covariate functions go and gi producing disparate post-treatment covariates Xo and X, when applicd to Z. Similarly, the SimOnly model suffers due to the distributional mismatch between the real and simulator data.
ACIC-All:
5.4 RQ3: Ablation of SimPONet Losses
We evaluate the impact of the di fi and S regularizers in SimPONet's objective 3 We experiment with the Lincar IHDP datasct (Sec. 5.2) and the Non-Linear Gaussian process datasot (Sec. 5.5 For the Lincar IHDP dataset, in the - case, we added an L2 penalty on w, for the alternating minimization to work. Table presents the difference in CATE errors of SimPONet and the ablation, averaged over five seeds with p-values. A negative entry mcans PONet does better than the ablation. We observe that S loss is very effective since SimPONet outperforms the SS in both datasets. For IHDP, removing di d(fe.ff) F5) loss helps. We could not set Xy weight to be small because both RealOnly and Sem, achieved zero factual error. Despite this,
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Table 4: RQ4: Results for linear covariate and GP-hased nonlinear outcome functions. We run each experiment 5 times and show the p-values. SimPONet outperforms others in many settings. RealOnly is strong contender.
d(fo.1)
d(r,r%)
SimOnly
RealOuly
Riml,
SinPONel
0.00
high
high
5.30 (0.09)
2.99 (0.11)
3.17 (0.09)
1.94 (0.00)
low
low
low
1.77 (0.15)
1.06 (0.73)
1.05 (0.77)
1.26 (0.00)
low
low
high
1.56 (0.08)
0.98 (0.65)
1.07 (0.44)
1.05 (0.00)
low
high
low
4.60 (0.04)
2.87 (0.07)
3.12 (0.04)
1.88 (0.00)
low
high
high
4.15 (0.04)
2.96 (0.02)
3.11 (0.01)
1.60 (0.00)
high
low
low
2.39 (0.14)
1.19 (0.68)
1.12 (0.77)
1.37 (0.00)
high
low
high
1.94 (0.11)
0.98 (0.83)
1.16 (0.58)
1.21 (0.00)
high
high
low
7.42 (0.09)
2.65 (0.38)
2.95 (0.25)
2.37 (0.00)
high high high
3.70 (0.07)
2.80 (0.17)
3.16 (0.09)
2.10 (0.00)
SimPONet with both regularizers comfortably outperformed the other proposals demonstrating it as a better candidate for our task.
5.5 RQ4: Linear covariate function g and non-linear outcome function 1
Now, we consider a more complex setup where the covariate functions gr and ge remain linear, but the outcome functions μl and e are nonlinear in Z. In particular, we sample the outcomes y and y using Gaussian Processes (GPs) Raasuussnn & Williams 2005). Tet GP(O,, denote a GP with RBF kernel of width 7, so higher y results in a more complex function. To sample the μr-ln such that their difference has gap 7, we follow: (1) Sample I using G PPt:T~GPPP.. GP(O,K,,) (2) Sample Ho GGP,KK,) (3) Set fi tT As before, we set 0.4. Now, to sample S uá such that d d(r,r")=7r: (1) Set 75 GPPGG (2) Sample vd GP(OKK,) (3) Set = +
We estimate js in Eq. 1 in closed-form, whereas we learn other parameters using gradient descent. We show the results in Table 4 w where we observe: (a) SimPONet outperforms the other baselines in five out of nine settings (b) In alignment with our theory, Reml,,Sim, performs better than others only when f(fe.f))) is small. In summarry, when the properties of the underlying DGP are unclear, SimPONel proves to be an effective approach for learning T.
5.6 RQ5: Comparing CATE methods when trained on Z vs X vs fX)
To address RQ5, we use the IHDP dataset to evaluate the baseline models when trained on post-treatment covariates X directly, and we compare these results with those from Table where the basclines were trained on simulated causal representations, f (a). We also evaluate the baselines trained on pre-treat.ment Z to explicitly show the detrimental impact of post-treatment covariates on the CATE error.
In addition, extend Table 2 by considering non-invertible, non-linear transformations with -ayer Perceptrons (MLPs) on the IHDP dataset to assess the robustness of SimPONel and baselines when the diffeomorphism assumption is violated. To this end, we used two-layer MLPs for and ge to generate covariates from 2. We present the results in Fig. 3 and make the following observations:
In the pre-treatment setting, causal representation extraction is unnecessary, making simulator supervision inconsoquential; thus, we omit the four post-treatment CATE methods introduced in this paper. For other baselines, Fig. 3 shows that they perform significantly worse with most-treatment X than with pre-trealment Z, underscoring the importance of causal representation recovery. DragonNet achieved the best performance with pre-treatment covariates.
For Normalizing Flow-generated covariates, CATE error consistently decreased when baselines used simulator- based causal representations, (x),, rather than X, validating the utility of simulators in extracting causal representations. However, the CATE error remains significantly higher compared to using pre-treatment covariates Z, highlighting that simulators, while helpful, are not ideal and exhibit a distributional gap between real and simulator distributions. This gap impacts all CATE methods. Notably, SimPONet specifically
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designed to leverage simulator supervision and incorporate regularizers using simulatoor-eneeraeed data, achieves the best results.
In the MLP-based covariate experiments with non-invertible covariate functions 90.91, SimPONet consistently outperformed all baselines, exhibiting trends similar to those observed with Normalizing Flow-ge Flow-generated covariates. This suggests that the diffeomorphism assumption, while necessary for our theoretical results, may be inconsequential in practice.
2.00
Input Pre-T
Input NF X
1.75
CATE Error
Input NF (x)
Input MLP ((
0.50
0.25
RNet
XNet
DRNet
CPRNet
FlexTENet
DragonNet
IPW
K-NN
PerfectMatch
PairNet
simonly
RealOnly
Real, Simr
SimPONet
Figure 3: Comparing CATE errors under pre-treatment Z, and MLP, Normalizing flow generated post-treatment covariates X.
5.7 Varying "/r in Arbitrary DGP Experiment
The results of the varying ?r experiment are presented in Fig. where we compare two approaches that leverage simulator data during training: SimOnly and SimPONet. Across all Y gaps, we observe that SimPONe consistently outperforms SimOnly in three out of four datasets, with ACIC-2 being the exception. This is expected, as ACIC-2 satisfies the condition T2 = T. The performance gains of SimPONet are particularly notable in the ACIC-7 and ACIC-26 datasets, where the CATE error for SimOnly escalates significantly at larger 7/r values. The exact error values for these cases are shown the inset subplot at the top-right. These findings underscore our argument: while SimOnly can perform well on simulators closcly aligned with the real world, struggles with real-world simulators that diverge from reality. In contrast, SimPONet's adjustment strategies strategies-enabled enabled by theoretically grounded regularizers derived from the CATE error analysis -ycllsssilllld much more reliable CATE estimates.
Varying Y, that controls d(
10
50
IHDP Simonly
40
30
IHDP SimPONet
10
20
ACIC-2 Simonly
CATE Error
6-
ACIC-2 SimPONet
ACIC-7 Simonly
ACIC-7 SimPONet
ACIC 26 Simonly
2.
ACIC- 26
SimPONet
0.0
0.2
0.4
0.6
0.8
1.0
Y
Figure 4: vary 7r which controls the gap between the synthetic CATE, 75, and the real CATF, т. Fach dataset is represented by distinct color, where the pale version of the color indicates SimOnly and the darker version denotes SimPONet. For ACIC-7 and ACIC-26, as 7, increases, the CATE error grows significantly. Therefore, we present these results inset figure in the top-right corner.
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Table 5: RQ:: Evaluating Simy performance with Z from SimPONet's f: vs. fi obtained from Daya- Real,,
with SimPONet's Jt significantly outperforms in many settings indicating SimPONet extracts better Z.
IHDP Linrar f.. Linear
GP Linear fr. Non-Lincur Pt
d(Jo.f3
d(fo.f"))
d(r,r")
Simy
Reel,,SinnPPO.. SiNePONett
Real, Suny
0.00
high
high
18.70 76 (0.05)
3.28 (0.00)
3.17 (0.32)
2.67 (0.00)
low
low
low
2.73 (0.00)
0.97 (0.00)
1.05 (0.81)
1.55 (0.00)
low
low
high
2.73 (0.02)
1.19 (0.00)
1.07 (0.78)
1.55 (0.00)
low
high
low
48. 76 (0.05)
3.48 (0.00)
3.12 (0:23)
2.53 (0.00)
low
high
high
48.70 76 (0.05)
3.21 (0.00)
3.11 (0.34)
2.73 (0.00)
high
low
low
2.73 (0.00)
0.98 (0.00)
1.12 (0.72)
1.29 (0.00)
high
luw
high
2.73 (0.02)
(0.00)
(0.76)
1.42 (0.00)
high
high
low
48.70 76.76 (0.05)
3.37 (0.00)
2.95 (0.57)
3.11 (0.00)
high
high
high
48.76 (0.05)
3.23 (0.00)
3.16 (0.42)
2.97 (0.00)
5.8 On the Quality of 2 Extracted by SimPONet
In this experiment, we evaluate the quality of representations learned by SimPONet. If SimPONet's regularizers in the losses enable the recovery of representations identifiable for CATE, then a RealOnly model trained on these representations should outperform the Real modee trained on representations extracted using contrastive training on the simulator dataset. We focus on the Linear IHDP and non-linear GP datasets, as they provide closed-form solutions for Objective [ and and gives fine grained control to set the gaps between real and simulated datasets and assess the performance across many settings of these gaps. Specifically, we train the RealOnly model using extracted from f, via SimPONet, and call it. as ,. SimmPNee,, y, and compare it to Sim, trained from Fs. Results in Table 5 show that SimPPet, significantly outperforms the baseline Sim aacross many DGP settings, with particularly strong results in the Linear IHDP case where p-values are consistently small. This demonstrates that SimPONet effectively learns high-quality representations Z.
5.9 Sensitivity to Loss Weights
Table This table assesses the sensitivity the loss weights on all three versions of the IHDP dataset considered in our
work - Synthetic Linear Sec. 5.2 Synthetic Non-Linear Sec. [.3 and Semi-synthetic Sec. 5.3 We consider nine different
settings of the loss weights in SimPONet's objective 3
The table shows mean + standard dev deviation the CATE Error.
X
A
Synthetic Linear
Synthetic Non-Linear
Semi-Syuthetic
0.1
0.1
3.4611.91
4.2312.21
0.72110.19
0.1
0.5
3.42:10.84
3.55
0.70
0.1
1.0
3.43
3.40
0.71
0.5
0.1
3.41+0.90
4.06+2.21
0.74
0.5
0.5
0.88
3.32+1106
0.73+0.19
0.5
1.0
3.37 0.82
3.23+ 1.00
0.71+0.20
L0
0.1
3.38 0.86
1.911
0.71110.0
1.0
0.5
0.89
3.24+1.02
0.72+1..19
1.0
1.0
0.88
3.14=0.93
0.71 0.22
In this experiment, we analyze the sensitivity of SimPONet's performance to the loss weights Ny and X, in its objective function. Using the IHDP covariates, performed this analysis in the three variants used in our study. For the synthetic datascts, we sct the real simulator gaps, as specified in Tables I and to high. In the Semi-Synthetlcc IHDP setting, we evaluate performance on five randomly selected datasel versions. The results, summarized in Table 6 report the mean CATE error along with its standard deviation across dataset seeds. All methods consistently performed poorly when the loss weights are small. This indicates that regularizers are important to impose on post- post-treatment covariates. Across other settings, overall, we observe that SimPONet is sensitive to the choice of hyprraarameter settings in non-linear and semi-synthetic versions, and remains
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fairly robust in the linear setting. However, tuning these hyperparameters is challenging in practice due to the absence of counterfactual data in real observational datasets. Despite this sensitivity, SimPONel outperforms achieves either comparable performance or manages to surpass the baseline methods across different loss weight configurations.
6
Conclusion
This paper addressed the challenge of estimating treatment. effects from post-treatment covariates setting not identifiable from observational data alone. We proposed to tackle this task using off-the-shelf simulators that synthesize counterfactuals, in contrast lo prior work that relied on real-world counterfactuals, which limited their practical applicability. Our theoretical analysis established a bound on the CATE error based on the distributional mismatch between real and simulated data. Notable, ours is the first work to systematically analyze the role of simulators in CATE estimation. We introduced Sim SimPONet, a framework that jointly learns from real and simulated samples enhance CATE estimates beyond what could be achieved from observational data alone. Extensive experiments across various DGPs demonstrated that SimPONet is robust and effective method for estimating CATE from nest-treatment data.
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A
Appendix supplemental material
A.1
Code
We have released the code in the URL htt.l//gg./ggggtb...o/////bl/elp
A.2
Learning Counterfactual Simulators
Here we discuss prior works that train generative models for synthesizing counterfactuals. general, to obtain counterfactuals in the real distribution, we need to follow three steps (Pawlowski et al. 20200: (a) abduction, inverting X to obtain Z, (b) action, applying new treatment, and (c) prediction generating a X under the new treatment. These steps require prior knowledge of the DGP specifications, which are often difficult to define and cannot be learned from obsorvational data alone (Pawlowski ct al. 2020). Consequently, many methods bypass the principled approach and use pre-trained models like Dilfusion models and Large Language models to generate pseudo counterfactuals from a related synthetic domain. Such simulators are proposed
across various modalities, including images (Pawlowski et al. 2020 Gu et al. 2023 Thiagarajan et al. 2021
Pan Bareinboim. 2024 Sauer & Geiger 2021
Jeanneret et (022), text (Madaan al.) 2020 Calderon
et al.
2022
Chen al. 2023 Dixit et al.
2022
Robeer et. al. 2021 and healthcare (Lang al.) 2023 Wang
et al.
2020
Dhinagar et al. 2024). Prior research (Gondal et al.
2019 shows that while such simulated data is
not directly usable for downsttremm lasks, they provide strong inductive biases that transfer well to the real distribution. Our method can incorporate any such counterfactual generators as simulators, provided they contribute to lcarning causal representations that are predictive of CATE.
A.3 SimPONet Architecture
We present an overview of the SimPONE model architecture in 5 Our model has four primary parameters: Jo and fi for extracting causal representations, and Po and i1 for predictinggoutcomes.. outcomes. Shared layers project So and into a common space.
go(2i) 2,(0) fo())
Ao(+)
(0(
Shared Layers
(1 = (1)
(1())
A
(1)
Figure 5: SimPONet's model architecture.
A.4 SimPONet Pseudocode
Here, we present the SimPONet pseudocode. The steps involved in our algorithm are:
line 1 First we use the simulator dataset Dayn to apply contrastive losses on the counterfactual covariates using Eq. This optimization gives us a Z extractor in the simulator distribution, which we denote as
line 2 We partition the training dataset into train, validation dataset using stratified split based on T. We then initialize the loss weigts A Ar to their defaults.
lines 4,5 We now decide upon the loss weight Ay. To do so, we train RealOnly and Rem,, models. We then assess the factual prediction errors of these models on the validation split of the training dataset. If RealOnly model performs much better than the Real,,Simy model, means that the f obtained from line2 above is of inferior quality. Therefore, we scale Ny that regularizes the SimPONet's f. based on f very small value, le-4.
line 6 We can now apply gradient descent algorithu on the SimPONel's objective Eq. parameters of the model.
train the Pr.
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Algorithm 1 SimPONel Algorithm
Require: Observational Data Dum: ((2),t)))))) Simulator Data Daya: {(a (0), (1).v/ (0).)) (0), (1))
Let [f(,,t)} Z extraction functions, and (fi(,.t) outcome functions for 0,1,1
Let fi + Eq. [Minimize Contrastive loss on Daya)
3: Set Dval SPLIT(Dum, 0.3, stratify=T)), and init default hyperparameters x5 +-1,1
4: RealOnlyt-min min (2.7) Dum (v. - ūt; (fe, (xx)))²; RealySimyt min; Siinype.. (yi - -n, (x,))))))
Set A le-4 if FactualErr (ReallOnly Dvai) >> FactualErr (Real,,imn, DDal)
( Eq. 3 (perform gradient descent SimPONeetss objective using D Dry while early stopping using
Factual Error on D
7: Return (f for 0,1
We present the pseudocode for SimPONet in Alg.
A.5 Theoretical Analysis
In this section, we present the proofs for our theoretical results.
A.5.1
Proof of Lemma 2
The CATE error is related to the factual and counterfactual error ECATE 2 +2E
Proof. We decompose the CATE error into factual and counterfactual estimation error as follows:
SCATE=
(xx(2,t)=fx(x.21))) (aala)=- J [((]()
WEN
ØEN
ZEX
Let t' =1-t denote the counterfactual treatment. We can then rewrite the above expression as:
CATE
JDEX
Now using the inequality <a=b)==2a" +2b2 we can separate the factual and counterfactual teris:
ECTTE $2 / [ / [o xex
#EE
=2 +
A.5.2 Recovery of f upto diffeomorphic transformation
Lemma 5. As D syml oc, contrastive training with paired covariates recovers f while paired outcome supervision recovers S S oh-1 where h is diffeomorphic transformation. Moreover, when the latent space Z C 1) (unit-norm hypersphere in R=s), ), h is a rotation transform by Extended Mazur- Ulam Theorem as shown in Zimmermann et al. 2021 (Proposition 2).
Proof. Theorem 4.4 of (Von Kügelgen et al. 2021 shows that contrastive training with covariate pairs (x)(())));) (0),z;((1) recovers Z upto diffeomorphic transformation h, i.c. for the simulator DGP our estimate ,, = f h(z)) (xf())) E T. Moreover for unit-norm latent representations, Z
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gdx-1, (Zimmerman et al. 2021) show that h is an isometric (norm-preserving) function and therefore, rotation transform by extension of Mazur-Ulan Theorem. Mazur-Ulan Theorem states that any smooth, invertible and isometric function is necessarily affine. Moreover, in our setting, the norm of as well as & is always one and thus, necessarily a rotation. Therefore, we recover f ho upto a rotation of the true inverse map f with sufficient paired samples from the simulator.
Next, we recover 7 from the following minimisation:
7 argminE
-
argginn
The above optimization gives 75 = ====8oh0 -shh-1 and hence we recover the CATE function S for the simulator DGP composed with h-1
Proof of Lemma
Assume r Ky-Tipschitz, and f and 7 are estimates from the simulator DGP obtained from the optimization in Eq. Then, the CATE error on the estimates fe and 7 admits the following bound:
(F. 12d 12 ( (12d. (122, 42
where d (x)) distance functions in Sec. and E is the factual loss.
Proof. We now construct at upper bound on counterfactual error Eor that relies both observational data and simulator estimates to motivate the SimPONel objective:
EOF=
JOEN
[4 P(x
MEX -f JEE
Where 1 when t=0 and zero otherwise, and thus, 12 = adjusting the sign of CATE terms. Now utilise the inequality (a+b+c)² ( +b2+2)+++++)) to obtain:
EO
(1t2=:=)||ddd00--000000000000010 P(ol)dd
DEX
3
( [
NEX
ZEX
+3
xEN
33
[T(2) [[())-
:E2
zEX
Here h denotes the unknown rotation transformation that relates the estimated simulator functions (75,5)) with the ground-trute simulator functions (f5,5)) S) as shown in Lemma Let K, be the Lipschitz constant for F. We can bound the second term in the above expression as follows:
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EOr 33 / (T(2) [ .EEZ
(3H +38
Now we can add and subtract simulator function estimates to bound the two distance terms as follows:
OF S3
(T(2) [x())--(0)++1)))-7M)))))PP)
ZEZ
+3K?
TEX
<6
(T(2)- (3))- (
ZEZ
+6K?
[M
DEX
TEX
the second term. Thus, de = de oh). Now making use of Definition 2 we can rewrite this as ( (f,75) 75) which a distance function defined on the space of rotated latents h(2). We also rewrite f as fs in the fourth term
Now, using Lemma we can rewrite rewrite?"= -5=75
Morcover, day ft, - df. since h is a rotation transform and preserves the distance between any two vectors. Thus, (x) (2)]l2 ho fe (x)) ho f (æ) Combining these results, we can evaluate the above bound to the following:
Elr S (0, +00,
A.6
Linear DGP Derivation
We derive expressions for CATE estimates (x,t,t) well as COTTE, for each of our proposed estimators in the linear setting below. Note that ground truth CATE Tx (x,t) R,, (wi - Wo). We consider factual treatment 1=1 1 to illustrate the errors.
A.6.1
SimOnly
For SimOnly, we use R S-1 and uZ = w which are obtained by training on simulator data. Thus, the CATE estimate Fx .t) = x² S. 1 - w The CATE error on a sample x", with treatment t= 1 is given by
[xx(x* -x (a =1)]==((>** (ST (wf w R, w
A.6.2 RealOnly
'ú. For RealOnly, the factual objective E 1,, rú v112 = ; x: xF, " 1l Thus, the closed form solution of the estimator 1 10 R w, ET. Since we can't decouple the terms R, and de the CATE
estimate given by Fx (x= -f)==x R "0 x RT ui =x R "w Rg "Wg.
CATE error on sample xx with treatment f is given by [ix (m=,1) 11) (a= 1)}² = ((* R x Ro'
wR [w) =
R, )wo)
A.6.3 Sim,
For Rial,,Simy, we first set R, - S. which is obtained by training on simulator data. Next, we train ie on the factual objective: I| -
1 uù; x: R, - :wx 1 I..S th. Rt 1 This, gives closed form solution for the
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minimising ui, = R, ww.. The CATE estimate (x(,
w.. Fixing treatment 1. 1= 1, this simplifies further: 7x x= x'S (S) 1 R w1 S. -
== S (ui 0b) = 5 (S W1 S wo)
wa) = x² (R
Si R. Ry woo). CATE Error is given by (x)(xx1)-TX") (a" 1111 lx" (R 11 S R wp) - -x R2 (w1-
un)]² x R We x = R wol wool==fa [10]===x((p) (R, - - R )wol?
Table 7: This table presents the predicted CATE and the corresponding CATE errors obtained from the three CATE proposals computed analytically for a test instance x" observed under treatment 1.
Method
Estimate for CATE Fx (1*,1)
CATE Error [Tx (a* ,1) -00---(111111
SimOnly
x ST w
x [ w/ S -S-1w:) w;
xx Ri =w1 1R Ro'wo) Simi
RealOnly
x (x* 1---PP Ri 1
x S S R, w1
S/-1 So Ro Wp
x (RRR" -1-
-ST S Ry )wo! 2
So
A.6.4 SimPONet We train both R₁ _60
"00 on the following objective jointly:
C((R,' -0,))=01))=
Σ 1| uē, 10--ate xR, We + 100.1
Σ t=01
RR, -xS; +1z11z(16 - -w) - =z(w1 - wo)l )
Here, 2 2 the latents for simulated covariates a (which are identifiable from Dsya). Due to the joint nature of this optimisation, it is not possible to derive closed form solutions for the optimum. However, one can compuet gradients of the objective with respect to R and vi, separately. This, gives us an alternating minimisation algorithm with closed form updates.
δ OR. i
Oc
|R ws v011111+ =112+AyllaR,, -
= 22x RR, (wir0, - yuu 2xx==2xx=
Setting the derivative to zero, we obtain the following update rule:
R P yu: + (,) T +\2
where x ===xxTx:)" ((xTx:)-1=" is the pscudoinverse of x.
aw; 0ú. || 1ú, -0112 A Ar||z(10e wer -vvs - )11)
ac 8
= -2 2)(; - 23t - y+2x (2" zú; =² (zurr + .15 - - ))) = -2( 2))===)]]]) 2)) 10,-010000 -2 y > +,y w())
Where 8= =x Setting the derivative to zero, we obtain the following update rule:
i -
-eT y >y (zur (vf - -v)))
For SimPONel, we perform alternating updates of wi, and πt fixing the other estimate.
A.7 Summary of Datasets
The Infant Health and Development Program (IHDP) is a randomized controlled trial designed to assess the impact of physician home visits on the cognitive lest performance of premature infants. The dataset exhibits selection bias due to the deliberate removal non-randon subsets of treated individuals from the training data. Since outcomes are observed for only one treatment, we generate both observed and counterfactual outcomes using synthetic outcome generation function based on the original covariates for both treatments, making the dataset suitable for causal inference.
IHDP.
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Table 8: Effect of varying training sizes on CATE using the IHDP dataset. We experiment with training proportions of 10%, 25%, 50% and 75% of the full training set. Results demonstrate how the performance of SimPONet and baseline methods evolves with varying amounts of training data.
Training Percentage
0.10
0.25
0.50
0.75
RNet
3.08 (0.08)
2.52 (0.07)
2.30 (0.12)
2.30 (0.14)
XNet
2.43 (0.09)
1.90 (0.12)
1.16 (0.19)
1.10 (0.35)
DRNet
3.28 (0.04)
2.01 (0.03)
1.05 (0.40)
1.05 (0.44)
CFRNet
1.41 (0.03)
1.06 (0.21)
0.95 (0.59)
1.02 (0.50)
FlexTENet
2.97 (0.08)
2.33 (0.07)
1.04 (0.40)
1.03 (0.49)
DragonNet
3.48 (0.04)
2.02 (0.04)
0.97 (0.53)
1.02 (0.50)
IPW
3.44 (0.03)
1.74 (0.06)
0.94 (0.60)
1.03 (0.49)
NearNeighbor
2.10 (0.19)
1.65 (0.05)
2.65 (0.11)
1.07 (0.47)
PerfectMatch
3.44 (0.03)
2.54 (0.03)
3.32 (0.01)
1.05 (0.45)
PairNet
1.64 (0.13)
1.18 (0.11)
1.09 (0.30)
1.09 (0.36)
SimOnly
1.45 (0.23)
1.45 (0.07)
1.45 (0.15)
1.45 (0.04)
RealOnly
2.28 (0.13)
2.92 (0.07)
0.97 (0.54)
0.86 (0.75)
Simm
3.35 (0.08)
2.01 (0.04)
1.07 (0.31)
1.02 (0.25)
SimPONNet
1.01 (0.00)
0.93 (0.00)
0.95 (0.00)
0.87 (0.00)
The IHDP dataset includes 747 subjects and 25 variables. While the original dataset discussed in (Shalit et al, 2017) had 1000 versions, our work uses smaller version with 100 iterations, aligning with the CATENets benchmark. Each version varios in the complexity of the assumed outcome generation function, treatment effect heterogeneity, elc. As outlined in (Curth el al. 2021), reporting the slandard deviation of performance across the 100 different seeds is inappropriate. Therefore, we calculate p-values through paired t-tests between our method (SimPONet) and other baseline methods, using SimPONet as the bascline for all experiments. We follow setting D of the IHDP dataset as mentioned in (Curth & van der Schaar 2021] where response surfaces are modified to suppress the extremely high variance of potential outcomes in certain versions of the THDP dataset.
ACIC. The Atlantic Causal Inference Conference (ACIC) competition dataset (2016 consists of 77 datasets, all containing the same 58 covariates derived from the Collaborative Perinatal Project. Each dataset simulates binary treatment assignments and continuous outcome variables, with variations in the complexity of the treatment assigmment mechanism, treatment effect heterogeneity, the ratio of treated to control observations, overlap between treatment and control groups, dimensionality of the confounder space, and the magnitude of the treatment effect.
All datasets share common characteristics, such as independent and identically distributed observations conditional on covariates, adherence the ignorability assumption (selection observables with all confounders measured and no hidden bias), and the presence of non-true confounding covariates. Of the 77 datasets, we selected a subset of three: versions 2, 7, and 26, aligning with the CATENets benchmark. These versions present non-linear covariale-to-ontcomee relationships and maximum variability in treatment effect heterogeneily. Version 2, notably, exhibits no heterogeneity, meaning the treatment effect constant across all individuals. However, accurately estimating outcome differences even for this version challenging due to the inherent noise in potential outcome realizations in the dataset.
A.8 Experiments with Limited Training Data
In this experiment, we evaluate the performance of CATE methods with varying training sizes. We use 10 randomly selected dataset versions of the IHDP dataset. For each version, the methods are trained using 10%, 25%, 50%, and 75% of the training data. Importantly, the simulator dataset is not subsampled, so the SimOnly model maintains the same CATE error across all training sizes. The results are presented Table S At
https:///jemiferhhiil wwwp:rhhihii......ttee vixsite. wwtps/cccceic/////ceccaaa 20t.iiccccotttoootttttiiio
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Published in Transactions on Machine Learning Research (01/2025)
extremely low training sizes (10%), SimPONet achieves the lowest CATE error with significant margin over all baselines. With 25% of the training data, the baseline methods improve, but SimPONel continues to deliver the best CATE error. At 50% training size, the baselines further improve and perform comparably to SimPONet. While SimPONet's CATE error increases slightly in this sctting, remains close, trailing the best-performing baseline by only 0.01. Finally, at 75% training size, SimPONed's CATE error decreases significantly, while many baselines platean performance. Overall, SimPONet demonstrates exceptional robustness in estimating CATE, particularly in limited training data scenarios.
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Published in Transactions on Machine Learning Research (01/2025)
A.9 Table of Symbols
Symbol
Definition
Real post-treatment covariates: Random Variable
Y
Real outcomes: Random Variable
X
Simulator post-treatment covariates: Random Variable
ys
Simulator outcomes: Random Variable
T
Treatment: Random Variable
Z
Latent (unobserved) pre-treatment representations: Random Variable Observatooal training dataset from Real DGP
Dun
D syn
Counterfactual dataset from Simulator DGP
Dua
Test dataset from Real DGP
Realisations of random variables X,X ,,T,Y,Y' respectively
X
Space post-treat covariate values: Set
T
Space of treatment values: Set {0.1}
Space latents: Set
y
Space of outcomes: Set
Rg
Dimensions of vector spaces which 2,X lie
(i(t)
Potential outcome for 8th unit under treatment f
(,(t)
Potential post-treatment covariate for 5th unit under treatment
Mapping from Z-*X, transforms latents to real post-treatment covariates under t Mapping from -+ transforms latents to simulated post-treatment covariates under Mapping from X'+ Z, transforms real post-treatment covariates under to latents Mapping X 2, transforms simulated post-treatment covariates under to latents Probability distribution latents
g
ft
Pz
Outcome function for real data under
Outcome function for simulated data under t
Conditional Average Treatment Effect for real data, fi Ao- A Mapping Z-y Conditional Average Treatment Effect for simulated data, / uh Mapping 2++
Composition of functions
Tx (w.t)
Conditional Average Treatment Effect for real data, fofe( Mapping X
Conditional Average Treatment Effect for simulated data, o (x=",, Mapping k Diffeomorphic transformation, arises due to contrastive learning
h
Sd
Unit-norm hypersphere dimension Subset (dd-1)
dzlt
Expected squared-distanee between two functions on P(X[T), see Section for definition
d,
Expected snuared-distanee between two functions on Pz, see Section 3 for definition
dnis)
d. under transformation f on see Section for definition
sim
Cosine similarity
Estinate for fe Estimate for f
74
Estimate for Pr
25
Estimate for p
f
Estimate for f recovered from contrastive learning
p
Estimate for μa on recovering Simulator DGP
ÉCATE
CATE estimation error
CCTTE El.
CATE estimation error on covariates under treatment
Factual error on treatment samples
Eor
Counterfactusl error on treatment i samples
K₃
Lipschitz constant for He-fi
K,
Lipschitz constant for 7.F
Lipschitz constant for /6 Pi f6
K K,a
Lipschitz constant 7 5 $7.5
R.
for linear DGP: Matrix
S.
of for linear DGP: Matrix
for linear DGP: Vector
"b for linear DGP: Vector
w.
T for linear DGP: Vector
for linear DGP: Vector
27


Reviews:
[{"review_id": "fNl7eRw8t2", "reviewer": "Reviewer_6Vao", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "*Overall Strengths*\n\nThe paper is clearly written (with a couple of minor exceptions, see below) and the idea is well-motivated and presented. The implementation of the idea itself seems sound and I appreciated the investigations concerning the use of simulated data in the context of CATE estimation. The authors evaluated on multiple datasets, including the well-known IHDP and ACIC benchmark variants, and highlight that the approach, as well as the variations of the approach, have results with are consistent with expectations and well explained. Furthermore, the authors have associated code which includes the evaluations. I think this is especially valuable given that evaluation of causal methods is somewhat variable (not falling into the traditional supervised learning paradigm) - in my view it is not enough to say which datasets were evaluated on with what metrics, and there are often details which one only discovers by reading the code itself.\n\n*Some other miscellaneous positives:*\n\nThe presentation of the three approaches in section 4 was appreciated - added a good intuition for the relative strengths and weaknesses before presenting SimPONet. \n\nFor RQ2 I think the use of normalising flows is a good choice in this instance.\n\n*Some questions / points of discussion:*\n\nRelatively minor point, and I’m not confident about the best solution, but a t-test seems inappropriate given the shared variance arising from the use of the same datasets for evaluation. I think this is talked about here: https://ieeexplore.ieee.org/document/6790639 and some solutions are also offered. Interested to hear the opinion of the authors here. I think the McNemar’s test is OK for paired comparisons, and possibly applicable here.  In any case, if the authors wish to take the differences between results seriously, it might be worth exploring the options a bit. \n\nOn a related (also minor) note, and I  very well might’ve understood something here, but you say that you ran a one-sided t-test, with SimPONet as the baseline where low p values favour SimPONet.  I just want to clarify that this presumably means that if the effect size of the comparison method is in the opposite direction to that which the test assumes, the p-value will be high even though the method is worse than SimPONet. \n\nI was also wondering whether the authors can speak to any possible consequences of assuming Y is drawn from N(.,.).\n\n*Some Weaknesses*\n\nI think my biggest concern relates to the benchmarks / evaluations. To clarify, this is not a serious weakness, and I feel like they are sufficiently convincing, but could be stronger. I believe Curth et al.,  discussed the limitations associated with these benchmarks in the same paper the authors reference with regards to the use of standard deviation metrics.  Sadly I don’t have a good alternative, except for adding more of them and thereby ascertaining some kind of consistency. I think the field is generally lacking a good solution here. Perhaps the authors could comment on why, for instance, they might or might not consider Jobs and Twins, for instance (I can think of good and bad attributes thereof).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors tackle the challenging problem of CATE estimation when only post-treatment covariates are available. Identifiability is not possible in this situation, since a backdoor path exists between treatment and outcome. They propose a solution SimPONet which involves the simulation of counterfactual data, and in turn they derive a generalisation bound showing how simulated data can affect CATE estimation error as a function of the degree of discrepancy between the simulated and real data. They show the degree to which the simulated data improves on observational data alone when being used to estimate the CATE, and provide a range of interesting related results to highlight, for example, the impact on the CATE error when only the post-treatment covariates are used.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It’s 90% clear to me what the numbers are in the results. e.g. for 2.82(0.27) in Table 1, so 0.27 is the p-value, and 2.82 is  the mean-squared error on the CATE? Maybe just add some words in the relevant table and/or figure  caption to confirm this would help. If the authors could also briefly explain the colour coding in the captions this would also increase clarify.\n\nFinally, I wonder if I misunderstood something, but if the method enables one to infer the pre-treatment Z and this can be used in other methods, I feel the authors could make a bigger emphasis of this strength - such a pre-treatment Z could be useful for other downstream tasks, perhaps?\n\nI’m not sure why related work is in the appendix. This is the opportunity to situate the contribution with respect to other peoples’ works, and also to build a strong motivation for what is missing in this prior work. Maybe there is, at least, a compromise to be made here.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "33uVuCcwWT", "reviewer": "Reviewer_D6nV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n1. The paper introduces a unique approach to post-treatment CATE estimation, highlighting limitations in existing literature and demonstrating how the proposed method addresses these challenges.\n\n2. Clear, step-by-step construction and rigorous proof showing how the proposed loss bounds the true CATE error.\n\n3. Intuitive demonstrations of the contribution of each regularization terms through data.\n\nWeakness:\n1. A potential overstatement of the scope of the paper. it said: $\\textit{In this paper, we address a more realistic scenario where both covariates and outcomes are gathered after treatment}$. While post-treatment covariates are challenging, substantial numbers of non-randomized trials and studies where selection bias arises from pre-treatment covariates that influence treatment choices.\n\n2.  Limited practical value: the method may improve CATE estimation on existing data compared to current approaches. However, in many real-world applications, we aim to use learned functions to guide treatment recommendations for patients or individuals. In these cases, predictions are typically based on pre-treatment covariates.\n\n3. The paper assumes a diffeomorphism of $\\( g(t) \\)$, making CATE identifiable for pre-treatment methods by establishing a bijective mapping $\\( Z \\leftrightarrow X \\)$. Many existing methods apply separate functions to treatment and control groups, allowing conditioning on $\\( X \\)$ to serve the same purpose as conditioning on $\\( Z \\)$. This assumption makes CATE identifiable within the paper’s data generation framework.\n\n4. For generality, pre-treatment can be considered a special case of post-treatment by assuming $X$ remains unchanged regardless of the treatment given. It’s promising that your proposed loss can bound the CATE. However, many causal methods that learn a latent space also establish bounds. Could you provide an apples-to-apples comparison showing that your theoretical bound is tighter in certain cases? For example, you could compare it with a classic method, such as CFRnet, which you cited in your paper.\n\n5. In many cases, the simulator seems overly ideal, as on page 10 where $\\tau^S(z) = \\tau(z) + (\\sigma(\\tau) \\gamma_\\tau \\cdot z^T \\omega_\\tau^S)$, which is also unbiased. To make the simulation more realistic, consider using a VAE approach to learn the observed data distribution. Otherwise, it seems like the model is being fed additional information, naturally giving it an advantage over other methods.\n\n6. On page 10, you mention that for other competing methods, which were not developed for post-treatment settings, you use a learned $f_t(x)$. However, this might create an unfair comparison since these methods are not designed for this purpose and may therefore perform suboptimally. Given that pre-treatment covariates are a special case of post-treatment covariates, could you compare your method with others using only pre-treatment covariates? While your method doesn’t need to outperform them, demonstrating its robustness would provide valuable insight.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper offers an intriguing new perspective on CATE estimation by addressing a previously overlooked challenge. While most prior work assumes access to pre-treatment covariates to model CATE, this paper focuses on the scenario where only post-treatment covariates are available, rendering traditional assumptions invalid and CATE non-identifiable. By incorporating an off-the-shelf simulator, the authors demonstrate how it can be combined with a novel loss function to address this challenge, presenting a unique data-generating and learning process tailored to this setting.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would like the authors to address my questions in the weakness sections.\nSome minor suggestions:\n\n$\\newline$\n\n1. Consider adding a footnote to your first table explaining the metric and the meaning of the brackets, as these details are currently only mentioned in the Neural Architecture and Hyperparameters section, which readers may have difficulty locating.\n2. Another advantage of your method may be its robustness with small sample sizes. By using the simulator to augment the sample size, it can improve performance and stability. It would be insightful to see this effect demonstrated by reducing the training data size to observe how your method's advantage over others changes.\n3. An interesting experiment would be to test how the model's performance is affected when using an entirely uninformative simulator, to see if it can remain robust. In practice, simulator quality isn’t always guaranteed, so understanding the model's behavior in worst-case scenarios could provide valuable insights.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "b1wK4kFsNg", "reviewer": "Reviewer_q6UZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n \n- The work is presented clearly, and it is easy to follow.\n- Clear description of the problem that is tackled, the challenges it presents. \n- I like how the authors start from naive strategies and show why these can be suboptimal, to then use these insights to motivate the proposed method.\n- A combination of theoretical and empirical results aiming to validate the method.\n\n## Weaknesses\n\n- The assumptions underlying the methodology need to be explained more clearly.\n- It is not clear ow the method and benchmarks were tuned (or should be tuned).\n- The experimental analysis needs to be improved.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper focuses on estimating treatment effects based on covariates that are collected after treatment. This setting can appear in reality and brings some unique challenges, but has not been analyzed in prior work on ML for CATE estimation.\n\nThe authors analyze this problem. Starting with a general impossibility result, they describe the necessary assumptions for solving it. Different general strategies are compared. The insights from this comparison lead to a novel method, SimPONet. The strategies and method is analyzed empirically and compared to other benchmarks.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Presentation\n- Some more practical, real-world examples of the considered setting would be appreciated. I feel like this would make the importance of the proposed work more clear.\n- The work makes several assumptions, but does not discuss the feasibility of these assumptions in reality. Some examples:\n  - $g_t$ being diffeomorphic is a strong assumption, which I could imagine not always being reasonable in practice. A discussion of some real cases where this assumption is violated would be useful. Additionally, an experiment analyzing your method under violations of this assumption would be insightful.\n  - Why do you assume a dirac-delta and normal distribution for the DGP (Section 3)? These assumptions seem arbitrary and are not discussed.\n\n## Tuning\n- In my opinion, the authors do not give practical advice for tuning the loss weights $\\lambda_f$ and $\\lambda_g$ in Section 4. The default values are chosen in what seems to be an arbitrary manner. At the very least, more experiments need to be performed to judge the sensitivity of your method towards these hyperparameters.\n- How were the neural network architectures and training procedures tuned (for your method and the benchmarks)? \n\n## Experimental results\n- Sensitivity analyses for the loss weights as described above.\n- Why are three iterations of ACIC selected, instead of running the methods over all iterations, as is standard practice? I would understand differentiating between constant CATE and heterogeneous effects for ACIC, but that does not exclude any iterations.\n\n___\n### Minor points\n- Abstract: \"More realistic scenario\": I think both collecting covariates pre- and post-treatment are realistic. There is no need to argue that one is more realistic than the other.\n- p. 19: \"Please refer [to] (Pearl, 2015) ...\"", "broader_impact_concerns": "/", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers were unanimous in leaning toward acceptance of this work. All appreciated the novelty of the formulation and approach, and the revisions made to the manuscript following the review.
Decision: Accept (poster)