Representation Balancing with Decomposed Patterns for Treatment Effect Estimation
Authors: Yiyan HUANG
WANG Siyi
Cheuk Hang LEUNG
Qi WU
Dongdong WANG
Zhixiang Huang
Venue: Submitted to ICLR 2023 (2023)
Keywords: Treatment Effect Estimation
Counterfactual Estimation
Representation Balancing
Selection Bias
Covariate Shift

Abstract: Estimating treatment effects from observational data is subject to a problem of covariate shift caused by selection bias. Recent studies have attempted to mitigate this problem by group distance minimization, that is, balancing the distribution of representations between the treated and controlled groups. The rationale behind this is that learning balanced representations while preserving the predictive power of factual outcomes is expected to generalize to counterfactual inference. Inspired by this, we propose a new approach to better capture the patterns that contribute to representation balancing and outcome prediction. Specifically, we derive a theoretical bound that naturally ties the notion of propensity confusion to representation balancing, and further transform the balancing Patterns into Decompositions of Individual propensity confusion and Group distance minimization (PDIG). Moreover, we propose to decompose proxy features into Patterns of Pre-balancing and Balancing Representations (PPBR), as it is insufficient if only balanced representations are considered in outcome prediction. Extensive experiments on simulation and benchmark data confirm not only PDIG leads to mutual reinforcement between individual propensity confusion and group distance minimization, but also PPBR brings improvement to outcome prediction, especially counterfactual inference. We believe these findings are heuristics for further investigation of what affects the generalizability of representation balancing models in counterfactual estimation.

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REPRESENTATION BALANCING WITH DECOMPOSED PATTERNS FOR TREATMENT EFFECT ESTIMATION
Anonymous authors Paper under double-blind review
ABSTRACT
Estimating treatment effects from observational data is subject to a problem of covariate shift caused by selection bias. Recent studies have attempted to mitigate this problem by group distance minimization, that is, balancing the distribution of representations between the treated and controlled groups. The rationale be- hind this is that learning balanced representations while preserving the predictive power of factual outcomes is expected to generalize to counterfactual inference. Inspired by this, propose new approach to better capture the patterns that contribute to representation balancing and outcome prediction. Specifically, we derive theoretical bound that naturally ties the notion of propensity confusion to representation balancing, and further transform the balancing Patterns into De- compositions of Individual propensity confusion and Group distance minimiza- tion (PDIG). Moreover, we propose to decompose proxy features into Patterns of Pre-balancing and Balancing Representations (PPBR), as it is insufficient if only balanced representations are considered in outcome prediction. Extensive experi- ments on simulation and benchmark data confirm not only PDIG leads to mutual reinforcement between individual propensity confusion and group distance mini- mization, but also PPBR brings improvement to outcome prediction, especially counterfactual inference. We believe these findings are heuristics for further in- vestigation of what affects the generalizability of representation balancing models in counterfactual estimation.
INTRODUCTION
In the context of the ubiquity of personalized decision-making. causal inference has sparked a surge of research exploring causal machine learning in many disciplines, including economics and statis-
tics (Wager& & Athey 2018 Athey & Wager 2019 Farrell| 2015) Chernozhukov et al. 2018) Huang
et al. 2021p, healthcare (Qian et al. 2021] Bica et al. 2202 b), and commercial applications (Guo al. 2020ble Chu [ The main problem of causal inference is the treatment effect timation, which is tied to a fundamental hypothetical question: What would be the outcome if one received alternative treatment? Answering this question requires the knowledge of counterfactual outcomes, but they can only be inferred from observational data, not directly obtained.
Selection bias presents a major challenge for estimating counterfactual outcomes (Guo et 2020a Zhang et al. 2020 Yao et al. 2021). This problem is caused by the non-random treatment assign- ment, that is, treatment (e.g., vaccination) is usually determined by covariates (e.g., age) that also affect the outcome (e.g., infection rate) (Huang et al. 20226). The probability of person receiving treatment well known as the propensity score, and the difference between each person's propen- sity score can inherently lead to covariate shift problem, i.e., the distribution of covariates in the treated units is substantially different from that the controlled ones. The covariate shift issue makes it more difficult to infer counterfactual outcomes from observational data (Yao et al. 2018 Hassanpour & Greiner 2019a).
Recently, a line of representation balancing works has sought to alleviate the covariate shift problem by balancing the distribution between the treated group and the controlled group in the representa- tion space (Shalit al. 2017: Johansson et al. 2022). The rational insight behind these works is that the counterfactual estimation should rest on the accuracy factual estimation while enforcing min- imization of distributional discrepancy measured by the Integral Probability Metric (IPM) between

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the treated and controlled units. However, there are two issues that remain to be resolved. First, Wasserstein distance (Cuturi & Doucet 2014) is the most widely-adopter metric for group distance minimization (Shalit al.] 2017 Huang et al. 2022a Zhou et all 2022). whereas H-divergence
has still received little attention in causa representation learning though it is an important distance metric in other fields (Ben-David et al. 2006 2010). Second, enforcing models learn outcome predictors with only balanced representations may inadvertently weaken the predictive power of the outcome function (Zhang et al. 2020 Assaad al. 2021 Huang et al. 2022a). We provide intuitive examples to illustrate the second issue in Section A.4
The aforementioned issues motivate us to explore approaches (i) improving factual outcome pre- diction without affecting learning balancing patterns or (ii) learning more effective balancing pat- terns without affecting factual outcome prediction. In this paper, we propose a new method, DIGNet, with learning decomposed patterns to achieve these two goals. The contributions are threefold: (1) We interpret representation balancing as concept of propensity confusion and derive correspond- ing theoretical results based on H-divergence to ensure its rationality; (2) DIGNet transforms the balancing Patterns into Decompositions of Individual propensity confusion and Group distance min- imization (PDIG) to capture patterns beneficial to representation balancing, and we empirically find that the PDIG structure enables individual propensity confusion and group distance minimization to reinforce each other without affecting factual outcome prediction; (3) DIGNet decomposes repre- sentative features into Patterns of Pre-balancing and Balancing Representations (PPBR) to preserve patterns contributing to outcome modeling, and we experimentally confirm that the PPBR approach brings improvement to outcome prediction without affecting learning balancing patterns.
1.1 RELATED WORK
The presence of a covariate shift problem stimulates the line of representation balancing works (Johansson et al.] 2016 Shalit et al.) 2017: Johansson et al. 2022). These works aim to balance the distributions representations between treated and controlled groups and simultaneously try to maintain representations predictive of factual outcomes. This idea is closely connected with domain adaptation. In particular, the individual treatment effect (ITH) error bound based on Wasserstein distance is similar to the generalization bound in BBnn-aavdd et al. (2010); [Lngg et al. (2014); Shen al.] (20T8). In addition to Wasserstein distance-based model, this paper derives a new ITE error bound based on H--iivereence (Ben-David et al. 2006 2010 Ganin et al.) 2016). Note that theoretical results (Section 3.2 and experimental implementations (Section 4.1p differ greatly from Shalit et al. (017) due to distinct definitions between Wasserstein distance and H-divergence.
Another recent line of work investigates efficient neural network structures for treatment effect es- timation. Kuang al. (2017); Hassanpour & Greiner (2019b) extract the original covariates into treatment-speciffc factors, outcome-specific factors, and confounding factors; X-learner (Künzel ett al., 2019) and R-learne (Nie & Wager 2 are developed beyond the classic S-learner and T-learner; Curth & van der Schaar (2021) leverage structures for end-to-end learners to counteract the inductive bias towards treatment effect estimation, as motivated by Makar et al. (2020).
The proposed DIGNet model is built on the PDIG structure and the PPBR approach. The PDIG structure is motivated by multi-task learning, where we design framework incorporating two spe- cific balancing patterns that share the same pal-bblanning patterns. The PPBR approach is inspired by Zhang et al. (2020); Assaad et al. (202T): |Hanng et al. (2022a), where the authors argue that im- properly balanced representations can be detrimental predictors for outcome modeling, since such representations can lose the original information that contributes to outcome prediction. Other rep- resentation learning methods relevant to treatment effect estimation include Louizos et al. (2017);
Yao et al. (2018): et al. (2018); Shi et al. (2019) Du et al. (202I).
2 PRELIMINARIES
Notations. Suppose there are the N i.i.d. random variable samples D [XX,, T,, Y)), with
observed realizations {xx, 44, y:)) J where there are Ni treated units and No controlled units. For each unit i, X, X C Rd denotes d-dimensional covariates and T; E {0, 1) denotes the binary treatment, with el e(x,) := p(T, = 1 X,= = x, defined as the propensity score (Rosenbaum & Rubin 1983). Potential outcome framework (Rubin) defines
defines the potential outcomes yo Y
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for treatment T 1 and T = 0, respectively. We let the observed outcome (factual outcome) be Y= = T- + (1 -) T) Y Fort E {0,1}. let 7' := E 2 X = x] be a function of Y w.r.t. X,
then our goal to estimate the individual treatment effect (ITE) T r(x) == IR [Y] - yo X = = =
T' (xx - (x)) and the average treatment effect (ATE) TATE == E [Y- y = Sx (x))p(x)dxx
2.1 PROBLEM SETUP
In causal representation balancing works, we denote representation space by R C Rd and + : X' + R is assumed to be a twice-differentiabl. one-to-one and invertible function with its inverse x. The densities of the treated and controlled covariates are denoted by pxx T-- p-ppp''''__ (x) == p(x T = 1) and pー= = - := p(x T = 0), respectively. Correspondingly, the densities of the treated and controlled covariates in the representation space are denoted by p$= T=1
: X such that V (x(x))
pt 1 (r) == (r T 1) and pi
of =0 pt (r) = pr T = 0),
respectively.
Our study is based on the potential outcome framework (Rubin) 2005). Assumption states standard and necessary assumptions to ensure treatment effects are identifiable. Before proceeding with theoretical analysis, we also present the necessary terms and definitions in Definition
Assumption (Consistency, Overlap, and Unconfoundedness) Consistency: If the treatment is t, then the observed outcome equals Y Overlap: The propensity score is bounded away from 0 to 1: 0 < e(x) < Unconfoundedness: Y IL T X, Vt E (0.
Definition Let h R {0, 1 > y be an hypothesis defined over the representation space such that hi (x()) t) estimates y². and L y x R be loss function (e.g. L(y, (y (y--1)²², If we define the expected loss for (x, 1) as (x, - Sy. L(y', (x((xx, !))p(s (x)dy, we then have factual and counterfactual losses, as well as them on the treated and controlled:
f (x,4(x, ()(xx) t) t)dxdt,
EF 2)
[ExtO.) lx, ())(xx t)dxdt,
ECF(h, Φ)
[xx(0.1)
(1-1(1,9) 1) = 1x (x 1)p""
--" (h, = / (xx=
6F
T=1 (h, b)- f (x,
[ (x.0(x, (x)==((x)dxx
TT (h, b)-
If let f(x,t) f) be h h(T(x), then the estimated ITE over f is defined as (x) ;= f(x, 1) f(x,, 0). Finally, better treatment
t), where f (0, + y is prediction function for outcome,
effect estimation can be reformulated as smaller error in Precision in the expected Estimation of
Heterogeneous Effect (PEHE):
EPEHE = -Sx L(fr(x);
(1)
Here, (PEHE(J) can also be denoted by <PENE(h, b) if we let f(x, 1) be h()((x),
3 THEORETICAL RESULTS
In this section, we first prove é épEhE is bounded by er and fCF in Lemma ! Next, we revisit the upper bound (Theorem concerning the group distance minimization guided method in Section Section 3.2 further discusses the theoretical results of the proposed individual propensity confusion guided method in Theorem 2 Proofs and additional thcoretical results are deferred to Appendix.
Lemma Let functions h and 4 be as defined in Definition 7 and L be the squared loss function. Recall that 7'(x) = E y X=x), x). Defining o² min oe (p(x, t)), (p(x, - -t)} Vt. E
{0, 1}, where oz (p(x,1) ()) = Sxx - (x))) [x)x)xxxxxxxxx,,,xxx1x)))))))xx' 1)dy' dxdl, we have
(F 4) < 2(ccr(h, Φ) cF ch, 1) 20
Note that similar results will hold as long L takes forms that satisfy the triangle inequality, but is not limited to the squared loss. For instance, we give the result for absolute loss in Lemma i Appendix. This extends the result shown in Shalit et al. (2017) that L only takes the squared loss.
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3.1 GNET: GROUP DISTANCE MINIMIZATION GUIDED REPRESENTATION BALANCING
Previous group distance minimization guided approaches seek representation balancing by minimiz- ing the distance measured by the Integral Probability Metric (IPM) defined in Definition Definition Let & be a function family consisting of functions g : + R. For a pair of distribu- tions p1 P2 over S, the Integral Probability Metric is defined as
IPMg(P1, P2) : =supgel g(s)(oi(s)) - pp(s))ds)
If G is the family of 1-Lipschitz functions, we can obtain the so-called 1- 1-Wasserstein distance, denoted by W (ass(p11.00 P2) (Sriperumbudur et al. 2012). Next, we give the bounds for counterfactual error ECF and ITE error ÉPEHE using Wasserstein distance in Theorem
Theorem Let h. D, V, po=1, and p/=0 be defined before. Let L be the squared loss, Pr(T- = 1), and G be the family of I-Lipschitz functions. Assuming that there exists a constant
B& 2 0 such that g (r, == 1 o (r), G for fixed E {0, then have
Bg
ccr(h, 4) S (1 .) ff (h, b) + tuu.c u.F ---0
(h, b) + BB Wassspps 4==1 Pb --0)
(2)
CPEНE (h, 4) & 2 - (h. 4) + cF
T (h, 4) + B W Wass(pp T=1 .p 20
(3)
Theorem D will be identical to Shalit et al. 2017 if L is the squared loss. Note, however, that similar results for Theorem still hold as long as L takes forms that satisfy the triangle inequality. For instance, we give the result for absolute loss in Theorem [ in Appendix. We refer the model as GNet (aka CFR-WaL Wass in Shalit et al. (017)) since it is based on group distance minimization.
3.2 INET: INDIVIDUAL PROPENSITY CONFUSION GUIDED REPRESENTATION BALANCING
The propensity score is recognized central treatment effect estimation because characterizes the probability that one receives treatment (Rosenbaum & Rubin 1983). Therefore, in addition to group distance minimization, the propensity score can be naturally used to identify if representations are adequately balanced, since representation balancing can be intuitively interpreted as propensity con- fusion, that is, when is hard to distinguish whether each unit in the representation space is treated controlled, the representations strate how minimizing H. H--ivergence (Definition is empirically related to propensity confusion. Below, we first derive ITE bound in Theorem based on H-divergence.
thought adequately balanced. In section 4.1 we will demon-
Definition Given a pair of distributions pl, P2 over S, and a hypothesis binary function class H. the H H-divergence between P1 and p2 is defined as
dup(1, P2) == 2supnen Prpp
= Prpa (n()) =
(4)
Theorem Let h. 4. L, P4 T=1 and ph = be as defined before. Let be the squared loss, u := Pr(T = 1), and H be the family of binury functions. Assuming that there exists constant K 2
such that fy L(y. 1/ dyy S K Vy' E V, we then have
K
(cr $)5() (1 - u) of (h. 1) + u OF
dx(p
T-1 T=0).
.P4
(5)
2
K
EPEHE D) S o (h, D)
0 (h, 4) + 2 dn Ep
T-1 -T-00 20
(6)
We can apply similar arguments in proving Theorem 2 to obtain the corresponding upper bounds as long as L takes forms that satisfy the triangle inequality. For instance, the result for absolute loss and the proof of Theorem 2 are given in Theorem 2 in Appendix. We refer to the model as INet
since it based on individual propensity confusion.
METHOD
In this section, we will demonstrate how Theorem 2 is associated with propensity confusion, and further suggest a decomposition network DIGNet based on GNet and INet. Section 4.1 presents the objectives of standard representation balancing models, GNet and INet. Section 4.2 introduces PDIG and PPBR components for the representation balancing with decomposed patterns scheme, and gives the final objective of the proposed model DIGNet.
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4.1 REPRESENTATION BALANCING WITHOUT DECOMPOSED PATTERNS
In representation balancing models, given the input data tuples (x, t, = {(xi, 4, M:) the original covariates x are extracted by some representation function ( and representations ((x) are then fed into the outcome functions h h'(.) == h(:, 1) and (-) == h(- 0) that estimate the potential outcome y¹ and y". respectively. Finally, the factual outcome can be predicted by (²(-))
(-) + (1 - (-), and the corresponding outcome loss is
t,
E ((
If model does not have decomposition modes like GNet and INet, both outcome prediction and representation balancing will rely on the extracted features (xx). Below we will introduce the objectives GNet and INet.
Objective of GNet.
GNet learns the balancing patterns over & by minimizing the group distance loss (x, t; 2 = Wass ((,()))),,, (0,,),t,, If the original covariates x are extracted by the feature extractor 1 (.),,, then the final objective of GNet is
min ggh
&x h²) + (x,
(8)
For the convenience of the reader, we illustrate the structure of GNet in Figure I aa).
Objective of INet. Next, detail how Theorem 2 is related to propensity confusion and give the objective of INet. Let I[(a) be an indicator function that gives a is true, and be the family of binary functions as defined in Theorem 2 The representation balancing seeks to ahieve smaller
empirical H-divergence (p( P4
T== such that
No [x((((x)) = 0J NO
N1 N Σ lx(((x))) -
-1) =l):
dx(r%= T=1 PB T=0) = 2
Σ Est,
min EEW
(9)
The "min" part in equation equation@iindiaateee that the optimal classifier 17" H minimizes the classification emorr between the estimated treatment 17" (*(()))) and the observed treatment 44, i.e., discriminating whether (xx;) is controlled (T 0) or treated (T = 1). As result, dx =f=")) will be large if ry" can easily distinguish whether ((x;) is treated or controlled, i.e., the optimal classification error small. In contrast, dx(p)== T=1 pT=0) will be small if it is hard for 17" to determine whether (xx,) treated controlled, i.c., the optimal classification error large. Therefore, the prerequisite of a small H-divergence is to find a map & such that any classifier E H will get confused about the probability of & (xx;) being treated or controlled. To achieve this goal, we first define a discriminator (r) R - [o, 1] that estimates the propensity score of The classification error for the ith individual can be empirically approximated by the cross-entropy loss between क (T(())) and t,:
(t nt (+)) [t, log N (#(x.) (1 t,) log(1 x (x(x))))
(10)
To minimize the classification error equation we aim find an optimal discriminator such that x² maximizes the probability that treatment is correctly classified of the total population:
max (b, RCM
max RCH
1
Σ ((lx)(((x))))
1 N1
Σ (((l,
(11)
No
Given the feature extractor E-), the objective of INet can be formulated as min-max game:
min max Gg.
t, ty; RE, + (x t; T)
(12)
As stated in equation 12 INet achieves the representation balancing through a min-max formula- tion. In the maximization, the discriminator π is trained to maximize the probability that treatment correctly classified. This forces π (xx)) closer to the true propensity score e(x;). In the min- imization, the feature extractor 4g is trained to fool the discriminator This confuses such
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M(Pa. 1)
1)
M
M
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D
hl
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(a) GNet (CFR-Wass)
(b) INet
(c) DGNet
(d) DINet
(c) DIGNet
Figure Illustrations the network architecture of the five models studied in Section 5
that (x,)(x cannot correctly specify the true propensity score e(x,). Eventually, the represen- tations are balanced as it is difficult for æ determine the propensity of .(x;) being treated or controlled, For the convenience of the reader, we illustrate the structure of INet in Figure Tb). Note that though INet follows the strategy of approximating H-divergence theoretical derivations are completely different due to the significant differences (c.g., definitions, problem settings, and theoretical bounds.) between causal inference and domain adaptation.
Ganin al. (2016), the
4.2 REPRESENTATION BALANCING WITH DECOMPOSED PATTERNS
PDIG. Previous demonstrations have shown that GNet is thriving and widely adopted, while INet meaningful and interpretable. Nevertheless, there is no consensus on the absolute best approach, as each method has its strengths and weaknesses. To this end, we expect to capture more effective balancing patterns by turning the balancing Patterns into Decompositions confusion and Group distance minimization (PDIG). More specifically, the covariates are extracted by the feature extractor & (.). and then (x) are fed into the balancing networks &aa(-) and (-)-)) for group distance minimization and individual propensity confusion, respectively. Finally, the losses for the two separate balancing patterns are
Individual propensity
min Lo(x, t; 2 4E),, min max Cr(x, 11 B.. 4G
(13)
4,
Here, o denotes the composition of two functions, indicating that D(-) in La(x, t; and Cx t; 4, T) are replaced by 2a (+))-)) and 4 (bb())), respectively.
PPBR. Motivated by the discussion in Section 1 we aim to design a framework that is capa- ble of capturing Patterns of Pre-balancing and Balancing Representations (PPBR). First, the rep- resentation balancing patterns &e ( and ())+x(x) are learned over 4G and Φ,, while ER is remained fixed as pre-balancing patterns. Furthermore, we concatenate the balancing rep- resentations (x))) and Φ (x)() with the pre-balancing representations (x as at- tributes for outcome prediction. As a result, the proxy features used for outcome predictions are @ &x ( ( (x))))))))))))))))))))))))())( 4 Φ, (x)),), where @ indicates the concatenation by column. For example, if = [1,2] and b- [3, 4]. then a @ = [,44
Although mainly investigate whether PDIG and PPBR are beneficial to treatment effect estima- tion in this paper, it would also be an interesting direction for future research to find out whether there exists any interaction or mutual reinforcement between them.
Objective of DIGNet. Combining with PDIG and PPBR, we develop new model architecture, DIGNet, as illustrated in Figure T)). The objective of DIGNet is separated into four stages:
min 46
a1 t: 46 o
(14)
max
ax D, fg.
(15)
min Dr
agL $g
(16)
min & t0.
f,(x t,; $ ① (0, Φx) (oo &2),
(17)
Within each iteration, DIGNet manages to minimize the group distance via equation 14| and plays an adversarial game to achieve propensity confusion through equation 15 and equation 16 In equa- tion 17 DIGNet updates both the pre- valabaann and balancing patterns 4g, Φ,, &G along with the outcome function h to minimize the outcome prediction loss.

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DGNet and DINet. For further ablation studies, also propose two models, DGNet and DINet. The two models can be considered as either DIGNet without PDIG, or GNet and INet with PPBR. The structures of DGNet and DINet shown in Figure Tc) and Figure Td), and the objectives of DGNet and DINet are deferred to Section A.6 in Appendix.
(guueeTd).
5 EXPERIMENTS
In non-randomized observational data, the ground truth of treatment effects is inaccessible due to the lack of counterfactuals. Therefore, we use simulated data and semi-synthetic benchmark data to test the performance of our methods and other baseline models. In this section, we mainly investigate two questions: QI. Compared to DGNet and DINet without the PDIG structure, can DIGNet with the PDIG structure achieve a better representation balancing task? Q2. Are DGNet and DINet that involve PPBR capable of improving the performance on outcome prediction compared with standard representation balancing models such as GNet and INet?
5.1 EXPERIMENTAL SETTINGS
Simulation data. Previous causal inference works assess the model effectiveness by varying the distribution imbalance of covariates in treated and controlled groups at different levels (Yao al.) 2018 Yoon et al., 2018, Du et al. (202)). As suggested in Assaad et al. (202T), we draw 1000 observational data points from the following data generating strategy:
p)Ip)), T. X, Bernouli(1/(1 +xp(->)x.))) X, ~0 o². 1
y, 3x + E.
++++& E::
& ~~0,0)
Here, 1p denotes the p-dimensional all-ones vector and I, denotes the identity matrix of size 3
[1.3,..., 1.3) and vary y (0.25, 0.5, 0.75, 1, 2. 3) to yield different level of selection bias. As seen in Figure 6 in Ap- pendix, selection bias becomes more severe with y increasing. For each 7, we repeat the above data generating process to generate 30 different datasets, with each dataset split by the ratio of 20%% 20%%
We fix p 10, 0. = 2. = 00. 0.3)
siininn/validaaioon/ess sets.
Semi-syathetic data The IHDP dataset is introduced by Hill 2011). This dataset consists of 747 samples with 25-dimensional covariates collected from real-world randomized experiments. Selection bias is created by removing some of treated samples. The goal is to estimate the effect of special visits (treatment) on cognitive scores (outcome). The potential outcomes are generated using the NPCI package Dorie (2021). We the same 1000 datasets as used in [Shalit al. (2017). with each dataset split by the ratio of 63%/27% /10% as training/validation///
sets.
Models and metrics. In simulation experiments, we perform comprehensive comparisons be- tween INet, GNet, DINet, DGNet, and DIGNet in terms of the mean and standard error for the following metrics: VEPEHE. VECF, and /EF with L defined in Definition I being the squared loss, as well as the empirical approximations of W aas((pp" T=1 .p T=") and dx Pt =²)= (denoted by Wass and dx, respectively). Note that as shown in Figure Wass is over 4f for GNet while over & for DGNet and DIGNet; du is over 4g for INet while over 4, for DINet and DIGNet. To analyze the source of gain, we fairly compare models by ensuring that each model shares the same hyperparameters, e.g., learning rate, the number of layers and units for (Φg. P.. 1 f*), and (11.a)). Note that we apply an early stopping rule to all models as Shalit et al. (2017) do. In IHDP experiment,
/EPEHE. E. well evaluate performances of various causal models (see them in Table 3). More descriptions of the implementation detailed in Section A.5 of Appendix.
additional metric CATE |FATE TaTEl to
5.2 RESULTS AND ANALYSIS
Varying selection bias. We first make general comparison between the models datasets when the degree of covariate imbalance increases, and the relevant results are shown in Figure 2 There three main observations: (1) DIGNet attains the lowest VEPEHE across all datasets, while GNet

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PERE
VFF
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dig
Wess
DINec
DGNei
3
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DGNNe
30
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DIGNA
DIGNei
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0.75
35
0.5
0.75
1.5
0.25
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0.25
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155
Figure 2: Plots of model performances test set for different metrics as Y varies in (0.25, 0.5, 0.75, 1, 1.5, 2. Each graph shows the average of 30 runs with standard errors shaded.
performs worse than other models; (2) DINet and DGNet outperform INet and GNet regarding Vecr and VEPEHE: (3) INet, DINet, and DGNet perform similarly to DIGNet on factual outcome estimations (Vep), but cannot compete with DIGNet in terms of counterfactual estimations (VECF); (4) DIGNet achieves smaller dy (or W Wass) than DINet and INet (or DGNet and GNet), especially when the covariate shift problem is severe (e.g., when 7 1).
In conclusion, the finding (2) reveals that (i) the PPBR approach improves the predictive power of outcomes, especially for counterfactual outcomes; and the findings (3) and (4) reveal that (ii) the PDIG structure conduces to making group distance minimization and individual propensity confu- sion complementary and nutually reinforcing.
Source of gain. To further investigate the above findings, we choose the case with high selection bias (7 3) to explore the source of gain for PDIG and PPBR. We report model performances (mean + std) averaged over 30 training and test sets in Table and plot specific metrics of 30 runs Below we discuss the source of gain in detail.
on test set in Figure 3 and Figure 4
(1) Ablation study for PDIG: The PDIG structure manifest be effective in cap- turing balancing patterns. According to Figure [ although DIGNet, DINet, and
DRGNNE 18T. BAE BiNer 10
EXGiNer BM 10 D.NG 1IST
DRGNGE 1-94-DO0
DICNN 0405- 0.00
DGNN: 0 10.000
DGNet have comparable estimates of fac- tual outcomes (VEF), DIGNet can achieve more balanced representations regardless of the discrepancy is measured by du or Wass. In particular, by comparing DIGNet with DGNet and DINet in Figure 3. we find that the PDIG structure does not affect the factual outcome estimation (VEF). Nevertheless, DIGNet achieves smaller du with a |1.94/1.96 - 1 = 1.0% reduction (or Wass with a |0.06/0.10 that PDIG enables group distance minimization and individual propensity confusion to complement and reinforce each other, thereby learning better balancing patterns. This bring benefits with regard to counterfactual estimation. In particular, DIGNet reduces /ECF by /2.89/2.99 2.992.99 1| = 2.0% and [2.89/3.08 1| = 6.2% for DINet and DGNet, respectively. Thanks to the efficacy of PDIG in capturing balancing patterns, DIGNet shows superiority in treatment effect estimation (VŪPEHE and (ATE) compared to DGNet and DINet, as seen in Table [
Plots of model performances on test set for VEF, VECF, dx, and W aas when = 3. Each graph plots the metric for 30 runs. Mean + std of each metric averaged across 30 runs are reported on the top.
Figure 3:
40% reduction) compared with DINet (or DGNet). This indicates
(2) Ablation study for PPBR: The PPBR approach plays an essential role in outcome prediction, especially counterfactual inference. From Figure 4 we gain an important insight that the difference in learned representation balancing patterns, measured by W asss (or dz).. between DGNet and GNet DINet and INet), is negligible. This means that PPBR has no impact on the representation balancing task. However, PPBR can improve the predictive power of factual outcomes, reducing /EF by 11.0/ 11.12 - 1| = 4.5% for GNet and [1.07/1.08 1 0.9% for INet. Such improvement is pronounced in counterfactual estimation, where /ECF is reduced by |3.08/3 008/3.5 - 13.2% for GNet and 2.95//3.47 - = 15.0% for INet. Benefiting from the advantage of PPBR, the treatment
effect errors (VEPEHE and eaTg) attained by DINet and DGNet are significantly smaller than those attained by INet and GNet, as shown in Table
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Under review as a conference paper at ICLR 2023
1071001 GNet 1.12+0.03
3.0810.077 GNol: 3.55H0144
0.10100000
DINet 107=001
DiNet: 2.95=0.07
DINer 196::000
GNet 0.1000.00
1.08:0001
3.47
1.96
133
100
Wiss
30
Figure 4: Plots of model performances on test set for VEF, VECF, dx, and W ass when 3. Left graphs compare DGNet with GNet, and right graphs compare DINet with INet. Each graph plots the metric for 30 runs. Mean = std of each metric averaged across 30 runs are reported on the top.
Table 1: Training- & test- set VEPEHE & EATE Table 2: Training- & test- set VEPEHE & EATE
when y = 3. Mean + standard error of 30 runs. on IHDP. Mean + standard error of 100 runs.
Training set
Test set
Training set
Test set
VEPEHE
CATE
VEPENE
CATE
V VEPEHE
CATE
VEPEHE
CATE
GNet
3.30+0.15
2.58:0.14
3.30110.16
2.59:1.114
GNet
0.71+0.11
0.12=0.01
0.7710.18
0.1510.02
INet
3,24=0011
2.46=0.09
3.22+20.12
2.47=0.10
INet
0.66:10.09
0.13=0001
0.72+0.11
0.15+0.02
DGNet
2.86:0.06
2.15=0.03
2.83100077
2.1540.04
DGNet
0.53+00.07
0.11=0.01
0.60=0.09
0.1340.01
DINet
2.70=0006
2.12:0.04
2.6940.08
2.1340005
DINet
0.57+0.12
0.13=0.01
0.60=0.11
0.14+0.01
DIGNet
2.66+0.07
2.04+0.05
2.63++0.77
2.03+00.44
DIGNet
0.42+0.02
0.11+0.01
0.45+0.04
0.12+0.01
Comparisons on IHDP benchmark.
Table Training- & test- set VEPEHE & CATE on IHDP. Mean + standard error of 1000 runs.
We first conduct an ablation study for PDIG and PPBR on 1-100 IHDP datasets and report the results in Table 2 Further, we undergo comparisons between DIGNet and other causal models on 1-1000 IHDP datasets and report the results in Table 3 Note that "_" indicates either the result is not reproducible or the original paper does not report relevant values. Table 2 shows that DINet and DGNet are superior to INet and GNet but inferior to DIGNet concern- ing treatment effect estimation, suggesting that each component of PDIG and PPBR is advantageous for treatment effect estimation. For example, on the test set, DINet reduces VEPEHE by |0.60/0.72 1 = 16.7% for INet, and DIGNet achieves |0.45/0.00 1..500.00- 1| = 25% error reduction regarding VEPEHE & for DINet. This is consistent with the findings before: PPBR and PDIG are ben- eficial to treatment effect estimation.
Triining scl
sct
1473
SATE
CEVAE
2013
2.7
34
35
22
GANITE
191.
05
2
Draguund Sirre 2m1
14-11 01
21 1
30 DS
BNN Lahnnsy 2011
12+
37
42 03
CFR-Wlas CoNe 2017
73
11
81
.15
DELITE 2lai Se
.65 00
CFR:15W Hasdimos Uile
Pu
20 9
100 68
RWCFFOOW Cescaare mi
1071
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18
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43 9
19
RWCFR--rrnn PPr Compine
19 01
MBRI Huung Foram 200e
52
12.4 0]
57
.13 00
DICING (Oun)
42 .0
.11
.48 111
.12:
SableBdemonstraite that models that involve either propensity score or representation balancing (e.g., DKLITE, CFR-X, BWCFR-X. X, and MBRL) attain VEPEHE and CATE of 0.57 0.70 and 0.13 0.19, respectively. Compared to the second-best best method, DIGNet improves performance by |0.45/0.57 - 1| = 21% and |0. 12/0.13 - 1| = 7.7% regarding VEPEhE and e ATE, respectively, revealing the prominent outperformance of the proposed method. Moreover, it is noticeable that DIGNet achieves the lowest errors overwhelmingly across datasets and metrics, indicating that the proposed method has the most robust performance.
6 CONCLUSION
In this paper, we derive a theoretical ITE bound based on H-divergence and connect representation balancing with the concept of propensity confusion. Furthermore, we propose the components of PDIG and PPBR, on which we construct a decomposition network structure DIGNet for treatment effect estimation. Comprehensive experiments verify that PDIG and PPBR follow different path- ways to achieve the same goal of improving treatment effect estimation. In particular, PDIG helps the model better capture representation balancing patterns without affecting outcome prediction, while PPBR preserves patterns predictive of outcomes to enhance the outcome prediction without affecting balancing patterns. We believe that our findings constitute important step towards the generalizability of representation balancing models in counterfactual estimation.
Promising directions for future work include discouraging redundancy of shared information of balancing patterns in the PDIG structure, improving the efficacy of optimizing DIGNet's objective, and exploring whether there exists an interaction between PDIG and PPBR.
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REFERENCES
Serge Assaad, Shuxi Zeng, Chenyang Tao, Shounak Datta, Nikhil Mchta, Ricardo Henao, Fan Li, and Lawrence Carin. Counterfactual representation learning with balancing weights. Interna- tional Conference on Artificial Intelligence and Statistics, pp. 1972-1980. PMLR, 2021.
Susan Athey and Stefan Wager. Estimating treatment effects with causal forests: An application. Observational Studies, 5(2):37-51, 2019.
Shai Ben-David, John Blitzer, Koby Crammer, and Fermando Pereira. Analysis of representations for domain adaptation. Advances in neural information processing systems, 19, 2006.
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wort- man Vaughan. A theory of learning from different domains. Machine learning, 79(1):151-175, 2010.
Ioana Bica, Ahmed M Alaa, Craig Lambert, and Mihacla Van Der Schaar. From real-world patient data individualized treatment effects using machine learning: current and future methods to address underlying challenges. Clinical Pharmacology & Therapeutics, 109(1)887-100, 2a.
Ioana Bica, Daniel Jarrett, Alihan Hüyük, and Mihaela van der Schaar. Learning "what-if" explana- tions for sequential decision-making. International Conference on Learning Representations, 2021b. URL httthhttt/peeeeeeee
net. Etrumr (errumm2dd-Odd3@ww.G
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural pa- rameters. The Econometrics Journal, 21(1):C1-C68, 2018.
Hugh A Chipman, Edward I George, and Robert E McCulloch. Bart: Bayesian additive regression trees. The Annals of Applied Statistics, 4(1):266-298, 2010.
Zhixuan Chu, Stephen L. Rathbun, and Sheng Li. Graph infomax adversarial learning for treatment effect estimation with networked observational data. In KDD, pp. 176-184, 2021. URL https:
Richard K Crump, V Joseph Hotz, Guido Imbens, and Oscar A Mitnik. Nonparametric tests for
treatment effect heterogeneity. The Review of Economics and Statistics, 90(3):389-405, 2008.
Alicia Curth and Mihaela van der Schaar. On inductive biases for heterogeneous treatment effect estimation. Advances in Neural Information Processing Systems, 34:15883- 2021.
Marco Cuturi and Arnau Doucet. Fast computation of wasserstein barycenters. In International conference on machine learning, pp. 685-693. PMLR, 2014.
Vincent Doric. Nonparametric methods for causal inference. https://gihubb.co//vorit
2021.
Xin Du Lei Sun, Wouter Duivesteijn, Alexander Nikolaev, and Mykola Pechenizkiy. Adversarial balancing-basee representation learning for causal effect inference with observational data. Data Mining and Knowledge Discovery, 35(4):1713-1733 2021.
Max H Farrell. Robust inference on average treatment effects with possibly more covariates than observations. Journal of Econometrics, 189(1):1-23, 2015.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Romain--dversarre training of neural net- works. The journal of machine learning research, 17(1):2096-200 20000, 2016.
Ruocheng Guo, Lu Cheng, Jundong Li, Richard Hahn, and Huan Liu. survey of learning causal- ity with data: Problems and methods. ACM Computing Surveys (CSUR), 53(4):1337, 2020a.
Ruocheng Guo, Jundong Li, Yichuan Li, K Selçuk Candan, Adrienne Raglin, and Huan Liu. Ignite: A minimax game toward learning individual treatment effects from networked observational data. In IJCAI, pp. 4534 4540, 2020b.
10

---

Under review as a conference paper at ICLR 2023
Ruocheng Guo, Jundong Li, and Huan Liu. Learning individual causal effects from networked observational data. In Proceedings of the 13th International Conference on Web Search and Data Mining,, pp. 232-240, 2020c.
Negar Hassanpour and Russell Greiner. Counterfactual regression with importance sampling weights. In IJCAI, pp. 5880-5887, 2019a.
Negar Hassanpour and Russell Greiner. Learning disentangled representations for counterfactual regression. In International Conference on Learning Representations, 2019b.
Jennifer Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statisticx, 20(1):117-240, 2011.
Yiyan Huang, Cheuk Hang Leung, Xing Yan, Qi Wu, Nanbo Peng. Dongdong Wang, and Zhixiang Huang. The causal learning of retail delinquency. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 204-212, 2021.
Yiyan Huang, Cheuk Hang Leung, Shumin Ma, Qi Wu, Dongdong Wang, and Zhixiang Huang- Moderately-balancee representation learning for treatment effects with orthogonality information. In Pacific Rim International Conference on Artificial Intelligence, pp. 3-16. Springer, 2022a.
Yiyan Huang, Cheuk Hang Leung, Qi Wu, Xing Yan, Shumin Ma, Zhiri Yuan, Dongdong Wang, and Zhixiang Huang. Robust causal learning for the estimation of average treatment effects. In 2022 International Joint Conference on Neural Networks (IJCNN 2022). IEEE, 2022b.
Guido W Imbens and Jeffrey M Wooldridge. Recent developments in the econometrics of program evaluation. Journal of economic literature, 47(1):5-86, 2009.
Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual infer- ence. In International conference on machine learning, pp. 3020-3029, PMLR, 2016.
Fredrik D. Johansson, Uri Shalit, Nathan Kallus, and David Sontag. Generalization bounds and representation learning for estimation of potential outcomes and causal effects. Journal of Ma- chine Learning Research, 23(166):1550 2022. URL http: jwwp.errg/apprr///2222
19- html
Kun Kuang, Peng Cui, Bo Li, Meng Jiang, Shiqiang Yang, and Fei Wang. Treatment effect estima- tion with data-driven variable decomposition. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 31, 2017.
Sören R Künzel, Jasjeet S Sekhon, Peter J Bickel, and Bin Yu. Metalearners for estimating heteroge- neous treatment effects using machine learning. Proceedings of the national academy of sciences, 116(10):156-41655 2019.
Mingsheng Long, Jianmin Wang, Guiguang Ding, Jiaguang Sun, and Philip Yu. Transfer joint matching for unsupervised domain adaptation. In Proceedings of the IEEE conference com- puter vision and pattern recognition, pp. 1410-1417, 2014.
Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. Causal effect inference with deep latent-variable models. Advances in neural information pro- cessing systems, 30, 2017.
Maggie Makar, Fredrik Johansson, John Guttag, and David Sontag. Estimation of bounds on po- tential outcomes for decision making. In International Conference Machine Learning, pp- 6661-6671. PMLR, 2020.
Xinkun Nie and Stefan Wager. Quasi-oracle estimation of heterogeneous treatment effects. Biometrika, 108(2):299-33 2021.
Zhaozhi Qian, Yao Zhang, Ioana Bica, Angela Wood, and Mihaela der Schaar. Synctwin: Treat- ment effect estimation with longitudinal outcomes. Advances in Neural Information Processing Systems, 14:3178-3118 2021.
11

---

Under review as a conference paper at ICLR 2023
Paul R Rosenbaum and Donald B Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1))41-55, 1983.
Donald B Rubin. Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American Statistical Association, 100(469):322-333) -31, 2005.
Shai Shalev-Shwarts and Shai Ben-David. Understanding machine learning: From theory to algo- rithms. Cambridge university press, 2014.
Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: gener- alization bounds and algorithms. In International Conference on Machine Learning, pp. 3076- 3085. PMLR, 2017.
Jian Shen, Yanru Qu, Weinan Zhang, and Yong Yu. Wasserstein distance guided representation learning for domain adaptation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
Claudia Shi, David Blei, and Victor Veitch. Adapting neural networks for the estimation of treatment effects. Advances in neural information processing systems, 32, 2019.
Bharath K Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Schölkopf, and Gert RG Lanckriet. On the empirical estimation of integral probability metrics. Electronic Journal of Statistics, 6: 6:1550-1599, 2012.
Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 1136523):1228-12222 2018.
Liuyi Yao, Sheng Li, Yaliang Li, Mengdi Huai, Jing Gao, and Aidong Zhang. Representation learning for treatment effect estimation from observational data. Advances in Neural Information Processing Systems, 31, 2018.
Liuyi Yao, Zhixuan Chu, Sheng Li, Yaliang Li, Jing Gao, and Aidong Zhang. A survey on causal inference. ACM Transactions on Knowledge Discovery from Data (TKDD), 15(5): 2021.
Jinsung Yoon, James Jordon, and Mihaela Van Der Schaar. Ganite: Estimation of individualized treatment effects using generative adversarial nets. In International Conference on Learning Rep- resentations, 2018.
Yao Zhang, Alexis Bellot, and Mihaela Schaar. Learning overlapping representations for the esti- mation of individualized treatment effects. In International Conference on Artificial Intelligence and Statistics, pp. 1005-1014. PMLR, 2020.
Guanglin Zhou, Lina Yao. Xiwei Xu, Chen Wang, and Liming Zhu. Cycle-balanced representation learning for counterfactual inference. In Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), pp. 442 450. SIAM, 2022.
A APPENDIX
A.1 PRELIMINARIES
We start by making some assumptions about the distribution concern, and give neces- sary definitions and results. We agree with the strong ignorability assumption, which assume that there exists a joint distribution p(X. T, yo y1) such that conditioning on covariate X, the poten- tial outcomes yo y1 are independent of T, i.e., (Y0,yl) 11 T X, and the propensity score e(x) = TT = X x) is bounded away from to i.e. 0 < e(x) 1. Recall that we also sume consistency, i.e., if the treatment is f, the observed outcome equals Y*. These assumptions are crucial conditions that make individual treatment effect identifiable (Imbens & Wooldridge 2009).
Definition The individual treatment effect ffTE)for unit is:
T(x) E E
yd x x=
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Let (') prediction function.
E| X = x|, we have 7(x) - r'(x) (x).. Let X x {0,1} y be
Definition The individual treatment effect estimate can be defined as:
{y(x) := fxx, 1) -f(x,0) 0). Definition (Hill 2011) Let y x y - R& be loss function. The expected Precision in Estimation of Heterogeneous Effect (PEHE) loss of is:
EPEHE 1 L(+y(xx) (
Definition 4 The covariates' distributions in the treated and controlled groups can be denoted by p T= (x) == p(x = and T (x) == p(x = 0), respectively.
In our causal representation balancing approach, we assume that the representation function + X' + R is a twice- diice-iffferentiable one-to-one function, where R C IR is the representation space. Then, we can denote V + X by the inverse & and the induced distribution of by pa.
Definition 5 The covariates' distributions in the treated and controlled groups over R can be de- noted by p4z =1 (r) = (r(r T and p4 (r) == ps (r T = 0), respectively.
Let h : R x (0. 1 + y be an hypothesis defined over the representation space R, such that f(x, 1) h (x).
Definition 6 The expected loss for the unit and treatment pair (x, () is
(x, t) [ Las h((((x) (x)dy'.
Definition 7 The expected factual loss and counterfactual losses of h and to are, respectively:
(h 4) - en f)p(x, t)dxdt,
- En (x, Op(x, 1 - l)dxdl.
ccr(h, ④)
Definition 8 The expected treated and control losses are:
/ (x, (10"
of (h,
/ (xx
T-, 4)
x
OF (h, ②)-
x ex, 1)p (x)")
T=0 CF (h, 1)
(x (xx (x)dx.
Let Pr([ = 1) be the proportion treated the population. We then have the result:
Lemma
cr 1) = u. u.
T- (h. + + (1 u) of ( Φ) T=
ccr (h, ②) u) CCF (h, b) + T (h, T).
Noting that p(x, (x) (1 u) p (x). the results can be easily obtained from the
Definitions 7 and 8
Definition Let G a function family consisting of functions g S RR. For a pair of distribu- tions Pt, P2 over S, define the Integral Probability Metric:
IPMg =sup] / (P)
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Let G be the family of 1 1-Lipschitz functions, we obtain the so-called 1- Wassersttin distance between distributions, which we denote W "aass(ppL P2) (Sriperumbudur et al. 2012).
Definition Given a puir of distributions pt, P2 over S, and hypothesis binury function class H. the H- H-divergence between P1 and P2 is
(p(P1, P2) 2 Prpp [n(s) = -] Pr In(s) = -]].-
Lemma 2 Let G in Definition 9 be the family of binary functions, we obtain half of H--divergence
Proof Let I()) denotes an indicator function.
du (P1.P2)
[ (pi(s) - p2(s))ds (()==1
=2 sup NEA
-2 sup NE
I[((() 1)(p((s) - P2 (2(2s)
S
[
22 sup E
-(s)(o(()))ppp())d pe(s))
(18)
The last equation is because an indicator function is also a binary function.
A.2 BOUNDS FOR CONTERFACTUAL ERROR ECF
We first derive the counterfactual error bounds when using Wasserstein distance. The following Lemma corresponding proof is identical to the Lemma in (Shalit et al. 2017].
Lemma Let 4 : X' + R be an invertible representation with V being its inverse. Let P4p (r), p = (r) be as defined before. Let h R 10,1) + V. u == Pr(T 1) and G be the family of 1-Lipschitz functions. Assume there exists a constant B 2 0, such that for t = 0, the function (r, B4 [ (r). E G. Then we have:
EC Φ) S (1 u) ef (h, 1) + u ET (h, Φ) BB W Wass(pl T-1
Po -0)
Proof
ECF (h. 4) - [1 x (h,
+ (h, +u.
Φ)
=[(1 u) fFr (h, 4) + u T= (h, 4)]
T=1
u) EF T- (h, 4) + uu- u. T- (h, 4)1
=(1 - u) |ETT (h, 4) cf r- (1, +
+u. (FF (h, 2) - 6f T= (h, D)]
-(1 u) / (x 1)(p (x) - (x) (x, ()(p T=1 (x) - pp" (x)=x
(19)
-(1 - u) /
1/d T=1 (r))drr + / (V)) (r)
p4 T=0 dr
(20)
<Bo (1 uu) u) / (noo 1)(1 (r) - 14 -
1
R Bo
+ BB
(1) T=1 (r) (r)-pp o T-0 (r) (r))dr
R Br
/ g(r)(1z (r)))
<Bo (1 - u) sup sup
T=1 -
gEg
R
+ B& u sup gEG
JR g(r)(15-1(6)) (r)) - p4 T-o (r)
(21)
-Bo Wass(pp T=1 p2 T=0)
(22)
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Equation 19) is by Definition (r) (r)
equation 20 is by the change of formula. pa (r) V (r)) (r), where Ju (r) is the absolute of the determinant
of the Jacobian of (r); inequality 2/ is by the premise that ( E 9 for t 0, and (2 is by Definition of an IPM.
The crucial condition in Lemma is that g0, (r, =- Be l.. E G. Bounds for B can be given to evaluate this constant when under more assumptions about the loss function L, the Lipschitz constants of p(y") [x), h, and the condition number of the Jacobian of Φ. These assumptions and the specific bounds for Bş can be seen in supplement Section A.3 of (Shalit et al. 2017).
Now we turn to derive the counterfactual error bounds for the H-diivergence case.
Assumption There exists a constant K such that for all y2 E y. .yy (1yy,38)ddy S K.
Lemma Let + X + R be an invertible representation with & being its inverse. Let P4p (r), (r) pb (r) be as defined before. Let h R [0,1] + V. u Pr(T 1) and H. be the family of binary functions. Assume loss function L obeys the Assumption 7 Then we have:
K
(c-(t)))((11x))) cf TT (h, 1) + u ep (,, 1) + dx (pb
P4 T=
2
Proof
(cr(h, 0)- - -111 (1)--)-- - (h, @) + cF T (h, +1]
=(1 un) / (V () ( (r) p (r) dr / ln (V().
p4 (r)))r
1R
(23)
<(1 - u)
Ln (V))) (p6- - p3 (r))
3/6 18-0006-1
+
0) T-1 (r) ps T-0 (r))dr
(24)
pb
<(1 u)K
(pg ( 1/
- + a K
(p-=1 (x)
pb -0 (r))drr
1/
(25)
=(1 u)K
I(p's 10
>
p4
T=
(p T-0 T (r)
T T-1 (r))
R
tu. K
T
po
T-=
(pb 1-1 (r)
po (r)
T T=0
<(1 K)KK sup
T=1 (r) ((]
T (r))dr
nEH
R
+ K sup
((()(p'
T=1 (r) - -p- 1-0 (r))dr)
(26)
nEH
R
<K sup <up / nEH
T-1 (r) ((F)
p4 T- (r)) (r)) drl
K
du (pb T=1 . p4 )
(27)
Equation 23 is same to equation 20 equation (24) is by ln 2 0 for all r and inequality
25
is by Definitin T and Assumption inequalit (66 is because an indicator function is also binary
function; equation (77 is by in Lemma
A.3 BOUNDS FOR THE PEHE LOSS &PHE
We first state two lemmas for EPERE e with respect to two different loss functions: the squared loss and the absolute loss. In fact, similar lemmas hold for loss functions that satisfy the (relaxed) triangle inequalities.
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Definition 11 The expected variance of xf with regard to p(x. is:
(p(x t)) = - (0' - (x)) (x)xxxx t)dy dxdt,
and define:
= (p(x, t)), o.y (p(xx, --t)] t))}.
Lemma 5 Let loss function L be the squared loss, Lyy1, 1/2) (y1 y2)2. For any function f:Xx x {0,1} + V, and distribution p(x, 1) over x {0, 13:
(PERE(h, +) - (1rcr(h, 3) + cr(h, +) 1 -207)
Proof We denote {PEHε(f) = <PEHE(h, P), ef(f) eF D). ECF(f) Ecr(h, D) for f(x,t)= h()(x),t)
εPEHE(S)
(1[(x, f(x, - (+' - -
$ f ((f(x, 1) -x)))² (f(x,0) (xx)) (pxx))x
(28)
-2 / (J(x, 1) px))) T 1)dx 2 / (f(x, 0) (xxx -O)dx
x
/ (f(x, 1) - - - (x)) T 0)dx + 2 (f(x. 0) (xx) p(x, T 1)dx
+ x
(29)
/ (f(x, t) - (x)) ()xxxl))xlllllll
-2
(f(x,1) 7' (x)))x(xx 1 l)dxdl.
xx(0,1)
xx(1111
Inequality (28) is because the relaxed triangle inequality, (x y)² < 2(x² + +));; equation 29 is
because p(x) = p(x, T 0) p(x,''11
cf(f)
=I
3xx(0.1xxX (f(x,1) P'y' (>)((())((xx t)dy' dxdt
=I (f(x,t) - (xx)) xx)(xxxxxxpp))))))) t)dy dxdt
+ I JEX(OI)A (F'(x) 2')²p((/' (x) (xx t) dxdt
+2 [xx(011XX (f(x,t) 7' (x)) (x) s y') (x)pxx))(xx))))))))))))))))))( t)dy dxdt
(30)
I xx
>xx)))) l)dxdl + of (p(,
(31)
Equation 31 is by Definition and last term equation
30
equals to zero, since 7'(x)
Jy Ly ply"
x Adyc A similar result can be obtained for 6CF:
S (xx(0.1)
cCF())
t)dxdl + oz 11 1)).
Combining these results and Definition
11 we have
CPEHE ②) < 2(cr(f) (p(x,t)) f))) + 2(ccr(f) -
oz (p(x,1 t)))
< 2(eor(h, @) +e) 2) - 20²)
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Under review as a conference paper at ICLR 2023
For the absolute loss (y1 92) 1311 V/l that satisfies triangle inequality, the upper bound in Lemma 5 will replace the standard deviation oy by mean absolute deviation Ag.
Definition 12 The mean absolute deviation of yf with regard to p(x, () is:
Ay 1)) S w (x))) (x)x"xx) (x)p(x, t)dy' dxdt,
and define:
Ag - min (p(x; ()), Ay (p(x, 1-)))
Lemma 6 Let loss function L be the absolute loss. (V)1 = 181 -22l- For any function f X x {0.1} y, and distribution p(x, t) over X {0, 1}:
EPEHE(,, 4) S ECr(h, 4) + E ②) -2Ay- Proof Recall that EPERE(F) = CPEHE(h, 2) eF(f) = Er(h, D), eCF(f) = ecF(h, D) for tx h(t(x),t).
{PEHE(f)
/ 1(f(x, 1) - f(x, - (x))))x)
$ f (1f(x,1) -1 +x(x) + 0) ( (xx)))x(x)
(32)
[ 1f(x, 1) - (x))xx(x T = 1)dx + / [f(x, 0) xxx))|x(x,T= T 0)dx
( [5(x, 1) -xxx)xx(xx T 0)dx + / 15(x,0) (x)))x(xx 1)dx
(33)
[ 1f(x,t) (xx))x((x t)xxdt + [ 1xx00.11 [f(x,t) t) -x(x))p((x t)dxdt.
xxx00.11
Inequality (22 because triangle inequalit, Jxr vl |ad| + lul: equation 33 is because p(x)
p(x, T =0 0) p(x
1).
cr(
/
xxx0111x f(x,t) (1pps/))p(xx ddyy dxdt
[(x,1) +'x))p(('' 1x)p(x, t)dy dxdl
1x11X,1
| 17'(x) s lp(s x))( ty dxdtt
(34)
(xx(o.1)
lf(x,t) - (x)) t) dxdtt + (p(x, t)).
(35)
Inequality (34) is also because I yl < |x| + vl. equation 35 is by Definition 12 similar result can be obtained for ECF:
ccr(f) = S [f(x - (x) t)dxdt + Ag (p(x, -t)).
Combining these results and Definition
12 we have
EPEHE 2) < EF())
Ag (p(xx, ()) (pxxx)
EcF(h, (h, v) EF(h, 2) -2Ay.
We summarize the upper bounds of Egr and ÉpEhE above, and give the final bounds for these two distance using the squared and absolute loss, respectively.
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Theorem Let ④ : X - be an invertible representation with Vi being its inverse. Let pa ph (r), (r) be as defined before. Let h RR x {0, + Y, == Pr(T = 1) and 9 be the family of 1-Lipschitz functions. Assume there exists a constant B& 2 0, such that for t = 0, 1, the function (I, Am (vr), E G.
Let loss function L be the squared loss, L(yi, 82) (y1 y2)² Then we have:
<PEHE $)
<2(ecr(h, +) er(h, 4) 20 -2)0)
(36)
52(EF- (h, 2) + - ②) + Bx W Wass(p T-1
Pb -) 20
(37)
Let loss function L be the absolute loss, Li y2) = 1y - y2l. Then we have:
<PEHE(h, $)
SecF (h, 4) eF (h, 4) -
(38)
(h, 4) + BB W Wass(pl T=1 PD4 .24-0) - -2 <ef <eT= (h, 1) + c T
(39)
Proof Inequality (36) is by Lemma [ inequality (37) is by Lemmal 7nnd Lemma Inequality (38
by Lemma 6 inequality (39 is by Lemma and Lemma
Theorem Let + X - R be an invertible representation with being its inverse. Let T (r), Ptz T= (r) be as defined before. Let : R x {0.1) - V. u u= Pr(T 1) and H be the family of
binary functions.
Let loss function L be the squared loss such that L(y1, 1/2) = (31 - 82)2². Then we have:
<PEHE(h, 4)
52(ccr(h, +) + cr :) -2
(40)
K
<2(eT (h, 2) + ( 2) + +
T=1 PP ,p/-0) 20 dx(pb=
(41)
Let loss function L be the absolute loss such that L(y1, 92)
81 821.
Then we have:
bPFnr(h, $)
SECF(h, 4) + Er(h, 4) - -2A,
(42)
-c (h, 3) +
①) + 2 dx (p$
T=1 T=0) 2A,,,
PP
(43)
Proof Inequality (40) is by Lemma 5 inequality (47) is by Lemma by Lemma inequality (43) is by Lemma and Lemma 4
1 Nand Lemma Inequality 422
Obviously, when using Wasserstein distance, there are various versions of bounds for different loss functions as long as they satisfy the (relaxed) triangle inequality and assumptions about f, in Theorem T Similarly, when using H-divergence, there also various versions of bounds for loss functions that satisfy Assumption and the (relaxed) triangle inequality.
For an empirical sample and family of representations and hypotheses, can further upper bound T- and of=1 by their respective empirical losses and a model complexity term using standard arguments (halev-Shwartz & Ben-David, 2014). Both the Wasserstein distance and H-divergence can be consistently estimated from finite samples (Sriperumbudur et al.) 2012 Ben-David et al. 2006 2010).
A.4 ILLUSTRATIVE EXAMPLES
Examples for the motivation for decomposed patterns. To explain the dilemma between repre- sentation balancing and outcome prediction, we give two intuitive examples below to help readers better understand the motivation and importance of involving decomposed patterns in representation balancing models.
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0.25
Treated Controlled
Treated
0.5
Controlled
0.20
Adjust covariates
0.4
0.15
0.10
0.05
0.00
0.0
90
90
Covariate
#: Adjusted Covariate
Figure 5: Example for illustrating the importance of decomposed patterns.
Example 1. Suppose there is vaccine to prevent some kind of disease. Let X denote the covari- ate (age). T denote the treatment (getting vaccinated), T 0 denote the control (not getting vaccinated), and Y denote the outcome (probability of getting the disease). Suppose that the vac- cine is assigned according to age, and we have found that the older, the higher the probability of getting the disease. The left graph in Figure 5 shows the distribution of pre-balancing covariate X for treated and controlled groups, which indicates that vaccines are more likely to distribute to older people. Technically, the pre-balancing data preserve the outcome-predictee to estimate Y using the covariate X, we are confident that people in the treatment/control group (orange/blue) are sussepttbbeluusssseppibbblee to the disease since they are older/younger. The right panel of Figure shows the distribution of the adjusted covariate X, over which the distributions of treated and controlled groups are highly balanced. In this case, however, the distribution of X is too balanced, making it hard to distinguish the treatment samples from the control samples. Conse- quently, if we want to estimate Y using X, we may get confused about which group is susceptible to the disease because the distributions of X are almost identical between the treated and controlled groups. Therefore, only considering balancing patterns can result in a loss of outcome-prediction information.
information: if we want
Following the example above, allow us to give a special but more intuitive instance. Imagine two are entirely identical other than age, of whom one is older (treatment, T) and the other is younger (control, C). So we can use the covariate age to distinguish between T and C. We also found that the older one is susceptible to some representations such that their representations are balance, even identical, then such representations can be useless distinguish who is T and who is C. As a result, will be difficult for such representations to be used for estimating who is susceptible to the disease. Therefore, over-balanced representations may lose outcome- preddccive information.
Example 2.
the disease. However, once their ages are mapped
In summary, on the one hand, involving representation balancing can benefit treatment effect es- timation. On the other hand, if p-=1 and p pf=0 are too balanced, model may fail to preserve pre-balancing information that is useful to outcome predictions. Such dilemma motivates us to incorporate PPBR and PDIG such that PPBR improves outcome prediction without harming rep- resentation balancing, and PDIG helps model to achieve more balanced representations without harming outcome prediction.
A.5 ADDITIONAL EXPERIMENTAL DETAILS
Hyperparameters. In simulation studies, we ensure fair comparison by fixing all the hyperpa- rameters in all datasets across different models. The relevant details are stated in Table In IHDP studies, to compare with the baseline model CFR-Wass (GNet), we remain the hyperparameters of INet, DGNet, DINet and the early stopping rule the same as those used in CFR- Wass Shalit et al. (017). Since DIGNet is more complex than other four models, we adjust the hyperparameters of @ g. 2g. 1,, a,, and ag for DIGNet as (Shalit et al. (2017) do. The relevant details are stated in Table 5
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Table 4: Hyperparameters of different models in simulation studies.
4xp
A²
681
batcheiic
auraliun
leaming
leaming pite for
Gnet
[100, 100. 150, 100)
(300. 100)
(100. 100)
0.1
100
300
lc
East
[100, 100, 150, 10D)
[100, 150, 100)
(100. 100)
(100. 100)
0.1
100
300
Lend
DCNN-
[100) 160, 160, 100)
(100) 100)
(100. 103)
(100. 100)
0.1
100
500
DINS
[100. 100, 160, 100)
(100, 100)
(1000 100, 100)
(300. 100)
(100, 100)
0.1
100
500
1sm3
DRGNet
100,
[100) 100)
(100, 186)
100)
(100, 100)
011 0.1
100
16
Table 5: Hyperparameters of different models in IHDP experiments.
Tramning mM
lramiing FMM
Clack
(09G, 298, 2900, L1K))
(200, 565, 194)
(206) 500, 500)
1101
(18)
Bedt
(100) 198. 199. (00)
(206 200 2013
(300) 143, 105)
1100 305 196)
1190
(00)
DINNe
[100:
100. 100)
(100) 1001
(356, 100)
(100 100 196)
1190
(106.
(160)
(103)
100, 100)
(336, 930
(306,
106)
100)
14"2
Device. All the experiments on Dell 7920 with 1x 16-core Intel Xeon Gold 6250 3.90GHz CPU and 3x NVIDIA Quadro RTX 6000 GPU.
Table 6: Additional comparisons on IHDP dataset.
Training set
Test set
VEPERE
VEPERE
OLS/R
Johansson al.
2016
.73
99
OLS/LR Rg
Johansson et ml 2016
24+
.14 001
2.5 +
31
02
Chimp 3r
.14 .01
.79
BART
Chipinan et ml
2010
2.1 .1
.23 1::1.00 001
23
34 02
Athey
.18
CEVAG
Loinzos et ml 201
2.7 1 1
311 01
2.6
466 02
Yao
2018
.22
244
GANITE
Yuon el al. 2018
1.9 10:11
43 00
2.4 4
.99 00
Johanson
.72
5.8
BNN Jolluns el ml
2016
2.2
.37 A03
2.1
42 03
TARNer SSanpl
26
CFR-Wass (GNNo (S SSaar e 2017
73 0
.1 001
81 + 0
15 01
Dragonnet (Shi
132.4
.14
DKLITE ZZaag of 2021
52
.65 05 03
CFR-ISW (Hasshiouur Gremerr 2019a
19
BWCFK-OW (Assin el an 2011
66 +
18= 01
BWCFR-M MW
.19-01
SWCFR-TrnnJPP AAsead 2021
63
19= 01
MRRI. (Huang
.52
.12
.57 + 0
11--00
DIGNNe (Ours)
.42
1
.
.11
111:00
.45 1:
.
.12 .01
Additional IHDP results. The selection bias of different simulated datasets is illustrated in Figure We compare DIGNet with more baselines in Table 6 Note that indicates either the result is not reproducible or the original paper does not report relevant values. We collect baseline methods that focus on treatment effect estimation, especially methods using deep representation learning techniques, from recent machine learning conferences (e.g., ICML., NeurIPS, ICLR, AISTATS, and PRICAD)
6
Analysis for training time and training stability. We record the time it took for different models to run through 100 IHDP datasets, and each model is trained within 600 epochs. Following [Shalit al. (2017), all models adopt the early stopping rule. We also record the average early stopping epoch on 100 runs and the actual time on 100 runs, where (actual time) (total time) x (average early stopping epoch)/600. Not surprisingly, GNet took the least amount of time with 3096 seconds since the objective of GNet is the simplest. However, it is very interesting that the proposed methods, DGNet and DINet, are the first two to early stop. As result, though DGNet and DINet have multi- objectives, they spent less actual training time but achieved better ITE estimation compared to GNet and INet. Since GNet and INet are actually DGNet and DINet with PPBR ablated, we find that PPBR component can help model achieve better ITE estimates with less time. In addition, find that DIGNet spent the longest time to optimize since it has the most complex objective. To further study the stability of the model training, we also plot the metrics EEp, Wass, dx- and /EPEHE for the first 100 epochs of each model on the first IHDP dataset. We find that the training process
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Under review as a conference paper at ICLR 2023
y 0.75
7=1.5
7=2
20
20
20
Figure 6: T-SNE visualizations of the covariates as y varies. Red represents the treatment group and blue represents the control group. A larger y indicates greater imbalance between the two groups.
of DIGNet is stable, even steadier than GNet and INet. From this perspective, haven't 't seen difficulty of optimizing DIGNet.
Table 7: Training time records on 100 IHDP datasets.
Model
Time for 600 epochs
Avg early stopping
Actual time
VEPEEE on test set
GNet
3096s
240.61
1241s
0.77-10.18
INet
4042s
254.19
1712
0.72
DGNet
3775s
169.17
1064s
0.60=:0.09
DINet
3212s
157.98
846s
0.60++0.11
DIGNet
4984s
226.76
1884s
0.45+t0.04
GNet
GNet
INet
INet
2.50
DGNet
DGGNet
4.00
DINet
DINet
DIGNet
DIGNet
2.00
3.00
1.50
2.00
1.00
1.00
60
80
20
40
60
GNet
INet
1.05
DGNet DIGNet
DINet
0.06
DIGNet
1.04
0.04
1.03
0.02
1.02
Wass
dx
0.00
1.01
20
40
60
80
20
40
60
Figure Training loss plots for the first 100 epochs on the first IHDP dataset.
A.6
OBJECTIVES FOR DIFFERENT MODELS
Objective of GNet.
min gg.h
(+) (x
Objective of INet.
max
min bgght
DE, 1²) azLi m).
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Under review as a conference paper at ICLR 2023
Objective of DINet. Note that similar to DIGNet, the pre- rre-balaniing patterns preserved by only updating @ but fixing &g in the second step.
max
a2L1 t; By Φg T); 0; 9F
min
min e Pr
L,(xtt,;; 1 k))
Objective of DGNet. Note that similar to DIGNet, the pre-balancing patterns are preserved by only updating &e but fixing yE in the first step.
min
an t; P6 FE),
ta
min 4g th
yy 2 @ ( o 2)),
Objective of DIGNet.
min
to 4E),
th
max
0
min
o
min o t
Ly(x,t,y; 2g 1 (0) 28) @ (Rc o &g),
22


Reviews:
[{"review_id": "L_HBIUq8aOw", "reviewer": "Reviewer_JQVc", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work seeks to improve upon representation learning modeling for inferring individualized treatment effects which employs a discrepancy constraint between the representations of the treatment and control groups. The authors propose to use H-divergences to achieve this. This is motivated by a bound on the ITE error by an H-divergence. Empirical results show gains relative to prior art. ", "summary_of_the_review": "As I mention above, I think this is a nice, simple idea with good execution. The authors do a nice job of explaining the relative merits of using an H-divergence and provide compelling empirical evidence for their claims as well. While this work may border on incremental in terms of novelty, I think it is still a valuable addition to the literature. ", "strength_and_weaknesses": "The strengths of this paper are in the relative simplicity and efficacy of the proposed approach. The use of H-divergences is both natural and well motivated with theory. I thought the authors did a nice job of presenting both the motivation and solution as well. I think the empirical results are also compelling. \n\nIn terms of weakness, the main issue for me is a relative lack of novelty, though I don't think that is enough to vote for rejection at this time. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "HNHVF6C-Jt3", "reviewer": "Reviewer_eA2v", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a framework for treatment effect estimation that consists of two components. The first is to balance representations by minimizing group distance and maximizing individual propensity confusion (i.e., PDIG). The second is to use both pre-balancing and post-balancing representations for outcome prediction (i.e., PPBR). ", "summary_of_the_review": "This paper proposes a method for treatment effect estimation. This method balances representations, and uses both pre-balancing and post-balancing representations for outcome prediction. This method is demonstrated to have good performance in simulations. ", "strength_and_weaknesses": "Strength: \n\nThe proposed method is supported by two bounds of the expected estimation error of heterogeneous treatment effects. The first bound of treatment effect estimation error uses the Wasserstein distance between treated and control groups. The second bound of treatment effect estimation error uses H-divergence as a measure of propensity confusion.\n\nWeakness:\n\nThe connection between the two components (PDIG and PPBR) could have been better elaborated. For example, how is PDIG helpful for PPBR? Furthermore, it would be nice to provide an example of why it is important to account for pre-balancing representations for outcome prediction. A simple numerical example could be useful to understand intuition. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "6qD3cI_O4F", "reviewer": "Reviewer_ncJx", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a framework that accounts for observed confounding in treatment effect estimation via (i) balanced representation learning of treatment groups and (ii) adversarial propensity score learning. Experimental results on synthetic and semi-synthetic data sets demonstrate improved treatment effect estimation.", "summary_of_the_review": "The paper combines previously proposed DANN and  CFR in a unified framework. However, the performance gain of the proposed approach seems marginal when compared to the  DANN framework alone (DINet). Additionally, several baselines are missing from the experimental evaluation.\n\n**Post Author (s) Response**\n- The author (s) addressed most of my concerns, and I have adjusted my score accordingly.\n", "strength_and_weaknesses": "**Strengths**\n- The paper is well-written and easy to follow\n- The proposed approach seems easy to implement\n- Leveraging both propensity score and balanced representation in a unified framework is interesting\n\n**Weakness**\n\n*Novelty*\n- The paper seems to be a straightforward combination of DANN (Ganin et al., 2016) and CFR (Shalit et al., 2017)\n- The paper should highlight how the proposed approach differs from related works \n\n*Unconvincing Experiments*\n- It may be challenging to optimize the four objectives, which also include adversarial learning\n- The paper should also compare with similar approaches incorporating balanced representation and propensity score learning, *e.g.*, Zhang et al., 2020; Assaad et al., 2021; Huang et al., 2022a;  Hassanpour & Greiner, 2019b; *etc.*\n- How are the $\\alpha$ weights selected?\n- The paper should provide an expanded discussion  on the chosen baselines and Table 3 results", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject