Learning Counterfactual Outcomes Under Rank Preservation
Authors: Peng Wu
Haoxuan Li
Chunyuan Zheng
Yan Zeng
Jiawei Chen
Yang Liu
Ruocheng Guo
Kun Zhang
Venue: NeurIPS 2025 poster ()
Keywords: Causal Inference
Nonparametric Identifiability
Individualized Treatment Effects
Primary area: probabilistic_methods
TL;DR: Learning the individualized treatment effects under the assumption of rank preservation.

Abstract: Counterfactual inference aims to estimate the counterfactual outcome at the individual level given knowledge of an observed treatment and the factual outcome, with broad applications in fields such as epidemiology, econometrics, and management science. Previous methods rely on a known structural causal model (SCM) or assume the homogeneity of the exogenous variable and strict monotonicity between the outcome and exogenous variable. In this paper, we propose a principled approach for identifying and estimating the counterfactual outcome. We first introduce a simple and intuitive rank preservation assumption to identify the counterfactual outcome without relying on a known structural causal model. Building on this, we propose a novel ideal loss for theoretically unbiased learning of the counterfactual outcome and further develop a kernel-based estimator for its empirical estimation. Our theoretical analysis shows that the rank preservation assumption is not stronger than the homogeneity and strict monotonicity assumptions, and shows that the proposed ideal loss is convex, and the proposed estimator is unbiased. Extensive semi-synthetic and real-world experiments are conducted to demonstrate the effectiveness of the proposed method.

Learning Counterfactual Outcomes Under Rank Preservation
Peng Wu', Haoxuan Li Chunyuan Zheng?, Yan Zeng', Jiawei Chen', Yang Liu Ruocheng Guo Kun Zhang 3.7
Beijing Technology and Business University 2Peking University
3 Mohamed bin Zayed University of Artificial Intelligence
*Zhejiang University
University of California, Santa Cruz " Carnegie Mellon University
"Intuit Al Research
Abstract
Counterfactual inference aims estimate the counterfactual outcome the indi- vidual level given knowledge of an observed treatment and the factual outcome, with broad applications in fields such epidemiology, econometries, and manage- ment science. Previous methods rely on a known structural causal model (SCM) assume the homogeneity of the exogenous variable and strict monotonicity between the outcome and exogenous variable. In this paper. we propose a prin- cipled approach for identifying and estimating the counterfactual outcome. We first introduce simple and intuitive rank preservation assumption to identify the counterfactual outcome without relying on known structural causal model. Build- ing on this, we propose novel ideal loss for theoretically unbiased learning of the counterfactual outcome and further develop kernel-based based estimator for its empirical estimation. Our theoretical analysis shows that the rank preservation assumption not stronger than the homogeneity and strict monotonicity assump- tions, and shows that the proposed ideal loss is convex, and the proposed estimator is unbiased. Extensive semi-synthetic and real-world experiments are conducted demonstrate the effectiveness of the proposed method.
Introduction
Understanding causal relationships is fundamental goal across various domains. such as epidemi- ology [1, econometrics [2], and management science [ Pearl and Mackenzie [4] define the three-layer causal hierarchy-assocee
intervention, and counterfactuals- distinguish three types of queries with increasing complexity and difficulty (5. Counterfactual inference, the most challenging level, aims to explore the impact of treatment on an outcome given knowledge about a different observed treatment and the factual outcome. For example, given patient who has not taken medication before and now suffers from a headache, we want to know whether the headache would have occurred if the patient had taken the medication initially. Answering such counterfactual queries can provide valuable instructions in scenarios such as credit assignment (6), roos- analysis (
attribution g 9 10 I1]. well as fair and safe decision-making making 12 B [3)
Different from interventional queries, which are prospective and estimate the counterfactual out- come in hypothetical world via only the observations obtained before treatment (as pre-treatment variables), counterfactual inference retrospective and further incorporates the factual outcome (as post-treatment variable) in the observed world. This inherent conflict between the hypothetical
Work done while at ByteDance Research, rguo.asu gmail.cc......m.....
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and the observed world poses unique challenge and makes the counterfactual outcome generally unidentifiable, even in randomized controlled experiments (RCTs) T3
For counterfactual inference, Pearl et al. 8 proposed a three-step procedure (abduction, action, and prediction) to estimate counterfactual outcomes. However, it relies on the availability of structural causal models (SCMs) that fully describe the data-generating process [18] 19]. In real-world applica- tions, the ground-truth SCM likely be unknown, and estimating it requires additional assumptions to ensure identifiability, such as linearity [20] and additive noise 221 23 Unfortunately, these assumptions are hard to satisfy in practice and restrict the applicability.
To tackle the above problems, several counterfactual learning approaches have been proposed with respect different identifiability assumptions. For example, Lu et al. (24), Nasr- Masr--sfahany et al. 251, and Xie et al. [19] established the identifiability of counterfactual outcomes based on homogeneity and strict monotonicity assumptions (3) 26). The homogeneity assumption posits that the exogenous variable for each individual remains constant across different interventional environments, and the strict monotonicity assumption asserts that the outcome strictly monotone function of the exogenous variable given the features. In terms of counterfactual learning, [24) and [25] adopted Pearl's three-step procedure that needs to estimate the SCM initially. In addition, 19 proposed using quantile regression to estimate counterfactual outcomes that effectively avoid the estimation of SCM. Nevertheless, relies on a stringent assumption that the conditional quantile functions for different counterfactual outcomes come from the same model and it requires estimating different quantile value for each individual, leading to a challenging bi-level optimization problem (27].
In this work, we propose a principled counterfactual learning approach with intuitive identifiability assumptions and theoretically guaranteed estimation methods. On one hand, we introduce the simple and intuitive rank preservation assumption, positing that an individual's 's factual and counter- factual outcomes have the same rank in the corresponding distributions of factual and counterfactual outcomes for all individuals. We establish the identifiability of counterfactual outcomes under the rank preservation assumption and show that it is slightly less restrictive than the homogeneity and monotonicity assumptions used in previous studies.
On the other hand, further propose theoretically guaranteed method for unbiased estimation of counterfactual outcomes. The proposed estimation method has several desirable merits. First, unlike Pearl's three-step procedure, it does not necessitate a prior estimation of SCMs and thus relies on fewer assumptions than that in 24 and (23. Second, in contrast to the quantile regression method proposed by [19), our approach neither restricts conditional quantile functions for different counterfactual outcomes to originate from the same model, nor does it require estimating a different quantile value for each unit. Third, we improve the previous learning approaches by adopting convex loss for estimating counterfactual outcomes, which leads to a unique solution.
In summary, the main contributions are follows: (1) We introduce the intuitive rank preservation assumption to identify the counterfactual outcomes with unknown SCM: (2) We propose a novel ideal loss for unbiased learning of the counterfactual outcome and further develop kernel-based estimator for the ideal loss. In addition, we provide a comprehensive theoretical analysis for the proposed learning approach; (3) We conduct extensive experiments both semi-synthetic and real-world datasets to demonstrate the effectiveness of the proposed method.
2 Problem Formulation
Throughout, capital letters represent random variables and lowercase letters denote their realizations.
Structural Causal Model (SCM, (28). An SCM M consists of a causal graph G and a set of structure equation models J = (f1, fp). The nodes in & are divided into two categories: (a) exogenous variables U = (U1, Up), which represent the environment during data generation, assumed to be mutually independent; (b) endogenous variables V (Vi.... Vp)., which denote the relevant features that we need to model in a question of interest. For variable Vj, its value is determined by a structure equation V, = (P U,), j = 1,,,, p. where PA, stands for the set of parents of Vy. SCM provides formal language for describing how the variables interact and how the resulting distribution would change in response to certain interventions. Based on SCM, we introduce the counterfactual inference problem in the following.

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Counterfactual Inference. Suppose that we have three sets of variables denoted by X,Y, C V, counterfactual inference revolves around the question, "given evidence
e, what would have
happened if we had set X to different value x'?". Pearl et al. [ propose using the three-step procedure to answer the problem: (a) Abduction: determine the value of U according to the evidence E = (b) Action: modify the model M by removing the structural equations for X and replacing them with = x', yielding the modified model Mar (c) Prediction: Use Ma and the value of U to calculate the counterfactual outcome of Y. In this paper, we focus estimating the counterfactual outcome for each individual. To illustrate the main ideas, we formulate the common counterfactual inference problem within the context of the backdoor criterion.
Problem Formulation. Let V (2,X,Y) Y), where X causes Y, Z affects both X and Y, and the structure equation of Y is given as
Y= fx,
Let Yz' denotes the potential outcome if we had set X = x'. The counterfactual question, "given evidence (X = z, Z 2, = y) of an individual, what would have happened had we set X x' for this individual". is formally expressed as estimating Ye', the realization of Yg for the individual. Here, we adhere to the deterministic viewpoint of [28] and [8) treating the value of Y, for each individual as fixed constant. According to Pearl's 's three-step procedure, given the evidence (X = r, Z 2, Y = y) for an individual, the identifiability of its counterfactual value ye can be achieved by determining the structural equation fv and the value of Ux for this individual. This is the key idea underlying most of the existing methods.
For clarity, we use Ya' to denote the realization of the counterfactual outcome Ya for a specific individual with observed evidence (X x, Z = = yy)
3 Analysis of Existing Methods
In this section, we elucidate the challenges of counterfactual inference. Subsequently, we summarize the existing methods and shed light on their limitations.
3.1 Challenges in Counterfactual Inference
The main challenge lies in that the counterfactual value yr is generally not identifiable, even in randomized controlled experiments (RCTs). By definition, Ya is a quantity involving two "different worlds" at the same time: the observed world with (X x. Z 2, Y y) and the hypothetical world where X = x'. We only observe the factual outcome Y,- but never observe the counterfactual outcome Yz which is the fundamental problem in causal inference 29 30
This inherent conflict prevents from simplifying the expression of Ya! to do-calculus expression, making it generally unidentifiable, even in RCTs B. Therefore, in addition to the widely used assumptions such as conditional exchangeability, overlapping, and consistency [I] counterfactual inference requires extra assumptions to ensure identifiability. Essentially, estimating ye' is equivalent to estimating the individual treatment effect Ye' Ya. while the conditional average treatment effect (CATE) E|y,' - Y2 - = 2] represents the ATE for a subpopulation with Z = z, overlooking the inherent heterogeneity in this subpopulation caused by the noise terms such Ux B3
34 35 36
3.2 Summary of Existing Methods
We summarize the existing methods in terms of identifiability assumptions and estimation strategies.
We first present an equivalent expression of Eq. using (Y,, Eq. be reformulated as the following system
== fy, Y fy (y,,2,UU,) U2),
where U2 and Ug. denote the values of Ux given X x and X = x'. respectively. The exogenous variable Ux denotes the background and environment information induced by many unmeasured factors [8) and thus U, and U,, account for the heterogeneity of Y, and Y, in the observed and hypothetical worlds, respectively. These two worlds may exhibit different levels of noise due to
unmeasured factors [32 37 For identification, previous work [19 24 25 relies on the key
homogeneity and strict monotonicity assumptions.
3

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Assumption 3.1 (Homogeneity). Up Uz.
Assumption 3.2 (Strict Monotonicity). For any given (z, 2), Y, = fr U2) smooth and strictly monotonic function of U2; or it is a bijective mapping from Up to Yz.
Assumption 3.1 implies that the value of Ux for each individual remains unchanged across r.
Assumption 3 implies that Y is strict monotonic function of U² in the subpopulation of (X x, 7 = 2). Assumption 3.2 the smoothness and strict monotonicity of fy (Zt, 2, Un) are akin to don't distinguish them in detail.
bijective mapping of Yg and U2 and serve the same purpose.
Lemma 3.3. Under Assumptions 3.7 3.2 Ya! is identifiabl.
and 25 initially estimate fy and Ux for each individual. However, estimating fy and Ux needs to impose extra assumptions, such linearity [20) and additive noise [22]. In addition, [19] demonstrate that yz corresponds to the - -' quantile of the distribution F x' Z = 2 where T" is the quantile of y in P(Y x; Z z) (See the proof of Lemma 3.3 Section 4.1 for more details). Based it, the authors uses quantile regression to estimate Yus, which avoids the problem of estimating fy and Ux. Nevertheless, this method fits single model obtain the conditional quantile functions for both the counterfactual and factual outcomes. Thus, its validity relies on the underlying assumption that the conditional quantile functions of outcomes for different treatment groups stem from the same model. In addition, it involves estimating distinct quantile value for each individual before deriving the counterfactual outcomes, posing a challenging bi-level optimization problem.
For estimation of Yu', following Pearl's three-step procedure,
4 Identification through Rank Preservation
We introduce the rank preservation assumption for identifying yz- From high-level perspec- tive, identifying Yuu essentially involves establishing the relationship between Y, and Yz for each individual. Pearl's s three-step procedure achieves this by estimating fr and Ux.
4.1 Rank Preservation Assumption
Our identifiability assumption is based Kendall's rank correlation coefficient defined below.
Definition 4.1 (Kendall [38). Let (11, ,1,,,.., (Im: yn) be set of observations of two random variables (X, Y), such that all the values of x; and y; are unique (ties are neglected for simplicity). Any pair of (rs, Vi) and (æj, U5), if (Itz x, (V; ys) >> O, they are said to be concordant; otherwise they are discordant. The sample Kendall rank correlation coefficient is defined
2
Pa(X,Y)= = ann 1)
Σ ISKKiS
sign((æ; x3)(1: -3)),,
where sign(t) = -1, -1, for t < 0, == 0, O. respectively. For any two random variables (X,, Y). we define p) = pm p. Y) for all integers 2 2.
The Pn (X, Y) also can be written as 2(N.. - -Na) /n(n 1), where N₂ is the number of concordant
pairs, Na the number of discordant pairs. easy see that -1 S pn, (X, < 1 and the agreement between the two rankings is perfect (i.c., perfect concordance), Pa (X, Y) = 1.
Assumption 4.2 (Rank Preservation). P(Y,, (2) 1.
Assumption 4.2 is a high-level condition that establishes a connection between Y, and Yg.. This assumption is satisfied many common scenarios, as illustrated below.
Causal models with additive noise: Y = g(X, 2) + U for an arbitrary function g.
Heteroscedastic noise models: Y = gXX, Z) h(X, Z)U for arbitrary functions g and h, with h(X, Z) denoting the conditional standard deviation of Y given (X, Z).
For the individual with observation (X = x, = 2, Y y), we denote (yz = y, Ma') as its true values of Y.r). Assumption 4.2 limlies that for this individual, its rankings of ya and ye are the same in the distributions of (Y22 = 2) and P(Y . = 2). respectively. Therefore, we have
P(Y S 1z 1-
P(Y S Va I 2).
(2)

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Since ys = is observed and the distributions P( = 2) and P(Y,.112 2) can be identified
as P( z, 2) and P(Y x'. Z 2), respectively, by the backdoor criterion (i.e.. (Ya, Yrr) 2) Therefore, we have the following Proposition 4.3 (see Appendix [A] for proofs).
Proposition 4.3. Under Assumption 4.2 Yu idemiified as the T -" quantile of P(Y)X x', 2), where 7" is the quanfile of y in the distribution of (Y = =)).
Proposition 4.3 shows that Assumption 4.2
can serve as a substitute for Assumptions 3.1 3.2 in
identifying Ye Unlike Assumptions 3.1 3.2 Assumption 4.2 is simple and intuitive, as it directly
links Y, and Yz' for each individual. To clarify the relationship between Assumption 4 introduced
by this work and Assumptions 3.1 3.2 from previous work, we present Proposition 4. below.
Proposition 4.4. The proposed Assumption is strictly weaker than Assumptions 3. 3.2
Proposition 4.4 2.45 is intuitive, as correlation (Assumption 4.2 does not necessarily imply identity To illustrate, consider SCM with X E (0. Y1 Z Z UU
(Assumption U = U. In this case, p(Y,. Z = 1, but U1 & Un- Nevertheless, Assumption 4.2 only slightly weaker than Assumptions 3.1 3.2 by allowing Us' * Un. Specifically, we can show that U, is
2/2 + Up.
strictly monotone increasing function of Uz,, Assumption 4.2 is equivalent to Assumption 3.2 see
Appendix A for proofs.
4.2 Further Relaxation of Striet Monotonicity
In Definition 4.1 we ignore ties for simplicity. However, when the outcome Y is discrete or
continuous variables with tied observations, p(Yz; Yzr) will always be less than 1. To accommodate such cases, we introduce a modified version of the Kendall rank correlation coefficient given below.
Definition 4.5 (Kendall (39]) Let (I11 91), ..., (I'; Yn.) be the observations of two random variables (X,Y), the modified Kendall rank correlation coefficient is define as
sign( (2, x1) yj))
(x(X)))=
E
Igicisn Vn(n 1)/2 T. Vn(n 1)/2 1 T
where T. is the number of tied pairs in .... In and T, is the number of tied pairs in {vi, mm Bn We define ő(X, Y) = 1. if pm. (X = 1 for all integers n 2 2.
Compared with Definition 4.1 can see that Y) adjusts p(X,Y) by eliminating the ties in the denominator, and p(X, YJ reduces to p(X, , Y) if there are no ties.
Assumption 4.6 (Rank Preservation). (YY,, 12) 1.
Assumption 4.6 is less restrictive than Assumption 4.2 as it accommodates broader data types of
Y. To illustrate, consider dataset with four individuals where the true values of (Y,, Y²) are 11, (2, 1.5), (2, 1.5), (3, 2.5). In this scenario, Eisisign sign((Vi.r - (iz) Vj.z') = 5,
Ty, Ty, = L, resulting in p(Y,, Y 5/6 and Yr) = 5/(v6- 16
Assumption a.6 guarantees the identifiability of yz- Proposition 4.7. Under Assumption 4.6 the conclusion in Proposition 4.3 also holds.
5 Counterfactual Learning
We propose a novel estimation method for counterfactual inference. Suppose that {(rrs 2x, ye) 1. ...., N} } is sample consisting of N realizations of random variables (X, Z, Y). For an individual, given its evidence (X = x, = z, Y = y), we aim to estimate counterfactual outcome yor, i.e., the realization of Y for this individual.
5.1 Rationale and Limitations of Quantile Regression
For estimating ye', Xie al. [19 formulate the following bi-level optimization problem
N
T" = arg min (f - 2l. f = arg min
Σ
( f(Ix:
N
5

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where (g) = = IiC5 2 0) + (T I(C < 0) is the check function [40], the upper level optimization is estimate T" the quantile of y in the distribution P(Y = Z 2), and the lower level optimization is to estimate the conditional quantile function q(0, 2; _ in P(Y x y|X = Z = 2) 2 )} for given Then y/ can be estimated using g(4", 22
We define two conditional quantile regression functions,
2x (x(2;
inf P(Y. S z| = 2), ge (x; ( inf P(Y. S y 2) 7).
y
2 Yur can be expressed as yn (2; with T" being the quantile of y in the distribution of P(Y, 2), i.e., P(Y, S 2 = =) = T". Lemma 5. (see Appendix [ for proofs) shows the rationale behind employing the check function the loss to estimate conditional quantiles.
By Eq.
Lemma 5.1. We have that
(i) qz Z; 7) arg min, BY, [(Z))) for any given x; (ii) (X,, T) = arg miny (Y - f(X, Z))].
There are two major concerns with the estimation method of [19] First, only fits a single quantile regression model for qX, T; to obtain estimates of qu (Z:T) and 9r (2; When the two conditional quantile functions 4 (2; and qu (Z: originate from different models, this method may yield inaccurate estimates. Second, it explicitly requires estimating the quantile T* for each individual before estimating the counterfactual outcome yu-
Inspired by 41 a simple improvement to estimate (2(2; and qar (2; separately. For example, for estimating (z; the associated loss function is given
(e(f.T) = 1 I(rx = - (yk f(zx))
(x()
where =(( the propensity score, (2(=) its estimate. Likewise, we could define (f. by replacing with x'. Then the estimation procedure for ys involves four steps: (1) estimating P> (z): (2) estimating q. (=; by minimizing R₂ (J. for a range of candidate values of 7; (3) identifying the r" in the candidate set of T, that corresponds to the quantile of y in the distribution P(Y = x, 7 2): (4) estimating Yr' using (2; r where qu (2;T*) is obtained by minimizing Rg (J,))) Despite this four-step estimation method that allows qu (Z;T) and To (Z;+)
to come from different models, it still needs to estimate a different 7" for each individual.
5.2 Enhanced Counterfactual Learning Method
To address the limitations mentioned above in directly applying quantile regression and improve estimation accuracy, we propose novel loss that produces an unbiased estimator of yr for the individual with evidence (X 7 2, Y = y). The proposed ideal loss is constructed as
RS (f| z.y) E [
t| Z = =] + E [sign - y) Z = =] -
which is a function of t and the expectation operator is taken on the random variable of (Yz, Yzr)
given 2. The proposed estimation method based on Theorem 5.2
Theorem 5.2 (Validity of the Proposed Ideal Loss). The loss R., (t/x,, 2, y) is convex with respect to t and is minimized uniquely at t", where t" is the solution satisfying
P(Y. 22 2) PY, S v/ = z).
Theorem 5.2 (see Appendix B for proofs) implies that given the evidence (X =x, = for an individual, the counterfactual outcome Ye satisfies yr arg min, Rw (1|rr 2, under Assumption 4.6 Importantly, the loss Re (f|2, 2, neither estimates the SCM a priori. nor restricts qu (2; and qu' (z; stem from the same model, and it does not need to estimate a different quantile value for each individual explicitly.
To optimize the ideal loss Rg (t; x y). we first need to estimate it, which presents two significant challenges: (1) R., (t) 12, 2, 8) involves both Y, and Y,,. but for each unit, we only observe one of them; (2) The terms [Y, 1] = 2] and [sign(Y, y) = 2] in R2 (fJx, 2, y) is conditioned on Z = and when Z is continuous variable with infinite possible values, it cannot be estimated

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Table 1: CPEE of individual treatment effect estimation the simulated Sim-m dataset, where the dimension of Z.
Sim-5
Sim-10
Sim-20
Sim-40
Methods
In sample
Out sample
In sample
Out sample
In sample
Out sample
In sample
Out sample
Tleammer
2,95 0.02
2.66
0.01
2.99
0.01
3.17
3.36 0.02
3.19
0.03
5.12
0.02
4.74 0.04
X X-leamer
2.94 + 0.01
2.66 + 0.01
298 + 0.02
3.19 + 0.02
3,31 + 0.02
3.21 + 0.02
5.08 + 0.04
4.77 + 0.03
BNN
2.91
2.64
0.11
3.08 0.12
3.21 0.13
3.13
4.81 0.10
4.54 1:44::0.00
TARNet
2.89 +.07
2.64 1.0000.000
3.16 2.04
3.18 + 0.07
3.11
+
4.56 0.07
CFRNet
2.88 0.07
2.62 0.06
2.07
3.15 0.08
3.15 0.07
0:07 3.08
4.71 0.12
4.45 0.11
CEVAR
0.22
2.65 0.21
+
+
3.16 0.17
+.11
4.88 0.23
4.53 + 0.20
DragonNet
0.00
2.63 0.08
0.07
0.08
3.16 0.11
0.10
4.78
0.11
4.50 0.12
DeRCFR
0.06
261 0.06
0.75
3.07
0.06
0.07
0.04
4.77
0.11
0.10
DESCN
2.13
2.66 0.09
1.21
1.79
3.22
3.06
4,91
1 0.37
4.59 0.35
ESCER
2.87
262 0.07
2.04
3.09
3.03 -
3.06
4.71
0.15
4.3
CPQP
2.91 0.09
2.67 0.11
0.30
0.37
3.21
0.12
3.18
0.11
4,93 0.14
4.55
0.13
Quantile Reg
2.80
1.001.0.06 0.06
2.54
0.05
2.78
0.08
3.05
0.09
2.9 1 0.07
3.01
0.08
4.39 0.13
4.12
0.10
Ours
2.45 0.17
2.28 0.23
2.25
0.07
2.33 + 0.07
251 0.07
2.46
0.06
3.74 0.26
3.66
0.21
Table 2: / CPEHE of individual treatment effect estimation on the simulated Sim-m dataset, where m is the dimension of Z.
Sim-80 (o
Sim-80 (p
Sim-40 (p 0.3)
Sim-40 (p = 0.5)
Methods
sample
Out sample
In sample
Out sample
samsae
Out sample
sa-sample
Out sample
TARNet
12.63 = 0.93
12.51 0.90
12.35 = 1.24
12.68
1.51
8.91 + 0.97
8
+
8,76 0.76
8.51 0.68
DragonNet
12.50 + 0.75
12.36 + 0.80
12.71
1.29
13.02
1.54
8.83 + 0.90
8.73+-0.72
8.62 + 0.70
8.39 0.53
ESCER
12.61 = 1.09
12.53 1.09
12.56
= 1.36
12.87
+ 1.64
8,76
103
8.65
8.76 0.79
8.50
X learner
12.82 + 0.91
12.68 + 0.95
12.74
1.22
12.99
+ 1.33
8.97
+ 0.87
8.81 +-064
8.91 + 0.75
8.61 0.58
Quantile Reg
0.94
11.59
0.73
Ours
9.28 = 0.72
9.28 1 0.72
9.03 1 1.09
9.27
= 0.97
7.07 1 0.39
7.05 1 0.41
7.07 + 1.23
6.98 1.08
by simply splitting the data based on Z. We employ inverse propensity score and kernel smoothing techniques to overcome these two challenges. Specifically, we propose a kernel-smoothing-biee estimator for the ideal loss, which is given as
k== Kn(zk 2)
M(x==)) (a)) Uk 11 E Kn(zk 2)
1( Pa(as) sign(yk- y)
Re (t(x,,,))
Σ
Kn -2
Σx Kn(ex- 2)
where h is bandwidth/smoothing parameter, (u) k(u/h)/h, and K(-) is symmetric kernel function [42 43 44 that satisfies f K(u)du = and (uK(u)du = 1, such as Epanechnikov kernel K (uu) = 3(1 - u²) I([ul= S 1)/4 and Gaussian kernel ((u) exp(-u² (2)(-u22)/// 2 for u E R. Then we can estimate yor by minimizing P.r (l; v) directly.
Proposition 5.3 (Consistency). If h + as - oc, P=(2) and p.. are consistent estimates of p,(22 and p (2), and the density function of Z is differentiable, then Ry (t/x, 2, y) converges to R2 24,,,,y) z) in probubility.
Proposition 5.3 (see Appendix B for proofs) indicates that Ra (t|ar, 2, y) is consistent estimator of Rg. 2, y), demonstrating the validity of the estimated ideal loss. The loss (1|x, z, y) is applicable only for discrete treatments due to the terms [[ =x)) x') and [[(x = x). However, it can be easily extended continuous treatments, detailed in Appendix
is well known that smootll-n based estimators suffer from scalability issues high- dimensional settings (i.c., the high oigh-dimensional covariates) 42). Therefore, for implementation, we avoid applying kernel functions directly to the original covariates. Instead, we first learn low- dimensional representation of the covariates, and then apply the kernel-smoothing- -based estimator to this representation to learn the counterfactual outcomes.
Experiments
6.1 Synthetic Experiment
Simulation Process. We generate the synthetic dataset by the following process. First, we sample the covariate Z ~ N' (0, Im) and the treatment X ~ (ern(x(Z)), where Bern(-) is the Bernoulli
7

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10
10
Ouartileffgg
Quandi Rog
Quardi
1.5
2.0
2.5
3.0
1.5
2.0
2.5
3.0
1.5
2.0
2.5
3.0
1.5
2.0
2.5
3.0
0: Haterogeneity Conffiitt
0r. Heterogeneity Confficiint
a: Hoterogenalty Coefficient.
4: Hetarogenilty Coafficient
(a) Sim- (In-sample)
(b) Sim-10 10 (Out-sample)
(c) Sim-40 (In-sample)
(d) Sim-40 (Out-sample)
Figure I: Estimation performance of individual treatment effects under varying heterogeneity degrees.
ESCFR
Quarcile
EGCFF
Quartile-
ESCPA
1.0
3.0
5.0
7.0
9.0
1.0
3.0
5.0
7.0
9.0
1.0
2.0
3.0
4,0
5.0
1.0
2.0
3.0
4.0
5.0
Karmel Bandwidth
Kernel Bandwicth
N:: Karnel Bandwidth
2r. Kernel Bandwwidth
(a) Sim-10 10 (In-sample)
(b) Sim-10 (Out-sample)
(c) Sim-40 (In-sample)
(d) Sim-40 (Out-sample)
Figure 2: The estimation performance with different kernels and bandwidths.
distribution with probability T(2) = P = 2) oo : o.) is the sigmoid function, and W Uni Unif(-) is the uniform distribution. Then, sample the noise Uo ~ N (0, 1) and U1 = a - Uo to consider the heterogeneity of the exogenous variables, where a is the hyper-parameetr
control the heterogeneity degree. Finally, we simulate Y1 W, 7 + U² and Yo = Wy, Z/a + Un with W, ~ (O, Im). We generate 10,000 samples with 63/27/10 train/validdtiiontest split and vary m (5, 10, 20, 40} in our synthetic experiment.
Baselines and Evaluation Metrics. The competing baselines includes: T-learner [45), X-learner 45
BNN 46). TARNet [7]). CFRNet [47), CEVAE 48) DragonNet [4), DeRCFR (50] DESCN
51
We evaluate the individual treatment effect estimation using the individual level Precision in Estimation of Heterogeneous Effects (PEHE):
ESCER (52). CFQP [8]. and Quantile-Rec
19
1 N CPENE = N ((((())-(000) - (X)() Y((0))
where Yi(1) and (0) are the predicted values for the corresponding true potential outcomes of unit i. It is noteworthy that <PEEE is tailored for individual-level evaluation and counterfactual estimation, which is different from the common metric 47 given by N > -i=1 (X,) fuo (xx))))
(MI(X)) (X,) (o((, ()))) where μn (X;) (X.) ) == (1)( - (0) are the true CATE, and (X,) - io (x) is its estimate. Both in iamsa and out- -ample performances are reported in experiments. In addition,
the Google Colab platform. For the representation model, we use the MLP for the base model and tune the layers in {1, 2,33. In addition, we adopt the logistic regression model as the propensity model. We tune the learning rate in (0.001, 0.005, 0.01, 0.05, 0.1). J. For the kernel choice, we select the kernel function between the Gaussian kernel function and the Epanechnikov kernel function, and tune the bandwidth in 1,,999
run all experiments
Performance Analysis. The results of estimation performance are shown in Table T Our method stably outperforms all baselines with varying covariate dimensions m, demonstrating effectiveness of the proposed method. In addition, we investigate our method performance with violated assump- tions on rank and uncorrelated covariates. Specifically, we modified the data generation process to explore the performance of our method under correlated covariates by sampling the covariate Z ~ N (0, ), where the Psy in Σ,, is max(0.01, pli-ál).. The results are shown in Table The results show that our method still outperforms the baseline methods. Moreover, we further explore the effect of heterogeneity degrees on the performance of the proposed method, as shown in Figure [] where one can see that as the heterogeneity degree increases, our method stably outperforms the Quantile-Reg in terms of PEHE, Finally, examine the effect of different kernels and bandwidths,
8

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Table 3: The experiment results on the IHDP dataset and JOBS dataset. The best result is bolded.
IHDP
JoBs
In-sumple
Out-sample
In-sample
Out- Out-sample
Methods
V/PEHE
Rpa
Npu
T Fllarner
1.49 0.03
0.37
0.05
1.81 0.04
0.49
0.04
0.31
0.06
0.16
0.10
0.27
0.08
0.20 1: 0.07
X-leamer
1.50 0.002
0.21
0.05
1.73
0.00
0.36
0.07
0.16 + 0.04
0.07 +0005
0.16 + 0.03
0.10 + 0.09
BNN
2.09 0.16
1.00
0.23
2.37
0.15
0.19
0.15
0.01
0.08
0.03
0.16
0.13 - 0.07
TARNet
1.52
0.22 0.13
1.78
0..88
0.17
0.17+00.0 0.06
0.06 0,08
0.18
0.10 0.06
CFRNet
0.06
0.11
1.77
0.22
0.17
0,05 0.03
0.10 0.19
CEVAE
4.08 0.88
3.67
1.23
4.12
3.75 1.23
0.05
0.09
0.10 0.20
DragonNet
1.99
0.22
0.14
1.06
0.29
0.19
0.06
0.07
0.11 0.20
DeRCER
148 0.06
0.25
0.14
1.69
0.25
0.14
0.15 0.02
0.04
0.16 0.04
0.15 0.11
DESCN
2.8
0.74 1.00
267
1.04
0.15
0,11
0.16 0,22
ESCFR
1.09
0.16
1.78
0.27
0.16
0.14
0.00
0.10 0.12
Quantile-Reg
1.05
0.09
0.14 0.09 0.03
0.06
0.15
0.07 0.04
CFQP
1.110
0.18
0.17
148
0.05
0.15 0.08
0.15 1 0.02
0.23 0.15
0.16 0.03
0.15 11510000 0.07
Ours
1.1 0.02
0.11
0.10
1.00
0.06
0.13 0.08
0.08 0.04
0.06 + 0.02
0.11 0.05
0,05 0.05
as shown in Figure 2 our method stably outperforms the Quantile-Reg and ESCER methods with different kernels and bandwidths.
6.2 Real- World Experiment
we conduct experiments on semi- semi-synthetic dataset IHDP and real-world dataset JOBS. The IHDP dataset (55] is constructed from the Infant Health and Development Program (IHDP) with 747 individuals and 25 covariates. The JOBS dataset (56) is based on the National Supported Work program with 3,212 individuals and 17 covariates. We follow 47 to split the data into raaniingvalidatioonessttgg set with ratios 63/27/10 and 56/24/20 with 100 and 10 repeated times on the IHDP and the JOBS datasets, respectively.
Dataset and Preprocessing.
Following previous studies 47
48
53
54
Evaluation Metries. Following previous studies 47 48
besides εPEhE, we also use the absolute error in Averuge Treatment Effect (ATE) for evaluation, which is defined as CATE = E (1))
We use /CPEHE and CATE to evaluate performance on the IHDP dataset. For the JOBS dataset, since one of the potential outcomes is not available, we evaluate the performance using the absolute error in Averuge Trealment effect on the Treated (ATT) as EXTT = |ATT
[T] Eier (Y;(1) Y;(0) with ATT = | Eier Y; [CNE| EiECnE Yil. We also the policy 0, = ()) > 0)+ (0) (1) <
risk Rp 1- (1 (1)-
0, 0] (( (1) (O) S 0)). where T, C, E are the indexes of treatment sample set, control sample set, and randomized sample set, respectively.
Performance Comparison. The experiment results are shown in Table 3 Similar to the synthetic experiment, the Quantile-Reg method still achieves the most competitive performance compared to the other baselines. Our method stably outperforms all the baselines both the semi-synthetic dataset IHDP and the real-world dataset JOBS, especially in the out-sample scenario. This provides the empirical evidence of the effectiveness of our method.
7 Related Work
Conditional Average Treatment Effect (CATE). CATE also referred to as heterogeneous treatment effect, represents the average treatment effects on subgroups categorized by covariate values, and plays a central role in areas such as precision medicine [57 58, 59 60). policy learning [6]] 62),
and recommender systems (63] 64 65) Benefiting from recent advances in machine Icarning, many methods have been proposed for estimating CATE, including matching methods [66]
67 [4] 68),
tree-based methods (69 70), representation learning methods 46 47 49
50 52]. and generative
methods [48] 53 Unlike the existing work devoted to estimating CATE at the intervention level for
subgroups, our work focuses counterfactual inference at the more challenging and fine-grained individual level.
Counterfactual Inference. Counterfactual inference involves the identification and estimation of counterfactual outcomes. For identification, 771 provided an algorithm leveraging counterfactual graphs to identify counterfactual queries. In addition, (72) discussed the identifiability of nested

---

counterfactuals within a given causal graph. More relevant to our work, 19 and [24] studied the identifiability assumptions in the setting backdoor criterion under homogeneity and strict monotonicity assumptions. Several methods focus on determining its bounds with less stringent
addition, [11] proposed a method for identifying the joint distribution of potential outcomes using multiple experimental datasets.
assumptions, such as [0 13 73 74
g introduced three-step procedure for counterfactaal inference. Many machine learning methods estimate counterfactual outcomes in this framework, such as 66 18
For estimation,
24
25
75 76
77.. Recently, [9] employed quantile regression estimate the counterfactual outcomes, effectively circumventing the need for SCM estimation. In our work, we extend the above methods in both identification and estimation. Recently, counterfactual inference methods have been extensively applied across various application scenarios, such as counterfactual faimeess [78
79
80
81
82
83 policy evaluation and improvement [14
86], reinforcement learning 24
90 91 92], imitation learning [93] 94], counterfactual generation 76
95| 96]
997. counterfactual
explanation [98 99 100, 101
102 counterfactual harm [33
14
103 15 physical audiovisual
commonsense reasoning [104), interpretable time series prediction [105, classification and detection
in medical imaging [106), data valuation [107], etc. Therefore, developing novel counterfactual
inference methods holds significant practical implications.
8 Conclusion
This work addresses the fundamental challenge of counterfactual inference in the absence of a known SCM and under heterogeneous endogenous variables. We first introduce the rank preservation as- sumption to identify counterfactual outcomes, showing that it is slightly weaker than the homogeneity and monotonicity assumptions. Then, we propose novel ideal loss for unbiased learning of counter- factual outcomes and develop a kernel-basee estimator for practical implementation. The convexity of the ideal loss and the unbiased nature of the proposed estimator contribute to the robustness and reliability of our method. A potential limitation arises when the propensity score is extremely small in certain data sparsity scenarios, which may cause instability in the estimation method. Further investigation is warranted to address and overcome this challenge.
Acknowledgments and Disclosure of Funding
This research was supported by the National Natural Science Foundation China (No. 12301370. 623B2002), the BTBU Digital Business Platform Project by BMEC, the Beijing Key Laboratory of Applied Statistics and Digital Regulation, Academy for Interdisciplinary Studies at BTBU, the BTBU Research Foundation for Youth Scholars (No. BRFYS2025), NSF Award No. 2229881, AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract ROIHLI 59805, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program, and the Al Deira Causal Education project.
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[98] Eoin M. Kenny and Weipeng Fuzzy Huang. The utility of "even if" semifactual explanation to optimise positive outcomes. Advances Neural Information Processing Systems, 2023.
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A Proofs in Sections and
One can show Lemma 3.3 by a similar argument of the proof of Theorem 1 in [9] For the sake of self-containedness, we provide novel proof of it.
Lemma 3.3 Under Assumptions 3. 3.2 Uz identifiable.
Proof of Lemma 3.3 First, the distributions IP(Ya) Z = 2) and PY 1Z = =) can be identified as P(Y x, 7 2) and P(Y x', 2), respectively, by the backdoor criterion (i.e.,
(Ya, ) 1 x of the setting.
Then, according the model (T. we can equivalently write
Yx = (2, 2, Un), Y =
and Y and Ux in model [ can be expressed as Y = Lmex IIXX = x). Y2 and Ux = Emee 1(X = r) U2 where is the support set of X and II()) is an indicator function. Assumption 3. implies that Ux = U2 = U2' conditional on Z, i.e., Y, = fr (z. 2, Ux), Y = 2x',,,,,xx)
Finally, for the individual with observation (X s, Z= Y, y), we denote (yer ys') ) the true values of (Yz, Yar) for this individual, For this individual, we can identify the quantile of in the distribution of IZ 2) P x, Z =), denoted by T" Let ut.- be the true value of Ux for this individual, it is the T q'qquntelle in the distribution 2 2- = z). then we have
T" P(Y, S ya 2 = 2) P(U, S 1 =
(by the definition of r)
(by Assumption 3.2)
P(U,, S ur17=
(by Assumption 3.1
=P(Y, S fr 2, 2, 11 1Z= =
(by Assumption 3.2
=PYY, S Ver IZ =
(by the definition of yzr),
which implies that for this individual, its rankings of yz and ya are the same in the distributions of 12 12 2) and P(Y Z 2). resepcctively. Thus, ya identified the -uuantie of the distribution P(Y Z = 2) = P(Y = x,, = 2).
Proposition 4.3 Under Assumption 4.2 Yo is identified as the T" -th quantile of P(( x', = 2).
==
where T" is the quantile of y in the distribution of IP(Y x, Z = 2).
For the individual with observation (X = x, = 2, = y). we denote (ya, Yz') the true values of (Yz, Y.r). Assumption 4.2 implies that for this individual, its rankings of y/ and ye are the same in the distributions P(Y = 2) and PY Z = 2). respectively. Therefore,
Proof of Proposition 4.3
P(Y S 1z = =) = PY S Yx 7 = z).
Since yr y is observed and the distributions PP 7 =) and PZ 2) can be identified as X = x. 2 = 2) and P(Y = x' 7 = =), respectively, by the backdoor criterion (i.e., (Y,, Y 11 Z22,, we can identify the quantile of yı the distribution of PY x, Z 2). denoted by r", Then
P(Y < Yor 2 2) =T", which yields that 0 is identified as the 7 -uaantll of P
The following Proposition
4.4 serves complement to Proposition 4.4
Proposition 4.4 Under Assumption 3.1
more generally, if U,, is strictly monotone increasing
function of Up,, Assumption
4.2 is equivalent to Assumption 3.2
Proof of Proposition
According the model I.. can equivalently write
Ye= fr Ux), Yo fx 2,U)
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Suppose that Up is strictly monotone increasing function of Ug (Assumption 3.1 i.e, U₁ = Up', , is special case of it). Under this condition, we next prove sufficiency and necessity, respectively.
First, we show that Assumption 3.2 implies Assumption 4.2 If Assumption [.2 holds, then Y is strictly monotonic function of Un., and Yz is strictly monotonic function of Uz Since Un is strictly monotone increasing function of Us', then Yz is a strictly increasing monotonic function of Yz', which leads to Assumption
4.2
Second, we show that Assumption 4.2 implies Assumption 3.2 Assumption 4.2 holds, then given Z = 2, Yg is a strictly increasing function of Yz'. When Uº strictly monotone increasing function of Up and note that
Y = 2xx, Y = (x', Ux),
which implies that fy is a strictly monotonic function of Ux, i.., Assumption 3.2 holds. This finishes the proof.
Proposition 4.7 Under Assumption 4.6
the conclusion in Proposition 4.3 also holds.
Proof of Proposition 4.7
This can be shown through proof analogous that of Proposition 4.3
B
Proofs in Section 5
Recall that (5) = T T((E 2 0) + (T - 1)g T(g < 0), and
[; & inf P(Y s y x, Z 2) r}
(0(2;7) A in PYY s u 2) 2)
(2; A inf P(Y s z/ 2) r)
Lemma 5.1
We have that
(i) (Z;r) arg min E[- f(Z))) for any given
(ii) q(X, Z; arg miny EE[, (Y f(X,
= We prove Qe (Z: arg miny E|1,(Y, f(Z)), and g(X, Z;r) arg ming [E[1, f(X, 2)] can be derived by an exactly similar manner. We write
Proof of Lemma 5.1
IE[, (Y, f(2)) EE - f(z)) 2H}.
To obtain the conclusion, note that 4, (Yz - f(Z)) is always positive, it suffices show that (=: = arg min EII-- f(Z)) = 2|
for any given 2 = z. Next, we focus on analyzing the term 1E[4(Y,, - -f(2)) | 6 = 2]. Given Z = 2, f(Z) is constant and we denote it by c, then
Ei4 (Y₁ f(Z)) 2 = 2]
= EEL - c) Z = z]
= TE| T(V= c)T(V, 2 c) + 1)(V, - c]T(V, < c) 2 =
=7 / (ye - c)g(val 22) dd + (T (T-1) JS (yz c)g(v/ 1z)dy
where g(y (2) denotes the probability density function of Y.. given Z = z.
25

---

Since the check function is a convex function, differentiating (Y, -e) Z = 2] with respect to and setting the derivative zero will yield the solution for the minimum
a BY - -2)
0
11v= - -)g((u) =)]dy= + (T -T- f
de U(v= c)g(v=l))) dy
T de
1 I g(velz)dyls) (1 (1-r) 7) / ((Vol))dy.
Then let 2 (V, c) 7 2] 0 leads that
[ g(ynl=)dy= =
that is, T)) This completes the proof of Proposition 5.1
Theorem 5.2 Validity of the Proposed Ideal Loss). The loss Ry (t; x, R, y) is minimized uniquely at t", where E is the solution satisfying
P(Y= <t" = P(Y S y/Z 2).
Proof of Theorem 5.2
Recall that
RS (1,x, y) IE. Y2 2 = 2 +E sign(V, v) 2 2
Let ((Uxlz) be the probability density function of Yv given Z = 2. By calculation,
N - Z 2 (.x (y= t)g(Ue (2)dy + S" (t- Ye' (lle)laa)))
(Ya' - 11 2 = z] =- 1" g(wa' (2)ddya)) + 1 a(uxr=))dy) = -P(Y S 5 = =) 1, -
Ø1.
and
sign(Yz y) 2= 2
-2I(Y2 50) +1 2 2) -P S 5 2) +1,
we have
R. (1,,, = 2P(Ya I 117 = 2) 11+E sign Y -) =
0t
-PP(Y I t = 2) -1 2P S y/2 = =2) +1
+1
= =2(p(y=- s f/2) - P(Y, s v1z)
Since
8 RS 2,1) = 2Y, S 51(2)/Ot 2g(y. t|z) 2
R2y (1,,, 2, is convex function with respect to 4. Letting @ R.r (1) xr, z, z, 0 yields that P(Y.. S f|z) P(Y. - y|2) = 0.
That is, R2,(1,,, 2, v) attains its minimum qu (2; where 7" is the quantile of y in the distribution PPY2 2).
26

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Proposition 5.3 If a N + co, Pz(z) and (z) are consistent estimates of pz(2) and Pz (2), and the densiry function of differentiable, then
(t; 2. + Ry (t; (,;,),
where- = means convergence in probability.
Proof of Proposition
For analyzing the theoretical properties of (1;,,,; we rewritten
R (t; x, 2, v) as
_ VE E (n - 2) Pa(Zx) KA (Z+ 2) (2)
[(X, sig - y) L
Ra (1;x,2,1)
Ekm1 (Z -
Σi=1 Kn (Zx (Zx-2)
where the capital letters denote random variables and lowercase letters denote their realizations. This slightly different from that used in the main text.
When (z) and p (z) are consistent estimates of Pa (z) and pa (z), to show the conclusion, it is sufficient to prove that
Sk=l K (Zx 2) I(X2 x²) (2))
IY4 t
+ E
1(X = x') Pxt (2)
Y
2 = = = =1(I 1 2 == -=
Σx Kn -2)
(5)
ΣE (z<
2) 1(Xi=0) D.(Ze)
sign(Vi: 1) P. E
I(X = x) (z(
siign( - -) 2 E
sign (Y y)
Z = 2
Σm (Z 22
(6)
We prove equation (5) only, as equation (6) can be addressed similarly.
Note that
EN Kn - 2) [(x 1r - 1 (2))
N E Kn 2) (( 1r 4
Ex Kn -2)
E (n - 2)
we analyze the denominator and numerator on the right side of the equation separately. For the denominator, it is an average of N independent random variables and converges to its expectation E|KK (2x =]] almost surely. Let g(=x) be the probability density function of Zz. and ())) (2+) ) is its first derivative. Since
1
1
K( 2k - )=x))
E (2x 2)]
/ K(u)g(=+ + hu)du
(let z4 = = hu)
K(u): {g(2) g (z)huu )d
(by Taylor Expansion)
= g(() / K(u)du + (z) / K(u)udu + o(h)
g(2) + o(h) (by the definition of kernel function),
when h - as N + xo, the denominator converges to g(z) in probability.
Next, we focus on dealing with the numerator, which also converges its expectation.
E K (Zx- 2) T(Xx x') 1Y t|]
Pe (Z+)
E Ka =)
1(Xx x¹)
Yv. 11 11(2 }
(by the law of iterated expectations)
Per (7k)
R m[Kn(Z> - =) I(Xx = x²)
Ysr 1| b
(write Ye as the form of potential outcome)
Pe. (Zx)
B Ka z)B Var. Zz
(by backdoor criterion Yo 11 Zx)
(8)
27

---

Define m(2) = E[[y - - and m (Z) is its first derivative, then the right side of equation ( m(z), and
Kn =2 B (Y 2x = =B - 2) ml ml(z
26 - m(=x) g )dzx / _K(
h h
ʃ K(u)- -K(()- ml= + hu)) g( + hu)du
(let 24 + hu)
= f K(u) {m(=) + m 0 ( + o {g(=) + ((h)dd
(by Taylor Expansion)
= mm{))(() + o(h).
Thus, when h + 0 as N + oo, the numerator converges to g(z) in probability,
Combining equations
8 and (9) yields the equality
This completes the proof.
C Extension to Continuous Outcome
When the treatment is continuous, we can estimate the ideal loss with the following estimator
(t)x, Ek= Kn - =) Ka 1yu - 41 Ek=1 Ka(zx =) Kn(ze-z) sign(ux - 1)) t, pe(ot)
(x))
ΣN (= =)
ΣN Kn(2x - 2)
which is smoothed version of the estimator
Kn(=* 2) Max==z) (=2)
Uk 1 ΣN Kn(26 2) M(r=a)) sign(yk y) 1 pr(aa)
Rs (1),,,,))
Σ Ka(za 2)
Σ Kn Kn -2
defined in Section 5 In addition, by a proof similar to that of Proposition Re R1; x. z, 3) Eg R2 (1; 21,,,,,))) z,
5.3 we also can show that
28


Reviews:
[{"review_id": "BoxugL16iH", "reviewer": "Reviewer_niUF", "summary": "This paper addresses individual-level counterfactual outcome estimation by introducing a new identifiability assumption called rank preservation, claiming it is weaker than the traditional homogeneity and strict monotonicity assumptions. Building on this, the authors propose an ideal loss function for unbiased counterfactual estimation and develop a kernel-based estimator for empirical implementation. Theoretical results, including identifiability proofs, unbiasedness properties, and consistency analysis, are provided. The paper also critiques quantile regression-based approaches and demonstrates empirical results on semi-synthetic and real-world datasets.", "questions": "see suggestions above", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The paper addresses a challenging problem in counterfactual inference where full knowledge of the SCM is unavailable, a realistic scenario in many applications.\n- Introducing the rank preservation assumption provides an alternative path to identifiability that appears, theoretically, to relax previous requirements like strict monotonicity.\n- The proposed ideal loss circumvents limitations of previous quantile regression methods, and the bi-level optimization.\n- The authors provide theoretical analysis for the proposed estimator.\n- Experimental results demonstrate consistent improvements over baseline methods.\n\n**Weaknesses**\n\n-  All identifiability assumptions (including this) are restrictive\n\nWhile the authors rightly critique homogeneity and strict monotonicity assumptions for being strong and hard to verify, the proposed rank preservation assumption is similarly unverifiable in practice. Whether ranks of factual and counterfactual outcomes align across individuals is fundamentally untestable, especially when counterfactuals are unobserved. The assumption may be mathematically weaker, but its practical plausibility remains speculative.\n\n*Suggestion*: The paper would benefit from a more candid discussion acknowledging that all identifiability assumptions in counterfactual inference, including rank preservation, impose unverifiable structure on unobserved quantities. Comparing empirical consequences of assumption violations (as partially done in Appendix E) is a good direction that could be expanded.\n\n- Reliance on propensity score estimation and kernel choices\n\nThe method depends on accurate estimation of propensity scores and bandwidth parameters for kernel smoothing. Theoretical sections highlight this, but practical guidelines for tuning and robustness checks are minimal.\n\n*Suggestion*: More systematic sensitivity analyses, especially in high-dimensional settings, would clarify the method's reliability. Exploring alternatives to kernel smoothing for scalability would also improve practical value.", "was_revised": true, "final_justification": "see discussion above.", "rebuttal": "Thank you very much for your positive evaluation of our paper. Below, we hope that the our clarification addresses your concerns. \n\n> **W1:** All identifiability assumptions (including this) are restrictive. The paper would benefit from a more candid discussion acknowledging that all identifiability assumptions in counterfactual inference, including rank preservation, impose unverifiable structure on unobserved quantities. Comparing empirical consequences of assumption violations (as partially done in Appendix E) is a good direction that could be expanded. \n\n**Response to W1:** Thanks for your helpful suggestions. We agree with you that all identifiability assumptions are restrictive and are unverifiable or untestable. We will add a discussion on it in the conclusion. \n\nWe also agree with you that comparing empirical consequences of assumption violations is helpful. **In Appendix E, we explored empirical performance with violated assumptions on rank preservation. As a complement, we may further theoretically investigate the impact of the violation of rank preservation.** Specifically, when the rank preservation is relaxed into the following Assumption R: \n- **Assumption R:** $| \\mathbb{P}(Y_{x} \\leq y_x \\mid Z=z ) - \\mathbb{P}(Y_{x'} \\leq y_{x'} \\mid Z=z) | \\leq \\alpha $ for $0 \\leq \\alpha < 1$. \n\nWhen $\\alpha = 0$, Assumption R reduces to the rank preservation assumption. **Then we could explore how the bounds on $y_{x'}$ varies with $\\alpha$ (a novel sensitivity analysis method). By theoretical analysis (similar to Theorem 5.2), we can show the following conclusion.**\n- **Proposition 5.7:** Let $$R_{x'}(t| x, z, y, \\beta) = \\mathbb{E}[ |Y_{x'} - t |  | Z=z ] + \\mathbb{E}[ \\text{sign}(Y_x - y)  | Z=z]  \\cdot t + 2 \\beta t$$\nUnder Assumption R, $y_{x'} \\in [ y^l, y^u ]$, where the lower bound $y^l$ is the unique solution of $R_{x'}(t| x, z, y, \\alpha)$, and the upper bound $y^u$ is the the unique solution of $R_{x'}(t| x, z, y, -\\alpha)$.  \n\n\n> **W2:** Reliance on propensity score estimation and kernel choices. More systematic sensitivity analyses, especially in high-dimensional settings, would clarify the method's reliability. Exploring alternatives to kernel smoothing for scalability would also improve practical value.\n\n**Response to W2:** Thanks for your constructive suggestions. **Empirically, we have explored several sensitivity analyses for various compoents of the proposed method, such as covariate dimensions (Tables 1 and 2), violation of rank preservation (Table 2 and Appendix E),  heterogeneity degrees (Figure 1), and  kernel choices (Appendix E).  \nTheoretically, we complement a sensitivity analysis for the violation of rank preservation (see our response to W1).** For more discussion on the high-dimensional settings, we leave it to future work. Thanks again for your kindful comments."}, {"review_id": "ccbyoLN0tK", "reviewer": "Reviewer_4AFD", "summary": "This paper proposes an approach for individual-level counterfactual inference without requiring structural causal models (SCMs). The approach is based on the introduction of the Rank Preservation Assumption which posits that confounders account for all the correlation between potential outcomes, with correlation defined in terms of Kendall’s rank coefficient. Under this assumption, the authors derive a convex loss function and an unbiased estimator for the counterfactual outcome. They  propose a kernel-based estimator and demonstrate its performance through theoretical analysis and empirical evaluation on both semi-synthetic and real-world datasets.\nCompared to existing methods, which typically require estimating an SCM along with a monotonicity assumption (i.e., the potential outcome is a monotonic function of unobserved variables) or assume equality of conditional quantile functions, this paper’s approach relies on  weaker identifiability conditions  and simpler estimation procedure.", "questions": "The authors claim that Assumption 4.2 holds for common models such as those with additive noise  $Y = g(X, Z) + U$ or heteroscedastic noise $Y = g(X, Z) + h(X, Z)U$. Could you elaborate on why this is the case?\n\n- How can practitioners assess or empirically validate the rank preservation assumption in real-world applications?\n\n- In the experiments, the proposed method appears to outperform all competitors consistently. This is somewhat surprising, as methods usually have specific regimes where they excel and others where they do not. Could you provide some intuition for these results? Have you explored other scenarios or settings? Did you use simple methods for estimating nuisance functions for the baselines? This is particularly surprising in settings where the assumption is violated, such as in Table 4 of the appendix, where some competitive methods are missing. Also, in non-simulated settings, how do you choose or tune the bandwidth parameter?\n\n- The reliance on kernel-based estimation could limit scalability in high-dimensional or large-sample settings. How do you handle scenarios with many continuous confounders? In particular, in your experiments, the estimator appears quite stable even when the covariate dimension reaches 40; could you clarify why?\n\nminor remarks:\n- Finally, the abstract states that rank preservation is “not stronger” than homogeneity and strict monotonicity, whereas the main text (e.g., Proposition 4.1) shows that it is in fact strictly weaker. Clarifying this language in the abstract would help avoid potential confusion for readers.\n- In line 120, the references cited do not seem directly relevant to the point made. Highlighting heterogeneity does not necessarily imply that one must estimate individual-level effects rather than conditional effects. One could, for instance, use an extreme form of stratification with very small group sizes instead.", "limitations": "The paper provides limited guidance on how to assess the plausibility of the rank preservation assumption in practice. As it is untestable from observational data, incorporating  a discussion or any diagnostic tools  would strengthen the method’s practical value.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper addresses the complex problem of counterfactual inference problem by introducing the rank preservation assumption. The authors prove that this assumption is strictly weaker than homogeneity and strict monotonicity, which are commonly used in prior work.  Based on this assumption, the authors derive a theoretically loss function that avoids SCM estimation and bi-level optimization, making their method  practically implementable. The proposed estimator also demonstrates strong empirical performance across multiple benchmarks and is backed by consistency, unbiasedness, and convergence guarantees.\n\nDespite these contributions, the paper could benefit from more details on the assumption, in particular a more intuitive explanation of the rank preservation assumption (Assumption 4.2). While the definition is mathematically rigorous, its practical implications may not be immediately obvious. A brief paragraph and example are provided, but this may not be sufficient to help readers build strong intuition. Including additional pedagogical material to discuss in depth the different assumptions would significantly improve accessibility.\n\nIn addition,  the exposition of the paper and in particular in Section 5,  the derivation and approximation of the ideal loss, could be streamlined. The writing is mathematically dense and is difficult to follow. Distinguishing between the intuition, formal derivation, and implementation steps could enhance clarity.", "was_revised": true, "final_justification": "The authors will include in the revised version more insight about the rank assumption as well as details on the estimation process which will reinforce the contribution.", "rebuttal": "Thank you very much for your positive evaluation of our paper. Below, we hope that the our clarification addresses your concerns.\n\n> **Q1:** The authors claim that Assumption 4.2 holds for common models such as those with additive noise $Y = g(X, Z) + U$ or heteroscedastic noise. Could you elaborate on why this is the case?\n\n**Response to Q1:** Thanks for your comments and we apologize for any lack of clarity. Consider the additive noise model ($Y = g(X, Z) + U$) as an example. This model satisfies both the homogeneity and strict monotonicity conditions (Assumptions 3.1 and 3.2). Then by Proposition 4, \\$\\rho(Y_x, Y_{x'}\\mid Z)=1\\$ (Assumption 4.2) holds.  \n\n> **Q2:** How can practitioners assess or empirically validate the rank preservation assumption in real-world applications? \n\n**Response to Q2:** Thanks for your comments. In response to your question, we would like to clarify the following two points:  \n\n**First, the rank preservation assumption is unverifiable empirically.** This challenge is not unique---all causal inference methods rely on fundamental assumptions that cannot be directly tested, such as unconfoundedness or the validity of instrumental variables. The plausibility of these assumptions must instead be evaluated through domain expertise and contextual knowledge. Consequently, the choice of appropriate causal methods and identifiability assumptions depends on the specific problems at hand. This is the key difference between causal methods and non-causal methods.  \n\n**Second, to assess the robustness of the proposed method, we typically perform sensitivity analyses under scenarios where key assumptions are violated.** In Appendix E, we explored empirical performance with violated assumptions on rank preservation. It indicates that the proposed perform stably well even if the n rank preservation is violated. \n\n> **Q3:** In the experiments, the proposed method appears to outperform all competitors consistently. This is somewhat surprising, as methods usually have specific regimes where they excel and others where they do not. Could you provide some intuition for these results? Have you explored other scenarios or settings? Did you use simple methods for estimating nuisance functions for the baselines? This is particularly surprising in settings where the assumption is violated, such as in Table 4 of the appendix, where some competitive methods are missing. Also, in non-simulated settings, how do you choose or tune the bandwidth parameter? \n\n**Response to Q3:** Thanks for your comments. **We first would like to clarify that as shown in Table 3, although our method overall stably outperforms the baselines, the performance of CFQP on out sample $\\sqrt{\\epsilon_{PEHE}}$ is better than our method on the IHDP dataset, and the performance of CFRNet on in sample $\\epsilon_{ATT}$ is better than our method on the Jobs dataset.** \n\n**In addition, on the synthetic dataset, as shown in Table 1, our method outperforms among all scenarios, which is due to the data generation process already satisfying the ranking assumption.** These results are intuitive: this is because all the baselines (except Quantile-Reg and Ours) are not designed for estimating counterfactual outcomes; they are designed to estimate the conditional average treatment effects $E[Y_{x'} \\mid Z=z]$. In addition, they do not use the information of $y_{x}$ when estimating $y_{x'}$. \nFor the estimation methods for counterfactual outcomes, incluing Quantile-Reg  and Ours. Since the data-generating mechanisms violate the homogeneity assumption (Assumption 3.1), and the Quantile-Reg rely on the homogeneity assumption and thus underprior our methods.   \n\n**Moreover, we explored other scenarios or settings as shown in Table 2 and Table 4.** Taking Table 4 as an example, compared to the original results in Table 1, the advantage of our method is smaller, and some results (3 out of 8) are not significantly outperformed the best baselines. \n\n**We do not use simple methods for estimating nuisance functions for the baselines; all methods are implemented as it should be and all the results are compared in a fair way.**\n\nFinally, in non-simulated settings, for tuning the bandwidth parameter. We treat it as a hyperparameter and use grid search for selecting the bandwidth. We will add these details in the revised version. Thanks again. \n\n> **Q4:**  The reliance on kernel-based estimation could limit scalability in high-dimensional or large-sample settings. How do you handle scenarios with many continuous confounders? In particular, in your experiments, the estimator appears quite stable even when the covariate dimension reaches 40; could you clarify why?\n\n**Response to Q4:** We thank the reviewer for the questions. Note that we do not use kernel function directly on the original covariate. Instead, we first learn a low dimension covariate representation, then use this representation as the input, see the code in the supplementary for more details (line 42 in Ours1.py file). This is a standard implementation in the community of causal machine learning. We will clarify this issue in the revised manucript. \n\n> **Q5:** Finally, the abstract states that rank preservation is “not stronger” than homogeneity and strict monotonicity, whereas the main text (e.g., Proposition 4.1) shows that it is in fact strictly weaker. Clarifying this language in the abstract would help avoid potential confusion for readers.\n\n**Response to Q5:** Thanks you for pointing this out. We will revise the abstract to make it consistent. \n\n\n> **Q6:** In line 120, the references cited do not seem directly relevant to the point made. Highlighting heterogeneity does not necessarily imply that one must estimate individual-level effects rather than conditional effects. \n\n**Response to Q6:** Thank you for your valuable comment. The cited references here all emphasize the distinction between individualized treatment effects and conditional average treatment effects, rather than asserting that one must estimate individual-level effects instead of conditional effects. We will revise this subsection to clarify this point further."}, {"review_id": "6uTeNDMLBC", "reviewer": "Reviewer_X9RM", "summary": "The authors introduce a novel condition (assumption 4.6) for counterfactual identifiability in RCTs that is a strict relaxation of the common \"homogeneity + monotonicity\" requirement. Roughly, this condition demands that \"for any two individuals sharing the same observable pre-treatment features $Z = z$, if one individual fares better under any treatment $x$ they fare better under all treatments\". The authors develop a consistent estimator for counterfactual learning under this condition.", "questions": "The authors did not benchmark their estimator in an example where \"homogeneity + monotonicity\" is satisfied. How would their estimation strategy would compare to the classical quantile-based estimator in this setting?", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "The authors' condition cuts explicit dependence on $U_X$, and, in my view, directly captures the intuition used when justifying the monotonicity condition. This is a significant improvement over a classical result and the authors clearly communicate the advantage of their approach through examples and experiments.\n\nAs for weaknesses, I have a few comments: \n- I do not agree with line 169 stating arbitrary heteroskedastic models satisfy the condition. If $h(X, Z)$ changes sign between $X = x$ and $X = x'$ for some $Z = z$ then the assumption is violated since we have $h(x, z) h(x', z) < 0$. This is an important distinction since the authors' assumption does not allow a treatment to be beneficial to some individuals with the same $Z = z$ and harmful to others. \n- The authors should make the non-technical statement of their assumption explicit, rather than describing it in solely in terms of the rank coefficient.", "was_revised": true, "final_justification": "I am impressed with the authors' submission and any concerns raised by the other reviewers have not significantly affected my score. \n\n\"Monotonicity + heterogeneity\" is a classical, heavily cited assumption often taught to undergraduates. Fundamentally, it is a functional assumption which looses its intuition (no defiers) for continuous treatments and/or outcomes. The authors show that a weaker assumption, based purely on counterfactual reasoning, can achieve the same identifiability results.", "rebuttal": "Thank you very much for your positive evaluation of our paper. Below, we hope that the our clarification addresses your concerns.\n\n> **W1:** I do not agree with line 169 stating arbitrary heteroskedastic models satisfy the condition.\n\n**Response to W1:** Thank you for pointing this out. We fully agree with you and we will revise it accordingly. \n\n\n> **W2:** The authors should make the non-technical statement of their assumption explicit, rather than describing it in solely in terms of the rank coefficient.\n\n**Response to W2:** Thank you for your helpful suggestions. **Below, we provide more non-technical statements for Assumption 4.2 (\\$\\rho(Y_x, Y_{x'}\\mid Z)=1\\$).** \n\nAssumption 4.2 implies that an individual’s factual and counter factual outcomes have the same rank in the corresponding distributions of factual and counterfactual outcomes for all individuals. To better understand this, imagine two counterfactual worlds: \n- In the first world, every individual receives treatment $x$;\n- In the second world, every individual receives treatment $x'$. \n\nFor a given individual $i$, their outcomes in these two worlds are $y_{i, x}$ and $y_{i, x'}$, respectively. Assumption 4.2  states that the rank of $y_{i, x}$ among all outcomes in the first world ${y_{j, x}: j = 1, ..., N}$, is the same as the rank of $y_{i, x'}$ among all outcomes in the second world ${y_{j, x'}: j = 1, ..., N}$.  In clinical medicine, we might consider a scenario where each individual’s health status ($U$) is fixed, and its rank is determined prior to treatment. Furthermore, this rank may influence—or depend on—the rank of potential outcomes after treatment.\n\n\n> **W3:** The authors did not benchmark their estimator in an example where \"homogeneity + monotonicity\" is satisfied. How would their estimation strategy would compare to the classical quantile-based estimator in this setting? \n\n**Response to W3:** Thanks for your insightful comments. We add experiments by modifying the data generation process to make the data satisfying \"homogeneity + monotonicity\". Specifically, we set $\\alpha = 1$ and add a constant in $Y_1$,  i.e., \n     $$Y_1 = W_y · Z + U_1 + 1 \\textup{ and }  Y_0 = W_y · Z + U_1  \\textup{ with }  Wy \\sim N (0, I_m).$$ \nThe corresponding numerical results are presented in the following tables. They show that under the 'homogeneity + monotonicity' setting, the proposed method performs less favorably than the quantile-based estimator. \n\n|Methods|Sim-5||Sim-10||Sim-20||Sim-40||\n|-|-|-|-|-|-|-|-|-|\n||In-sample|Out-sample|In-sample|Out-sample|In-sample|Out-sample|In-sample|Out-sample|\n|Quantile-Reg|1.772±0.903|1.759±0.926|1.356±0.648|1.354±0.654|1.607±0.507|**1.600±0.503**|**1.563±0.595**|**1.585±0.593**|\n|Ours|**1.405±0.110**|**1.393±0.110**|**1.347±0.274**|**1.347±0.292**|**1.593±0.753**|1.606±0.748|1.611±0.813|1.620±0.823|\n\n\n|Methods|Sim-5||Sim-10||Sim-20||Sim-40||\n|-|-|-|-|-|-|-|-|-|\n||In-sample|Out-sample|In-sample|Out-sample|In-sample|Out-sample|In-sample|Out-sample|\n|Quantile-Reg|1.188±0.906|1.162±0.896|0.876±0.630|0.873±0.647|**1.243±0.718**|**1.231±0.714**|**1.074±0.730**|**1.091±0.725**|\n|Ours|**0.984±0.157**|**0.967±0.164**|**0.779±0.460**|**0.771±0.499**|1.537±0.949|1.532±0.944|1.138±0.793|1.157±0.800|\n|"}, {"review_id": "f9B1NkzmF3", "reviewer": "Reviewer_5kW7", "summary": "This paper presents a new approach for individual-level counterfactual inference that does not require a known structural causal model (SCM). The authors introduce a \"rank preservation\" assumption to identify the counterfactual outcome. Building on this, they propose a convex ideal loss function and an unbiased kernel-based estimator. Theoretical analysis shows their assumption is no stronger than those used in prior work, and experiments on semi-synthetic and real-world data demonstrate the method's effectiveness.", "questions": "- The proposed loss function is a key component of the method. Could the authors provide more intuition and discussion behind its design? \n- The paper claims that the method can be extended to continuous treatments and that Proposition 5.3 remains valid in this setting. However, a formal proof for this claim is not provided. Could the authors please discuss on this point?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n- The paper is well-written, and its arguments are clear.\n- A significant advantage of the proposed method is its ability to perform counterfactual inference without relying on a known SCM.\n- The approach is supported by theoretical results that formally establish its properties.\n\n**Weaknesses:**\n- The paper's core contribution relies on the rank preservation assumption. While the introduction briefly explains it, it lacks a discussion or illustrative examples to motivate why this assumption is appropriate and practical in real-world scenarios. Adding this would significantly strengthen the paper's introduction.\n- There are some minor typographical and organizational issues that should be addressed:\n    - On line 184, \"a SCM\" should be corrected to \"an SCM\".\n    - Colons appear to be missing in several places where definitions are introduced (e.g., lines 94, 159, 194).\n    - The placement of Section 6 after Table 1 disrupts the flow; it would be more logical to place it before the table.", "was_revised": true, "final_justification": "I think the authors' rebuttal addresses some of my concerns.", "rebuttal": "Thank you very much for your positive evaluation of our paper. Below, we hope that our clarification addresses your concerns.\n\n>**W1:** The paper's core contribution relies on the rank preservation assumption. While the introduction briefly explains it, it lacks a discussion or illustrative examples to motivate why this assumption is appropriate and practical in real-world scenarios. Adding this would significantly strengthen the paper's introduction.\n\n**Response to W1:** Thank you for your helpful suggestions. **We first provide more explanations for Assumption 4.2 (\\$\\rho(Y_x, Y_{x'}\\mid Z)=1\\$).** \nAssumption 4.2 implies that an individual’s factual and counterfactual outcomes have the same rank in the corresponding distributions of factual and counterfactual outcomes for all individuals. To better understand this, imagine two counterfactual worlds: \n- In the first world, every individual receives treatment $x$;\n- In the second world, every individual receives treatment $x'$. \n\nFor a given individual $i$, their outcomes in these two worlds are $y_{i, x}$ and $y_{i, x'}$, respectively. Assumption 4.2  states that the rank of $y_{i, x}$ among all outcomes in the first world ${y_{j, x}: j = 1, ..., N}$, is the same as the rank of $y_{i, x'}$ among all outcomes in the second world ${y_{j, x'}: j = 1, ..., N}$. \n\n**Then, for an illustrative example in clinical medicine,** we might consider a scenario where each individual’s underlying health status ($U$) is fixed (here we don't consider $Z$ for simplicity), and thus its rank is determined prior to treatment. If all patients received the same treatments, then the rank of outcomes for each individual remain the same rank of the individual underlying health status. \n\n>**W2:**: There are some minor typographical and organizational issues that should be addressed. \n\n**Response to W2:** Thank you for pointing this out. We will revise it accordingly. Thanks again. \n\n\n>**Q1:** The proposed loss function is a key component of the method. Could the authors provide more intuition and discussion behind its design?\n\n**Response to Q1:**  Thank you for raising this interesting question. Actually, the design of the loss function is more mathematical than intuition. Inspired by Koenker and Bassett (1978) [reference 32 of the manuscript], we found that \n$$\\frac{\\partial \\mathbb{E}[ |Y_{x'} - t|  \\mid  Z=z ] }{ \\partial t} =  2 \\mathbb{P}(Y_{x'} \\leq t \\mid Z=z)  - 1$$\nTo construct a loss function whose first derivative satisfies \n $$2 \\\\{ \\mathbb{P}(Y_{x'} \\leq t \\mid Z=z) -  \\mathbb{P}(Y_{x} \\leq y \\mid Z=z) \\\\}  = 0.$$\nIt is natural to define the loss function as \n  $$ R_{x'}(t| x, z, y) =  \\mathbb{E}[ |Y_{x'} - t |  | Z=z ] + (1 - 2\\mathbb{P}(Y_{x} \\leq y \\mid Z=z)) t  = \\mathbb{E}[ |Y_{x'} - t |  | Z=z ] + \\mathbb{E}[ \\text{sign}(Y_x - y)  | Z=z]  \\cdot t.$$ \n\n\n>**Q2:** The paper claims that the method can be extended to continuous treatments and that Proposition 5.3 remains valid in this setting. However, a formal proof for this claim is not provided. Could the authors please discuss on this point?\n\n**Response to Q2:** Thanks for your comments. In Appendix D, we have presented the loss function for continuous treatments along with a brief proof. A similar derivation can also be found in Theorem 1 of [1]. We would be happy to provide a more detailed proof if the current presentation requires further clarification. \n\n---\n[1] Nathan Kallus and Angela Zhou (2018), Policy Evaluation and Optimization with Continuous Treatments, AISTATS"}]

Author rebuttal:

Meta-review: This paper introduces a new assumption of rank preservation for counterfactual outcome estimation. 
The authors propose a learning framework that shows both theoretical guarantees and strong empirical performance.

The main strength of the paper is its novelty in relaxing traditional assumptions, providing a more flexible path for counterfactual inference, which was highly valued by the reviewers, while the main weakness is that rank preservation cannot be tested in practice. 

Overall, the contribution is clear and significant. With stronger discussion of the practical meaning of rank preservation, the impact of the paper would be even greater.
Decision: Accept (poster)