A Relative Error-Based Evaluation Framework of Heterogeneous Treatment Effect Estimators
Authors: Jiayi Guo
Haoxuan Li
YE TIAN
Peng Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Causal Inference
Conditional Average Treatment Effect
Relative Error
Robust Evaluation
Primary area: causal reasoning
TL;DR: We propose a robust relative error-based evaluation framework for heterogeneous treatment effect estimators.

Abstract: While significant progress has been made in heterogeneous treatment effect (HTE) estimation, the evaluation of HTE estimators remains underdeveloped. In this article, we propose a robust evaluation framework based on relative error, which quantifies performance differences between two HTE estimators. We first derive the key theoretical conditions on the nuisance parameters that are necessary to  achieve a robust estimator of relative error. Building on these conditions, we introduce novel loss functions and design a neural network architecture to estimate nuisance parameters, thereby obtaining a robust estimation of relative error. We provide large sample properties of the proposed relative error estimator. Furthermore, beyond evaluation,  we propose a new learning algorithm for HTE that leverages both the previously HTE estimators and the nuisance parameters learned through our neural network architecture. Extensive experiments demonstrate that our evaluation framework supports reliable comparisons across HTE estimators, and the proposed learning algorithm for HTE exhibits desirable performance.

Published as a conference paper at ICLR 2026
A RELATIVE ERROR-BASED EVALUATION FRAME- WORK OF HETEROGENEOUS TREATMENT EFFECT ESTI- MATORS
Jiayi Guo', Haoxuan Li', Ye Tian", Peng Wu
Pekingg University, Northeast Normal University. BBeijing Technology and Business University
ABSTRACT
While sigmificant progress has been made in heterogeneous treatment effect (HTE) estimation, the evaluation of HTE estimators remains underdeveloped. In this article, we propose a robust evaluation framework based on relative error; which quantifies performance differences between two HTE estimators. We first derive the key theoretical conditions on the nuisance parameters that are necessary to achieve a robust estimator of relative error. Building on these conditions, we introduce novel loss functions and design a neural network architecture to estimate nuisance parameters and obtain robust estimation of relative error, thereby achiev- ing reliable evaluation of НTE estimators. We provide large sample properties of the proposed relative error estimator. Furthermore, beyond evaluation, we propose a new learning algorithm for HTE that leverages both the previously HTE estima- tors and the nuisance parameters learned through our neural network architecture. Extensive experiments demonstrate that our evaluation framework supports reliable comparisons across HTE estimators, and the proposed learning algorithm for HTE exhibits desirable performance.
INTRODUCTION
The estimation of heterogeneous treatment effects (HTFs) has attracted substantial attention across range of disciplines, including economics (Imbens & Rubin 2015). marketing (Wager & Athey 2018b), biology (Rosenbaum 2020), and medicine (Heráán & Robins 2020), due to its critical role in understanding individual-level treatment heterogeneity and supporting personalized, context-specific decision-making (Wu et al., 2024b ch. Various methods have been developed to estimate HTES; see Kunzel al. (2019); Caron et al. (2022) for comprehensive reviews. Despite their growing popularity, the evaluation of HTE estimators remain relatively underexplored (Gao 2025). Assessing estimator performance crucial in real-world applications, as a reliable evaluation framework can identify the
most suitable methods (Curth & Van Der Schaar 2023), directly impacting downstream tasks.
Evaluating HTES is inherently challenging. as the ground truth is not available: only one potential outcome observed for each individual, while HTEs are defined as the difference between two. To address this, researchers often rely on stringent model assumptions (Saito YathiAu 2023
Mahajan al. 2024 Wu & Wang 2026) preprocessing methods (e.g.. matching, Rolling & Yang (2014) to approximate the unobserved counterfactuals, and obtain an estimated treatment effect. Our work is motivated by Gao (202)), who introduced relative error to quantify the performance difference between two estimators, thereby reducing the bias caused by using inaccurately estimated treatment effects as ground truth.
Despite the significant contributions of Gao 2025) a notable limitation remains unaddressed. Their estimator requires that all nuisance estimators (propensity score and outcome models) are consistent at rate faster than n- 1/4 to achieve consistency and valid confidence intervals for the relative error, which may be too stringent for real-world applications. In practice, the outcome models for potential outcomes heavily rely model extrapolation. These models are trained separately within the treated and control groups, yet their predictions are applied across the entire dataset. When
"Corresoonding author; pengwu@hthu.eddan

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there exists significant distributional difference between the treated and control groups (Jeong & Namkoong/ 2020; Jing Qin & Huang 2024), the extrapolated predictions from these models are prone inaceuracy, potentially leading to unreliable conclusions. Therefore, it is desirable develop methods that reduce reliance on such extrapolation to ensure more robust and trustworthy evaluations.
To address this limitation, we propose reliable evaluation approach for HTE estimation that retains the desirable properties of the method Gao (2025). while relaxing the requirement for consistent outcome regression models. We show that the proposed estimator of relative error /n-consistent, asymptotically normal, and yields valid confidence intervals, provided that the propensity score model consistent at a rate faster than n -111/, even if the outcome regression model is inconsistent.
This robustness is achieved by carefully exploring the relationships between nuisance parameter models. We first derive the key conditions necessary for robustness and then design novel loss function for estimating outcome regression models. Moreover, because our method still relies consistent propensity score model, we introduce new balance regularizers that reduce this dependence by encouraging the estimated scores to satisfy the balance property (Imai & Ratkovic, 2014), ensuring that inverse- inverse-propensile propensity-weightee covariate functions have equal expectations across treatment groups.
Furthermore, by combining the novel loss function with balance regularizers, we design a new neural network architecture that more accurately estimates outcome regression and propensity score models, enabling more reliable relative error estimation and, in turn, more robust HTE evaluations. It should be mentioned that the proposed method does not require sample splitting and is numerically more tractable, which is discussed in Section 4.4
The main contributions are summarized as follows.
We reveal the limitations of existing methods and, through theoretical analysis, derive key conditions for estimating the relative error that mitigate these limitations.
We propose reliable HTE evaluation method by designing novel loss functions and introducing new neural network, enabling more robust estimation of relative error.
We conduct extensive experiments demonstrate the effectiveness of the proposed method.
2 PRELIMINARIES
2.1 PROBLEM SETTING
We introduce notations to formulate the problem of interest. For each individual i, let A, E {0, 1} denote the binary treatment variable, where A, 1 and A, 0 denote treatment and control. Let X; E X C IR be the pre-treatment covariates, and Y E R be the outcome. We adopt the potential outcome framework in causal inference, defining Y2()) and Y;(1) the potential outcomes under A; = and A; 1, respectively. Since each individual receives either the treatment or the control, the observed outcome Y, satisfies Y, (1) + (1 (0).
The individual treatment effect (ITE) is defined as Y₁(1) - Y2(0). which represents the treatment effect for specific individual However, since only of (Y) Yi(1)) is observable, ITE is not identifiable without imposing strong assumptions (Pearl) 2009 Wu et al. (025). In practice, the heterogeneous treatment effect (HTE) is often used characterize "individual" treatment effects.
(xx (E(((1) Y((O)X; x/.
which captures how treatment effects vary across individuals with different covariate values. Assumption (Strongly Ignorability, [Rosenbaum & Rubin 1983)) (i) (Y:(0), Y())) IL A, Xx: (ii) 0 < e(x) A P(A 1 X; x) for all X, where e(x) is the propensity score.
Under the standard strong ignorability assumption, HTE is identified as μu (a) (1)))~p(()) (0( where paa (z) = E|Y; X; = A; = a) for a = 0, 1 are the outcome regression functions, and various methods have been developed for estimating HTE (Wager& & Athey 2018a| Shalit et al. 2017a). Suppose we have a set of candidate HTE estimators trained select the estimator with the highest accuracy of HTE using a test dataset {(,,, Y)), 1, 1,...,n), which is of size and sampled from the super-population IP, and is independent of the training set.
training set, denoted by (z), ... fx (z)) We aim to
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2.2 EVALUATION METRICS: ABSOLUTE ERROR AND RELATIVE ERROR
For a given estimator f(x), its accuracy is typically evaluated using the MSE defined by
((f) e E[(f(X) -(X))²).
For any two estimators A (z) and (2)), the difference in their MSE is
,() - ((+)) ( (x) (x) - - (2())
Gao (025) refers to (f+) and 8(f1, f2) as the absolute error and relative error, respectively. In
practice, absolute error used much more frequently than relative error. However, [ao (2025) established, through theoretical analysis and experiments, that relative error is preferable to absolute error; details are provided in Section 3 Intuitively, one can see that the key advantage of using relative error over absolute error lies in that it only relies on the first-order term of the unobserved which reduces the impact of estimation error in 7.
Several studies (Gutierrzz & Gérardy 2017 Powers et al. 2017 have used ((((()))(()=+(()) to evaluate the estimator F(x). However, its estimator requires knowing the values of (Y()), that are not observable in real-world applications. We note that
(I( (0) - +(x))²
- - + E[Var(Y (1) (O))X)),
where the second term on the right-hand side is independent of F(x). Thus, this metrie is essentially equivalent to the absolute error 6(f), and we will not discuss it further. For clarity, we provide notation summary tab, but due limited space, we present it in Appendix
3 MOTIVATION
In this section, we briefly discuss the advantages of using relative error over absolute error, and then analyze the limitations of the method in Gao (2025), which motivate this work.
The key theoretical advantage of relative error over absolute error demonstrated through its semiparametric efficient estimators. A semiparametric efficient estimator is considered optimal (or gold standard) in the sense that it has the smallest asymptotic variance under regularity conditions (Newey) 1990: van der Vaart 1998) given the observed test data. Let (ē(x),, (æ), Au(x) } be the which are the nuisance functions to construct semiparametric efficient estimators of absolute error and relative error. Denote F(x) = fi (a) (o(x).
estimators of (e(x), jii (I), oo (x)
Absolute Error. Given f(x). an estimator of ((+) is constructed as
[Ax(Y; - (X.)) - 1-=A)(Y - (x.)) ((X))
((F)
71 Σ (F(x)) +(x))) +
F(X.))
1 1- -(x()))
Under Assumption I ((+)) /n-consistent, asymptotically normal, and semiparametric efficient, provided that the estimated nuisance parameter satisfy the key Condition
Condition 1. [[(((()) (x))) or 1/22), (X) pa (X)) (n 1 for a 0,
Relative Error. Likewise, we can construct the estimator of 8(f1, f2) given
(2) = l (Zz; C), where 2, (A,, X,, Yi),
p(Ze F2, A (xx())) (X.)) - 2(f) -xx)) fx(x))
A. (X,))) + (X.) (1-A0) A.) (X.)) po(xi))
e(Xi)
1 1- E()(Xi)
Under Assumption 1 the /ii-cooniistency. asymptotic normality, and semiparametric efficiency of 8(f2. T2) rely on the key Condition 2 below.
Condition 2. (X) ma (X)]l((X)
e(X)] (1122
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Condition is strictly weaker than Condition Moreover, the estimator 8(f1, f2) offers several additional advantages over ((f), see Appendix B for more details.
Motivation. Although 6( f2) has several desirable properties, a notable limitation that Condition 2 requires all nuisance parameter estimators to be consistent (as ē(x) and (z) generally converge at most at the rate of n n= 1/2), which may be overly restrictive in practical settings. In practice, the outcome regression model faa (x) is learned from the data with A = a and then applied to the entire data. It heavily relies on model extrapolation, as there is often significant distributional difference between the data with A = a and A = a (Jeong & Namkoong 2020 Jing Qin & Huang 2024). As result, (x) is likely to be inaccurate and biased, violating Assumption it is beneficial and practical to develop methods that rely less on model extrapolation. In contrast, estimating the propensity score does not involve any model extrapolation, as the score is learned from the full dataset. As result, the model is not required to predict outside the support of the observed covariates, making propensity score estimation substantially less susceptible to extrapolation errors and more robust to flexible machine learning specifications.
Therefore,
natural and practical question arises: Can we develop a method for estimating ö(fi, f2), that retains all the desirable properties of 8f f2), while allowing for bias in (z (relaxing Condition 2 In this article, we show that this is achievable by carefully exploiting the connection between the propensity score and outcome regression models, and by designing appropriate loss functions.
4 PROPOSED METHOD
In this section, we propose novel method for estimating the relative error 8(1,, f2) that retains the desirable properties of 8(f1, f2) while simultaneously being robust to bias in (a(x) for 0, 1. We consider the following working models for the propensity score and outcome regression functions,
exp(((X)T>)
e(X) = P(4 - x) c((0())())))))))
1 1 exp((((T))))
(X) = E( = 0bx()) a 0,1,
(2)
where (XX is representation of X adaptively learned from data. It facilitates theoretical analysis of the connections between models and helps establish more desirable theoretical properties and is widely used in the literature (e.g. (Shi al.) (019b)) and usually yield more stable numerical results. That said, to illustrate the role of representation sharing more concretely, we also report in Appendix F.2 the empirical performance of the method when shared representation is used.
To quantify the bias of (z). it is crucial to distinguish between the working model and the true model. We say a working model is misspecified if the true model does not belong the working model class, and it is correctly specified if the true model is within the working model class. For clarity, we provide a misspecified example in Appendix B
For models 1 and (2). let and By denote the estimators of y and Ba, respectively. Define & and B as the probability limits of and 3a. and denote e(X) e (x)((X), (X).)) 5) and (X) pa (xx), B.). If model (I) is specified correctly, e(X) = e(X); otherwise, e(X) + E(X) and their difference represents the systematic bias induced by model misspecification. Similarly, if model 2 is correctly specified, (X) Po(X), otherwise, (X) + (X)... It is important to note that (5, Bo. 31) always converges to (7, Bo, B1), regardless of whether models ( and (2) are correctly specified.
4.1 BASIC IDEA
Before delving into the details, we outline its basic idea to provide an intuitive understanding.
First, to retain the semiparametric efficiency, the proposed estimator 6(f1, i2) preserves the same form of 8ff1, 12), which is given
1 n 72
8(1, 7. Bo. B1)
Σ
(Z₁; Po: (2,
where é(X) = e($(X),)) and p.a (X) ( B.) for a = 0, Although 8(f,, f2: Bo. Br) shares the same algebraic form 8(f1, F2), its construction is fundamentally different, which
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makes the resulting estimator substantially more robust to bias in (X). This distinction is key methodological contribution of our work.
Second, we analyze the key conditions necessary to achieve robustness to biases in [a() By Taylor expansion of Öff 12: 9, Po. B1) with respect to (7, Po. 31), we have that
6(f1 92:7. Po. B1) f2; B1) AT - A (Bo Bu) + 1 (31 32) -))((--)))++ 5) - + Bo)):
where
A:(1 - E(XI))X (xx))
(1 A-)e())) - po((i))) (xx)))
A, = (x,) - ((x)))
(x()))
((Xi)
E(X.)
(1 1 A
Σ
2( (Xi) - (xx.))
1 e(X:) ((()),
i=1
A e(X.) ((Xi).
Σ (X.) +x())))
Under mild conditions (see Theorem 1. the last term of above Taylor expansion is op(n-1/2). We note that 8(f1, f2; n, 3o. , 31) is a /n-consistent and asymptotically normal estimator of 8(1,, i2) if (#o(x)) Pi (a)) is correctly specified. Thus, it is robust biases in joa (x) for a 0, 1 and is the ideal estimator we aim obtain. To ensure that 8(f1, f2; 9, Bo. B1) has the same asymptotic properties 8(fi, f2: 7, Bo B1), we require that
either ((x: is correctly specified,
AT 7)
(3o Bb) 3, (Bi B1)) or(n-1/2)
even when (jo (1), ; fi (1)) is misspecified. Note that (7, Bo. 3,) always converges to (5, 3o, 31). To satisfy Eq. 3 it suffices for A,, A Ba and Ag, to converge to zero at certain rate. By the central limit theorem, A E[A,] On(n 1/2), Apo B[Ag,] Or(n-1/2), and As, E[A,, Op(n-1/2) Thus, Eq. (3) holds provided that
E[A,] = 0, E[Ap2] = O, E[A,,] = 0,
which is equivalent to the following equations:
(X,) (x()))
(x(1-=(()))(
(1- (1-A))0(x)00 (X.)) ((X.) =0, 1-2(X,)
((X+)
E (fr(),) (x()))
1 A.
2(X,) =0,
(4)
a(X.)
E (f)()) - (x()))
(1 (1-1-a I-((X))
1-A
((X,)
= O.
4.2
NOVEL LOSS FOR NUISANCE PARAMETER ESTIMATION
To ensure that the first term in Eq. 4 holds, we design the weighted least square loss function for (Bo, B1) as follows:
(1 A.)((X(0G 4(X)T Adl - -(X))( 4(X)&
Con(30. B1:5)
Σ (X.) -
( (xx.))
E(X))
e(X,)
These loss functions imply that 31) A arg ming El 31; 7)]. By setting [E(Lwt 31; 31)50 a== = O and OE WE(L (Bo. 31; 1 s. = 0, one can see that the first term in Eq. (4) holds even if (z), A (x)) misspecified.
For learning 7, observe that Eq. (4) specifies 2d linear constraints, whereas 7 E Rd contains only d free parameters, making the system inherently over-constrained. To obtain feasible solution, we adopt a soft relaxation strategy inspired by the soft-margin formulation of support vector ma- chines (Murphy 2022): we introduce slack variables & n RRd that permit controlled deviations
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from the constraints, and penalize their magnitudes in the optimization objective. Formally, solve:
1 7 Σ [A. log(e(X.)) - (1 A.) - e(X.))) (5) + n²)
min 7.-.9
exp(((X::) 7)
e(X.) + exp(((X,) 7)²
i= 1=1,...,,
Σ (x.)))) -x((..)) (1- (1-o(x))
A
(y((.) Sti Vi.
n
c(X.)
Σ (x.) - f2(x.)) 11-
I- (1-
e(x.) --(((x))
Sbb, Vi. 1 -
A,
(x(x))
n
n, j-L...,d
where e is a given hyperparameter. In practice, we convert the above constrained optimization into two unconstrained loss terms, including the cross-entropy loss
Lee
[A log(c(X.)) + A:) log(1 c(x()-] c(X:)))
and the balance regularizer
max
E. (F) - -2((x)) (1
E. 0f T-o(X,) x(X.)
o(x
A
((Xi)
max D (f - (x.))) (1
1-4
17 0}
Leunst
C + 1b: +p-
max E, 0)
max (77 0)
where p 0 is penalty parameter encouraging the satisfaction of the original constraints. As we show in Appendix F:4, this relaxation is effective in practice, and the resulting unconstrained formulation still enforces the original conditions to high degree of accuracy.
4.3 CONSTRUCTING NEURAL NETWORK
Building the constraint loss from Section we adopt neural architecture derived from the Dragonnet framework (Shi et al. 2019a). The proposed network takes input features x E RR,, and first passes them through multiple fully connected layers to produce the shared representation (xx) R This representation then fed into three separate heads: a control outcome head (o()), predicting the potential outcome under control; treated outcome head μ4 (x). predicting the potential outcome under treatment; a treatment head e(z), estimating the propensity score via a sigmoid activation.
The control outcome head and the treated outcome head contribute to the weighted least square loss Lwls, while fae and Leons are computed by the treatment head and the shared representation. During training, we minimize the total training loss given by:
L = Lwla + Acece + A2Cooust-
This formulation encourages the propensity model e(X) and the outcome model (a(X) to satisfy Eq. (4). providing a reliable estimation that can be used computing the estimated relative error mentioned in Section 3 For clarity, we provide schematic illustration of the network architecture in Appendix D
4.4 THEORETICAL ANALYSIS
We analyze the large sample properties of the proposed estimator ö(f1, 92; 5 Bo, B1).
Theorem 1. If the propensity score model is correctly specified, and 4, Bo as well as Br converge to their probability limits rate faster than n- 1/4, then we have
Vntōlfi, t2; Bo, B1) - +2) 4 o(0,0
where x² Var(yp(Z; un, U1, 3)} and 4, means convergence in distribution.

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Theorem 1 Ishows that the proposed estimator is /n-consistent and asymptotically normal. These properties hold even when the outcome regression model is misspecified, as long as Po. and & converge to their respective probability limits at rate faster than n n-1/4, This condition is readily satisfied, as (7, So: 31) always converge to their probability limits (7, Po. B2), and a variety of flexible machine learning methods can achieve the required convergence rates (Chermozhukov al. 2018
Semenova & Chernozhukoy (02TT.
Based on Theorem we can obtain valid asymptotic (1 (1-1) .7) confidence interval of 8(f1, f2).
Proposition 2. Under the conditions in Theorem a consistent estimator of o²
0²
1 n
(zlZz uo ún 8) 8(Fi, f2; 31)3
an asymptotic (1 11) confidence interval for 8(f1, f2) is ,o, B1) + 2n/2 2n/22/22// / where 2n/2 is the (1 n/2) quantile of the standard normal distribution.
9,
Proposition shows that a valid asymptotic confidence interval for 8(1, f2) is achievable even with a misspecified outcome model, unlike previous methods that require correct specification. This further indicates the robustness of the proposed method. We also emphasize that, unlike (Gao) (2025). our proposed methodology does not require sample splitting. The key derivation in Section 4.1 as well as the proofs of Theorem and Proposition in this section, are conducted using the full dataset without sample splitting.
Besides, we clarify that the correct specification of propensity score in Theorem and Proposition mild condition. First, ((X) can be adaptively learned from the data, are likely gain the true working model using flexible neural networks. Despite this, we provide a sensitivity analysis on the correct specification on (xX (Appendix FF3).. In addition, we could adjust the model specification iteratively by balancing checking as follows: (1) Model specification and estimation: specify the model and estimate the propensity scores; (2) Covariates balance checking (Imai Ratkovic 2014). If the covariates balance is inadequate, adjust the model specification and re-estimate the propensity scores; if the balance satisfactory, proceed with the estimated propensity scores.
ENHANCED ESTIMATION OF HETEROGENEOUS TREATMENT EFFECTS
In this section, building on the evaluation framework proposed in Section A develop learning method for HTE. In general, reliable evaluation method naturally serve as a basis for developing learning method. In our proposed approach, for any given pair of HTE estimators Fx (x) and Fx. (z)). the proposed neural network architecture introduced in Section 4.3 can output the corresponding estimates of outcome regression functions. We denote them as (a 7k. fx) and ýı (æ; tR. ), emphasizing their dependence on (x) and (x)). This leads to a new HTE estimator, defined
we extend the idea to
F(x; fx. fw) = ju (x; fr, fw) - noo (z; nk. fw).
Clearly, the performance of the estimator heavily depends on the choice of HTE estimators F(2; fk. ). However, due to the fundamental challenge in evaluating HTE (i.c., the absence of ground truth), it difficult to develop direct strategy for selecting them. To mitigate this issue, we propose the following aggregation strategy for estimating HTE,
2
F(x: ICIII 1)
Σ KKKKK
(n (k, fi) po (x;; fR, Pwr
where K. = 21, } is the index set for the candidate HTE estimators. This aggregated estimator aims to stabilize and improve the estimation of HTE by averaging over all pairs of candidate estimators. When K is large, averaging over all pairs can be computationally burdensome. In such cases, randomly select subset of pairs f(x; fr. f and compute their average instead. Surprisingly, our experiments show that this estimator performs exceptionally well, even surpassing the performance of any single candidate estimator.
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Coverage
Twing Coverage
JHDP-Seleeee Selection Accuracy
Twins Selection Accuracy
1.0
1.0
0.9
0.0
0.8
0.7
0.7
0.8
0.0
0.6
0.
TN C
vs
Figure 1: Coverage rate IHDP and Twins.
Figure 2: Selection accuracy on IHDP and Twins.
Table I: HTE estimation performance on the IHDP and Twins datasets (in-sample and out-of-sample). The best results are bolded.
IHDP
Twins
Method
v Goraa:
km
V Sran
<AT:
LanDLL
1,053 : 0.134
0
+ 0.152
1,885
:: 187
0.574
= 0.176
0.295 = 0.005
0.013
:: 0.009
0.296 :: 0.008
0.013 3:13:::00 0.010
SpaDML
0.832 0.119
0.252
1: 0.185
0,866
0.112
0.280
0.183
0,300
0.008
0.066
0.030
0.303
0.010
0.016
0.033
0.419
0.903
0.127
0.403
0.005
0.012
0.908
0.008
0.013
Leamme
0
0.178
0.196
1: 0.137
0.987
0.196
0.207
4: 0.141
0,293
0.005
0.022 2: 0.014
0.294 1: 0.008
0.024 0.016
S-1 Leanne
0.920 0. 0.102
0.212
+ 0.100
0.950
0.111
0.205 + 0.117
0.298
0.011
0.057
+ 0.042
0.299
0.010
0.059 0.022
TARNet
0.895 0.054
0.279
0.084
0.920
0.070
0.266
0.117
0,292
0.011
0.090
0.047
0.294
0.019
0.091
0.045
Dragonnet
D.840
0.046
0.124
0.089
0.867
0.087
0.134
DM02
0.292
0.004
0.000
0.00
0.290
0.007
0,002 0.011
DRCFR
0.741 0.068
0.186
0.138
0,760 1 0.090
0.185 1 0.135
0.290 = 0.001
0.075
1: 0.007
0.288 1: 0.007
0.076 1 0.010
SCIGAN
0,898
0.374
0,358
:: 0.509
0.919
369
0,358
0,502
0,296
0.037
0.041
0,044
0.393 0.039
0.040 1 0.047
DESCN
0.793 0.187
0.835
0.197
0.060
0.059
0.063
0.058
ESCFR
0,502 0.041
0.111 1: 0.070
0.074 0.841
0.135 1 0.076
0,290
0.001
0.075 2: 0.007
0.288
0.007
0.076
0.010
Ours
0.638 0.138
0.090 + 0.087
0.670
0.150
0.105 + 0.099
0.284 0.005
0.009 + 0.005
0.286 + 0.007
0.009 0.006
EXPERIMENTS
6.1 EXPERIMENTAL SETUP
Datasets and Processing. Following previous studies YYoon et al. 2018 Yao et al. 2018 Louizos et al. 2017). we choose one cymi-synthetic dataset IHDP, and two real datasets, Twins and Jobs, to conduct our experiments. The Twins dataset is constructed from all twin births in the United States between 1989 and 1991 (Almond et al., 2005), owning 5271 samples with 28 different covariates. The IHDP dataset is used to estimate the effect of specialist home visits on infants' future cognitive test scores, containing 747 samples (139 treated and 608 control), each with 25 pre-treatment covariates, while the Jobs dataset focuses on estimating the impact of job training programs on individuals' employment status, including 297 treated units, 425 control units from the experimental sample, and 2490 control units from the observational sample. We provide more dataset details in the Appendix F.I We randomly split each dataset into training and test sets in 2:1 ratio, and repeat the experiments 50 times for the Twins, 100 times for the IHDP, and 20 times for the Jobs.
Evaluation Metrics. We consider two classes of evaluation metries below.
We assess the proposed relative error estimator using two key metrics: (i) the coverage probability of its confidence interval (named coverage rate), and (ii) the probability of correctly identifying the better estimator (i.e., selecting the true winner, named selection accuracy). In practice, we only pick the winner when the confidence interval for the relative error does not contain zero, otherwise, no selection will be made. We calculate the coverage rate of the targeted 90 % confidence intervals.
For evaluating the performance of HTE estimation of our novel network, following previous stud- ies (Shalit et al. 2017a; Shi et al. 2019b Louizos et al. 2017), we compute the Precision in Estima-
tion of Heterogeneous Effect (PEHE) (and 2011). where /CPEEE = V Σ (f(x;) - +
and the absolute error on the ATE, ATT ATE
ATE| where ATE 1 EM Di (vt -13), in
which yf and vf are the true potential outcomes.
Baselines and Experimental Details. To evaluate the performance of relative error estimation, we select three representative estimators from different methodological families: Causal Forest (tree-based) (Athey & Wager 2019). X-Learner (meta-leamer) (Künzel et al. 2019). and TARNet (representation learning) (Shalit et al. 2017a). We estimate their pairwise relative errors and evaluate the estimation performances.
For HTE estimation, the baselines include Causal Forest (Athe & Wager 2019), meta-learners (X-Learner, S-LLearner) (Künzel et al. 2019), double machine learning (Linear DMI., Sparse Linear
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Table 2: Inference with Different Nuisance
Table 3: Running Time under Different Settings
IHDP
Twins
Sample Size
Time (s)
Candidate Est.
Time (s)
Nuisance
Coverage
Selection
Coverage
Selection
30
2.527
1,0780
400
2.6688
3.1321
Regression
0.94
0.44
0.94
0.88
Boosting
0.95
0.48
0.94
0.86
2.739
6,1955
600
3.063
12.2401
Ours
0.96
0.80
0.94
0.94
700
3.134
TARNet
2,0306
Table 4: Sensitivity analysis the hyperparameter X2 (weight of constraint loss) for IHDP and Twins datasets. The best hyperparameter values and results are in bold.
IHDP
Twins
Value
VEHE
V fesse
Coverage
Selection
Value
-ATE
ATT
Coverage
Selection
0.01
0.860
0.216
0.902
01238
0.85
0.50
0.005
0.319
0.331
0.027
0.82
0.38
0.1
0,800
0.142
0.837
0.158
0.91
0.61
0.05
0.289
0.016
0.292
0.015
0.82
0.84
0.714
0.099
0.747
0.118
0.95
0.78
0.297
0.018
0.297
0.020
0.86
0.638
0.090
0.670
0.105
0.96
0.80
0.5
0.284
0.009
0.286
0.009
0.94
0.94
5
0.715
0.099
0,748
0,116
0.94
0.77
2.5
0.285
0.011
0.287
0.012
0.94
0.92
0.157
0.172
0.289
0.290
100
0.801
0.156
0.836
0.170
0.90
0.60
50
0.287
0.024
0.288
0.023
0.84
0.88
DML) (Chernozhukov et al. 2024),
TARNet (Shalit et al. (017a), Dragonnet (Shi et al., ).).
DR-CFR (Hassanpour & Greiner 2020), SCIGAN [ica et al. 22020), DESCN (Zhong et al. 2022)
and ESCFR (Wang et al. 2023). See Appendix F.10 for training details of hyperparameter tuning.
6.2 EXPERIMENTAL RESULTS
Quality of Relative Error Estimation. We first evaluate the performance of relative error estimation, comparing different pairs of HTE estimators. In Figures [ and 2 we present the coverage of the 90% confidence intervals and the accuracy of selecting the better HTE estimator on the test sets, respectively, where TN stands for TARNet, CF stands for Causal Forest, and X stands for X-Learner, and the red dashed line marks the target level of 90%. From these two figures, our method successfully achieves the target coverage, and provide trustworthy advice on the selection across different pairs of HTE estimators. These results demonstrate the validity of our uncertainty quantification and estimator selection.
Accuracy of the HTE Estimation. We then evaluate the performance of HTE estimation learned by our novel network and compare with competing baselines. We average over 100 realizations of our networks in IHDP and 50 realizations in Twins. The results are presented in Table [T] Our proposed method achieves the best performance across all metrics, with the lowest VEPEhE and EATE on both datasets. This demonstrates its ability to accurately estimate HTE. In addition, we report results on the Jobs dataset in Appendix [55 due to limited space. We also examine how varying the number of candidate HTE estimators affects the final performance, and report these results in Appendix [F.6)
Comparison with Gao's Method. Although Gaols work does not propose a concrete learning method, we follow their choice of nuisance estimators (Linear Regression, Boosting) to compute relative errors for reference. We provide the results on the IHDP and the Twins. When plugging conventional nuisance estimators (linear regression and gradient boosting) into the relative error framework, the resulting procedures do achieve nominal coverage (Table 2 Nevertheless, the corresponding variance so large that the confidence intervals frequently include zero, making it essentially impossible to tell which candidate estimator is superior. These baselines therefore serve valid but uninformative references. contrast, our proposed method not only maintains well-calibrated coverage but also delivers much higher selection accuracy, producing confidence intervals that are substantially tighter and practically useful for identifying the winner.
Running Time under Different Settings. As the proposed HTE estimator indicates high level of complexity, we demonstrate experiment results for assessing the computational complexity and scalability of it. We vary sample size and the number of candidate estimators, and a more detailed description is provided in Appendix F.7 It can be seen from Table 3 that our estimator exhibits a runtime that grows approximately linearly with the sample size, but increases super-linearly the number of candidate estimators in the system expands. Nevertheless, when the system contains only small number of estimators, method remains faster than the baseline TARNet.

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Table 5: Ablation study results on the IHDP and Twins datasets.
IHDP
Twins
Training Loss
ViEE
<TE
AFE
Coverage
Selection
PENE
<XTE
V
CAfE
Coverage
Selection
Ceb feust
0.725
0.101
0.758
0.122
0.92
0.71
0.284
0.013
0.287
0.013
0.94
0.92
Exb & fex
3.495
2,879
3.531
2900
0.88
0.14
0.319
0.028
0.328
0.026
0.82
0.14
Full (Ours)
0.638
0.090
0.670
0.105
0.96
0.50
0.284
0.009
0.286
0.009
0.94
0.94
Table Sensitivity Analysis on Propensity Score
Metric
(0.05, 0.12)
(0.1, 0.13)
0.13)
(0.05, 0.39)
(0.1, 0.39)
(0.2, 0.32)
no noise
Coverage
0.88
0,94
0.96
0.88
0,94
0.80
0.96
Selection 0.74
0.78
0.82
0.76
0.78
0.74
0.84
Sensitivity Towards Propensity Score Misspecification. To test if our evaluation method is sensitive towards the misspecification of propensity score, we conduct sensitivity analysis. Because the true propensity score is required for this experiment, use simulated data here, and the data- generaenng process is provided in Appendix F.9 To evaluate the sensitivity of our method to the propensity score, we fix the t-head of our proposed neural network to the input propensity score while keeping all other computations unchanged. The input propensity score is constructed by adding Gaussian noise with varying means and variances the true propensity score. As shown in Table (we provide (p, of the gaussian noise), injecting noise leads to degradation in the accuracy and validity of the relative-error confidence intervals, but the decline is not substantial. Overall, our method appears to be reasonably robust to perturbations in the propensity score.
Sensitive Analysis. The hyperparameter A2 before the constraint loss Leonst, X before the cross entropy loss fie and the penalty weight p in the constraint loss play important roles in our training. In order to explore under which parameters our method has the best performance, we conduct sensitivity analysis experiments. We present the results of X2 in Table 4 We observe that both the performance of HTE estimation and the relative error estimation remain relatively stable across a range of X2 values from 0.5 to 5, indicating robustness to this hyperparameter. However, when À is extremely small (c.g., A2 = 0.01), the performance of the proposed method degrades significantly, indicating the importance of the constraint loss Leoma- Also, perform sensitive analysis for A and p. the associated results are provided in the Appendix F.
the proposed method involves three loss functions: Lils; Lee. and Leonst. We conduct an ablation study to assess the impact of Lee and Leonst on overall performance. The corresponding results are reported in Table Specifically, removing Leons results in notable drop in the accuracy of both outcome and relative eтror estimation, whereas removing Lee only causes a moderate decline. These findings highlight the importance of the proposed novel loss Lcomst, which not only improves HTE estimation accuracy but also facilitates the construction of narrower and more precise confidence intervals for relative error. Besides, the method (Tuet&& Lec) can be seen as method of (Gao 2025), where the proposed neural network degenerates to TARNet and serves as a conventional nuisance estimator to be used in Gaols structure. From the result of the coverage rate and selection accuracy of relative errors, we can see that our method significantly outperforms this baseline.
Ablation Study. As shown in Section
7 CONCLUSION
In this work, we tackled the challenge of evaluating HTE estimators with reduced reliance on strong modeling assumptions for nuisance components. Building the relative error framework, we proposed a new loss together with balance regularizers, and embedded them in a Dragonnet- inspired neural architecture to enable more stable and accurate nuisance learning. A remaining limitation is our use of simple uniform averaging scheme over all estimator pairs blue in the enhanced HTE estimator, which, while stabilizing performance, may underutilize the heterogeneous strengths of individual estimators. Future work includes developing adaptive weighting strategies for the estimator and incorporating "worst-case performance" perspective to improve robustness in evaluation, potentially by relaxing Assumption [ although HTE characterizes treatment effect heterogeneity with respect to observed covariates, differs fundamentally from individual-level treatment effects (Wu et al. (202a)). Complementary investigation of ITE (Wu et al.] 2025) or joint distribution of potential outcomes (Wu & Mao 2025)
along the lines of Huang et al. (2024). Moreover,
would be desirable to provide more comprehensive evaluation alongside HTE.
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ACKNOWLEDGEMENTS
This research was supported the National Natural Science Foundation of China (No. 12301370), the BTBU Digital Business Platform Project by BMEC, the Beijing Key Laboratory of Applied Statistics and Digital Regulation, and Academy for Interdisciplinary Studies at Beijing Technology and Business University.
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A
NOTATION SUMMARY
summarizes the notation used the main text.
Table 7: Notation and their meanings.
Symbol
Meaning
Binary treatment variable
X
Pre-treatment covariates
Outcome
T(x)
Individual treatment effect
e(x)
Propensity score
(x)
Outcome regression function, i.e., (a F[ X 2, A= a) for 0,
8(f1, 72)
Relative error between estimator 51 and f2
8(f1, f2)
Estimated relative error between estimator f and f2
é, Pa 9.30
Nuisance estimators for propensity score, conditional outcomes and their coefficients
Par 3.30 (X)
Probability limits of propensity score, conditional outcomes and their coefficients The shared representation of X, defined in Eq. (1) & (2)
B MERITS OF RELATIVE ERROR
There are several advantages of using relative error over absolute error.
(1) Weaker condition. Condition is strictly weaker than Condition [ Condition Canditionnrequute requires that all nuisance parameter estimators converge their true values at rate faster than n n1111 In contrast, Condition 2 imposes weaker requirement-onl that the product of the bias, (x) - ((()) e(x)), converges at a rate of order (1/2) as well as the nuisance function estimators being consistent. This allows for cases where ((x) converges at a rate op (1/5) and (z) converges at rate of op (n 1133).
(2) Easier to compare multiple estimators. When comparing two estimators ř (x) and 12 (x) in terms of absolute error, although both (fr,) and ((fg) are asymptotically normal, we cannot directly construct confidence interval for @(fi) o(f2) due to their dependency (as they use the same test data and share the same nuisance parameter estimates). In contrast, 8 f2) does not suffer such a problem.
(3) Double robustness. When we replace ve(n-1/2) in Conditions and 2 with or(1), both ((fi) and 8(f1, f2) are consistent (asymptotically unbiased) under their respective conditions. Thus, from Condition 2 8(A,, exhibits the property of double robustness, meaning it is consistent estimator if either ē(x) is consistent (a(x:) for 0, are consistent. However, d(+t) dose not possess this property.
C A MISSPECIFIED EXAMPLE
Example (A misspecified model). Consider X as a scalar and assume that the true model is (X) A a) = X2 which represents the true data-genent generating mechanism of Y given (X,A= == a). However, if we learn X X3, 3. IR, we introduce an inductive bias, meaning
a) using a linear model, i.e. Pa(X, Ba) :=
never reach the true value of 3", even though the estimator may converge. Specifically. denote 3. the least-square estimator of Ba. By the property of trast-square estimator, it converges to 3n := E[XX] - E{Y,, regardless of whether Ma(X, (X, Baa) is misspecified, Ba # 8%-
Ha (X, Ba) is correctly specified not. Since pa
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Total Loss
minimine
C Lub - Ag Coontt
Weighted Least Square Lous
Constraint
Eutropy Cor
Cxla
Eccus
pu Head
Head
Pt Head
Shared Representation ((X) gd
Hidden Layer
Hidden Layer
Input
Figure 3: Neural Network Structure
D ILLUSTRATION OF NEURAL NETWORK STRUCTURE
Figure shows the schematic structure of proposed network. The input covariates X E RRP passed through fully connected hidden layers to obtain a shared representation x(X) Rd. This representation is fed into three heads: the control outcome head (XX),, ), the treated outcome head Nn (X), and the treatment head e(X). The outcome heads contribute to the weighted least square loss Lwis: the treatment head contributes the cross entropy loss Lee, and the shared representation regularized by the constraint loss Lconst- The total objective is given by
L = Lwlss + A1han + A2Coomst
E PROOF OF THEOREM
Theorem 1. If the propensity score model is correctly specified, and 5, 30 well B converge to their probability limits rate faster than n -144 then have
Vūlōlfi, te; 7, Bo. B.) - (2) N(0,
where o² Var[yp(Z; uo 31, ) and d means convergence in distribution.
Proof of Theorem 1. As discussed in Section we first show that
8(f1, f2: 3o. 31) 5(A1, f2: 8.. B1) or 1/2). (1/2).
(A.1)
By a Taylor expansion of 8(f R2i 7, 3n B,) around (7, Ro, 31), we obtain
8(f,, te: 3o. 31) 8(f f2: 3o, B1)
= =T((1 - 7) -)) -4 Bo) + + (3) - -B1)
20(((1-))2 ())))))) - Bi) (5 (30 Bo)):
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where
A
2( (X2) (x()))
71
i=1
A,(1 ((Xi))(X, -I
+ (1 - A,),(X,) - (1 (X.),
X
((Xi)
1-((X))
Σ (x,) - +2(X.)) (1- 1- Exx) (2(()),
A
A%
n
i=1
1 Σ ((x((X)) +x(())) (1 Ai n
A9
E(Xi)) (2())..
By the condition that 9 9 + op (n 8o BB + op(n 1144 and 3n : + oo(n-1/4) we obtain Op((7 7)2 + (5- - (3) 31) + (5 185--1 (30 - (Bo) - (p(n=112). Thus, we only need to
deal with (1 7), 41, (Bo 3o) and 47, (3) - B.).
Since 7. 3o and B1 are the probability limits of 7, Bo and Bj. respectively, we obtain ý - = =o(()) 3. 8 = o1( and 31 - 31 = (1).
Then, it sufficies to show that Dy Ox(n-1/2). 4s = Op(n 1122 and Dg, Op(m- By CLT, A, - ((,,) = Op(n 1122), As. - ((A.)) = -Op(n -1/2) and As. - E(A,,) = Oz(n 1122) then, only need to show that E E(A,) = E(Ap.) E(As,) O.
We first deal with (AA,).
1
A1 - - (I()))
(1 (X) - - )0,(
B(A,)
=E
(x.) - -x())))
((X.)
1 1(x(x)
i=1
A(11 - -(())) - (x()))
(1 (1))) - -Ro(X)) ) (X)
(fi(X - +x())
2E
E(X)
1 1-((X)
(R1( - (x)) (() - 1n (X)) (x()))
-2E
E(X)
+ 2E (f1(x)) - xx() (1 (-)))(X)(( - (x)) 21 (X)
1 -((X)
-0.
The last equation holds by the definition of 3o and 31 and the fact that D1 (x() is sub-vector of (X).
We then deal with B(A))).
Σ 2( (X.) r2(X))) (1 (1-1- 2(x,)
Ai
) x2((i))
B(A R6) E
(1- 1- A
2E
(f(X) fx()))
D2 (x((x)
1 E(X)
-2Ex
IE
(x) x2()))
(1- (1-1-
1 A
)(X) 0x(x3/ x(xx)/x)) x))
-0.
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The last equation holds since the PS model is correct. Finally, we handle E E(Ag,) ag )-
(1-
A
B(A,,) = -
(x)) - (2(x)))
$2((i1)
E(Xi)
1=1
(fi(X) +z(()) (1 (1- E(X)
A
=2
(2(())
- +2(x)) (1
E(X) (xx(X) xx) x))
A
=2Ex
E
0.
The last equation holds since the PS model is correct. Therefore, equation A.1 holds.
We then want to show that
(2: B1) 8(f1 (2)) - - (00,
(A.2)
By definiton,
8( 7, 7, 7..0..00 31)
(xi(X.) (x,():)
- e(X.)
+ (x.)))) (1- A,)Y, -Po(x)) =o(x)))
- (
1-)(X;)
If model] 1 is correct,
E - [X9,)
E
(2))((x)- - -x())))
AHY -
+ (X,)
(1- A-)Y1
- [o(()))
- no(xx)))
E(Xi)
1- -((x)))
-E [())(X) - [x)(1)]]
Ex,E E {( (xx) - (x())
ArYY - -X,()) e(Xi)
(1- Ar)Y;; - [oo(x))) - Ro(x))) x:
+ +X.)
1 =((x))
-E (X))(X) F2 (x)]]
Ex. E {{ (x,) - (xx,))
[A(Y(1)) - (X())) (x.))
e(X,)
(1 - (.)))((0)) (xo(
-ox(..) ()(x X.
1 e(x))
=E (x)
((V(1) + (X,) - (Y:(0) (X.)) (X.))) X.)
=E [,0-0]]0EB.(((21100---0000000010
=E - -
f2)
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Published as a conference paper at ICLR 2026
If model is correct,
B (ff(x.) - (X(H)
(2()(x)) - - (x()))
AAY. -x(
(1- A,)Y,, - (x)(
- Ro((x))
-B
+ (X,,))
E(X,)
1 1-E(X,)
=R [(f(() - [x](1]]]
Ex, L K( {{2200 (x,
A:1%= - (X,(
(1 Ad)Y; -(o(X.)) Ro(x)) x
- (2()))
+ +x,())
8(X;)
1 =((.))
-E [(+)(X) - (x)]]
Ex, B {{ -fx(X)))xx(x))
A.(Y(1)) - (x(()))
((X,) + (x,))
(1 A.1Y((()) (O())))
-Ro(X.))
((x(X)
E (x) [x)]]
Ex, K
Astūn - + (Xi))
(x,) - (x()))
((Xi)
(1 - A.)1o(()))) - (X())
- -Ro(X)) 1)} Xi
1 1- -(x,))
=E (X)((X) (x))) Ex, (xx (xx))))(
-E (()((X) +3(()) -x) - (xx))
=8(f1, fz).
7, 7,80, Bi) 10( f2) the average Therefore, when at least one of models or is correct, 8(f), 7, of i.i.d. observations with mean 0 and variance Var uo E), By CLT, A.2 holds with x² = Var (Z: Uo ū1, 2)).
E.1 PROOF OF PROPOSITION
Proposition 2. Under the conditions in Theorem
| a consistent estimator of o² is
1 n
2²_
{p(Zz do, ūn, 8(f f2: (6.
an asymptotic (1 11) confidence interval for 8(f1, f2) is 8(A 2, B1)
where
2n/2 is the (1 n/2) quantile of the standard normal distribution.
Proof of Proposition 2.
7
02-02
(z; 20, u1, 8,. (zp( To, E) - E.p(Zi; To. uj, E)}
B,)3
T1
==1
(p(Ze 0o ش1 0) -
(p(Zz; ūo ū, Ep(Z;; ūo ūi, E)}
n
n
{ Zi; Co 0), Ep(Zz; u, 0))² E (zp ū1, Ep(Z;; ūo, ã,, E)
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Published as a conference paper at ICLR 2026
By the law of large numbers (LLN), ② S O, we only need to deal with Dr:
{0(Z; ūo. ū1 : 3o, 31.)))))
n
(z(Zz; úo, 8) (2zz uo, ūn 8) (0 to, u1 8) , Bo, B1)}
n
(Z; üo, 0) (Z no. 11 E))2+
Σ
{s Zz Wo, u1, - 8(f), R, Ao,
Zz; üo. ū1 8) - (2 uo 1 2) (0(Zg uo ū1 8(f f2 n, (0,3,,1)) 31)
17
By LLN, @ 4 0, and C 0. Similarly, we can obtain
(z; Ng. u1 2) Ep((, uo 11 2))² + (1).
Therefore ① 2 0, which leads to 0² 2 0².
The asymptotic confidence interval constructed by the standard theory.
F EXPERIMENTAL DETAILS
F.1 DATASET DETAILS
IHDP. The IHDP dataset is based on randomized controlled trial conducted as part of the Infant Health and Development Program. The goal is to assess the impact of specialist home visits on children's 's future cognitive outcomes. Following Hill (and) (011), subset of treated units is removed to introduce selection bias, creating semi-synthetic evaluation setting. The dataset contains 747 samples (139 treated and 608 control), each with 25 pre-treatment covariates. The simulated outcome is the same as that in Shalit et a2017) (Shalit et al. (017a), by setting "A" in the NPCI package DDorie 2016).
Twins. The Twins dataset is constructed from twin births in the U.S.. For each twin pair, the heavier twin is assigned as the treated unit (1, = 1), and the lighter twin as the control (4; = O). We extract 28 covariates related to parental, pregnancy, and birth characteristics from the original data and generate an additional 10 covariates following (Wu et al.) 2022). The outcome of interest is the one-year mortality of each child. We restrict the analysis same-sex twins with birth weights below 2000g and without any missing features, yielding a final dataset with 5,271 samples. The treatment assignment mechanism is defined as: f I; Bern (a(w X n)). where (-)) is the sigmoid function, w T (((-0.1, 38x1) 1), and n 0.100.1).
Jobs. The Jobs dataset is standard benchmark in causal inference, originally introduced by LaLonde (1986) (ndde 1986). evaluates the impact of job training on employment outcomes by combining data from randomized study (National Supported Work program) with observational records (PSID), following the setup of Smith and Todd (2005) (A. Smith & E. Todd (005). The dataset includes 297 treated units, 425 control units from the experimental sample, and 2490 control units from the observational sample. Each record consists of 8 covariates, such as age, education, ethnicity, and pre-treatment earnings. The task is framed as binary classification problem predicting unemployment status post-treatment, using features defined by Dehejia and Wahba (2002) (Dehejia & Wahha 2002).
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Table 8: Ablation Study on Removing the Shared Representation (IHDP)
Method
CATY
Coverage
Selection
separate propensity
1.659 0.224
0.167
1.576
0.187 0.145
0.88
0.25
all separate
7.177 0.119
7.074 + 0.119
7.133
0.003
7.057
0.067
0.83
0.21
Ours
0.638 0.138
0.090 + 0.087
0.670 0.150
0.105 + 0.099
0.96
0.80
Table 9: Ablation Study on Representation Depth and Width (IHDP)
Setting
REEE
V VEEE
Coverage
Selection
4 layers
0.784 + 0.249
0.126 + 0.099
0.820 = 0.279
0.146 + 0,121
0.90
0.75
2 layers
0.705 + 0.083
0.116 + 0.095
0.739 = 0.103
0.136 + 0.113
0.94
0.73
2x width
0.695 +: 0.156
0.093 0.074
0.734 0.188
0.113
0.099
width
+: 0.139
0.112
0.174
Ours (original model)
0.638 + 0.138
0.090 0.087
0.670 0.150
0.105 + 0.099
0.96
F.2 JUSTIFICATION FOR SHARED REPRESENTATION
To further examine the role of the shared representation, we conduct an ablation experiment in which the shared feature extractor is removed and the three task-specific heads (two outcome heads and one propensity head) are trained independently. We evaluate two aspects on the IHDP dataset:
1. the estimation accuracy of the nuisance components (measured by in- in-sample and out-of- sample PEHE and ATE errors), and
2. the validity of the resulting relative-error inference (coverage and selection accuracy).
Table 8 shows that removing the shared representation leads to substantial degradation in both nuisance estimation and inferential performance. contrast, our full model consistently achieves the lowest estimation errors and the highest inferential validity. These results highlight that the shared representation crucial for stable nuisance learning and is an essential component of our method.
F.3 SENSITIVITY ANALYSIS SHARED REPRESENTATION
Although (XX) data-driven, once the architecture is fixed, a neural network still constitutes a parametric model, and therefore it is important to demonstrate the robustness of our method when the network specification changes substantially.
To assess this, conduct an additional experiment on the IHDP dataset under the same setting as Table 1. Specifically, we alter the representation network by (i) changing its depth from layers to and 4 layers, and (ii) doubling or halving the hidden dimension of each layer. These modifications create clear and significant deviations from the original architecture.
As shown in Table 9 despite these substantial changes the model specification, the performance for both nuisance-estimation accuracy and relative-error inference accuracy is remarkably stable. In particular, the validity and efficiency of the relative-error inference exhibit no meaningful deterioration. This provides strong empirical evidence that our method is robust even when (XX is misspecified.
F.4 CONVERGENCE ANALYSIS OF THE LOSSES
We report the procedural results under the same settings as Table the main text. Here, constraint penalty (CP) refers to the first two terms of the penalty in Lconst, which penalize violations of the inequality constraints shown in lines 280-284. Negative penalty (NP) corresponds to the last two terms in Lears which penalize violations of the inequalities in lines 285-286. From Table 10 it is evident that the original conditions are well satisfied: the vast majority of constraints hold strictly, and the remaining few exhibit only negligible numerical discrepancies.
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Table 10: Constraint Penalty Statistics Across Datasets
Dataset
Mean CP
Mean NP
% Zero CP
% CP < 10-5
IHDP
1.511 x 10-5
72%
95% 100%
Twins
2.604 x 10-8
84%
Table 11: Performance on the Jobs dataset (in-sample and out-of-sample).
R
Rau pol
cut
Method
AATT
LinDML
0.158 + 0.015
0.019 + 0.015
0.183 + 0.040
0.053 + 0.051
SpaDML
0.150 + 0.024
0.131 0.118
0.165 + 0.046
0.144 0.134
CForest
0.016
0.018
0.155 = 0.028
0.058 = 0.047
X X-Learner
0.169 0.037
0.015
0.173 = 0.034
0.053 = 0.050
S-Learner
0.040
0.160 + 0.027
0.115 + 0.070
TarNet
0.141
0.447
0.145 - 0.009
0.190 = 0.074
Dragonnet
0.230+0.001
+
0.143 + 0.009
0.172 + 0.039
DRCFR
0.122
0.218 = 0.022
0.048 + 0.032
SCIGAN
0.144
0.112
0.220 = 0.026
0.049 = 0.034
DESCN
0.192
0.098 0.029
0.011
0.065 = 0.046
ESCER
0.202 + 0.023
0.023
0.086 + 0.028
0.145 = 0.011
0.076 + 0.045
Ours
0.112 0.019
0.018 + 0.012
0,131 = 0.030
0.053 + 0.039
F.5 RESULTS ON JOBS
Evaluation Metries.
For the Jobs datasets, there are counterfactual outcomes, report the true Average Treatment Effect on the Treated (ATT) and the Policy Risk (Rpol) recommended by Shalit al.(Shalit al., (017b). Specifically, the policy risk be estimated using only the randomized subset of the Jobs dataset:
11-(11A
S/ |AINE) E
Rpol 11 (141 [AINTIEE (IANAAE
Σ
Σ
+
vo
A
Tu E
|E|
DE
2,EAOTTONE
where E denotes units from the experimental group, A1 = {z. o (1)
0}, Ao = {zz
01 (3) 360 < 0}. and Ti, To are the treated and control subsets, respectively. Since all treated units T belong to the randomized subset E, the true Average Treatment Effect on the Treated (ATT) can be identified and computed as:
ATT
Σ U: iET
Σ y1 iECNE
CNE
where C denotes the control group. We evaluate estimation accuracy using the ATT error: EATT
ATT 1 Eier (f(z,, f(z, 0))
Accuracy of the HTE Estimation. and compare it with baselines mentioned in Section We average over 20 realizations of our network, and the results are presented in Table IT One can clearly see that our proposed method achieves the best performance across all metrics, having the lowest Rpol and carr in both training sets and test sets. Sensitive Analysis and Ablation Study. We explore which value of λ₁ \2 and p can achieve the best performance. The results are demonstrated in Table 12 We can see that our model is not sensitive to the change of hyperparameters. That is, the performance of HTE estimation remains relatively stable across range of hyperparameters. For the ablation study presented in Table [3] as that in IHDP and Twins, taking Cee off only causes a moderate decline, while removing Cconst brings whole disaster.
We evaluate the performance of HTE estimation by our network
F.6
THE EFFECT OF THE NUMBER OF CANDIDATE HTE ESTIMATORS
We use the same setting as Table T on IHDP. The candidate estimator list is: TARNet. Causal Forest, X-leareer, S-learner. It can be seen from Table 14 that increasing the number of candidate estimators
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Table 12: Sensitivity analysis on the Jobs dataset with respect to λ₁, X2, and p.
A
Ag
Value
ATT
Rjo
Value
R
ATT
Value
ATT
R
ATT
0.1
0.113
0.018
0.132
0.054
0.1
0.109
0.021
0.129
0.055
10
0.124
0.020
0.141
0.051
0.113
0.022
0.130
0.058
0.109
0.020
0.128
0.054
50
0.112
0.019
0.131
0.052
0.112
0.018
0,131
0.053
0.112
0.018
0.131
0.053
100
0.112
0.018
0.131
0.053
2
0.115
0.020
0.135
0.054
2
0.117
0.019
0.135
0.050
200
0.115
0.020
0.132
0.053
0.121
0.027
0.140
0.060
0.123
0.027
0.144
0.060
1000
0.020
0.133
0.052
Table 13: ablation studies Jobs
Training Loss
R
within "TT
Real
out-ots.
AFT
Dusa Lemns
0.114
0.023
0.134
0.053
Cun, & Loe
0.121
0.029
0.141
0.055
Full (Ours)
0.112
0.018
0,131
0.053
does not necessarily improve estimation accuracy, possibly because some of the pairs perform poorly and dilute the overall effect.
Table 14: Effect of Varying the Number of Candidate Estimators
# Estimators
Veiru AATE
out é PEEHE
FATE
2
0.708
0.107
0.740
0.112
3
0.638
0.090
0.670
0.105
4
0.6703
0.1107
0.6980
0.1227
TARNet (baseline)
0.896
0.279
0.920
0.266
F.7 DETAILS OF COMPLEXITY ANALYSIS
We conduct two experiments regarding sample size and number of candidate estimators, respectively.
We varied the sample size by randomly subsampling the IHDP dataset, and measured the running time of our method under different data scales. This demonstrates how the method scales with input size.
We varied the number of baseline HTE estimators t&s used construct the aggregated CATE estimator f(x)) as described in Section 4.5. Specifically, for each pair (k, k'), we used the shared neural network architecture to obtain outcome regressions fa (æ; t )x) and f fx, fer), and then formed F(z; fx, fr.). The final output F(z) is average over all (or a subset of) such pairs. Since this pairwise process increases computational cost quadratically with the number of baseline estimators, report running time to illustrate the method's s scalability. We also include the runtime of TARNet as baseline for comparison.
F.8 EXTENDED SENSITIVE ANALYSIS
In this section present the results of the sensitive analysis of hyperparameter λι and in the IHDP and Twins dataset. One can see from Table 15 and Table [6 that our model is robust to the change of X1 and p. remaining good performance in the HTE estimation as well as relative error prediction.
F.9 SIMULATED DATA- GATTAGENNRATING PROCESS
The data-generatin generating process is described follows: We sample latent covariates L ~ split them into I, C, D. Nonlinear structure is introduced via + sin(I) - C, A (A). The treatment assignment depends IC (T, through x (Z) (TC +6), e A(0,
(m0, and
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Table 15: Sensitivity analysis of X on IHDP and Twins datasets.
IHDP
Twins
Value
Fizz
+AZE
V PPIzE
-XIE
Coverage
Selection
Value
V<EE
"ATE
"ATE
Coverage
Selection
0.1
0.678
0.096
0.709
0.112
0.93
0.74
0.1
0.286
0.009
0.288
0.010
0.96
0.94
0.25
0.693
0.096
0.724
0.113
0.93
0.75
0.25
0.285
0.010
0.287
0.010
0.94
0.94
0.090
0.105
0.80
0.284
0.286
1
0.712
0.103
0.746
0.115
0.96
0.79
0.285
0.013
0.287
0.014
0.94
0.92
2.5
0,262
0.284
Table 16: Sensitivity analysis of on IHDP and Twins datasets.
HHDP
Twins
Value
V +Paa
V +Vii::
Coverage
Selection
Value
Coverage
Selection
10
0.698
0.108
0.735
0.125
0.96
0.78
0.299
0.015
0.306
0.015
0.92
0.62
0.098
0.116
0.79
0.289
100
0.638
0.090
0.670
0.105
0.96
0,80
100
0,284
0.009
0.286
0.009
0.94
0.94
200
0.737
0.103
0.772
0,123
0.76
200
0.286
0.288
1000
0.751
0.111
0.785
0.130
0.93
0.76
1000
0.284
0.010
0.285
0.011
0.92
0.94
with clipping to 0.91, and T Bernoulli (Z)))) Potential outcomes follow fo [C.A)) wn. (G,
Potenaaa
mo+ma
VJ, fi
+ Th, where to,7h ~ (0, 0.5), yielding No fo and μu f1-
mo+ma
F.10 MODEL. IMPLEMENTATION
We implement all models using PyTorch and optimize them with the Adam optimizer. The key hyper- parameters include the size each hidden layer, learning rate, the loss coefficients Å₁. A2. the penalty coefficient p. and the number of training epochs. These hyperparameters
manually tuned through empirical trials. The search ranges are as follows: hidden layer size in (30, 40, 50, 60, 70), learning rate in (5 10-4, 10-300000000 x 10 3 x 10 Å₁ A2 in {0.1, 0.25, 0.5, 2, p {10, 50, 100, 200). and number of training epochs in {700, 800, 900, 1000, 1100].
G THE USE OF LARGE LANGUAGE MODELS
We used large language models (LLMs), specifically, ChatGPT 5 to polish the writing of our paper. All ideas, methods, experimental designs, and analyses are our own, and the use of LLMs was limited to language editing.
24


Reviews:
[{"review_id": "JZfm1b1S4U", "reviewer": "Reviewer_UJPQ", "summary": "This paper aims to address a very important and critical challenge of CATE estimator evaluation. The authors design a novel loss function and neural network architecture that produces a robust relative error estimate, which remains root-n-consistent.", "questions": "1. What does the robust evaluation exactly mean? After reading this paper, I guess the author wants to claim the relative error estimator is \"doubly robust\", so it is robust to nuisance parameter estimation. \n\n2. What's the difference between your loss and R-loss (R-loss is also doubly robust)?\n\n3. The problem of robust evaluation of the CATE estimator has been studied in [1], where they also consider the worst-case performance of the CATE estimator selected by the proposed evaluation metric. Does this paper also provide any information on the worst-case performance? I guess Figure 1 provides such information, and I suggest emphasizing this point as it can strengthen the efficacy of \"robust\".\n\n4. A brainstorm: I think the whole framework can be extended to policy/estimator adaptation, e.g., following the setting in [2]. Maybe we can design a new policy evaluation metric that is doubly robust to nuisance parameters, which is very useful when distribution shift presents, and it has not been discussed in previous literature. So this is also a good point of this paper, as it has the potential to be extended to other problems.\n\n[1] Unveiling the Potential of Robustness in Selecting Conditional Average Treatment Effect Estimators\n\n[2] Optimal Policy Adaptation under Covariate Shift", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The idea of solving the fundamental and practical CATE evaluation problem is interesting. Evaluating CATE is not easy unless we impose additional structural assumptions. Instead of just using a standard doubly-robust estimator, the authors first derive the specific theoretical conditions required for their estimator to be robust to outcome model misspecification, and then they design a new loss function and a neural network architecture to force the nuisance parameter estimates to satisfy these conditions. This is, somehow, follows the idea of the design of Dragonnet but gives an interesting solution for CATE evaluation.\n\n2. Theoretical results are solid. They establish a theoretical asymptotic property for their proposed estimator, under some assumptions (conditions).\n\n3. The empirical evaluation is through. The experiments strongly support the paper's claims, showing that the proposed method achieves the target 90% confidence interval coverage and high \"selection accuracy\"", "weaknesses": "1. Can strengthen the connection with other studies (see below question 2&3).\n\n2. The asymptotic property relies on some critical assumptions, for example, $\\check{\\gamma}, \\check{\\beta}_0, \\check{\\beta}_1$ should converge to the true one at a rate faster than $n^{-1/4}$. It is strong but also reasonable. It is reasonable because orthogonal ML literature always assumes the convergence rate of nuisance parameters. It is strong because we are treating them as \"plug-in\" quantities, instead of the nuisance parameters that the estimator should be doubly robust to. It might be useful to provide some explanations to justify that the condition is realistic.\n\nOverall, I think these are minor weaknesses. The pros outweigh the cons.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pyvFiHzvWD", "reviewer": "Reviewer_eY6Y", "summary": "This paper extends the work of (Gao 2025) on the use of relative error for heterogeneous treatment effect estimation. Its main contribution is that it offers a relaxation of the requirement that the nuisance parameter estimators must be consistent. Instead, it is shown that robust estimation can be achieved if the a correctly specified propensity score estimator is provided. Moreover, the theoretical results lead to the development of an HTE method, while both theoretical and experimental results support the arguments of the paper.", "questions": "- Is there any theoretical or empirical evidence of what happens when the propensity score estimation model is misspecified?\n- How do results depend on the number of candidate estimators?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- This paper addresses a very important yet understudied problem, and an effective practical solution for the evaluation of HTE estimators is very valuable\n - The paper offers significant theoretical contributions offering theoretical guarantees on estimator consistency even with misspecified regression models\n - In addition to theoretical results on the evaluation of HTE estimators, the paper offers a learning algorithm and method for HTE estimation\n - The empirical evaluation on datasets available in the literature is comprehensive", "weaknesses": "- The need for a correctly specified propensity score model still remains a strong assumption, which may not hold or be guaranteed in practice in many cases.\n - One would expect a deeper evaluation on the comparison of the proposed method against the method by Gao 2025, as well as an empirical evaluation of what happens when the propensity score estimator is misspecified.\n - There is some confusion regarding the terminology, since there is mention of \"balance regularizers\" in the beginning but later this terminology is abandoned, where $\\mathcal{L}_{\\text{const}}$ is mentioned", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wCZp2zkkJu", "reviewer": "Reviewer_M3dQ", "summary": "The paper proposes a way of making an estimator of the relative error of two given CATE estimators robust against the misspecification of the mean outcome nuisance \\mu_a, relaxing the constraints of existing methods. Furthermore, the by-product of their algorithm could be used to craft a stronger CATE estimator.", "questions": "1. Your asymptotic CI relies on the correct specification of propensities. What happens when this assumption is violated?\n2. How is the ground truth $\\tau(x)$ constructed for Twins and Jobs\n3. Why does the method not require sample-splitting, a standard procedure in orthogonal estimators?\n4. Theorem 1 is only correct when Eq.(4) is satisfied, right? But in your method, you cannot make Eq.4 satisfied. Is there any bound analysis?\n5. Why is propensity score easier to estimate than the outcome? (line 172 - 175)\n6. The complexity of the method seems high. What is the average time cost of achieving an enhanced estimator using your method? And how does it compare to the time cost of training a baseline CATE estimator?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "- The paper successfully identifies a problem, which is that the existing estimators require asymptotic rates of nuisances no less than n^-¼. \n- Through Taylor expansion, the paper shows the conditions under which higher error rate of \\mu does not affect the asymptotic rate of n^-½ of the relative error estimator. This provides interesting theoretical insights.\n- The paper designs a novel loss that reformulates the equation-based condition as a minimization problem which allows batch gradient descent.\n- The paper shows that the method has practical significance in getting stronger CATE models.", "weaknesses": "1. The paper has limited novelty compared with existing DR estimator Gao (2025). The fundamental form of estimator remains the same as Gao. And the paper seems to just reconstruct the nuisances to meet certain constraints. \n2. The parametrization of e(X) and \\mu_a(X) in (1),(2) seem quite arbitrary and lack a decent justification on why these nuisances should share representation. \n3. There is NO solution to the moment-based conditions. To this, soft penalties are introduced for finite-sample optimization. Without the correct specification of propensity (which is never met in real world applications) or sensitivity analysis on the propensity score, I doubt the complete formulation might just go meaningless.\n4. Misspecification can also come from the representation $\\Phi(X)$.\n5. Nuisance overfit could be a problem without sample splitting.\n6. Typo: line 1017 should be L_const.\n7. Experiment: 1) The paper does not perform sensitivity analysis (empirical) on the propensity score, which makes the practical utility doubtful. 2) The paper does not explain how they compute the \\tau in real world datasets Twins and Jobs.3) The paper does not provide convergence analysis of the losses, especially L_const and L_ce. The whole method would only make sense if the paper could show, at least empirically, that under finite samples the two losses successfully approximate the original condition.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "va7E5RdEJB", "reviewer": "Reviewer_GfzX", "summary": "This paper introduces a new *evaluation framework* for HTE estimators, extending a recently proposed relative error–based framework that compares the performance of two estimators. The proposed approach relaxes a key assumption of the prior method, which required all nuisance parameter estimators to be consistent at a rate faster than n-1/4. In contrast, the new framework only requires the propensity score model to meet this condition, allowing the outcome regression and propensity model to be misspecified. Building on this evaluation framework, the authors further develop a new HTE estimation method and evaluate it on some very often used semi-synthetic datasets, including IHDP, Twins and Jobs, demonstrating improved performance compared to several baseline estimators. The paper also provides sensitivity and ablation analyses to support the empirical findings.", "questions": "- The paper focuses on HTE estimation but only provides formal definitions for ITE and CATE in the problem setup. For clarity, please provide a precise definition of the HTE you are targeting.\n\n- There appears to be a typo in Assumption 1(i). It should likely be the standard unconfoundedness assumption: $(Y_i(0), Y_i(1)) \\perp A_i \\mid X_i$. Please clarify and correct this.\n\n- Choice of Evaluation Objective: Line 109 states that the goal is to select the estimator with the highest accuracy on a given test dataset. Could the authors elaborate on why this objective was chosen over the more common goal of finding an estimator that accurately models the CATE function over the marginal distribution of covariates, $P(X)$?\n\n- Example 1 illustrates model misspecification, which is a well-understood concept in statistical machine learning. To improve conciseness, this example could be simplified or moved to the appendix.\n\n- The neural network architecture described in Section 4.3 appears to be the Dragonnet architecture. Could the authors clarify the novelty of their proposed network, or explicitly frame it as an adaptation of Dragonnet for their specific loss functions?\n\n- Experimental Comparison: The paper's main theoretical contribution is relaxing the assumptions of Gao (2025). It would be valuable to include a baseline that represents the Gao (2025) approach or can you clarify how the proposed estimator outperform it?\n\n- Minor Typos: Line 173: \"violating Assumption 2\" should likely refer to \"violating Condition 2\". Line 216: The text seems to have a typo and should probably read \"...the proposed estimator of... \".", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This is a solid paper that addresses an important and interesting problem. The main strengths are:\n\n- Valuable Problem: The paper focuses on the reliable evaluation of HTE estimators, which is a critical issue for real-world applications. The idea of using \"relative error\" to compare models is novel and meaningful.\n\n- Strong Theoretical Contribution: The highlight of this paper is its theoretical work. It successfully relaxes the strict assumptions from prior work (Gao, 2025), notably by removing the requirement for the outcome model to be correctly specified. This makes the proposed framework much more robust and practical for real-world scenarios, which is a significant and elegant theoretical improvement.\n\n- Clear and Solid Method: The paper is very clearly written and easy to follow. The proposed new loss function and network architecture are well-motivated and tightly connected to the theoretical derivations. The overall approach feels very solid.\n\n- Thorough Experiments: The authors provide convincing results from experiments on several standard datasets. The experimental design is comprehensive, including ablation and sensitivity analyses, which effectively validate the proposed method's effectiveness.", "weaknesses": "- The paper's primary weakness is the unclear practical motivation for using \"relative error.\" While the metric is theoretically interesting for comparing two estimators, its real-world applicability is not well-established. The paper would be significantly stronger if it could provide clear use cases where practitioners would prefer this comparative measure over standard absolute performance metrics like PEHE. This is particularly important for the ICLR audience, which values practical impact.\n\n- The paper is mathematically dense, and while this contributes to its theoretical rigor, it also appears to have led to several typos and minor inconsistencies. To ensure the paper's core contributions are communicated accurately, a thorough proofreading of all definitions, assumptions, and equations is highly recommended.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)