Direct Doubly Robust Estimation of Conditional Quantile Contrasts
Authors: Josh Givens
Song Liu
Henry Reeve
Katarzyna Reluga
Venue: ICLR 2026 Poster (2026)
Keywords: Heterogeneous Treatment Effect
Conditional Quantile Treatment Effect
Quantile Regression
Doubly Robust
Primary area: causal reasoning
TL;DR: We create the first direct estimator of the Conditional Quantile Comparator, a quantile based summary of the treatment effect.

Abstract: Within heterogeneous treatment effect (HTE) analysis, various estimands have been proposed to capture the effect of a treatment conditional on covariates. Recently, the conditional quantile comparator (CQC) has emerged as a promising estimand, offering quantile-level summaries akin to the conditional quantile treatment effect (CQTE) while preserving some interpretability of the conditional average treatment effect (CATE).
It achieves this by summarising the treated response conditional on both the covariates and the untreated response. Despite these desirable properties, the CQC's current estimation is limited by the need to first estimate the difference in conditional cumulative distribution functions and then invert it. 
This inversion obscures the CQC estimate, hampering our ability to both model and interpret it. To address this, we propose the first direct estimator of the CQC, allowing for explicit modelling and parameterisation.
This explicit parameterisation enables better interpretation of our estimate while also providing a means to constrain and inform the model. We show, both theoretically and empirically, that our estimation error depends directly on the complexity of the CQC itself, improving upon the existing estimation procedure. Furthermore, it retains the desirable double robustness property with respect to nuisance parameter estimation. We further show our method to outperform existing procedures in estimation accuracy across multiple data scenarios while varying sample size and nuisance error. Finally, we apply it to real-world data from an employment scheme, uncovering a reduced range of potential earnings improvement as participant age increases.

Published as a conference paper at ICLR 2026
DIRECT DOUBLY ROBUST ESTIMATION OF CONDITIONAL QUANTILE CONTRASTS
Josh Givens & Song Liu
Henry W. J. Reeve
School Mathematics
School of Artificial Intelligence Nanjing University
University of Bristol
joshgivens@hotmall. mail..aaill.. .o...ouu
Katarzyna Reluga
School of Business and Economics Humboldt University of Berlin
ABSTRACT
Within heterogeneous treatment effect (HTE) analysis, various estimands have been proposed to capture the effect of a treatment conditional on covariates. Recently, the conditional quantile comparator (CQC) has emerged as a promising estimand, offering quantile-level summaries akin to the conditional quantile treatment effect (CQTE) while preserving some interpretability of the conditional average treatment effect (CATE). It achieves this by summarising the treated response conditional on both the covariates and the untreated response. Despite these desirable properties, the CQC's current estimation is limited by the need to first estimate the difference in conditional cumulative distribution functions and then invert it. This inversion obscures the CQC estimate, hampering our ability both model and interpret it. To address this, we propose the first direct estimator of the CQC, allowing for explicit modelling and parameterisation. This explicit parameterisation enables better interpretation of our estimate while also providing a means to constrain and inform the model. We show, both theoretically and empirically, that our estimation error depends directly on the complexity of the CQC itself, improving upon the existing estimation procedure. Furthermore, it retains the desirable double robustness property with respect nuisance parameter estimation. We further show our method to outperform existing procedures estimation accuracy across multiple data scenarios while varying sample size and nuisance error. Finally, we apply it to real-world data from an employment scheme, uncovering a reduced range of potential earnings improvement participant age increases.
INTRODUCTION
As data becomes more and more readily available, the demand for personalised treatments and inter- ventions has increased dramatically. The statistical field addressing this challenge is heterogeneous treatment effect (HTE) analysis in which one aims to learn the effect of treatment outcome or response conditional on key covariates (Hirano and Porter 2009: Collins and Varmus 2015
Obermeyer and Emanuel 2016 Lei and Candès) 2021).
A core strategy for the analysis of HTE data is to estimate key estimands that quantify the effectiveness of treatment given the covariates. The two commonly used estimands the conditional average treatment effect (CATE) (Abadic and Imbens 2002 Imbens] 22004] Semenova and Chernozhukov
2021) and the conditional quantile treatment effect (CQTE) (Abadie al., 2002; Autor al.) 2017]]
Powell 2020) which represent the difference in the conditional mean and quantile le of the response respectively for the two treatments given the covariates. Both approaches have advantages: the CQTE
yields more granular treatment-effect summaries and is less sensitive to extreme values (Firpo) 2007
Bitler et al. 2006), while the CATE provides a more interpretable estimand with stronger estimation
guarantees Kennedy al. 2023 Kennedy 2023b Nie and Wager 2020).
recently introduced estimand, the conditional quantile comparator (CQC) (Givens et al.) 2024). aims bridge the gap between the CATE and the CQTE, The CQC does this by providing transport

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map between the conditional treated and untreated response distributions in quantile preserving manner. The definition of the CQC more naturally aligns with how treatment effects are discussed as they often talked about as either improving the response by a fixed amount or scaling the response (e.g. a medicine increased life expectancy by 2 years or by 50%). This scaling can be expressed naturally function of response while we would need to transform the input via the conditional cumulative distribution function of the untreated response in order to express it as a function of the associated quantile. Therefore in this case the CQC would be able directly capture this effect helping better understand the treatment and its efficacy while the CATE and CQTE would likely have much more complex relationship for the CATE and CQTE. As the CQC shares properties with the CQTE it also shares its strengths. Namely it is useful in settings where our distribution is heavily skewed, such as platform use or income, as allows us to make effective decision on which treatment is better without being heavily affected by a small number of extreme samples (Firpol 2007| Belloni ctt al. 2017). In relation to this, it can also help with decision making in cases where we want to evaluate our treatment only for certain response values. For example if we want to evaluate some employment intervention on income for those on lower incomes (see our example in Section 5)
In summary, this leads to the CQC being able to give information on the relationships between the treated and untreated distributions at all levels similarly to the CQTE, while having a more direct interpretation the response level. The current estimation method for the CQC, introduced in Givens et al. (2024). involves estimating an intermediate estimand and then inverting this to obtain a CQC estimate. Despite having some strong theoretical guarantees, this framework does not enable direct modelling of the CQC itself. This in turn prevents the use of informative parameterisations and limits our ability to constrain or inform the model structure, such as by enforcing smoothness in nonparametric settings. This approach also hinders interpretability of the estimate as it can only be examined via evaluating it at various samples, a procedure which itself can be computationally costly,
In this paper, we provide the first direct estimator of the CQC which addresses these limitations. Crucially, our new approach allows the CQC to be explicitly parameterised. This enables us to enforce assumptions on the CQC via flexible techniques including linear parameterisation, neural networks, kernel bandwidth choice in nonparametric settings, and regularisation. This also enhances interpretability by allowing greater flexibility in model inspection. Finally, because our approach models the CQC directly, the estimation error depends on the complexity of the CQC itself, rather than that of an upstream intermediate function. Meanwhile, retains the doubly robust property, ensuring accurate estimation of the CQC even when all nuisance parameters are estimated suboptimally.
To summarise, in this paper we:
Provide the first direct CQC estimation procedure.
Provide finite sample bounds on this estimation procedure.
Illustrate the robustness of our estimator theoretically, and through numerical experiments.
Show it to empirically outperform existing procedures in terms of estimation accuracy along various axes directly highlighting the advantage given by our explicit parameterisation.
Illustrate its interpretability by applying it to real world problems and analysing the results.
2 PROBLEM FORMULATION
We first introduce the general HTE setting. Let Y, X, A be random variables each representing information about an individual in our treatment setting. Specifically we take Y (on y C RR to give their univariate outcome/response: X (on X S (Rd) to give their covariates of interest e.g. age, height, etc.; and A (on {0, 11) to give their treatment assignment with 1 = Treatment and 0 = Control. Our overall aim is to understand the effect of treatment, A, on the response, Y, given the covariates, X.
We define Z = (Y, X. A) and let D = ( [Z() copies of Z representing our data sample with i indexing each sample/individual and 2n used for notational convenience. We assume that we are in the potential outcome framework so there exists (1)., (0) representing an individuals response both on and off treatment such that = (A)). To allow results to translate back to these potential outcomes we make the unobserved confounding assumption given by the identity (()) 1 A|X. Crucially this means that Y(a) X and Y|X ær, A are identically distributed for X.a a (0,1) 2005).
i=! ((Y(), (1), x AA))) for n E N denote III
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For E N. let [n] == (1..... n.). For vector E I2 let W represent the 7th component of w and let |ww|| be the Euclidean norm of w unless otherwise specified. For a function f : R x X' + R, we let (y denote the partial derivative f æ). Finally, as convention, for a < b we take Jh f(=)dr= = Sa f(z)dr= = - Soo.s) f(z)dz. With this notation and basic treatment effect set-up introduced, we can now define key estimands used in our framework.
Remark 1. For simplicity, we will assume that response, Y, is continuous with strictly positive density when conditioned upon any covariate, X, and treatment, A.
2.1 NUISANCE PARAMETERS AND KEY ESTIMANDS
We first define various muisance parameters, which are additional distributional objects necessary for the estimation of our estimand. The three nuisance parameters of interest are the propensity score, : X - (0. 1), conditional cumulative distribution function (CCDF) of YIX, A, Fa, and the conditional quantile function of YIX, A, F each defined as
<(x)) = P(A 1|X x) (v(l))) = P(Y - 2
(1)
(2)
Fr (alx) = inf E ( a
(3)
for all x E X and a E {0, }} and with π assumed to be continuous and bounded away from {0,1}.
The propensity score can be thought of as the probability of an individual being assigned to treatment given their covariates. Finally we take Pa (-|=)) to represent the probability density function (pdf) of
Y|X = x, A = a. We can now introduce the core HTE estimands.
Definition (CATE, CQTE, CQC). The CATE, CQTE and the CQC of the triple Z = (Y, A) are given by 7 X R, Tqg [0, x X R. and 9²: y x X + y respectively with
T(x)= = E|y = x. A = 1] E = A
(a/x)) FT (a
Fo (a|x),
g (1o (z) FT- [Fo(vol=))=z)
Both the CATE and the CQTE aim to summarise the effect of the treatment by examining the difference the outcome for the treated and untreated patients given specific covariate values. The CQTE offers added granularity by allowing the effect to be examined at specific quantiles rather than providing single summary statistic per covariate value.
The CQC is the central focus of our work and differs from previous estimands by instead mapping an untreated response and covariate value to a treated response value (Givens et al.) 2024). Specifically, it defines transport map from the distributions of the untreated response to the equivalent quantile value of the treated response via conditional on the covariates. Previous work has demonstrated the CQC's s ability to provide granular quantile level summaries of the treatment effect similarly to the CQTE while framing the input more naturally in terms of an untreated response value as opposed to quantile level. The CQC achieves this by providing summaries over multiple quantiles similarly to the CQTE and in fact has the relation that Ty (Fo [)] = g xz) yo-
key strength of the CQC working specifically in the response space, is that this more naturally mimics how the impact of a treatment or intervention is often characterised. Specifically the effect of a treatment often expressed in terms of either the absolute effect (additive effect) effect on the response itself (multiplicative effect.) If this impact is deterministic, the CQC will be able to represent these effects in a simple manner either of these effects while the CQTE may not. Figure
scaling
[ provides an example of this when the treatment doubles the response. We plot the CATE, CQTE and CQC and show that both the CATE and the CQTE contain complex high frequency changes not present in this treatment effect while the CQC does not. Specifically, the CQC will be g" (wo x = 2yo regardless of the marginal distributions. This relative simplicity of the CQC not only improves interpretability but can also lead more accurate estimation.
Optimal estimation of the estimands in Definition I has been the focus of much previous work (Robins et al.) 2008 Shalit et al. 2017 Foster and Syrgkanis 2023 Melnychuk al. 2025 Sun and Xial 2025). To achieve their optimal estimation, it is first necessary to estimate nuisance parameters, such as the propensity score and conditional cumulative distribution functions (CCDFs) in case
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1.0
9 2VV MM
A. 1...
0.0
0.5
-1.0 -0.5 6.0
-1. -03
0.0
x
x
r(x))
(b) To (a(())
(c) g" (ulz)
Figure 1: Surface plots for CATE (panel (a)), CQTE (panel (b)), and CQC (panel (c)) where Y|X x. ((sin(10x)) 1), YIX,
(1n00))) 4). We can see that CATE, and CQTE have high-frequency changes in while the CQC does not depend on z instead simply representing the doubling the response as g (ylz) = 2y.
of CQC estimation. Consequently, prior work has focused on developing methods that are robust to inaccuracies in these nuisance estimates. A notable class of such methods, known as doubly robust methods, can attain the desired overall convergence rate even when nuisance parameter estimates converge slower rates. Doubly robust methods have been introduced for each of the CATE (Kennedy et al.] 22033 Kennedy 2023b) Nie and Wager 2020). CQTE (Kallus and Oprescu )023), and CQC (Givens et al. 2024). We now introduce the existing doubly robust CQC estimation method, which serves as point of comparison for our proposed approach.
2.2 CURRENT CQC ESTIMATION
In Givens (024) CQC estimation method was introduced which involved estimating an intermediary function called the CCDF contrusting function defined as
h(v,, yo. .)) Fi (3)læ)) Fo az).
To obtain estimate of g (Jol (x) one would then have to estimate h(y1, Vo. x)) over a large number of y1 samples, isotonically project, and then choose the yi sample which gave h closest to This approach has three main shortcomings:
1. Its lack of explicit form for our CQC estimate, 9. makes harder to interpret and constrain.
2. Its estimation quality depends upon the difficulty of estimating h rather than our parameter of interest, g"
3. Its evaluation is computationally expensive especially when the estimate of h is expensive evaluate (see Appendix D.3.3 for further exploration and experimental validation of this.
Remark 2. We view simplicity of the CQC as more natural and easily satisfied notion than that of the CCDF contrast function. See Appendix C.I further discussion and an illustrative example.
We now introduce our approach which directly estimates the CQC, thereby addressing these issues.
THE DIRECT CQC ESTIMATOR
Similarly to the existing approach,
frame our estimation problem as finding y1 for given yo. x such that h(y1, you x) Fi (y1 (x) Fo (x) = O. While we could treat this as a Z-estimation problem, in order to extend this learning function over all 30 x. it is instead helpful to view it through this lens of M-estimation. To this end, since h is an increasing function of yı, any loss function & satisfying Ö, E(y1 No. x) h(y1, No. x) will be minimised at the value y/1 such that (111 No. x) = 0. our desired goal. Using this idea we now introduce our loss in Definition 2 justify via Equation 4 and demonstrate its direct relation to CQC estimation error Proposition
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Definition 2. For parameter space @ C R", let Ge = {ge : x V10 O be the of parameterised CQC estimates. Additionally, for yo E ) a E X. θ, and Yo a RV over y, define
(1y, No, x) =
h(t, yo. x)
E(O, yo. x) = {ge (30 Φ). yo. x)
00 (vulm)
L(0)= = EE|E(O, Yo X)]
0 argmin L(
ege
In summary, evaluating l at the CQC estimate, go (Valzz), yields the pointwise loss, E(0, Noo 2), whose expectation guides the estimation of 0. Specifically we then have that
9" (yol x)) argmin E(y1, Vo, æz).
(4)
91
This result follows from a simple application of the Fundamental Theorem of Calculus. A detailed proof is provided in Appendix A.I Now, suppose there exists unique 0" E θ such that go and supp(Yo) X = x) y for all æ E then, as 0* minimises & {ge (No [z), yo. x) pointwise for all yo, 2, we have that = 0" i.e. our minimiser is the true parameter.
To further aid in the interpretation and justification of the loss function in Definition including in cases where Go does not contains the true CQC, we will provide various bounds on the loss function in Proposition[] We do this via three different avenues, each requiring separate assumptions on the distribution of our treated response with varying levels of generality. While these bounds are helpful and illustrative, our loss is still justified even when none of these bounds hold.
Proposition 1. For any E V, x E X, and E 2 we have the following upper bound on the loss: 0, yo: a)) Io (volar) 9" (vol ax) Fi (0o(vol=))==) - Fi {g" (yn [ar)]ar]]
Under various conditions we have the following three lower bounds the loss:
(a) Suppose that pı (yl [x) < for all y. x, then
(Fi (go x)) } Fi (ya (2) |x))) 2[2 yo. x))
(b) Suppose that P1 (y/ x)) 2 f all y, x. then ( - (yolx)}² < 20(0, yo. x).
(c) Suppose that pı (y/ x) is decreasing function of y. then
[yo[ x)) 9" (yol xx) Fi (g0 x)) x - Fi [g (yo x)) x S 26(0, yo x).
The proof is given in Appendix A.I. A.I.I
Error terms involving both |ge (vo (a)) g" (wol and [Fi (go(yo [>)=> - Folg [ (vo [a)] are natural the first represents the error
estimator while the second is the error of our estimator when mapped on to probability space. The assumption in (a) covers many common distributions with densities bounded above. The assumption in (b) applies many bounded-support distributions such as the Beta. The final case less common but holds for some distributions, e.g., the exponential, and has been studied in density estimation (Birge 1989).
3.1 OUR ESTIMATOR
While the above results justify our loss & in Definition 2 they do not give us any approach to evaluate or even approximate
achieve this we return back to the derivative of E (also given Definition 2 with which we initially motivated our approach. To this end, with (y. x.a),, a), define
(0, yo. 2) = (xol())
{1 S go } Fi (vo [x)[=}}
(5)
TΦx)
{1 S yo) - Fo(unla)) + Fi (x) Fo(volar)
(ιx)
((0) = B (0, Z)}.
(6)
We then have following proposition. Proposition 2. For yo Y. X. and 0 θ we have that
((Ca,(0, yo: Z)| æ) Vol(O, yo, x)) and J(0) VoT(().

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The proof can be found in Appendix A.1
Remark 3. While an inverse probability weighting approach could instead be used to approximate Vol. this form of C provides the desirable double robustness property, we will demonstrate later.
Remark 4. While this only gives us gradient of a loss function rather than the loss function itself we discuss how an estimate of the loss itself can be derived via ID quadrature for validation and hyper parameter selection purposes in Appendix B.2
This result allows us to use Çar and samples from Z to perform gradient descent on the sample version of L(0). In practice,
do not have access to Fas π and will replace these with estimates given by Fas म. We use Čar represent the version of Çár with Fay replaced by Fa, R. With data, D = {2 and testing points (Y-()),, we define our Monte-Carle estimate of the gradient to be
(O, 10%)0 20)) 1 Σ ((, 2(0).
This finally allows us to define our estimation procedure which is presented in Algorithm
Algorithm 1 Doubly robust, direct CQC estimation algorithm
Require: D (2 2m Ge, ((0), T N. μ
1: Define I == {1..... n), J {n + 1,..., 2n} and split D into Dr (2(0) Jiez. ( Dj
(Z JEJ.
2: Use DI to estimate कt Fo. Fi
3: Set 0 = On-
4: for t= 1 to do
5: For E I sample yf (potentially dependent upon X()).. See Remark 6 for more detail.
6: Obtain our Monte-Carlo estimate (( given by (0, (()()) 2(e) in
7: Update 0 by ((++1) = g(t) (0(t)).
8: end for
9: return (T))
replace step 7 of Z any exclusively gradient-based (stochastic or otherwise) optimisation procedure such as Adam (Kingma and Ba (013)-
Remark 5. In practice,
Remark 6. We can choose our distribution over Yo relatively flexibly as this simply defines the test points for our CQC function (similarly to choosing the quantile level a in CQTE estimation). We commonty take Yo Y with Yo Z by simply choosing random untreated responses for each sample. Thus testing our CQC at typical Yo values. An experiment testing this choice is given in Appendix DD.5
Due to its more direct nature, this estimation procedure solves all three problems of the previous inversion approach discussed Section 2.2 Crucially, its explicit parameterisation of the CQC allows to inform and constrain significantly faster to sample from. In addition, since the estimation procedure operates directly on go. might naturally suspect its accuracy depend upon the complexity of the underlying CQC. We might also hope it retains the double-robustnes property present the previous approach. Below, we show that both of these properties hold.
model, well making
model more interpretable and
3.2 ACCURACY RESULTS
As we intend to use gradient descent for our minimisation, a natural question is when is this procedure guaranteed to converge and at what rate does this convergence occur. We now make some restrictions on our model architecture which allow us to achieve this.
Assumption 1. For all yo E y, E X,0 E O:
(a) a < (xx) < a for some a > O.
(b) 90 is of the form go (volz) 0 f(vo, ax) for some feature function f y X RR.

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(c) [ w||ا p for some p > 0.
Assumption I a) assumes that we can bound our estimated propensity away from {0, 1]. this is common assumption within HTE literature and is not very restrictive due to the true propensity
already being assumed to be bounded away from {0, 13. Assumption (b) enforces convexity of
our loss function w.r.t. 0 and bears similarity to the linear smoother framework used in Kallus and
Oprescul (2023): : Kennedy (2023a). Importantly, this assumption does not confine us to linear CQC functional estimates as the form of f can be chosen freely, enabling the use of kernel methods via random Fourier features (Avron et al. 2017 Liu et al. [0222] Rahimi and Recht (007) and other
general architectures. Assumption ()) is required in order to control the rate at which our CQC estimate changes with respect to our parameter 6.
Theorem 3. Let Ô be the minimiser of our population loss as given in Definition 2 Suppose that Assumption 1 holds and that ||0||5 B for some B > 0. For [n]. define ( inductively by ((1) 0. (++)) (c+) mex(()) and ((++1) argming.jon | ((t-
1>11, with Me 2p//"'
Be
and (t)) (()()) y 2 Finally, define our parameter estimate as i Ei= 0(t), Then, if म. Fa are independent of (y ,2(0) ) we have that
BIL(0) -(() S Ci (1/V e(f,, (o.,Po.Fi))
with
(8)
E(f, Fo. A) -
-xx))
E
sup LEE.).E(0.1)
(Fa(solx)-Fadaee Falvolx)))
Mo
where C₁ is a constant depending upon, B, C, p. Suppose further that the assumption in Proposition b)) holds and that Eff((Y.. fYo, X) 722 If we instead take μ Earan
1 then have that
E L()) < C2 (log(n)/n + =(( Fft,Fo. Fi)
(10)
where C2 is constant depending upon, B, C, p. 72.
The proof is provided in Appendix A.2 An additional result giving high probability bounds of the same rate as (8) is given by Proposition [ Appendix A.2.4 The requirement for the nuisance parameter estimates to be independent of the data used for fitting the CQC motivates the sample- splitting procedure in Algorithm I One could instead use a cross- fooss-fitting approach after sample- splitting and average the two CQC estimates which would lead to comparable theoretical results.
Regarding the result, first we see that in both (8) (TO) we have double robustness. This is because both of the nuisance parameter estimators can converge slower than the leading term in the error while not affecting the overall convergence rate due said errors multiplying. This is similar to other doubly robust approaches which have been presented for the CATE (Kennedy] (2033b). CQTE (Kallus) and Oprescu 2023), and CQC (Givens et al.) 2024) which all also derive their robustness results via product of errors over the nuisance parameters. For the second result our requirement the nuisance parameter estimation is stronger however as we need to obtain log(n)/r /n convergence on the product of the nuisance parameter estimates.
We also note that when our density is bounded below as in the second result, if θ 0" where go- = _² have (using Proposition 1) that [[[g((Y Hence if the nuisance term converges at the same rate as the leading term we get a convergence rate on the root mean square error (RMSE) of our CQC estimate of order 1/vn which desirable.
9" (Yo X)}2]= < E[I,(0) T())).
Furthermore from Assumption (c) this gives convergence of 0 to Ô of 1/vn as well.
4 SIMULATED RESULTS
We now illustrate the advantages of our approach by comparing it to two alternatives across mul- tiple dimensions. First, we evaluate it against the previously proposed inverting CQC estimation method from Givens et al. (2024) (labelled "Inv. DR") to highlight the benefits of our direct CQC parameterisation. Second, method, where Car is replaced by its IPW counterpart (labelled "IPW": see Appendix [B. for details), to demonstrate the gains from our double robustness. For each method we present an oracle version
compare it to an inverse probability weighting (IPW) variant of our
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which uses the exact nuisance parameters (Fa, It) and estimated version that uses their estimated equivalents. Further details can be found in Appendix C.2 Further comparisons to the S-Learner approach, where Fo, Fi are used to directly produce our CQC estimate are given in Appendix D
Throughout each experiment, we take X NO, Ia) for d 10, Y|X - N ((sinTTv) x)) ayy'' T x 1) and π (2x) o æ:) where o a random vector in Rdd with || vd, a the sigmoid function, and 7 > 0 can be varied. The sine term represents complexity in the marginal distributions as this an oscillating nonlinear change in the distribution. We thus have that the CCDFs contain the oscillating sine dependency over while the CQC itself does not, simply being (1o (x) y x.
We test two distinct models for the CQC. The first, "DR-Lin", is a correctly specified linear model where we take ge () (z) = (0 + 8sc.u) ( (1o) + [0 + 00 that Psc. ash represent the scaled and shift components of the CQC respectively. The second, "DR-NN" is a full connected Neural Network (NN) with ReLU activations and 2 hidden layers each of width 20.
We fit the propensity score via logistic regression and the CCDFs using kernel CCDF estimation in order to effectively model the sine terms. For each of the following experiments, 100 runs are repeated and mean absolute error of our CQC estimate alongside 95% confidence intervals are presented. Code to reproduce all experiments is provided the Supplementary Materials. Further experiments with different distributional settings are given in Appendix D.I aad experiments exploring sensitivity of performance to hyperparameters are given in Appendix D.3
4.1 INCREASING STEEPNESS OF THE CQC
For the first experiment we increase 7 increase the slope of the CQC. As our current approach is able to model the CQC directly as a linear function, it should be minimally affected by the increase in slope while methods which cannot model this linearity will struggle. Figure 2a shows that our directly parameterised approach (Est. DR-Lin) does indeed perform stronger especially at larger slopes. We see that our NN approach also performs comparably to the linear model. While our estimated versions (Est. DR-Lin/NN) are somewhat worse than their oracle counterparts, they still outperform the oracle inverting method.
4.2 INCREASING THE ERROR OF NUTSANCE PARAMETERS
We further investigate how errors in nuisance parameter estimation affect our estimator's accuracy. To do this, we add increasing levels of biased, random noise to the logits of the original nuisance parameter estimates. Results are shown in Figure 2b We observe that both parameterisations of our method (Est. DR-Lin, Est. DR-NN) perform strongest with the linear model performing marginally better. We also see that the inverting approach (Est. Inv. DR) performs well under increasing nuisance parameter error. Interestingly, the inverting estimator appears somewhat less sensitive to this error than our approach. Nonetheless, our gradient-based approaches (Est. DR-Lin/NN) perform comparably better across almost all levels of added noise, Additional experiments estimating each nuisance parameter separately is given in Appendix D.4
4.3 INCREASING SAMPLE SIZE
Finally we plot the error of these estimation procedures over various sample sizes which can be found in Figure [c] We can see that, once again, our approach performs best, achieving the lowest mean error across all sample sizes and demonstrating consistent improvement as sample size increases.
To summarise, across all our results we see that our approach the strongest for both a linear and NN based CQC model with substantial gains over the existing inverting approach especially when the slope of the CQC is larger. We see that the linear CQC model is marginally stronger than the NN model throughout which we would expect due to it encompassing the true CQC while being simpler model. Overall, these results are promising as they suggest that not only is our approach strong, but maintains much of this strength even when we do not know the explicit parametric form of the CQC.
REAL WORLD SETTING
We also apply our results to real world data to demonstrate their interpretability. Here we look an employment example which has been studied in multiple heterogeneous treatment effect examples
All code and data for the experiments presented can be found at https: :/ttps://iii comb wttphiiiii////rrrttttttssee ono 1 1eContraats
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Est. DR-Lin (Ours)
Est. DR-NN (Ours)
Est. Inv.
Est. IPW
Oracle DR-Lin
Oracle DR-NN
Oracle Inv. DR
Oracle IPW
10a
10a
10a
Mean Absolute g Error
10¹-
10'
10
10a
10a
10a
0.2
2400
Slope Gradient
Additional Logit Nuisance Error
Sample Size
(a) Varying CQC slope steepness
(b) Varying nuisance parameter er-
w.t. with sample size 500.
with sample size 500 and
(c) Varying sample size with ?
Figure 2: Mean absolute error of CQC estimate for various methods with 95% C.I.s over 100 runs.
(Autor and Houseman 2010 Autor et al.) 2017 Powell 2020 Givens et al. 2024). Here, the intervention (A 1) corresponds to enrolment in an employment programme, and the outcome (Y)
represents total earnings in two-year period in thousands of dollars.
For our estimation, we use the linear CQC model described Section We then subtract yo from 9g- to estimate A(yolz) [x) == g" (>) Yoo This enables easier interpretation positive and negative values of are associated with benefit and detriment of the intervention respectively.
10.0
8
Non-intervention Wage
17.5
7.5
6
15.0
5.0
12.5
2.5
4
20
10.0
20
15
y
-10.0
30
40
30
x 40
35
Age
Figure 3: Surface and heat plot of (y(y)z) for our employment data with =Age, Y=Income.
Figure shows this estimate for various values of (y, .). From these results we see an interesting pattern. Across all ages, the intervention had the most impact for those with high non-intervention earnings. The change in wage improvement as a function of on-intervention wages seems to decrease as age increases however. In other words for younger participants, the distribution of wages seems to multiplicatively scale while for older participants, the impact of treatment seems be better represented by a more uniform shift. examine the parameters of our estimate directly in Appendix D.6 Another example examining the effect of treatment on colon cancer remission is presented in Appendix D.7 where we use neural network (NN) to model nonlinear CQC function.
LIMITATIONS AND FUTURE WORK
One limitation of our approach is that while our direct estimator performs best overall, there is evidence to suggest it is practically more sensitive to nuisance parameter estimation error than the existing inversion based estimation approach. This is somewhat mirrored Theorem 3 where our double robustness is with respect error CQC. Future work could investigate these two properties and their relationship more thoroughly, with the potential improve upon them further.
loss function rather than directly
of the

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Additionally, while our estimator is direct in terms of exclusively estimating our estimand of interest, it does not have the form of estimating the estimand through a conditional expectation as is common for other estimators (e.g. Kennedy (2023b); Kallus and Oprescu (2023).) Such an estimator then has the advantage of being estimable by various non- non-parametric procedures for conditional expectation estimation while also being estimable parametrically via least squares. It also has the advantage of giving accuracy results directly in terms of the estimand of interest which we are only able to do under certain settings, As such, a future direction would be to explore whether a doubly robust estimator of this form could be produced for the CQC.
Finally, while our current convergence results apply a good number of parametric and nonparametric CQC models, later work could expand these results to CQC estimates which are not linear with respect their parameters, such as NNs (Shi al. 2019) or Bayesian additive regression trees (Hill 2011
Green and Kern, 2012; Künzel et al. 2019).
7 CONCLUSION
To conclude, have proposed the first direct estimation procedure for the CQC, an estimand which aims to bridge the gap between the CATE and the CQTE. We have demonstrated the efficacy of this estimation procedure both theoretically and empirically, showing it to outperform existing approaches. Furthermore, we have highlighted its ability to allow for direct parameterisation of the CQC and demonstrated its benefit in terms of both empirical performance and interpretability in real-world scenarios. Overall, this represents an improvement over existing CQC methods, further enhancing the utility and real- weal-world applicability of this emerging treatment effect estimand.
REFERENCES
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Givens, J., Reeve, H. W. J., Liu, S., and Reluga, K. (2024). Conditional outcome equivalence: quantile alternative to CATE. In Globerson, A., Mackey, L.,. Belgrave. D., Fan, A., Paquet, U., Tomezak, J., and Zhang, C., editors, Advances in neural information processing systems, volume 37, pages (02634-102671 Curran Associates, Inc.
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A ADDITIONAL THEORY AND PROOFS
A.1 LOSS JUSTIFICATION PROOFS
Proof of equation
As a reminder the identity of interest is
g" (yolæ) argmin ( yo. v))
F1
First define the intermediary loss function
By the fundamental theorem of calculus we have that oy (111, No. x) = ((V1, No. œ). Therefore, as Oy ((v,, No, æ) is increasing in y/1 for any æ, yo. we have that
V/ argmin {(vi Mo, w) = 0y Evvi No. 20)
v
+
Fi (Jn (x) Fo(uulæ)) = 0
Fi (y1 (x)) Fo(uolz)).
Hence by definition of g", we have that
g" (yo|z) = argmin ((11.80/22)
9n
Proof of Proposition
Firstly define Č(v1, 30 by
1
[ ==
(1 S (0(v)(=)) Fi (100(g)))))z))
{1 S vol [o(()()))
π x
T②x)
(1 xx) Fo(
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So that (00.3o.2) yo x) (e yo. x) = ( x jd do (volm) (30 x) x))
[z) (2)) a)) By the chain rule have that
Hence all that left to show is that EE (1 uo. 2) x = 2 Elvn, Volz)). To this end we can use the tower property to get that
A
vo. (11, Vo. 2)|X = x)
[B[L(Y S Sin A = -
n x
A
[E[TY S 30 un)) A 0 Fo(volz))
π (xx)
+ Fi (y (r) (uol
(y (x) Fo xx)
o, 20,, x²)
y0.
A.1.1 LOss BOUND PROOFS
We now provide the proof for our result bounding the loss in various circumstances. First however we provide proposition with various upper and lower bounds on integrals which will inform our upper and lower bounds on the loss.
Proposition Let F y + R be an arbitrary increasing function with F(a) F(b) 3 for < b E V. Also define (y) Fly) and = S F(y)dy. We then have that
1. 1 S (8) 6 al.
2. If fw) 2 n for all y E ja, b). 1 2 [b -)
3. If fy) S & for all y E [a, b). 1 -28)-1882
4. If f(y is increasing on [a, 6] then 1 2 1181 a/.
As a convention we allow for the possibility that >> b and take [a, b] in this case mean [b, a).
Proof. All results are proved under the case a S b. The results for the case a b follow an identical argument with signs and equalities reversed. The first result follows directly from the fact that Fly) < F(b) for all y E [a,b]. b).
For the second result we have that
Fly) '" f(s)ds+ F(a)
I" f(s)ds
2 (v a)n.
Therefore
b
I 2 Ja
(y - a) a)mdy
" (b a)2.
2
For the third result define F: |a, b] + } by
00
if ja,b- 3/E], Ev cb+ + 3 if E (b - 3/E, b).
F(y)
Then F' is, non-negative, continuous and increasing with F(b) = , and maximum gradient g. Furthermore we claim that F lower bounds any other functions with this property which also has continuous derivative.
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This is trivially true for [a, b 3/5). Otherwise suppose there exists function G ssatisfying all these assumptions excluding the gradient bound with G(y) (y) for some y (b - B/C, bj. Then we have that ((b)-G(m) < ç, hence by the mean value theorem we must have that GyG(y" < c for b-
some y' in ly. b). Thus by the contrapositive, F is the minimal function satisfying all these conditions on (b - B/E,
As such we can now get the following bound on /
I 12 Fly)dy
fe
5y Ady 33/5.
bb-//
For the final result note that (1) increasing implies that F(y) is convex. Therefore we have that
=
Fluidy
y (F(b) F(a)))
F(a)
dy b-a
a
<
3dy =
| 16 al.
0.
Proof of Proposition [-1, 1] given by
For notational convenience we introduce the function h : x x x X
h(y1, yo, x)) = Fi (y11 (z)) Fo (x))
so that E(y1, No. .²) = f (vola) h(t, yo, x)dt. Remember that ((0,, yo. x²) (gg( yo. æ). We
can then notice that h311, you x) satisfies the conditions of Proposition as a function of y/1 with
g" (yo, 2²) and go an) and Fi (go ))) [ar) (y ar))
Furthermore oy, E(vi, No. æ) (v xr). Therefore the assumptions in Proposition 1 [c) corre- spond to the assumptions in results 2-4 of Proposition 4
Therefore we can simply directly apply each result of Proposition prove our required results.
A.2 ESTIMATION ACCURACY THEORY AND PROOFS
A.2.1 CONVEX CONVERGENCE
Lemma 5. Let L(0) be convex function and define = argming L(0) with ||0||< B for some
B > O. Define (11) 0 and inductively take
f ) = g()) mp
( argmin 0 ( p(++))
0:00|5B
with and u1,-.., Un sequence of RVs with [|000]]| Finally, take our parameter estimate
p/m
be n Σm= (t).
Then we have that
Bp
11 (0))) 0,0(0)
L(O) - L(Ō) <
n
n
where e(t) p(() -V
Voc L(B(t))
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Proof of Lemma 5
Define (
Vic L((+)) so that E| ( + te()) where (
L((())
represents the unbiased gradient estimate and e(t) represents the bias.
From Shalev-Shwartz and Ben-David (2014) section 14.4.1 we have that
11
0 0¹, (,,000)
Bp Vn.
E
n
Additionally we have
L(O) L(00)= S
L(O -(00))
by Jensen's inequality.
t=1
(0 0" Vio)
by convexity of I, and definition of (1)
(0 0. .000)
0
0", c(0)
11
n
B
(e())
0" e,(())
from our prior result
72
1=1
Lemma 6. Suppose that assumption 7 holds. For arbitrary fixed O, Define
8 = C(O, Yo, 2) (0))
Then have that
2p
Fx - Fa 1x)
|||||| s
E ((X)
=(X)
E
sup Mo (0.1)
Proof. To do this first define
b(e, Yo, X) E(((0, Y.. 2) <(0, Yo, Z), Yo,
To bound b(0, Yo, Z) firstly have that
E[1lY < v}{{A a}|-] = EY A=) = alX) = FFA
We can then use the fact that IPAA = 1|X) = (X) to get
(Fi(00(0)x)IX - ((00611111)) (
E(Č(0, Yo. 0, Yo, X] =Veg0
- T(X)
(Folvoli)- - Fo(r)(x))
1 #(X)
Hence
x)) (Voo {
T(X)
7(x) - -1) Fi (F)10(())x))) -i(o0(Y6
1 11- x(X) (1- -1) (Fo(Y61))-F((x6x))) )
1 1- x(x)
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Now by the tower property and linearity of expectation, have that E|{] Elb(e, X) In turn we then get
[e]] E
1 Yo, x ( by Jensen's inequality.
Now using our bound on f in assumption using our bound on f we get that
we get that Vef fyo. x ك||ا p for all y, a. Additionally
π ( 1 H (x)
[(x)
Combining these we get
E |||||| - LE c
(T()) ( -x
Fi (Y - (F Fo(%/x) Fo(%ox)) |
< 2p
E
(x(x)= =xx)]
E
sup 9o (E,11))
-
(vol
Proposition 7. Suppose that ussumption define ( inductively by
holds and that ||0||S B for some B > O. For E [n].
e (()) (1x(1)
0 arggmnn 10 (
0:110
with ((1) 0, 14 2p//n' and (00) = ((0()), Y6 2(0). Finally, define the parameter estimate as
Ô n Σイ= ((1) Then, if ti Fa are independent of (ro1) z Jim. we have that
Vn (T(x)
1
27
(Fa(oo/X) )
- < Ci
E
sup Luo.a
(11)
where C1 = 4Bp/c.
Proof. First that
((((0()) r 200)) L(O())
= E Vor (Yo)x) (Fr (Y/xx))) Fo(%) [0(]]
We now aim to show that we are in the scenario of Lemma with
(1) (()))) ( z(t) (l(( Yo. 2) (0)
First we show that under Assumption [b), L(0) is convex as function of 0.
To this end we note that E(y), yo. 2:) convex w.r.t. y1 as, by construction,
By Evn. Vo, x)) F (v1 Fo(uolzz)
which is increasing in y1 for any Hence, as the composition of an affine function and convex function convex, we have that
Furthermore for any 9o ge is by construction affine in O.
(0, yo. x)) ((go (0) yo. xx)
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convex w.r.t. 0. Hence integrals of convex functions convex, L(Ø) E[ Yo. Z)] is also convex w.r.t. 0.
We also have that from Assumption aaa that C(y1, yo, x) - 1 + 1/c for all y/, yo. combining this with Assumptions [b)&(C) we have that
|| s sup [ x ))² f 2.), yo. 2) 0,yo.z
< p (1 (1+1/c) 2p
Meaning that we are in the setting of Lemma 5
Taking expectations on over the result the Lemma gives
2Bp c/n n Σ ((00) - -0,0(t))
([T.(O) - -
[((() 0 s(()) )|
and all we have remaining do is bound -E
For this we have that
-E { (000 - 0 = E
( ((0)) 8,E
(
< E { (00) Ø
E (0) (*)
by the Cauchy-Schwartz inequality
- E
000
- 0
E
by Jensen's incquality
$
2p
E
x(X)
7(x)/ /
E
sup yoEV. aa
Fa(volX) Pa Puol)
[0()) ق
E
by Lemma 6
4Bp
Fa Fa - Fa |x0
$
E
(X)
#(x)
E
sup E. (oE(O.1)
with the final line coming from our projection step. Combining this with Lemma 5 gives our desired result.
A.2.2 STRONGLY CONVEX CONVERGENCE
Lemma 8. Let L(0) be strongly function w.r.t 0 with strong convexity parameter y and define argming L( Assume that || for some B > Define e and inductively take
(+++ (()) (t)
(+++)) argmin ||0 o 0:100|SB
(n) a sequence of RVs satisfying ||o00||< P almost surely. Finally, take our parameter estimate to be = 1 E (())
with p4 1 and (
Then we have that
L(O) L(0) <
p + log(n)
Σ
(0()) 0" e(t))
2m
1=1
where e(t) = p(t) Voc L(O(t)).
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Proof of Theorem Define ( V git) (((()) so that E(u @(*) (()) +e(0) where 4
represents the unbiased gradient estimate and e(t) represents the bias.
From Shalev-Shwartz and cen-David (2014) section 14.4.1 we have that
| 0-1 1 11
-0, 0-, v(()) = 2
1100 +
2/4r
Additionally we have
L( - ((0)) 1 L(B -(
by Jensen's ' inequality.
(0()) 0", v(1))
!
( 1-
by strong convexity of
n
((() 0¹, v000)
" 1-11
|
Σ t=
g(() 0
e())
2
then from our prior result we get
|0(0) 0-
0-111100(+))
0011 (1 10
11
2/1r
(1)
0
e())
(
1-11
1
( - 00, s(0))
2nyl
2/1r
1=1
t=1
0 + log
(0()) 0" (e(())))
2n)
n1
t=1
Proposition 9. Suppose that assumption holds and that 10 B for some > 0. Ad- ditionally now suppose that Pi ()))))) &2 for all y, and that the minimum eigenvalue of E [f (Y,, X] is greater than 1/2-
Define ( = O and inductively take
((++)) =0()) (0)
((++1)
argmin |0 0 e:|le|<?
with 14 mzEan and (1)) = ((((()))))) y/",, 2(0). Finally, take our parameter estimate to be 1 D (
Then we have
[[ SC₂
1 + (ogg(())) (
B (xxx))(x)))
sup (Fa(vol)) -Fa(vol))) 2x) yoEy [oE(0.1)
with C2
1Be
croos
Proof. This is almost identical to the proof Proposition strong convexity of T((O).
7
The only additional step to prove
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We have that
VoL(O) [Vo(go())z) (4i (yo Fó (Mfo [c]]
= f X) (Fi (90 x)
- Fo [Bo [=]]
(O) E [ (Y,, R (V)1 (4) (26 Fo [2|-]]]
[Y,, f),, x) pı fgo (Y,
2 Eal [f(Y, (Y,,
which by our assumptions has minimum Eigenvalue greater than E2722 Hence L(0) is strongly convex with parameter == E27
Now we can proceed as in Theorem 3 to obtain
P 1 log(n)
( - )
B[L(O) L(O)]=
<
E
en252
Then using Lemma and following an identical approach to Proposition we get our result.
A.2.3 PROOF OF THEOREM 3
Proof of Theorem
This result is simply the concatenations of Propositions 7
A.2.4 PROBABILITY BOUNDS
We first state version of Hoeffding bound which will be useful for our work. This Lemma slight modification of the version found in Wainwright
Lemma 10 (Azuma- Hoeff--o. Forn N. let W() ..... w(() be Martingale difference sequence with respect to filtration (F()) }
Suppose also that [W()| I p a.s. for all E [n]. We then have that for any
n
2 2log(1/8)
P
W
2 8.
n2
n
Remark 7. For W(tt) to be a martingale difference sequence we must have that W(I) is F(t)
measurable, [+]] co, and E[ [[tt)) = a.s.
We now get finite sample probability result the setting of the first part of Theorem 3 For clarity restate this setting in the result.
Suppose that assumption holds and that ||0|| B for some B > O. For I [n]. define ( inductively by
Proposition 11.
(((+b)= ((() (t)
((++)) argmin 0 (
0:1|0|5B SB
with (11) Pt 22p/n'
Be and ((() = ((0()), 9 2()). Finally, define the parameter estimate as
i Σi= 0 Then if Fa are independent of (Y 7(0) S.m have that for any ö > with probability at least 1 5,
L( ((0) C3 + / Tog(1/8)
(() (xx)
sup Mo E.
Fv - Fa(vo
vn
a (0,1)
with Cg == B2
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Proof. Again are in the case of Lemma
5 with ý 2p/c meaning we have that
2Bp L(() L(0) - c/n
( - -0,(())
(12)
n t=l
Now for & [n]] define the filtration ( i=1 by F(0) (10(1) )/=1: Paj. Additionally define RVs W ( e E| (()) T,, Fal)
Then we have that ( )== is martingale difference process with respect (F(0)² J_
Furthermore we have that |0:00/ s 2g. Additionally | ato [(O()))|
- 2p. Hence
|
4p
c
((0)) E (+)) म,, Fa
8p
|(()
16Bp
<
As such we can apply the Azuma- Scuma-Hoeffding inequality stated in Lemma 10 to get that
log(1/6)
IP
w < &Os/
2 6.
with C3 16v2 Be Furthermore we have that
n
log(1/0)
w(t) S C3p
n
n
12
(1g(1/))
n
(1)) 0,5(1) S <Ca
0 0 EE (0() n, FF.F)))
71
n
71
(00) 0, S <C3V
log(1/8)
000
61 |1||
00 म. Fal
71
by the Sauchy-Schwarts inequality. By Lemma and the fact that (e(0) 0 < 2B this gives that
(1) 0,5(0) SOs log(1/5) + T(X) - #(X)
Fa(volX) - Fd66o1x00)
sup No EV (0.1)
Hence by equation 12 we have that w.p. at least -
L(0) L()) Cs 1 +
/og(11/))
x(X) #(x)
sup
(vol - Fa(wul
vn
(E(0.1)
B ADDITIONAL METHODS
B. IPW APPROACII
Alternatively to our doubly-robust gradient estimator we can define an arguably simpler estimator which only depends on the propensity function æ. This is done by defining
Spo (0, yo. 2)
Vogo (yol
ly S go g l=))
1-
By Yo-
π (x:
(Tx))
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We then have that EiSip yo: Z) =)] Vol(0, Vo. æ). Meaning that Proposition holds for Sipw well. From this we can define Gipw analogously to Čar and also use it in Algorithm This precisely the IPW procedure presented in results.
In these results we see that the performance of this very poor due it's over reliance on inverse probability weighting which can be quite unstable.
B.2 DIRECTLY EVALUATING THE LOSS
For validation purposes can be useful to approximate the sample loss directly rather than its gradient. To obtain this from the gradient Ç this can split the objective into parts, one involving all terms of Fi (v1 (<) and all other terms.
As such we re-write Ç as
1 1-
(ar Vo, 2) π. a
(y 5nn}
(n S vo Fo(vola:) Fo |a)
1 ((x)
+
a π (x)
Fi (1(
Now for the first term (11) we know that an anti-wweak)drrivative is which keeps the loss continuous w.r.t. y/
(31 8)
a (x)) {1 S vil)
1 1 a
(1 S yo Fo(vol=) - Fo(volz))
1- ((x)
For the second term (which is continuous as function of yı) we can use the FTC to get antiderivative of
T(x)) ) ( (tlx))dt
π a
In fact we can also view the antiderivative of I as the integral of 1 between V1, Combining these we thus get
(ar. No. x) = y)
a (xx)
(1f S 91
1-a a
(1 S vo) - Fo(uulæ)) - Fo(uolx))
1 - T④x)
+ (m(x))
(t(r))d
((x)
A
EI Yo, 2)] =E (Yo(Y) Y)
(1 S (Y((x)))
(X)
1-x(X) (1 < Yo} Fo(Yo))) F6(Y61))
1 A
(T(X)- - A) T(X)
A)
(8(%)1X)
+
(
We can then approximate the expectation via samples and the 1D integral via quadrature to get an approximation for the loss.
Remark 8. The choice of y for the lower bound of the integral simply chosen to keep the size of the integral reasonable and to give the first term simple form. Any choice of lower bound not depending upon yn would be valid.
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C ADDITIONAL DETAILS
C.I COMPLEXITY OF THE CQC VERSUS THE CCDF CONTRASTING FUNCTION
While not strictly weaker notion, we do believe that simple CQC function a more natural notion than simple CCDF contrasting function.
As general case suppose we are in the potential outcomes framework so that = YA with Yo: Yi representing our unobserved outcomes for individual were the off or on treatment respectively. Suppose now that given Yo. one can determine Y, as the following Y, f,,, X) with an increasing function of Yo (a natural notion wherein those who perform better off treatment also perform better on treatment.) We then have that the CQC is given by f. in other words g"(M" (yo (z) f(v, æ). Hence simplicity of translates directly to simplicity of the CQC.
Alteratively, for the CCDF contrasting function we get that
h(y1.80 yo: x) = Fi (vi æ:) Fo(uolar) (z) Fi (yn x)) Fi (f x))x)
which does not necessarily cancel out to give a function of f for all yo, V1- In fact the only case where we know this cancellation to occur is when Y|X x, A are certain cases of uniform distributions.
C.2 EXPERIMENTAL DETAILS
Here we provide additional details for our experiments. For our training we used 1,000 iterations of Adam with learning-rate of 0.1 for any optimisation based approach. For estimation of the propensity score we used logistic regression with L,2 regularisation.
For estimation of our CCDFs, used kernel CCDF estimation. Specifically for a kemel ls / X X' and sample {(Y(), x 1im1> take
Falyla)) En k(x X ( < y)
Σ xxxx(() X()
For our kernel we used an RBF kernel with bandwidth parameter chosen via grid- testing on separate data against the true CCDF.
For hyper-paramette optimisation of our CQC model, with the linear and ML. models with our approach, the only hyperparameter that tuned was the learning rate. This was set using 80-20 splits for training and validation from half the data used our training (the other half being used for nuisance parameter estimation.) As our validation loss we used the sample loss given Appendix B.2 A choice was made to take the trimmed mean removing the top and bottom 5% of samples in order avoid small number of large samples dominating the loss. For the pre-existing inversion based method, the kernel bandwidth was chosen on validation data when comparing to the true CQC when the CQC was trained on balanced data so that no nuisance parameter estimates are required. While not possible in practical examples, this was done to ensure the inverting method was not hampered by poor hyperparameter selection.
Each experiment was ran single 4 core CPU with 16Gb of ram and took no longer than 240 minutes to run (less than 1-minute per iteration).
The employment scheme data used in Section 5 was originally provided in Autor and Houseman (2010) with a Creative Commons Auributiion 4.0 International Public License found here: https www
The colon cancer data used in Appendix D.7 is provided as part of the R package survival and first introduced in Laurie et al. 1989 with no Licence provided.
D ADDITIONAL RESULTS
D.1 1-DIM EXAMPLES
In this example our data set-up is as follows X ~ N(,,,)) Y|X = x, = (6c)) 1). Y|X x, A 0 ~ (62cos(6z)) + 7x, 4). Again in this case the marginal distributions contain
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(Ours)
+ Oracle DR
Est. DR
++- Oracle Inv. DR
-4-- Oracle
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Additional Logit Nulsance Error
Sample Size
(a) Varying CQC slope steepness
(b) Varying nuisance parameter er- ror with sample size 500 and ạ = 2. (c) Varying sample size with y =
w.r.t æ with sample size 500.
Truncated mean absolute error of CQC estimate for various methods with top and bottom 2.5% of runs removed alongside 95% C.I.s over 100 runs. Lower is best.
Figure 4:
"complexity" via the high frequency sine term which persists into the CATE, CQTE, and CCDF contrasting function however the CQC is simple, being given by g" (y/lx) 2y + yx. As in Section 4 we test estimation of this example with varying levels of (representing steepness of our CQC), varying logit error on our nuisance parameter, and varying sample sizes. Due small number of outlier runs, for ease of interpretability, we present the truncated mean (where the largest and smallest 2.5% of results for each method removed) alongside 95% confidence intervals in figure For transparency, we also present the standard mean with 95% confidence intervals in Figure
Here we see identical patterns to our previous 10-dimensional example presented in Figure 2 with our approach (Est. DR) performing strongest in almost all cases. We again see that as the CQC gets steeper (Figure 4a our estimation error stay relatively unchanged while the estimation error of the inverting approach gets worse.
As vary nuisance parameter estimation error (Figure 4b), observe that Est. DR performs best all levels. Despite this, we again observe that there is no discernible difference between Est. Inv. DR and Oracle Inv. DR whereas Est. DR does seem perform marginally worse than Oracle DR. This does seem to support the hypothesis that Est. Inv. DR more robust to nuisance parameter estimation error. We do still see evidence of robustness in Est. DR however as is still minimally affected by nuisance parameter estimation error when compared to Est. IPW (which is not doubly robust.)
In Figure 4c we see our approach, Est. DR, having the smallest Mean absolute error across all sample sizes.
+ Est. DR (Ours)
- Oracle DR
Est. Inv. DR
Oracle Inv. DR
Est. IPW
-+- Oracle IPW
10P
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(a) Varying CQC slope steepness w.r.t. z with sample size 500.
(b) Varying nuisance parameter er- for with sample size 500 and
2.
(c) Varying sample size with y
Figure 5: Mean absolute error of CQC estimate for various methods with 95% C.I.s over 100 runs. Lower is best.
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D.2 10-DIM EXPERIMENT
Here we present additional results from our 10 dimensional experiment introduced in Section
D.2.1 S-LEARNER AND CQTE APPROACH
Here we introduce additional comparators specifically in the form of an S-Learner and the CQTE estimator of Kallus and Oprescu (2023).
S-Learner The S-Learner works by finding the value of y1 which sets h(y1, yo, x) = Fi (v1lkx) (o(volæ)) This can equivalent be thought of as taking our estimator to be F (Fooolozz))
where F-1 is computed by inverting Fi.
The results are presented in Figure 6 As we can see that Separate approach performs comparably to the DR approach in most settings except for the case when the slope parameter set to O. We can potentially understand this in terms of the derivative of our CQC w.r.t. We have that V V=gg" (yol2) yo. Alternatively we see that V Fo (yn ae) ( - sin(Tv x)) = [y sin(Tv x)) TO where f are the CDF and density of 0 means standard deviation Gaussian. As such while the CQC is simpler function, its derivative can be on larger scale than that of the CQC making it more difficult estimate from the perspective of Nadraya-Was -aatsnn (NW) estimation. As such an approach which estimates the CQC using NW estimation (as the inverting approach does) will get minimal benefit over estimating the two CCDFs separately and using this as its estimate.
Est. DR-Lin (Ours)
Est. DR-NN (Ours)
Est. Inv. DR
Est. IPW
Est. Separate
Oracle DR-Lin
Oracle DR-NN
Oracle Inv. DR
Oracle IPW
10a
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10
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(b) Varying nuisance parameter w.r.t æ with sample size 500. ror with sample size 500 and 7 (c) Varying sample size with ?
(a) Varying CQC slope steepness
Figure 6: Mean absolute error of CQC estimate for various methods with 95% C.I.s over 100 runs.
CQTE Estimator We also compare to the CQTE estimator of Kallus and Oprescu (0233) For estimation of each nuisance parameter and the final regression we use the same approach as used for the inverting estimator of Givens et al. 2024). The CQTE also requires estimation of the conditional density of Y || as the quantiles. That is for a E 10, PrI (F= (alz)) our specified quantile level. To rule nuisance parameter we use its exact value for both the oracle and estimated approach. To compare this estimator to our CQC estimate we use the identity
for a given value of a poor performance due poor estimation of this additional
g x) = (, + (y x))
to transform the CQTE estimate using the exact CCDF. Additionally as the CQTE estimator is constructed to learn the CQTE for specific quantile, for each run we fix the quantile that will test the estimator on. We do not change the training procedure of the other estimators.
we can see the CQTE approach performs Results for this experiment are given in Figure 7
comparably to the inverting approach and performs significantly worse than our direct estimator in almost all settings. Interestingly the Oracle and estimated approaches appear indistinguishable we could be due to using exact estimator of the conditional probability density function in both cases.
D.3 VARYING HYPERPARAMETERS & COMPUTE TIME
D.3.1 VARYING LEARNING RATE
Here we explore the effect our choice of learning rate on our performance for our 10-dimensional experiment in Section 4 The results are presented in Figure
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Est. DR-Lin (Ours)
Est. Inv. DR
Est. IPW
Est. CQTE
Oracle DR-Lin
Oracle Inv. DR
Oracle IPW
CQTE
10
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10"
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(a) Varying CQC slope steepness (b) Varying nuisance parameter er- ror with sample size 500 and
w.r.t. a with sample size 500.
2.
(c) Varying sample size with =
Figure Mean absolute error of CQC estimate for various methods with 95% C.I.s over 100 runs.
(Ours)
Oracle DR-Lin
Oracle IPW
Moan (Log Scale)
10a
Mean Error Scale)
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10'
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Leaminng Rate (with Decay)
Figure 8: Mean absolute error of CQC estimate for various methods as learning rate increases. 95% C.I.s included. For the right figure a learning rate decay was also introduced
As we can see, for our DR method, higher learning rates can hamper performance although the method does not seem excessively sensitive to learning rates. By contrast the IPW approach gets drastically worse as learning rate increases. We also see that adding learning rate decay can further mitigate the effect of the learning rate performance. For our main experiment chose learning rate via a validation procedure using the test loss discussed in Appendix B.2
D.3.2 VARYING ITERATION NUMBER
He we explore the rate at which our method converges. Figure 9 we plot the convergence of our method for the IPW and DR approaches with oracle and estimated nuisance parameters.
We see that our DR approach converges within about 150 iterations while the IPW approach doesn't 't seem to converge at all or if does converges to an incorrect value. note that while 1000 iterations is very conservative, this still takes around second with 1000 samples and so is reasonable to perform. In the following section we illustrate the time take for our new approach, demonstrating to have more desirable dependence upon sample and test size.
D.3.3 TIME TAKEN
In Figure TO we plot the time take to train and evaluate various models for various number of training samples (left plot) and evaluation samples (right plot). We see that for small training and evaluation samples the previous inverting approach is quicker due to not having distinct training sample however we can see that overall has less desirable dependency on the training and evaluation samples, with the computational cost being O(n² m) compared to O(nT + m) for approach with n =sample size, m =evaluation size, =iterations. Throughout we kept iterations fixed at an overly conservative 1000.
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Est. IPW
Est. (Ours)
Oracle IPW
Oracle DR-Lin
10a
Mcan g Error (Log Scale)
10a
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600
Iteration
Figure 9: Mean absolute error of CQC estimate for various methods over iteration number. 95% C.I.s included.
Oracle DR-Lin
Oracle DR
Taken (seconds)
(secands)
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Mean time taken for training and evaluation of our gradient approach and the inverting approach for varying number of training and evaluation samples. 95% C.I.s included.
Figure 10:
D.4 NUISANCE PARAMETER DEPENDENCE
Here we explore the dependence of our approach on the accuracy of our estimates. Specifically we fix either the propensity the CCDFs their true values and estimate the other alongside various levels of additional error. These results are presented Figure
Est. DR.Lin (Ours)
Est. Inv. DR
Est. IPW Oracie IPW
Oracle DR-Lin
Oracle Inv, DR 10"
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Mean (Log Scale)
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CDF Error
Figure 11: Mean absolute error of CQC estimate for various methods nuisance error for Propensity and CCDF estimates increases separately. 95% C.I.s included.
We see that both CDF error and Propensity error have some effect on performance for our method. Interestingly with additional error propensity performs comparably to the oracle however additional propensity error can have a notable impact when gets too large. Interestingly for the CCDFs, additional error doesn't 'tt seem impact performance but estimated approach performs
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significantly worse than oracle estimator suggesting that estimate for the CCDFs is already quite poor. For the inverting approach, increased propensity error seems to have no effect while the effect of the estimated CCDF is small but statistically significant.
D.5 Yo SAMPLING МЕТIОО
Here we explore the impact of our sampling choice on Y, as discussed in Remark 6 Specifically we sample Y in 3 different ways. Firstly we sample Y, uniformly from the range of the 5%-95% % quantile of Yo and call this method "Uniform". Secondly
sample Yo uniformly with replacement from our samples with A to approximately sample from Y Y|A = and call this method "Unconditional", Finally we sample exactly from Y = X A = 0 for each ( using the true inverse CDF and call this method "Conditional", Performance over various sample sizes are presented in Figure [2 As we can see the sample choice seems to have little impact on performance with the "Uniform" approach potentially performing marginally worse although this is not statistically significant for all sample sizes.
Yu Sample Type
Uniform
Unconditional
Conditional
0.9
Mean Enor
0.6
0.3
600
800
1000
1200
1400
Frequency
Figure 12: Mean absolute error of CQC estimate for various Y, sampling choices sample size increases. 95% C.I.s included.
D.6 EMPLOYMENT SCHEME EXAMPLE
Here we provide the parameters themselves for our aforementioned employment example.
Table 1: Table presenting the covariates from our CQC estimate plotted in Figure 3 The mode is
go (y) (x)) = Oint, shift + Oage, shin
(Oint, scale + Onge, scale)) y
Parameter Type
Covariate
Shift
Scale
Intercept
1.43
1.74
Age
0.032
-0.017
We can see the overall shape the CQC represented in the parameters. Firstly we see that the scale term is significantly larger than at the intercept and will continue to be larger than I for all values of age thus representing an increase earning improvement as non-intervention earnings increase. We also see this increase in carning improvement decrease scale parameter is negative. We can easily see how one could generalise this to multiple covariates. For interpretability it perhaps makes comparable scale and give the intercept more natural interpretation.
function of age as the age
normalise both and for all parameters to be
D.7 COLON CANCER EXAMPLE
We additionally apply our trial to data from a clinical trial on the the effect colon cancer treatment survival time/time to remission. This dataset originally introduced in Lau ri et al. (1989) and can be found in the "survival" package in R and loaded with the line data (co 1 lon, package="survives "). It was also previously studied via the CQC Giveess)
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et al. (2024). The dataset consists of 929 patients who are randomised to receive either treatment or control. The time until their death, recurrence of their cancer, or the end of the trial was then recorded alongside which one of these outcomes occurred. The longest recorded time individual participated the trial was 3329 days. We take our response (Y) to be the time until their event/end of the trial and 1-dim covariate (X) of the participants age upon trial entry.
As previous analysis of this trial showed the CQC to be distinctly nonlinear, here we fit the CQC using a fully connected Neural Network (NN). This NN takes you z as two separate features and then consists of two fully connected hidden layers of 20 nodes each and tanh activation functions. One again we estimate g and then use this to estimate A(ylæ)) = g (y (+) y. The results this
estimation are given in Figure 13 For comparison we provide the estimated CQC via the existing inversion procedure in Figure
1000
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to //
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Figure 13: Surface plot and heat plot of (yy over y,, æ for colon cancer trial data with X =Age, Y=Time to Event.
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Figure 14: Surface plot and heat plot of Ayy|z) over y, x for colon cancer trial data with X =Age,
Y=Time to Event.
Here we see a very interesting pattern in which for the a reasonable range of the untreated response, the treated response no difference and then there is sudden increase the treated response. This seems to suggest relatively binary treatment outcome in which some people do not respond at all to treatment while others see marked improvement. Interestingly, we also see that individuals younger than 50 seem to be most likely to see an improvement in their outcome while the strongest improvement seems to come for smaller number individuals between the ages of 56-66. This could partially be result of the censoring the largest values present the graph are over 1,000 days larger than the untreated survival time 1,000 days which, in total is reaching the longer end of follow-up. up. All of this aligns closely with the estimate CQC via the existing inversion approach presented in Figure 14 with the newer version providing a smoother and more readable estimate of the CQC.
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E LLM USAGE
An LLM was used for minor editing of the papers prose. This was done solely for the purposes of conciseness and clarity.
29


Reviews:
[{"review_id": "yo1H1Tqoys", "reviewer": "Reviewer_MqaP", "summary": "This paper introduces a new direct doubly robust estimator for the Conditional Quantile Comparator (CQC). Unlike prior approaches that estimate the CQC indirectly by inverting a contrast of conditional cumulative distribution functions, the authors present a direct parameterization and gradient-based estimation procedure that preserves double robustness while improving interpretability and computational efficiency.\nTheoretical results include finite-sample bounds, convergence guarantees, and robustness proofs. Empirical validation shows improved estimation accuracy across simulated and real-world datasets (e.g., employment outcomes). The method also generalizes to both linear and neural network parameterizations.", "questions": "1. Could the authors expand the real-world analysis by including quantitative comparisons to other estimators? Specifically, how does the proposed method perform relative to both inversion-based CQC estimators and standard CATE-based models such as TARNet, DragonNet, or BART? Even though these approaches estimate different quantities, wouldn’t such comparisons help clarify whether modeling conditional quantile contrasts offers practical advantages over mean-based causal estimators? If you have reasons for not comparing those, could you explain reasons?\n\n2. I think ablation studies isolating the contributions of each model component (e.g., nuisance models, parameterization choices) would strengthen the empirical claims - are there plans to include these analyses, or are there some reasons that this kind of study is not considered?\n\n3. Are there some possible ways to formally quantify or evaluate interpretability beyond qualitative visualization?\n\n4. Could the authors provide computational complexity comparisons (runtime, memory) versus the inversion method?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper presents the first direct doubly robust estimator for the Conditional Quantile Contrast (CQC), offering a method that effectively connects theoretical causal inference principles with practical implementation. It contributes new finite-sample and convergence bounds, extending robustness theory in heterogeneous treatment effect estimation. The proposed algorithm is clearly articulated through a gradient-based optimization procedure (Algorithm 1), making it both conceptually transparent and computationally feasible. The authors validate their approach with comprehensive experiments that vary sample size, noise level, and functional complexity, and further demonstrate interpretability through a real-world employment dataset. Overall, the work is grounded in a solid mathematical foundation and builds meaningfully on existing causal inference literature, particularly the frameworks of Kennedy (2023) and Kallus (2023).", "weaknesses": "1. The real-world data analysis, while effectively illustrating the interpretability of the proposed estimator, lacks quantitative comparisons to other causal inference methods. The employment dataset experiment focuses on qualitative visualization of treatment heterogeneity but does not benchmark performance against either inversion-based CQC estimators or widely used CATE-based models such as TARNet, DragonNet, or BART. Including such comparisons would provide essential empirical context, clarifying whether modeling full conditional quantile contrasts yields measurable advantages over standard mean-based causal estimators in practical applications.\n\n2. The introduction and abstract clearly present the statistical motivation behind estimating the Conditional Quantile Contrast (CQC) but do not effectively convey its practical significance. They could better highlight how CQC interpretation informs real-world decisions—such as in policy analysis, where understanding which subgroups benefit most or least from interventions is critical. Without a clear link to applied impact, the estimator’s broader relevance to practitioners and policymakers remains underemphasized.\n\n3. There are some grammar errors:\nLine 214: “AppendixA.1” → “Appendix A.1”\nLine 331-332: “Proposition 1b)” → “Proposition 1(b)”\nLine 361-362: “which their estimated equivalents” → “that uses their estimated equivalents”\nLine 398: “effected” → “affected”\nLine 409-410: “perform well” → “performs well”", "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": "Bf796CkeNs", "reviewer": "Reviewer_2ZCv", "summary": "This paper presents a novel approach for estimating the CQC, which removes the procedure of estimating an intermediate estimand followed by inversion, enabling the CQC estimate to be explicitly parametrized with enhanced interpretability. The authors cleverly transform the optimization scheme into a convex optimization problem by introducing\na loss function whose derivative with respect to y1 is the contrasting function. The upper bound of the loss is derived, and an estimator of the gradient is proposed. The authors illustrate that the estimated parameters obtained via gradient descent are doubly robust with respect to the loss, and demonstrate empirical results on simulated and real-world datasets.", "questions": "* Related to the second comment in \\emph{Weaknesses}, does the $\\sqrt{n}$ loss-consistency of $\\hat{\\theta}$ guarantee the consistency of $\\hat{\\theta}$ at sound rates? While the authors suggest that the result can be extended for a limited class of densities that are bounded below (lines 349--353), no guarantees are presented for a general class of probability densities.\n\n Minor typos:\n\n- Line 117: $A = 0 \\Rightarrow A = a$\n- Line 142: provides an example of this by showing an example of this (redundant phrase)\n- Line 201: $\\mathcal{Y} \\times \\mathcal{X} \\to \\mathcal{X} \\Rightarrow \\mathcal{Y} \\times \\mathcal{X} \\to \\mathcal{Y}$\n- Line 234: $g^*(y_0 | x) | x \\Rightarrow g^*(y_0 | x)$\n- Line 246: $g_\\theta(g \\mid x) \\Rightarrow g_\\theta(y_0 \\mid x)$", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The proposed framework enables parametrization of the CQC function,\nproviding a means to enforce structural assumptions on the model and\nto represent the estimation error in terms of the complexity of the CQC\nitself.\n2. The idea of transforming the optimization scheme into a convex optimiza-\ntion problem by introducing a loss function whose derivative with respect\nto y1 is the contrasting function was fascinating.\n3. Empirical results demonstrate improved performance.", "weaknesses": "1. The term ”direct CQC estimator” seems to be somewhat misleading, as the proposed estimator in Section 3.1 is defined with respect to the gradient rather than the estimand of interest, $g^∗$.\n\n2. Also related to the point made in 1, and as the authors acknowledged in the limitations already, the doubly robustness proposed in Theorem 3 holds with respect to the loss function rather than the CQC estimate $g_\\hat{\\theta}$. As the estimand of interest is $g^∗$, it seems imperative to demonstrate the convergence rate of  $g_\\hat{\\theta}$. Also, the derivation of the doubly robust estimator appears fairly standard, given that the loss function is smooth; in fact, it closely mirrors existing results. Personally, I do not find much technical novelty in the manuscript, though I do acknowledge the conceptual value of the proposed framework itself.\n\n3. The proposed method heavily depends on the previous work by Givens et al (2024), limiting its novelty and contribution.", "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": "TNcyGLRle8", "reviewer": "Reviewer_A3DT", "summary": "This paper proposes the first direct, doubly robust (DR) estimation method for the Conditional Quantile Comparator (CQC), an HTE estimand that bridges the gap between CATE and CQTE. The authors develop a novel loss function whose DR gradient can be estimated from data, allowing the CQC to be explicitly modeled and optimized, rather than being solved for indirectly via functional inversion.", "questions": "1. For Weakness 1, is it possible to derive any model evaluation metric, like those proposed in [1, 2], for evaluation? This is not the aim of this paper, but discussing this might bring some new insights for future research.\n\n2. For Weakness 2, the loss $L(\\theta)$ is an expectation over this $Y_0$ distribution. The paper suggests sampling from the control distribution ($Y|A=0$), but it's unclear how this choice affects the estimator's accuracy for $y_0$ values in the control group. \n\n3. For Weakness 2, what is the exact sample algorithm? Is it random or dependent on some information? How the sampling algorithm would affect the final result?\n\nOverall, I think this is a solid paper, and providing more explanations for the above questions might improve the quality.\n\n\n[1] Unveiling the Potential of Robustness in Selecting Conditional Average Treatment Effect Estimators\n[2] Empirical Analysis of Model Selection for Heterogeneous Causal Effect Estimation", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This work provides a practical, direct, and more efficient alternative, making the CQC a much more usable tool.\n\n2. The core method is novel. The idea of framing the CQC estimation problem as an M-estimation task by defining a loss $l$ such that $\\partial_{y_1}l = h$ (with $h=F_1 - F_0$) is very interesting. Deriving the doubly-robust gradient $\\zeta_{dr}$ (Proposition 2) provides a new class of estimators.\n\n3. The method is theoretically solid. The paper provides finite sample bounds that formally demonstrate the estimator's double robustness.\n\n4. The empirical study is thorough and shows that their direct method's error is low and constant, while the indirect method's error (which depends on $F_a$) is high and unstable.", "weaknesses": "1.  Practicality of model selection. The paper notes as a limitation that there is \"no natural definition of test loss\". The method optimizes based on an estimated gradient of the population loss, not a sample-based loss (like MSE). This makes standard validation and hyperparameter tuning very difficult. The paper suggests an approximation via quadrature (Appendix B.2), but this is complex and a significant practical barrier.\n\n2. The algorithm (Algorithm 1) requires sampling test points $Y_0$ to compute the gradient. But there are some unclear points, as shown in the Questions 2&3.", "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": "iAhZUmwHgN", "reviewer": "Reviewer_YMsC", "summary": "This paper provided an estimation framework for the conditional quantile comparator (CQC). The CQC estimand offers another dimension of heterogeneity compared to other HTE estimand (e.g. CATE) into it allows interpretation along the level of the untreated potential outcome. The proposed estimator reframe the Z-estimation problem into a loss minimization problem, and constructs a doubly robust pseudo-outcome for the gradient. The estimator is learned by taking a series of gradient update steps without explicitly evaluating the loss. The authors provided convergence analysis for the proposed estimator and demonstrated improved performance over existing baselines.", "questions": "- $\\tilde{\\theta}$ in Theorem seems to be undefined, is it the optimal solution within the radius B?\n- Is the proposed method sensitive to hyper-parameters (especially since there is clear evaluation to guide when to stop the gradient updates) like the step size?\n- What was the main technical difficulty when proving convergence rate for the CQC it self?\n- What causal quantity does the CQC correspond to? Is it $\\mathbb{E}[Y(1)|X,Y(0)=y]$? What are the identifying assumptions?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Provided theoretical guarantee on the convergence rate, and show that the difference in loss is doubly robust in nuisance estimation error.\n- Provided detailed overview/comparison with prior works. \n- The authors discussed some key limitations in the paper.", "weaknesses": "- It is more sensitive to nuisance estimation error compared to the plug-in/invert approach as the doubly robust rate is proven for the loss. \n- Since the only the gradient is calculated (the loss is never evaluated), it is hard to asses whether taking the gradient steps are sufficient. \n- Would be stronger if the paper also included experiments on hyper-parameters like step size or number of gradient update steps.", "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)