GCF: Generalized Causal Forest for Heterogeneous Treatment Effect Estimation Using Nonparametric Methods
Authors: Shu Wan
Chen Zheng
Zhonggen Sun
Mengfan Xu
Xiaoqing Yang
Jiecheng Guo
Hongtu Zhu
Venue: ICLR 2022 Submitted (2022)
Keywords: Heterogeneous Treatment Effect
Causal Inference
Double/Debiased Machine Learning
Continuous Treatment

Abstract: Heterogeneous treatment effect (HTE) estimation with continuous treatment is essential in multiple disciplines, such as the online marketplace and pharmaceutical industry. The existing machine learning (ML) methods,  like forest-based modeling,  either work only for discrete treatments or make partially linear or parametric assumptions that may suffer from model misspecification. To alleviate these problems, we extend causal forest (CF) with non-parametric dose-response functions (DRFs) that can be estimated locally using kernel-based Double/Debiased ML estimators.  Moreover, we propose a distance-based splitting criterion in the functional space of Partial DRFs to capture the heterogeneity for continuous treatments. We call the proposed algorithm generalized causal forest (GCF) as it generalizes the use case of CF to a much broader setup. We show the effectiveness of GCF compared to SOTA on synthetic data and proprietary real-world data sets.

Under review as a conference paper at ICLR 2022
GCF: GENERALIZED CAUSAL FOREST FOR HETERO- GENEOUS TREATMENT EFFECT ESTIMATION USING NONPARAMETRIC METHODS
Anonymous authors Paper under double-bling review
ABSTRACT
Heterogeneous treatment effect (HTE) estimation with continuous treatment is sential in multiple disciplines, such as the online marketplace and pharmaceutical industry. The existing machine learning (ML) methods, like forest-based model- ing, either work only for discrete treatments or make partially linear or paramet- ric assumptions that may suffer from model misspecification. To alleviate these problems, we extend causal forest (CF) with non-parametric dose-response func- tions (DRFs) that can be estimated locally using kernel-based Double/Debiased ML estimators. Moreover, functional space of Partial DRFs to capture the heterogeneity for continuous treat- ments. We call the proposed algorithm generalized causal forest (GCF) as it gen- eralizes the use case of CF to a much broader setup. We show the effectiveness GCF compared to SOTA synthetic data and proprietary real-world data sets.
propose distance-based splitting criterion in the
INTRODUCTION
Heterogeneous treatment effect (HTE) estimation has been of growing interest for decision-makers in wide spectrum of contexts. It uncovers the effect of interventions at sub-group levels, thereby providing highly tailored suggestions rather than a mne-size-fits-ll policy. When comes to pre- cision medicine, HTE provides the leveraged information that physicians need to precisely treat different patients with proper dosages of drugs depending on their genes, living habits, and EHR history. Recently, with the emerging big data that exhibits astronomical complexities, HTE has been essential for online marketplaces (Syrgkanis et al.) 202 Du et al. 2019: Ye et al. 2018). For large- scale data that presents huge challenge, algorithms that adapt the booming machine learning (ML) techniques to HTE estimation are proposed (Künzel et al. 2019 Zhao et al. 2017 Chernozhukov et al. 2018 Hill 2011). However, these methods largely focus binary discrete treatments and thereby being not widely applicable, given the fact that continuous treatments are prevalent in prac- tice. Examples include targeting customers with continuous incentives, customizing the duration of ads to improve user engagement, and design optimal interest rates to maximize revenue while control for default.
For HTE estimation with continuous treatments, it could be feasible to discretize the treatments and utilize the above approaches, but the limitations are two-phase and bring non-trivial challenges to researchers. First, ordinal treatments like different levels of education cannot be treated as cate- gorical otherwise the order information across treatments will be lost, and secondly, discretizing or bucketing has the limitations of model inaccuracy and cannot recognize the right patterns of the outcome across treatments. To elucidate the complex relationship between treatment and outcome, researchers put forward ideas of dosoonesponse function (DRF).
The estimation of DRF in the presence of high dimensional features can be viewed as a natural extension of regression problems with additional complexities. The success of random forest (RF) for regression problems motivates the development of causal forest (CF) (Athey et al.) 2019) and orthogonal random forest (Oprescu et al. 2019) and the modification of Bayesian Additive Random Forest (Woody et al.] 2020; Hahn et al. 2020). These methods partition the feature space with splitting criterion fitted on treatments that can provide accurate estimations for HTE. But often

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Linear DRF
Nonlinear DRF
Random Forest
Causal Forest
Random Forest
Causal Forest
20-
20-
15-
Dakcame
15
5-
10
15
0
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Treatment
Treatment
Average Entimate
Ground
Figure ADRF estimates random forest and causal forest in both linear and nonlinear DRF specifications. Blue line denotes model average estimations. Confidence bounds of model estimates generated by multiple simulation run are marked light-blue shade. Ground truth of testing data set is represented in red dash-line.
they posit linear or partially linear models DRF which are vulnerable model misspecifications. Figure D is an illustration of model misspecification of random forest and CF.
Doubly Robust estimators (DREs) allows for misspecification of either the treatment or outcome regression. Recent work (Kennedy et al., 2017) combines the estimation of generalized propensity score with regression models of the outcome that can further reduce the bias and use kernel regres- sion for non non-paramann parametric estimation of DRF. However, it may suffer from the curse of dimensionality when the feature space is high-dimensionnl given large-scale instances. To advance, (Colangelo & Lee 2020) provide kernel-based DML estimators that can easily deal with high-dimensional data. Nevertheless, the localization and weighting mechanism of CF not taken into account, which is essential for capturing heterogeneity. It remains unexplored that how to effectively combine ML methods with localized estimations under no model assumptions.
In this paper, propose generalized causal forest (GCF) by extending CF to generalized one with a brand new splitting criterion, and herein the generalization is two-fold. First, we generalize the local linear model of CF to non-parametric one with DRF. Meanwhile, employ the distance metrics in the functional space of Partial DRF as the splitting criterion and use kernel-based DML. estimators the DRF approximation.
When comes to implementations, many open source packages integrate recent research with the trending ML methods that facilitate НTE estimation, such EconML (Oprescu al.), CausalML (Chen et al. 2020) and GRF. EconML, CausalML offer a suite of high-performance ML algorithms like DML, while GRF specially supports CF. However, they are only compatible with single-maching machine systems and may not be efficient when it comes to large-scale instances. Apache Spark (Meng et al.) 2016) is integrated for handling large-scale data with distributed and in-memory computing, while inc inclusively provides APIs of trending MIL algorithms with the built-in MILib. Therefore, as one of our contributions, we build our algorithm on Spark that enables large-scale data processing and easy use of any ML techniques. To summarize, the main contributions are:
We are the first to generalize CF's partially linear model to a non-parametric one with DRF and apply kernel-based DML estimators. The proposed algorithm pictures the non- parametric behaviors of how outcome varies with treatments.
We propose a distance-based splitting criterion with various distance metrics in the func- tional space of Partial DRF that generalizes the differences of CAPE in CF.
Our proposed GCF has been tested on both synthetic and large-scale real-world datasets. It improves on SOTA in terms of multiple evaluation metrics. We implement GCF on Apache Spark MLLib and achieves higher computatinal efficiency by distributed computing.
The rest of the paper is organized as follows. In Section we present related works in the literature. In Section 3, introduce some notations and backgrounds. In Section 4, we introduce the Gener- alized Causal Forest. In Section we examine the empirical performance of GCF by applying it to both synthetic and real-world data sets. We conclude the paper with some discussions in Section 6.
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2
RELATED WORKS
growing amount of literature has been devoted address the problem HTE estimation with continuous treatments. The algorithms in the context of DRF include methods targeting at con- founding bias, kernel-based or ML-based methods for regression bias reduction, and techniques of DRE DML that balance the trade-off between the two biases. DRF combined with IPW (Zhu et al. 2015 Graham et al. 2015) can achieve consistent estimators by weighting the estimation with probability density of treatment. For the regression bias, (Galagate & Schafer 2015) employ para- metric estimations and (Flores et al.] 2007) model DRF as non-parametric functions using kernel regression. To advance, (Kennedy al. 2017) propose doubly robust estimators for DRF by combine the estimation of generalized propensity score with the estimation of outcome using kernel regression. Thus far, these approaches only provide global estimations and cannot handle massive amounts of high-dimensione data.
Recently, the great efficiency of ML methods motivates their generalization to the problem of HTE estimation. Towards that end, estimating DRF with ML-based algorithms is developed and among which tree-based models are great candidate since it partitions the feature space for dimension- ality reduction and maximizes the heterogeneity as well. Causal Forest (CF) proposed in (Athey jet al. (0T9) utilizes a subset of training samples for growing trees by recursively partitioning via splitting criterion. Then HTE estimation is given by weighted average over the outcomes of the maining training samples, known as honesty principle. The final estimator obtained from CF further exhibits a lower degree of bias by tree ensembles. However, their splitting criterion relies linear model assumption and is formulated as the difference between the slopes of linear models. The same limitation applies Orthogonal Random Forest (Oprescu et 2019) and Bayesian Additive Regression Tress (Hill 2011 Woody et al. 2020: Hahn et al. 2020). Nevertheless, the complexity of HTE estimation may not fully captured by linear or general parametric models, which brings the necessity of non- con-aramettii ML models.
Kernel-based DML (Colangelo & Lee 2020) estimates the nuisance functions with cross-fitting, constructs a non-parametric DML estimator by Gaeutax Derivative. This motivates the utilization of DML and non- non-parametric estimation in our work, though only provide global estimators with the limited capacity of localization. A fully non- parameeetrii DML with locally weighted estimations for continuous treatments has not been considered yet.
Building on prior art, we point out that our integration of DRF estimation into CF overcome the challenges aforementioned by using local naramaraet DRF to constructing splitting criterion. We introduce the conceptual partial DRF component of the splitting criterion and employ the distance in the functional space of Partial DRF as a proxy for heterogeneity instead of the difference of slopes. Moreover, the partial DRF can be estimated precisely by the robust kernel-base DML estimators in our splitting criterion.
PREREQUISITES
3.1 NOTATIONS AND ASSUMPTIONS
We formally introduce the notations for HTE estimation with continuous treatments. Following the potential outcome framework in (Neyman 1923 Rubin 1974), let be the continuous treatment, X = { be the px -x- confounding variables, U the outcome-specinn covariates, Z be the treatment-specific covariates that are independent of U, and Y be the outcome of interests. The population Ω (X, U, Z, T, E RPx +pu +1++++11+111- satisfies
Y = (T, where v noises of mean and g: :
+ TT= f(X, +
R R.
{(X,, U,, Z., Y, T I.....72 n. are i.i.d. samples drawn from the population Ω2 In practice, the decomposition of X. U, from observed covariates is difficult. Therefore, without additional spec- ification, we use X to represent observed covariates throughout the paper to simply notations. The potential outcomes under treatment is Y(e)- Recall that Propensity Score (PS) (Rubin) 1974) for discrete treatment is P(T t[X), the probability for unit receiving treatment given the covari- ates X. For continuous treatments. (Rubin 1974 Hirano & Imbens] 2004) introduce Generalized
Propensity Score (GPS) which is the probability density function n (7 -
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The estimand of interests, CATE {(t, X), is formally defined
((t, X) () X
Eo) XX]
To identify {t, X), common assumptions as in (Holland 1986 throughout the paper.
Kennedy et al. 2017) are made
Assumption 1. Consistency: E|Y = ETT t). i.e. the outcome of any sample solely
depends on its treatment;
Assumption 2. Ignorability: The potential outcomes Yery is independent of treatment T given
covariates X.
Assumption 3. Positivity: The GPS x [T = x > Pmin 0. Vt, X, i.e. the density for any
sample receiving any treatment is bounded away from 0.
Under the above assumptions, we have
(tt, X) = (E\Y -
E = -
E| = X
=
= X] - O
where the first equality holds by Assumption | and the second equality holds by Assumption Positivity is indispensable for the conditional expectation in the last line to be well-defined whereas being often too strong. In practice, it can be reduced to the following.
Assumption 4. Weak Positivity: The variance of GPS (T)) = Se a(T = x|
(J, (7 x(X)d))² Tmin > 0, V.
3.2 DOSE-RESPONSS FUNCTION
For continuous treatments, the treatment effect can be characterized by DRF, formally defined as
(11) = E -
Generally speaking, DRF can be either parametric or non-parametric, with parametric ones being applied to CATE estimation as (Robins) 2000) Van der Laan al. 2003). non-parametric ones can better explore the functional space of ADRF out of flexibility and hence being the tool of our proposed algorithm.
The target estimand of DRF heavily depends on the contexts
(Galagate 2016), such as treatment ef-
fect (E]{o) - (),),, partial effect (1)) E|Y (t)) and the elasticity (olog (o((
Olog(() Specifically, conditional DRF (CDRF) p(t, X) E| = t, X characterizes the treatment ef- feet at individual levels and thereby leading to CATE estimator as O(t, X) u(t, X) 110, X), which is the target estimand hereafter.
1 4
3.3 KERNEL REGRESSION AND DOUBLE/DEBIASED MACHINE LEARNING
For non cormmeracctiic modeling, keree regression Fan & Gijbels 2018) works with solid theoretical
foundations. Specifically, given non-parametric model y g(x) + with data (X,, Y2). kernel
regression with linear kernel smoothers gives the estimator ((x)) >x 2i=1 KA (xx X;) Di where
E. K3 (a. X2)
Kz X2) is scaled kernel density with h being the bandwidth for smoothness. Typical choices of kernel include Uniform, Epanechnikov, Biweight, Triweight and Gaussian.
The kernel-based DML estimator proposed in (Colangelo & Lee 2020) performs 2-stage DRF estimation maintaining asymptotically normal. The estimatto for the DRF (u() is proposed as follows. First, we estimate the CDRF u(t, X) with A(t, X) and the GPS T(T = x] with A (t)
Then, with the nuisance functions plugged in, kernel-basce double/debiased estimator gives
Σ p(t, xi) + (T - -t) (Y, -Y, X.)
((t)= =
*(t)X,)
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4 GENERALIZED CAUSAL FOREST
In this section, we formally present the details of our algorithm, GCF. Building CF, GCF relaxes partially linear assumption by adapting can-aarmetric DRF the splitting criterion that can be estimated using kernel-based DML estimator described in the previous section. In what follows, we give to workflow GCF at both the training stage and prediction stage, followed by the elabo- rations on the details of splitting criterion, estimations, and practical tweaks for implementation.
Algorithm 1: Generalized Causal Forest
Input Data set = (X, Tú Y:), 1...., ...,...., n; number of trees B; number of features sampled for growing tree mtry; the minimum sample size on each leaf node miin.odee.size honesty fraction a; tolerance т; positivity threshold Ç;
Trainings begin
Data Oj and data O2 by a (a,1 - a)) split rule applied to O for honesty: Pre-train outcome regression model f and treatment density estimation on sample 12
b 1- 1;
while b B do
Randomly sample feature set of size mtry X from X; for The stopping rule is not satisfied do
Identify each parent node P:
Compute the splitting criterion ((-) (3) over the samples in O with X Grow the tree Tr by splitting at the parent node P according to the A(-); end
Assign samples in O2 leaf nodes based on To and let b <- b b+ 1;
end end
Output Causal forest with B trees by recursive partitioning on X; node assignments for samples in O2;
Predictions begin
CDRF estimation (t, x) by local weighted average over the outcomes of samples in U2
that falls into (x) as Ex= (x,e(((x) 1T,
Yi CATE estimation
[(((z))
(t, (t, مo x) and ((t,x)) E (t(t,
end
We introduce the partial derivative of DRF, PDRF treatment with slightly abuse of notations
(T) (t) == ØT
and (tt, Op(T, X T==
OT
T=
Intuitively, it tells about how pa-) varies with treatments and characterizes the "slope" along t. Different leaf nodes on shall have different slopes as the heterogeneity of covariates X.
Let the number of trees be B and then GCF grows trees Tr, b = 1,..., b.....,,, B, by repeating tree-growing process B times with bootstrapping. At the training stage, with PDRF we construct the tree To by recursive partitioning based on a splitting criterion A (CC1,2)). It is proportional to the discrepancy between the PDRF the left and right child nodes, maximize the heterogeneity. We elaborate on this splitting criterion in the next part.
Our algorithm is implemented on Apache Spark for large-scale data processing and the mechanism of the tree-gre gre-ggrowig process is different from that in CF. To elucidate on that, the data is stored at the master machine and trees are copied to each branch machine. Then data is randomly distributed to branch machines for the computation and collected back to the master machine for the final criterion that is used for updating a tree. The updated tree will be copied to each branch machine again and we repeat the process until the tree stops splitting. This distributed computation and clustered storage leverage the computational power of multiple machines for speeding up the training process. We refer readers the appendix for more information GCF's Spark implementation.
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The splits or the tree-growing process terminate when various pre-specified stopping criteria met, for example, the sample size of child nodes is smaller than the parameter min.node.siz size or the distance D between PDRF is smaller than
т.
For prediction, we also follow the honesty principle proposed in (Athey et al., 2019) by partition- ing the training samples into parts where one is for constructing the tree and the other is for estimating CATE on the leaf node. That is to say that each training data can either be utilized to es- timate CATE contribute growing tree. At the prediction stage, the leaf node of given samples a on each tree To is denoted by (z).. The estimation of CDRF (l, x) on Ls (a(x)) takes a local
(x,e(s(m) 1r Y; and
weighted average on training samples in Lg (x)) which is given by Ei=1
[5x(3]] (l, = jus(l, a) (0, at). The final CATE estimation for X is the average of Ls(x)) over B trees, (tt, x)
1 Li x)).
4.1 SPLITTING CRITERION
Given parent node P and training samples with covariates X., our splitting criterion A(ci, e2) is proposed as
Acc nhe nea D( Xw), a (t, X.)),
nip
h.y $ are the PDRF estimators, respectively. The sample sizes parent node, left child node and right child node np, Re, and Reg, respectively. The ratio To in A(c1, 02) n
where $
is to balance the sample sizes between the two child nodes. Distance metric D measures the distance between &e, (t. X and &., (t. Xw) in the functional space of PDRF, which is to be maximized in the splitting so as optimize the heterogeneity. Some commonly used metrics are I,, I.2 and Too as in (Dttee et al. 2018) D1 So $ Xw) &. (1. Xw))dl, D2 = Se |& (1, X.)) (t,(t, ) Doo mar, Φ., (1, x oo X.)) We next show how to estimate PDRF Pe, and Pez using kernel-based DML estimator (Colangelo & Lee 2020).
4.2 PDRF ESTIMATION
The estimation for DRF is a precondition for the estimation of PDRF In this paper, point out kernel-based DML estimator in (Colangelo & Lee 2020) can be adapted to the DRF estimation in GCF and guides the tree splitting.
Recall that, the DMI. estimation for DRF is given by
+ (T, - t) #([[x,)
A(t)=
Σ iEC
p(t,
(Y; - ((t,
where (t,,, X;) is the pretrained estimator for ptt, X)) X,, (t) the pretrained estimator for GPS at X; and C' denotes the child node.
When adapted to our framework, additionally derive the following estimator for CDRF by inno- vatively combine kernel regression with the DML estimator.
CDRF estimator
The estimator is given by
(T; t) *((x,))
x(t, ji X.)+
(Y; - 1-p(tt X:))
PDRF estimator The PDRF estimator is given by (())
af(t) 5t
and & x)) @p(t..)
dt
In estimating PDRF, the smoothness condition of DRF plays an important role. For smooth DRF, can explicitly derive the closed-form PDRF, For smooth Gaussian kernels, +b() is
ap(() 06
E (T; -
Σ KT; -))
(())
(T. t) K. - [)
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However, for general DRF, the PDRF may be discontinuous ill-defined. For example, non- differentiable kernel functions like uniform, epanechnikov, biweight and triweight, d can only be estimated via numerical approximations of (() Elu(t + 5) jutt]
4.3 PRACTICAL CONSIDERATIONS
In the work of (Kennedy et al. 2017). they claim that the choice of bandwidth h for kernels weighs more than the choice of density functions, since deals with the trade off between bias and variance. Usually, small h avoids a large variance while large h reduces the bias. The empirical ways of choosing an optimal are Cross Validation and Rule-of-thumb (Dehnad 1987)
When utilizing kernel densities for weighting, often suffer from the boundary bias if without normalization. To this end, we normalize estimators with the cumulative density function of range [7min; T Tmnn of treatments. More specifically, the estimators are divided by f/mme (T t)dl.
Regarding the positivity assumption in practice, we use a hyperparameter Ç to regularize for the boundary of (ạ)) that the variance GPS. This is also a guarantee on the weak assumption as stated in Assumption 4 Formally, our proposed splitting criterion will be a regularized one
(2)) = The Ree D( (t, Xx), be (t. Xx)) + Ç (
nip
where स is the estimated GPS. With this additional term, the splitting will tend to maximize the variance GPS and hence taking the variety of treatments across samples into account.
4.4 ASYMPTOTIC PROPERTY
Let Assumptions 1-3 in (Colangelo & Lee 2020) hold and thusfar the convergence of kernel-based DML estimator is naturally guaranteed at a rate of /n as long as the outcome is unbiased or the generalized propensity score is unbiased. That is say that
utilize subsample of order /n to achieve converged kernel DML estimator. Then s satisfies that s/n + and s - oo which is stated Specification in (Athey et al. 2019). Under the above assumptions and suppose one additional condition be that the score function is lipschitz with a bounded support of treatment, then Assumption 1-6 in (Athey et al.) 2019) automatically hold for the score function. Throughout our algorithm, we also follow the honesty principle. Then by Theorem 3 (Athey et al. 2019). our final estimator asymptotically unbiased. This shows the doubly robustness our estimator by combining kernel DML and Causal Forest under slightly stronger assumptions on the score function.
5 EXPERIMENT
In this section, we validate GCF by showing that 1) GCF can provide CATE estimations in a variety of contexts such as simulated data with true labels and real-world data without the ground truth 2) GCF is flexible yet effective compared to SOTA which provides enough evidence for its practical use. More specifically, we present the detailed experimental results of our proposed algorithm and demonstrate its effectiveness compared to the prior art on synthetic data and real-world data sets.
5.1 EVALUATION
In what follows we introduce the commonly used metries for model evaluation on both synthetic data and real-world data sets, named as model errors and Qini score and Qini curve. respectively. For real-world data sets, the model errors are not available since the counterfactual outcomes in the potential outcomes cannot be observed.
Synthetic Data n be the number of treatment groups, 0g and Of the predicted and true treatment effect of in sample at treatment t, 0
We adopt PEHE and RMSE as the measure for model precision. Formally, let
i- 01 Ei=1
1.. Ø! be the reference outcome that is pre-determined.
WEHE,RMSE, ADRF are Σm (10' O'D)
Σn (0 0')2 v(t) = f,(, x x)
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Real-world data sets We refer readers to (Gutierrez & Gérardy 2017) for the formal definition of Qini Score and Qini Curve. In practice, a larger Qini Score indicates that model has better performance in identifying HTE.
5.2 MODEL SETUPS
The baselines for GCF include RF, CF, and Kennedy (Kennedy et al. 2017). 7. Both RF and CF use num.trees equals 500, mán.node.siz code of function cls.ef from R package npcousal available in (ehkennedy) (017). Superl learer library is set to ST.rraner and S..gmm We made small tweaks ST..ranger make share same hyperparaters with RF and CF for comparability.
50. For Kennedy (Kennedy et al. 2017). we use the
GCF We implement the proposed GCF on Apache Spark MLLib (Meng et al. 2016) with kemne-- based DML estimator, denoted by GCF For the implementation of GCF, we fix the distance metric D to be d2 out of simplicity. The number trees, the minimum node size and other hyperparameters set to match those in RF and CF.
RF is forest-ensemble method without controlling confoundedness. CF makes the partially lin- assumption on DRE. Kennedy is a non-parametric doubly robust algorithm but uses normal splitting criterion. The comparisons inclusively validate our algorithm for both non- -on-parmettiic DRF estimation and specially designed splitting criterion.
5.3
SIMULATION
Here show comprehensive simulation study synthetic data. First, elaborate the data generating process. is followed by the details model implementations. Finally we illustrate the numerical results that consistently show the superior performance of our GCF.
5.3. DATA GENERATING PROCESS
Let denote the number of samples and p px + Pu + pz be the dimension of covariates. The covariate matrix is (x (XI, Xn) E 1nxpx Data generating process (DGP) is
Y (T) + 0 X2 x T X 0x + U Bu +
T = 20 ( X 0x Z Bz))
where & is the sigmoid function and V is the pdf of Beta distribution. Here we set DRF to be polynomial(Poly), exponential(Exp). and sinusoidal funetions(Sinus).
The above DGP gives us total of 6 datasets by taking the possible combinations of DRFs (t() and setups of covariate Σ. Noises 6, follow Unii(- (-), Covariates X and coefficients 3 follow N (0, Lpx ). We also allow sparsity in covariates by randomly setting some of the coefficients to 0. Testing data sets are generated with randomly assigned treatments to allow unbiased evaluation.
Random Forest
Causal Forost
Kansedy
GCF
25
20
Outrome
10-
10
15
0
5 10 15
Figure 2: Comparison of ADRF estimates for RF, CF, Kennedy and GCF. DRF follows sinusoidal pattern. Blue line denotes model average estimations. Confidence bounds of model estimates gen- erated by multiple simulation run are marked in light-blue shade. Ground truth of testing data set represented in red dash-line.
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5.3.2 SIMULATION RESULTS
Since some models output the final prediction of (t) and some output the treatment effect ((t), we deliberately compute the pseudo outcome Y(t) (1) ((0) as the label in the purpose of the comparability across basclines. The simulation results pf 3 datasets are summarized in Table and those of the other datasets are in Appendix. Here PEHE and RMSE are averaged and standard deviations of the metrics across simulation runs are also reported. Overall, GCF significantly out- performs baseline methods, which again validates the superiority of the newly proposed algorithm. More specifically, GCF exhibits the smallest biases across multiple DGP setups which we defer details to the supplementary.
Additionally, we compare the ADRF curve given by different models to the ground truth. An exam- ple of ADRF under a DGP specification is shown in Fig. 2
Table 1: Simulation results datasets with different DRFs where number of samples n 1000, = 50, py 5, 5. Standard deviations are in parenthesis over 100 simulations.
Polynomial
Sinusoidal
Exponential
Methods
PEHE
RMSE
PEHE
RMSE
PEHE
RMSE
RF
5.63(0.4)
4.61(0.3)
4.27(0.4)
3.21(0.2)
3,57(0.4)
2.62(0.2)
CF
14.09(0.4)
12.58(0.4)
5.15(0.4)
3.96(0.2)
4.34(0.3)
3.37(0.2)
Kennedy
4.37(0.5)
3.36(0.5)
4.14(0.5)
2.78(0.3)
3.86(0.4)
2.54(0.2)
GCF
4.14(0.3)
2.88(0.2)
4.05(0.4)
2.7(0.3)
3.85(0.4)
2.48(0.2)
5.4 REAL-WORLD DATA SETS
Our method is also implemented on real-world data sets with size of 10,698,884 8884 which sourced from a leading ride-hailing company. A discount that randomly sampled (one of different values {do, d1, d2, d3, d1. ds do d1 < ... ds }) is assigned to all trips in given origin-destination-- time tuple. The effect of discount on the trip complete rate is primary interests. We apply GCF to this data set and evaluate the performance of models by calculating Qini Scores.
Table Qini scores of models under different treatments
Methods dg
d4
d3
d2
dj
Xgboost
0.253
0.171
0.177
0.206
0.177
CF
0.253
0.194
0.202
0.272
0.300
GCF
0.309
0.248
0.305
0.444
0.780
The Qini scores of different models summarized in Table The performance of GCF is superior than the company's current models as GCF has the highest Qini scores across all levels of discounts. We further create an incentive policy based on GCF output and deploy for online purpose. Online A/B testing shows significant lift in KPI for GCF-driven policy.
CONCLUSION
In this paper, we propose a novel forest-based nor-parametric algorithm address the problem of HTE estimation with continuous treatments. Under the situation that a fully non-parametric and localized ML-based algorithm has not been proposed yet, we extend Causal Forest with a DRF- based splitting criterion computed by the distance in the functional space of PDRF. To estimate this DRF, We use the kernel-based DML estimator to guarantee double robustness that mitigates model misspecifications. We implement GCF on Spark to leverage the computational efficiency and test on both synthetic data and real-world world data sets and compare it with the prior art approaches. The numerical results demonstrate that method significantly outperform competing methods.

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Under review as a conference paper at ICLR 2022
REFERENCES
Susan Athey, Julie Tibshirani, Stefan Wager, et al. Generalized random forests. The Annals of Statistics, 47(2):1 148-1178, 2019.
Huigang Chen, Totte Harinen, Jeong- Yoon Lee, Mike Yung, and Zhenyu Zhao. Causalml: Python package for causal machine learning, arXiv preprint wrXiv:2002.11631, 2020.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural pa- rameters, 2018.
Kyle Colangelo and Ying- Ying Lee. Double debiased machine learning nonparametric inference with continuous treatments. arXiv preprint urXiv:2004.030336 2020.
Khosrow Dehnad. Density estimation for statistics and data analysis, 1987.
Holger Dette, Kathrin Möllenhoff, Stanislav Volgushev, and Frank Bretz. Equivalence of regression curves. Journal of the American Statistical Association, 113(52))771-7299) 2018.
Shuyang Du, James Lee, and Farzin Ghaffarizadeh. Improve user retention with causal learning. In Proceedings of Machine Learning Research, volume 104 Proceedings of Machine Learning Research, pp. 34-49, PMLR, 05 Aug 2019. URL UR//htttps////rreeeddnngg ml press/ w104/dul 1a. html
chkennedy, npcausal. httt:s///sstabb...oo//eeeeeeee/p
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Jianqing Fan and Irene Gijbels. Local polynomial modelling and its applications: monographs on statistics and applied probability 66. Routledge, 2018.
Carlos A Flores al. Estimation of rese-response functions and optimal doses with continuous treatment. University of Miami, Department of Economics, November, 2007.
Douglas Galagate. Causal Inference With a Continuous Treatment And Outcome: Alternative Esti-
mators For Parametric Dose- Respone Functions With Applications. PhD thesis, 2016.
Douglas Galagate and Joseph Schafer. Estimating average dose response functions using the package causaldrf, 2015.
Daniel Graham, Emma J McCoy, and David A Stephens. Doubly robust dose-response estimation for continuous treatments via generalized propensity score augmented outcome regression. arXiv 2015.
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Pierre Gutierrez and Jean- vves Gérardy. Causal inference and uplift modelling: A review of the literature. In International Conference on Predictive Applications and APIs, pp. 1-13, 2017.
Richard Hahn, Jared Murray, Carlos M Carvalho, et al. Bayesian regression tree models for causal inference: Regularization, confounding, and heterogeneous effects (with discussion). Bayesian Analysis, 15(3):965-1056 2020.
Jennifer L. Hill. Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1):217-240, March 2011. ISSN 1061-8600. doi: 10.1 198/jegs. 2010.08162. Funding Information: The author gratefully acknowledges financial support for this project from NSF grant 0532400. thank Andrew Gelman, James Robins, Kosuke Imai, and anonymous referees for helpful comments the article and the methods.
Keisuke Hirano and Guido W Imbens. The propensity score with continuous treatments. Applied Bayesian modeling and causal inference from incomplete-date perspectives, 226164:73-84, 2004.
Paul W Holland. Statistics and causal inference. Journal of the American statistical Association, (396):945-96) -60, 1986.
Edward H Kennedy, Zongming Ma. Matthew D McHugh, and Dylan S Small. Nonparametric meth- ods for doubly robust estimation of continuous treatment effects. Journal of the Royal Statistical Society. Series B, Statistical Methodology, 19(4):122, 2017.
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Sören R Künzel, Jasjeet S Sekhon, Peter J Bickel, and Bin Yu. Metaleamnrs for estimating heteroge- neous treatment effects using machine learning. Proceedings of the national academy of sciences, 116(10):4156-116) 2019.
Xiangrui Meng. Joseph Bradley, Burak Yavuz, Evan Sparks, Shivaram Venkataraman, Davies Liu, Jeremy Freeman, DB Tsai, Manish Amde, Sean Owen, et al. Milib: Machine learning in apache spark. The Journal of Machine Learning Research, ((()::135--1241 2016.
Jersey Neyman. Sur les applications de théorie des probabilités aux experiences agricoles: Essai des principes. Roczniki Nauk Rolniczych, 10:1-51, 1923.
Miruna Opreseu, Vasilis Syrgkanis, Keith Battocchi, Maggie Hei, and Greg Lewis. Econml: A machine learning library for estimating heterogeneous treatment effects.
Miruna Oprescu, Vasilis Syrgkanis, and Zhiwei Steven Wu. Orthggonal random forest for causal inference. In International Conference on Machine Learning, pp. 4932-4941. PMLR, 2019.
James M Robins. Marginal structural models versus structural nested models as tools for causal inference. In Statistical models in epidemiology, the environment, and clinical trials, pp. 95-133. Springer, 2000.
Donald B Rubin. Estimating causal effects of treatments randomized and nonrandomized studies. Journal of educational Psychology, 66(5):688, 1974.
Vasilis Syrgkanis, Greg Lewis, Miruna Oprescu, Maggie Hei, Keith Battocchi, Eleanor Dillon, Jing Pan, Yifeng Wu, Paul Lo, Huigang Chen, et al. Causal inference and machine learning in practice with econml and causalml: Industrial use cases at microsoft, tripadvisor, uber. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 4072-4073, 2021.
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Spencer Woody, Carlos M. Carvalho, P. Richard Hahn, and Jared Murray. Estimating heterogeneous effects of continuous exposures using bayesian tree ensembles: revisiting the impact of abortion rates on crime. arXiv: Applications, 2020.
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Yan Zhao, Xiao Fang, and David Simchi-Levi. Uplift modeling with multiple treatments and general response types. In Proceedings of the 2017 SIAM International Conference on Data Mining, pp- 588-596. SIAM, 2017.
Yeying Zhu, Donna Coffiman, and Debashis Ghosh. A boosting algorithm for estimating gen- eralized propensity scores with continuous treatments. Journal of causal inference, 3(1):25 40, 2015.

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Under review as a conference paper at ICLR 2022
7 APPENDIX
The appendix runs as follows. First, we list all the notations used in the appendix. Then we dive into the details of the data generating process for the simulation study and present additional simulation results which further show the effectiveness of our proposed method. Subsequently, we briefly outline GCF's Spark framework, which enables large-scale instances processing and faster model training. Finally, we demonstrate how to apply our method to an online schema and achieve superior performance as another validation.
7.1 NOTATIONS
We restate the notations that consistent with the main paper.
Following the potential outcome framework (eyman 1923 Rubin 1974), we let 7 the con- tinuous treatments, X (X*) be the px-dim confounder, U be the py-dim outcome-specific adjustment variable, Z be the Pz-dim treatient-specific adjustment variable, and Y be the ob- served outcome. The potential outcomes under treatment t is Y(y)- The population Ω = (Σ (x, U, 2,, Y,) E R2x +pu+p2+1 satisfies
= g(, x, U)+ +E T f(x, Z) +
where c, are noises of zero mean,
and 1 +PC x R + R and f PPx + R.
((E,, Y,, 2;), .n) are i..d samples drawn from the population Then the covariate matrix Σ = (E)) (E,.... En) E Rexp where p px + pv + pz. The Generalized Propensity Score is T(T = t X), which is the probability density for unit receiving treatment t given the covariate X.
7.2 DATA GENERATING PROCESS
Recall that the covariate matrix Σ
UI.,U,,..., Xx, Zi......, Zn)
Rex(xX+pl +pU' and treatment T I where denotes the number of observations and p is the dimension of covariates. Data generating process (DGP) works as
Y ((T) 0.2 X 1 + X Bx + U. Bu -E
T = 20 ( 3 Z Bz)) +
where & is the sigmoid function and T is the pdf of Beta distribution with shape parameters set to and 3. Here we set DRF μs as pPlinomial(Poly), exponential(Exp), and sinusoidal functions(Sinus),
p(t) 0.2 (t (t-5) 5)2
5, Polynomial
p(t) log + (t - log(11), Exponential
p(t) sin(x)) (x) xr Sinusoidal
The above DGP gives variety of data sets by taking the combination of DRF su()) and covariate E that are specified by n.p and DRF. Parameter e, v are noises following N 1). Covariates X, U, Z follows N (0, I) and coefficient vector 3 follow [Unif(-1, 1)]. Following this rule, generate 100 rounds. Meanwhile, in test data, we randomly assigned the treatments to make sure unbiased evaluations.
7.3 SIMULATION RESULTS
In addition to the results provided in the main paper, here provide the inclusive results of simulation study, which covers all 6 data sets from the Data Generating process.
The results summarized in Table Compared to CF (Athey et al. 2019) that is the benchmark of our generalization, our GCF outperforms it with significantly smaller errors. With the baseine fixed to be Kennedy (Kennedy et al. 2017), our algorithm almost enjoys smaller error except dataset
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Table Simulation Results Different data sets
Setup: Ik, 1,,1000,00,1000 100, 10
Polynomial
Sinusoidal
Exponential
Methods
PEHE
RMSE
PEHE
RMSE
PEHE
RMSE
RF
8.08(0.4)
7(0.4)
4.89(0.4)
3.74(0.2)
4.28(0.4)
3.3(0.2)
CF
13.58(0.4)
12.04(0.4)
5.13(0.3)
3.96(0.2)
4.27(0.3)
3.25(0.2)
Kennedy
4.36(0.4)
3.27(0.3)
4.16(0.4)
2.75(0.2)
3.9(0.4)
2.56(0.2)
GCF
4.3(0.3)
3.01(0.2)
4.19(0.4)
2.86(0.2)
3.91(0.4)
2.53(0.2)
Setup: 1k,100,50.5,5
Polynomial
Sinusoidal
Exponential
Methods
PEHE
RMSE
PEHE
RMSE
PEHE
RMSE
RF
5.63(0.4)
4.61(0.3)
4.27(0.4)
3.21(0.2)
3.57(0.4)
2.62(0.2)
14.09(0.4)
12.58(0.4)
5.15(0.4)
3.96(0.2)
4.34(0.3)
3.37(0.2)
Kennedy
4.37(0.5)
3.36(0.5)
4.14(0.5)
2.78(0.3)
3.86(0.4)
2.54(0.2)
GCF
4.14(0.3)
2.88(0.2)
4.05(0.4)
2.7(0.3)
3.85(0.4)
2.48(0.2)
where DRF is sin. Overall, these experimental evidence demonstrates that our GCF superior than the algorithms for continuous treatments.
Meanwhile, ADRF curves also inclusively presented here validations of our proposed GCF. Figure 3 are ADRF curves of baseline methods and GCF on different synthetic data sets. Across all the data sets, our proposed GCF performs the best by being the closest the ground truth.
7.4 SPARK IMPLEMENTATION
Apache Spark (Meng al. 2016) has the power of large-scale data processing and provides APIs of any machine learning algorithms. Consequently, we build our proposed algorithm on Spark to do parallel computing and distributed model training that leverage the resources of multiple machines simultaneously.
The workflow of Spark with GCF is shown in Figure which include integrated data and distributed computation. More specifically, with the parallel structure of spark, the tree- tree-growing process runs and the process of GCF is depicted in Figure I0 which significantly different from that in CF (Ahhe et al. 2019) by distributing tasks for computations
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Kennedy
5
10
15
5
10
15
5
10
15
5
10
15
Treatment
Figure 3: ADRF estimates. Blue line denotes model average estimations. Red dashline represents
25
20
10
10
15
0
10
15
15
5
15
Treatment
Ground
Figure 4: ADRF estimates. Blue line denotes model average estimations. Red dashline represents
1.-
0
10
15
0
10
15
5
15
5
10
15
Treatment
Figure 5: ADRF estimates. Blue line denotes model average estimations. Red dashline represents
15
10-
10
0
10
15
25
5
15
Treatment
Ground
Figure 6: ADRF estimates. Blue line denotes model average estimations, Red dashline represents
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Foreat
Kennedy
0-
5
10
15
5
10
15
5
10
15
5
10
15
Treabment
age Truch
Figure 7: ADRF estimates Blue line denotes model average estimations. Red dashline represents
Random Forest
Causal Forest
Kennedy
GCF
10
15
5
Treatment
Aucrage Estimate
Ground Truth
Figure ADRF estimates. Blue line denotes model average estimations. Red dashline represents
Node Stack
14023 Pockes
1977 Pooks
Foetled Bark
Nodm of
Fartale/ foss
Nodel and ei
Sal Aii PARS
Fartbee Tast
Nodel and gar
SAE'S 2PIS FORES
2000
Patiako
Netiier Tank
Me in
Selt alaliraii
Tack3
Suit chatrobte CIAITIONE
Takk
Figure 9; Parallel architecture of GCF Spark Implementation. Data are partitioned the nodes level and distributed across executors to achieve high performance
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Samples
Search for nodes for splitting
Search for nodes for splitting
Honesty Tree Splitting
Create queries for the search
Create queries for the search
Stacks spitting candidates
For all nodes do feature splitting and broadcasting
For all nodes do feature splitting and broadcasting
Initialize the stack by the root nodes
Statistics aggregator for each node
Statistics aggregator for each node
Repeat for all the samples and update features and statistics
Yes
the stack is empy
Repeat for all the samples and update features and statistics
Compute the gain at each feature and splitting candidate
Compute the gain at each feature and splitting candidate
Left and right nodes into the stack
Yes
Splitiin
No
Generate a leaf node
Figure 10: Diagram of GCF algorithm workflow
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7.5 ONLINE DEPLOYMENT AND EXPERIMENTS
The deployment of models in the online marketplace is twofold as illustrated in Figure 11 On the hand, the offline model that is trained historical data provides HTE estimation that decision- makers need to customize incentives at the origin-destination-iins level. On the other hand, the outcome response the customized incentives contributes the data prepared for the subsequent model training. The following is an example of an online experiment with the GCF-driven policy.
We conduct online A/B testings with the model on equal-size non-overlapping groups, i.c. the cohort. The evaluation metric of the online experiment is the finish order (FO) increment. The numerical results of GCF and CF are shown in Figure 12 GCF improves on CF by 15.1%, ben- efiting from the relaxation of the assumptions on the model specification. It shows that the more complicated or more general the treatment is, the more efficient the model is, which again shows the superiority of our novel combination of DRF and CF for continuous treatment.
Training
Predictio
Model Serving
Pricing Polky
Distrbuttdd Caussl Learning and Serving System
Experiment System
Interiace
Onlic
Figure 11: Online Experimental Design
FO Lift
Days
Methods
GCF-lneer
GOT-kemel
Figure 12: GCF vs CF on online experiment
17


Reviews:
[{"review_id": "ixQx4BmA2T", "reviewer": "Reviewer_X6oe", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Weaknesses:\n- The paper does not contain new methodology; it is purely a combination of existing techniques (generalized random forest and doubly robust estimator). \n- The paper does not provide a theoretical analysis of the proposed method.\n- The authors oversell their contribution: Generalizing the Generalized Random Forest (Athey et al. 2019) from a partially lienar model to a full nonparametric one is not very creative. \n- It is unclear to me how the estimator $\\hat{\\mu}(t, X_i)$ is constructed. It is the weighted average of something (?), where the weights w_i(x) are presumably the frequency with which the ith training example falls into the the same leaf as x (see Athey et al. 2019).\n- The simulation study is not reproducible because the authors do not include values for sample size $n$ and dimension $p$. \n- The simulation study and real data analysis are very small scale.\n- Unsupported claims: In the introduction the authors claim that their method allowed for faster hyper-parameter tuning than competing methods. However, they do not provide any evidence to back-up this claim.\n\nAdditional comments:\n- Poor notation: For example, on page 3 $\\Omega$ is defined as a single vector, on page 4 the authors draw samples from $\\Omega$, in Algorithm 1 on page 5, $\\Omega$ denotes the dat set.\n- The exposition is unorganized and contains many abbreviations which are hard to keep track of.\n- The language is unnecessarily pompous and riddled with grammatical mistakes to an extend that makes it difficult to follow the authors ideas.", "summary_of_the_paper": "The authors extend the generalized random forest by Athey et al. (2019) from partially linear models to nonparametric ones and combine it with a doubly robust estimation step. The paper contains a small scale simulation study and a small real data analysis.", "summary_of_the_review": "The paper does not contain new methodology or theoretical result. The simulation study and real data analysis are weak.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "X3TdR2prjht", "reviewer": "Reviewer_akBL", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strength:\n\n1. The paper is concerned with an interesting and important problem in causal inference, which is heterogeneous treatment effect estimation with continuous treatment. Modeling the nonlinearity of the dose-response function is also an important and challenging aspect of this problem.\n\n2. The paper presents solid empirical results on real-world data. The paper also presents a real-world a/b test to demonstrate the effectiveness of the proposed method in practice.\n\n3. The paper also presents a scalable implementation of the proposed method, which can be used to handle large-scale (production-scale) data.\n\nWeakness:\n1. Presentation is not clear. Important concepts such as score function, splitting criteria, etc are not specified. The presentation of the proposed method is also difficult to follow. For example, it is not clear to me what are the components of CF, and how GCF is derived and differs from CF in section 4.\n\n2. The key component of the proposed method is to model the DRF in a non-parametric, doubly robust fashion. It is not clear to me whether such a contribution is significant enough to motivate acceptance. This is also because the proposed method performs comparably with Kenndy on synthetic data.  So, it is not clear what might be the additional value brought compared to Kenndy. The author may consider providing further explanation and justification on the complexity and technical difficulty of replacing a linear DRF with the proposed one.\n\n3. There is a lack of theoretical justification of the proposed method. However, the empirical results on synthetic data and real-world experiments help to assuage this concern.", "summary_of_the_paper": "The paper proposes a generalized version of the causal forest for heterogeneous treatment effect estimation for continuous treatment by non-parametric modeling of the dose-response function. To provide non-parametric modeling, the proposed method makes use of kernel-based double/debiased estimators. Experiments of the proposed methods are run on assorted synthetic datasets as well as real-world datasets to demonstrate the effectiveness of the proposed method compared to competing methods.", "summary_of_the_review": "Correctness: I did not identify major issues in terms of correctness.\nNovelty and significance: I have concerns about the significance of the proposed method in terms of modeling the RDF through non-parametric approaches. I appreciate the authors running real-world experiments with a/b test to demonstrate the performance of the proposed method.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "91W47_xtt-u", "reviewer": "Reviewer_kG5h", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I like the idea of generalizing the CF because it's impactful by allowing users to leverage any flexible modern machine learning tools. However, I think the paper is hardly understandable to most readers, and the theories are not deep enough. \n\nHere are some details. \n1. \"However, these methods largely focus on binary ...\"  may mislead, because this statement is only for a few recent estimators developed by Chernozhukov et al., Zhao et al., and Kunzel et al. For example, a BART estimator developed by Hill (2011) can deal with continuous treatments. I suggest conducting a more extensive literature review to provide more clear motivation. \n2. What is the meaning of \"local splitting\" in the 3rd paragraph in Section 1? Please don't assume that the authors are familiar with the details of the causal forest algorithm. \n3. This paper doesn't answer or motivate why we have to use the proposed GCF instead of the Kernel DML (Colangelo and Lee, 2020) or the doubly-robust estimator provided by Kennedy et al. (2017). It would be great if authors can provide more extensive literature review on the HTE estimators for the continuous treatment case. \n4. In Section 2, what is the meaning of the \"global\" estimator? \n5. I think the most important prerequisite to understanding the paper is the notion of the causal tree and causal forest, developed by Athey et al. Please add this in Section 3. Without it, it's indeed impossible to understand the paper. For example, what is the meaning of 'honesty' in Algorithm 1? \n6. It's not clear what the contribution of the paper is. Even if I wrote the contribution as \"replacing the partial linear model to the kernel DML,\" I only guessed so because it's not explicitly written. \n7. I expected that the GCF would achieve the doubly robustness property, following the properties of the kernel DML. Please consider adding the formal convergence analysis exhibiting the doubly robustness property. Showing the empirical performance is not sufficient to promote the proposed estimator. \n8. In the experiment, why don't you compare with the Kernel DML (Colangelo and Lee, 2020)? \n9. There are typos -- \"Table ??\" on page 9, and \"First, We\" on page 2. ", "summary_of_the_paper": "This paper provides a generalized causal forest (CF) by replacing the partial linear model in the CF as a kernel DML estimator. Empirical evidences show that the proposed generalized CF (GCF) outperforms the existing estimators. ", "summary_of_the_review": "- This paper implicitly assumes that readers are familiar with the details of the causal forest algorithm. Without this prerequisite, it's impossible to understand the paper. \n\n- This paper doesn't motivate to use the GCF over existing estimators (Kernel DML, or the one of Kennedy et al. (2017)). \n\n- No theoretical analysis on the convergence or bias analysis for the GCF.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper introduces some interesting ideas on how use causal random forests for conditional average treatment effects (CATE), with respect to some baseline treatment level ("0"), when the treatment variable is continuous. Figure 1 summarises the scope of the paper neatly. Scalability issues are also considered.

I think this *is* a paper "nearly there" in terms of a impactful contribution. The main issues are some presentation kinks and extra steps in the theory. I think the very low scores from the reviewers are not quite representative of the overall quality (I would be more generous). However, I'm afraid I'm also inclined towards a reject. The paper neglects some other developments on ML for CATE with continuous treatment e.g. Bica et al.'s "Estimating the Effects of Continuous-valued Interventions using Generative Adversarial Networks" (NeurIPS 2020) and the references within. A focus on the theory would help to differentiate it, but I'm not that confident that the results are currently mature enough to claim them.

Although I suggest a rejection, let me make clear I strongly encourage the authors to further pursue their ideas. You are doing good work, and the next iteration might nail it. As you found out in the discussion, emphasise the continuous aspect of it. I'd also emphasise the fact that you have a clear setup of the problem in terms of the contrast wrt to a baseline treatment effect instead of some generic contrast function. People in ML tend to be oblivious to such a setup, but I'm not convinced you are properly exploiting it.
Decision: Reject