Original: GCF: G ENERALIZED C AUSAL F OREST FOR H ETEROGENEOUS T REATMENT E FFECT E STIMATION USING N ONPARAMETRIC M ETHODS
Revised: GCF: G ENERALIZED C AUSAL F OREST FOR H ETEROGENEOUS T REATMENT E FFECT E STIMATION USING N ONPARAMETRIC M ETHODS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Heterogeneous treatment effect (HTE) estimation with continuous treatment is essential in multiple disciplines, such as the online marketplace and pharmaceutical industry.
Revised: Heterogeneous treatment effect (HTE) estimation with continuous treatment is essential in multiple disciplines, such as the online marketplace and pharmaceutical industry.

Original: 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.
Revised: 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.

Original: 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
Revised: 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

Original: ML estimators.
Revised: ML estimators.

Original: Moreover, we propose a distance-based splitting criterion in the functional space of Partial DRFs to capture the heterogeneity for continuous treatments.
Revised: Moreover, we propose a distance-based splitting criterion in the functional space of Partial DRFs to capture the heterogeneity for continuous treatments.

Original: We call the proposed algorithm generalized causal forest (GCF) as it generalizes the use case of CF to a much broader setup.
Revised: We call the proposed algorithm generalized causal forest (GCF) as it generalizes the use case of CF to a much broader setup.

Original: We show the effectiveness of
Revised: We show the effectiveness of

Original: GCF compared to SOTA on synthetic data and proprietary real-world data sets.
Revised: GCF compared to SOTA on synthetic data and proprietary real-world data sets.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Heterogeneous treatment effect (HTE) estimation has been of growing interest for decision-makers in a wide spectrum of contexts.
Revised: Heterogeneous treatment effect (HTE) estimation has been of growing interest for decision-makers in a wide spectrum of contexts.

Original: Itnot only uncovers the effect of interventions at sub-group levels,but also takes care of the discrepancy among groups, thereby providing highly tailored suggestions rather than a one-size-ﬁts-all policy.
Revised: It uncovers the effect of interventions at sub-group levels, thereby providing highly tailored suggestions rather than a one-size-ﬁts-all policy.

Original: When it comes to precision 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.
Revised: When it comes to precision 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.

Original: Recently, with the emerging big data that exhibits astronomical complexities, HTE has been essential for online marketplaces (Syrgkanis et al., 2021; Du et al., 2019; Ye et al., 2018).
Revised: Recently, with the emerging big data that exhibits astronomical complexities, HTE has been essential for online marketplaces (Syrgkanis et al., 2021; Du et al., 2019; Ye et al., 2018).

Original: For large-scale data that presents a huge challenge, algorithms that adapt the booming machine learning (ML) techniques to HTE estimation are proposed (K ¨ unzel et al., 2019; Zhao et al., 2017; Chernozhukov et al., 2018).
Revised: For largescale data that presents a huge challenge, algorithms that adapt the booming machine learning (ML) techniques to HTE estimation are proposed (K ¨ unzel et al., 2019; Zhao et al., 2017; Chernozhukov et al., 2018; Hill, 2011).

Original: However, these methods largely focus on binary or discrete treatments and thereby being not widely applicable, given the fact that continuous treatments are prevalent in practice.
Revised: However, these methods largely focus on binary or discrete treatments and thereby being not widely applicable, given the fact that continuous treatments are prevalent in practice.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: First, ordinal treatments like different levels of education cannot be treated as categorical 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.
Revised: First, ordinal treatments like different levels of education cannot be treated as categorical 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.

Original: To elucidate the complex relationship between treatment and outcome, researchers put forward ideas of dose-response function (DRF).
Revised: To elucidate the complex relationship between treatment and outcome, researchers put forward ideas of dose-response function (DRF).

Original: The estimation of DRF in the presence of high dimensional features can be viewed as a natural extension of regression problems with additional complexities.
Revised: The estimation of DRF in the presence of high dimensional features can be viewed as a natural extension of regression problems with additional complexities.

Original: The success of random forest (RF) for regression problems motivates the development of causal forest (CF) (Athey et al., 2019) and orthogonal random forest (ORF)
Revised: 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

Original: (Oprescu et al., 2019).
Revised: Forest (Woody et al., 2020; Hahn et al., 2020).

Original: These methods partition the feature space with alocal splitting criterion ﬁtted on treatments that can provide accurate estimations for HTE.
Revised: These methods partition the feature space with a splitting criterion ﬁtted on treatments that can provide accurate estimations for HTE.

Original: But often they posit linear or partially linear models of DRF which are vulnerable to model misspeciﬁcations.
Revised: But often they posit linear or partially linear models of DRF which are vulnerable to model misspeciﬁcations.

Original: Figure 1 is an illustration of model misspeciﬁcation of random forest and CF.
Revised: Figure 1 is an illustration of model misspeciﬁcation of random forest and CF.

Original: Doubly Robust estimators (DREs) allows for misspeciﬁcation of either the treatment or outcome regression.
Revised: Doubly Robust estimators (DREs) allows for misspeciﬁcation of either the treatment or outcome regression.

Original: 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 regression for non-parametric estimation of DRF.
Revised: 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 regression for non-parametric estimation of DRF.

Original: However, it may suffer from the curse of dimensionality when the feature space is high-dimensional given large-scale instances.
Revised: However, it may suffer from the curse of dimensionality when the feature space is high-dimensional given large-scale instances.

Original: To advance, (Colangelo &
Revised: To advance, (Colangelo &

Original: Lee, 2020) provide kernel-based DML estimators that can easily deal with high-dimensional data.
Revised: Lee, 2020) provide kernel-based DML estimators that can easily deal with high-dimensional data.

Original: Nevertheless, the localization and weighting mechanism of CF is not taken into account, which is essential for capturing heterogeneity.
Revised: Nevertheless, the localization and weighting mechanism of CF is not taken into account, which is essential for capturing heterogeneity.

Original: It remains unexplored that how to effectively combine ML methods with localized estimations under no model assumptions.
Revised: It remains unexplored that how to effectively combine ML methods with localized estimations under no model assumptions.

Original: In this paper, we propose generalized causal forest (GCF) by extending CF to a generalized one with a brand new splitting criterion, and herein the generalization is two-fold.
Revised: In this paper, we propose generalized causal forest (GCF) by extending CF to a generalized one with a brand new splitting criterion, and herein the generalization is two-fold.

Original: First, We generalize the local linear model of CF to a non-parametric one with DRF.
Revised: First, we generalize the local linear model of CF to a non-parametric one with DRF.

Original: Meanwhile, we employ the distance metrics in the functional space of Partial DRF as the splitting criterion and use kernel-based DML estimators as the DRF approximation.
Revised: Meanwhile, we employ the distance metrics in the functional space of Partial DRF as the splitting criterion and use kernel-based DML estimators as the DRF approximation.

Original: When it comes to implementations, many open source packages integrate recent research with the trending ML methods that facilitate HTE estimation, such as EconML (Oprescu et al.),
Revised: When it comes to implementations, many open source packages integrate recent research with the trending ML methods that facilitate HTE estimation, such as EconML (Oprescu et al.),

Original: CausalML (Chen et al., 2020) and GRF.
Revised: CausalML (Chen et al., 2020) and GRF.

Original: EconML, CausalML offer a suite of high-performance ML algorithms like DML, while GRF specially supports CF.
Revised: EconML, CausalML offer a suite of high-performance ML algorithms like DML, while GRF specially supports CF.

Original: However, they are only compatible with single-machine systems and may not be efﬁcient when it comes to large-scale instances.
Revised: However, they are only compatible with single-machine systems and may not be efﬁcient when it comes to large-scale instances.

Original: Apache Spark (Meng et al., 2016) is integrated for handling large-scale data with distributed and in-memory computing, while inclusively provides APIs of trending ML algorithms with the built-in Mlib.
Revised: Apache Spark (Meng et al., 2016) is integrated for handling large-scale data with distributed and in-memory computing, while inclusively provides APIs of trending ML algorithms with the built-in MLlib.

Original: Therefore, as one of our contributions, we build our algorithm on Spark that enables largescale data processing and easy use of any ML techniques.
Revised: 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.

Original: To summarize, the main contributionsof this paper are:
Revised: To summarize, the main contributions are:

Original: • We are the ﬁrst to generalize CF’s partially linear DRF speciﬁcation with a non-parametric one  and applythe local kernel-based DML estimator to estimate the DRF.
Revised: • We are the ﬁrst to generalize CF’s partially linear model to a non-parametric one with DRF and apply kernel-based DML estimators.

Original:  • We propose a distance-based splitting criterion with various distance metrics in the functional space of Partial DRFs to capture the heterogeneity for continuous treatments. This generalizes the differences of CAPEscalar in CF.
Revised: The proposed algorithm pictures the nonparametric behaviors of how outcome varies with treatments. • We propose a distance-based splitting criterion with various distance metrics in the functional space of Partial DRF that generalizes the differences of CAPE in CF.

Original: • Our proposed GCF has been tested across multiple scenarios including syntheticdata and  real-world data sets.
Revised: • Our proposed GCF has been tested on both synthetic and large-scale real-world datasets.

Original: The signiﬁcant improvement on SOTA demonstrates its superior performance and numerical efﬁciency.
Revised: It improves on SOTA in terms of multiple evaluation metrics.

Original: • For implementation, our GCFis built on Apache Spark MLib. Compared to the existing packages, we achieve much higher performance and faster hyper-parameter tuning bythe distributedand in-memory computing.
Revised: We implement GCF on Apache Spark MLlib and achieves higher computatinal efﬁciency by distributed computing.

Original: The rest of the paper is organized as follows.
Revised: The rest of the paper is organized as follows.

Original: In Section 2 , we present related works in the literature.
Revised: In Section 2 , we present related works in the literature.

Original: In Section 3 , we introduce some notations and backgrounds.
Revised: In Section 3 , we introduce some notations and backgrounds.

Original: In Section 4 , we introduce the Generalized Causal Forest.
Revised: In Section 4 , we introduce the Generalized Causal Forest.

Original: In Section 5 , we examine the empirical performance of GCF by applying it to both synthetic and real-world data sets.
Revised: In Section 5 , we examine the empirical performance of GCF by applying it to both synthetic and real-world data sets.

Original: We conclude the paper with some discussions in Section 6 .
Revised: We conclude the paper with some discussions in Section 6 .

Original:  R ELATED WORKS
Revised:  R ELATED WORKS

Original: A growing amount of literature has been devoted to address the problem of HTE estimation with continuous treatments.
Revised: A growing amount of literature has been devoted to address the problem of HTE estimation with continuous treatments.

Original: The algorithms in the context of DRF include methods targeting at confounding bias, kernel-based or ML-based methods for regression bias reduction, and techniques of
Revised: The algorithms in the context of DRF include methods targeting at confounding bias, kernel-based or ML-based methods for regression bias reduction, and techniques of

Original: DRE or DML that balance the trade-off between the two biases.
Revised: DRE or DML that balance the trade-off between the two biases.

Original: 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.
Revised: 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.

Original: For the regression bias, (Galagate & Schafer, 2015) employ parametric estimations and (Flores et al., 2007) model DRF as non-parametric functions using kernel regression.
Revised: For the regression bias, (Galagate & Schafer, 2015) employ parametric estimations and (Flores et al., 2007) model DRF as non-parametric functions using kernel regression.

Original: To advance, (Kennedy et al., 2017) propose a doubly robust estimators for DRF by combine the estimation of generalized propensity score with the estimation of outcome using kernel regression.
Revised: To advance, (Kennedy et al., 2017) propose a doubly robust estimators for DRF by combine the estimation of generalized propensity score with the estimation of outcome using kernel regression.

Original: Thus far, these approaches only provide global estimations and cannot handle massive amounts of high-dimensional data.
Revised: Thus far, these approaches only provide global estimations and cannot handle massive amounts of high-dimensional data.

Original: Recently, the great efﬁcacy of MLand DML methods motivates the generalization to thecentral problem of HTE estimation.
Revised: Recently, the great efﬁciency of ML methods motivates their generalization to the problem of HTE estimation.

Original: Towards that end, estimating DRF with ML-based algorithms is developed and among which tree-based modeling is a great candidate since it partitions the feature space locally which serves as heterogeneity as well.
Revised: Towards that end, estimating DRF with ML-based algorithms is developed and among which tree-based models are a great candidate since it partitions the feature space for dimensionality reduction and maximizes the heterogeneity as well.

Original: Causal Forest (CF) proposed in (Athey et al., 2019) address the concern by recursive partitioning via alocal splitting criterion that is  the difference of scalar estimators, which exhibits a lower degree of bias by local weighting and ensembles.
Revised: Causal Forest (CF) proposed in (Athey et al., 2019) utilizes a subset of training samples for growing trees by recursively partitioning via a splitting criterion. Then HTE estimation is given by weighted average over the outcomes of the remaining training samples, known as honesty principle. The ﬁnal estimator obtained from CF further exhibits a lower degree of bias by tree ensembles.

Original: However, their splitting criterion relies on a linear  assumptionabout score functions andhence the heterogeneity is  the difference of the slopes of linear models.
Revised: However, their splitting criterion relies on a linear model assumption and is formulated as the difference between the slopes of linear models.

Original: Similarly, Orthogonal Random Forest(ORF) (Oprescu et al., 2019) studies the modeling of general non-parametric score functions under the local orthogonality condition.
Revised: The same limitation applies to Orthogonal Random Forest (Oprescu et al., 2019) and Bayesian Additive

Original: But it suffers from model misspeciﬁcations when applied to HTE estimation where again the score function is assumed to be linear.
Revised: Regression Tress (Hill, 2011; Woody et al., 2020; Hahn et al., 2020).

Original: The complexity of HTE estimation may not be fully captured by  parametric models, which brings the necessity of non-parametric
Revised: Nevertheless, the complexity of HTE estimation may not be fully captured by linear or general parametric models, which brings the necessity of non

Original:  ML models.
Revised: -parametric ML models

Original: Kernel-based DML (Colangelo & Lee, 2020) estimates the nuisance functions with cross-ﬁtting, constructs a non-parametric DML estimator by Gaeutax Derivative.
Revised: Kernel-based DML (Colangelo & Lee, 2020) estimates the nuisance functions with cross-ﬁtting, constructs a non-parametric DML estimator by Gaeutax Derivative.

Original: This motivates the utilization of DML and non-parametric estimation in our work, though it only provide global estimators with the limited capacity of localization.
Revised: This motivates the utilization of DML and non-parametric estimation in our work, though it only provide global estimators with the limited capacity of localization.

Original: A fully non-parametric DML with locally weighted estimations for continuous treatments has not been considered yet.
Revised: A fully non-parametric DML with locally weighted estimations for continuous treatments has not been considered yet.

Original: Building on prior art, we point out that our integration of DRF estimation into CF overcome the challenges aforementioned by using local non-parametric DRF to constructing splitting criterion.
Revised: Building on prior art, we point out that our integration of DRF estimation into CF overcome the challenges aforementioned by using local non-parametric DRF to constructing splitting criterion.

Original: We introduce the conceptual partial DRF as a 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.
Revised: We introduce the conceptual partial DRF as a 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.

Original: Moreover, the partial DRF can be estimated precisely by the robust kernel-based DML estimators in our splitting criterion.
Revised: Moreover, the partial DRF can be estimated precisely by the robust kernel-based DML estimators in our splitting criterion.

Original:  P REREQUISITES 3. N OTATIONS AND A SSUMPTIONS
Revised:  P REREQUISITES 3. N OTATIONS AND A SSUMPTIONS

Original: We formally introduce the notations for HTE estimation with continuous treatments.
Revised: We formally introduce the notations for HTE estimation with continuous treatments.

Original: Following the potential outcome framework in (Neyman, 1923; Rubin, 1974), we let T be the continuous treatment,
Revised: Following the potential outcome framework in (Neyman, 1923; Rubin, 1974), we let T be the continuous treatment,

Original: X =
Revised: X =

Original: ( X j ) p X j =1 be the p-dim confounding variables, U be the outcome-speciﬁc covariates, Z be the treatment-speciﬁc covariates that are independent of U , and Y be the outcome of interests.
Revised: ( X j ) p X j =1 be the p X -dim confounding variables, U be the outcome-speciﬁc covariates, Z be the treatment-speciﬁc covariates that are independent of U , and Y be the outcome of interests.

Original: The population Ω =
Revised: The population Ω =

Original: ( X , U , Z , Y, T ) ∈
Revised: ( X , U , Z , Y, T ) ∈

Original: R p X + p U
Revised: R p X + p U

Original: + p Z +1+1 satisﬁes
Revised: + p Z +1+1 satisﬁes

Original: Y = g ( T, X , U ) + (cid:15) ; T = f ( X , Z ) + ν where (cid:15) ν are noises of zero mean and  R → R and f : 
Revised: [Equation] [Equation] ν where (cid:15), ν are noises of zero mean and g : R p × R → R and f : R p

Original: → R .
Revised: → R .

Original: { ( X i , U i , Z i , Y i , T i ) , i = 1 , . . . , n } are i.i.d.
Revised: { ( X i , U i , Z i , Y i , T i ) , i = 1 , . . . , n } are i.i.d.

Original: samples drawn from  Ω .
Revised: samples drawn from the population Ω .

Original: In practice,  decomposition of X , U , Z from observed covariates is difﬁcult.
Revised: In practice, the decomposition of X , U , Z from observed covariates is difﬁcult.

Original: Therefore, without additional speciﬁcation, we use X to represent observed covariates throughout the paper to simply notations.
Revised: Therefore, without additional speciﬁcation, we use X to represent observed covariates throughout the paper to simply notations.

Original: The potential outcomes under treatment t is Y ( t ) .
Revised: The potential outcomes under treatment t is Y ( t ) .

Original: Recall that Propensity Score (PS) (Rubin, 1974) for a discrete treatment is P ( T = t | X ) , the probability for a unit receiving treatment t given the covariates X .
Revised: Recall that Propensity Score (PS) (Rubin, 1974) for a discrete treatment is P ( T = t | X ) , the probability for a unit receiving treatment t given the covariates X .

Original: For continuous treatments, (Rubin, 1974; Hirano & Imbens, 2004) introduce GeneralizedPropensity
Revised: For continuous treatments, (Rubin, 1974; Hirano & Imbens, 2004) introduce Generalized

Original:  Score (GPS) which is the probability density function π ( T = t | X ) .
Revised: Propensity Score (GPS) which is the probability density function π ( T = t | X ) .

Original: The estimand of interests, CATE θ ( t, X ) , is formally deﬁned as
Revised: The estimand of interests, CATE θ ( t, X ) , is formally deﬁned as

Original: [Equation]
Revised: [Equation]

Original: To identify θ ( t, X ) , common assumptions as in (Holland, 1986; Kennedy et al., 2017) are made throughout the paper.
Revised: To identify θ ( t, X ) , common assumptions as in (Holland, 1986; Kennedy et al., 2017) are made throughout the paper.

Original: Assumption 1.
Revised: Assumption 1.

Original: Consistency: E [ Y | T = t ] =
Revised: Consistency: E [ Y | T = t ] =

Original: E [ Y ( t ) | T = t ] , i.e. the outcome of any sample solely depends on its treatment;
Revised: E [ Y ( t ) | T = t ] , i.e. the outcome of any sample solely depends on its treatment;

Original: Assumption 2.
Revised: Assumption 2.

Original: Ignorability:
Revised: Ignorability:

Original: The potential outcomes Y ( T ) is independent of treatment T given covariates X .
Revised: The potential outcomes Y ( T ) is independent of treatment T given covariates X .

Original: Assumption 3.
Revised: Assumption 3.

Original: Positivity: The GPS π ( T = t | X ) > p min > 0 , ∀ t, X , i.e. the density for any sample receiving any treatment is bounded away from 0.
Revised: Positivity: The GPS π ( T = t | X ) > p min > 0 , ∀ t, X , i.e. the density for any sample receiving any treatment is bounded away from 0.

Original: Under the above assumptions, we have
Revised: Under the above assumptions, we have

Original: [Equation]
Revised: [Equation]

Original: where the ﬁrst equality holds by Assumption 1 and the second equality holds by Assumption 2.
Revised: where the ﬁrst equality holds by Assumption 1 and the second equality holds by Assumption 2.

Original: Positivity is indispensable for the conditional expectation in the last line to be well-deﬁned whereas being often too strong.
Revised: Positivity is indispensable for the conditional expectation in the last line to be well-deﬁned whereas being often too strong.

Original: In practice, it can be reduced to the following.
Revised: In practice, it can be reduced to the following.

Original: Assumption 4.
Revised: Assumption 4.

Original: Weak Positivity: The variance of GPS σ ( π ) =
Revised: Weak Positivity: The variance of GPS σ ( π ) =

Original: (cid:82) t t 2 · π ( T = t | X ) dt − (cid:0)(cid:82)
Revised: (cid:82) t t 2 · π ( T = t | X ) dt − (cid:0)(cid:82)

Original: t t · π ( T = t | X ) dt (cid:1) 2 > σ min > 0 , ∀ t, X .
Revised: t t · π ( T = t | X ) dt (cid:1) 2 > σ min > 0 , ∀ t, X .

Original: 3. D OSE -R ESPONSE F UNCTION
Revised: 3. D OSE -R ESPONSE F UNCTION

Original: For continuous treatments, the treatment effect can be characterized by DRF of T , which is formally deﬁned as
Revised: For continuous treatments, the treatment effect can be characterized by DRF, formally deﬁned as

Original: [Equation]
Revised: [Equation]

Original: Generally speaking, DRF can be either parametric or non-parametric, with parametric ones being applied to CATE estimation as in (Robins, 2000; Van der Laan et al., 2003).
Revised: Generally speaking, DRF can be either parametric or non-parametric, with parametric ones being applied to CATE estimation as in (Robins, 2000; Van der Laan et al., 2003).

Original: Non-parametric ones can better explore the functional space of ADRF out of ﬂexibility and hence being the tool of our proposed algorithm.
Revised: Non-parametric ones can better explore the functional space of ADRF out of ﬂexibility and hence being the tool of our proposed algorithm.

Original: The target estimand of DRF heavily depends on the contexts (Galagate, 2016), such as treatment effect (cid:0) E
Revised: The target estimand of DRF heavily depends on the contexts (Galagate, 2016), such as treatment effect (cid:0) E

Original: [ Y ( t ) ]
Revised: [ Y ( t ) ]

Original: − E
Revised: − E

Original: [ Y (0) ] (cid:1) , partial effect(cid:18)
Revised: [ Y (0) ] (cid:1) , partial effect

Original: [Equation]
Revised: [Equation]

Original: t
Revised: and the elasticity (cid:18) ∂log [Equation] E [ Y

Original: [Equation]
Revised: [Equation]

Original: (cid:19) and the elasticity (cid:18) ∂log ( E [ Y ( t ) ]) ∂log ( t ) (cid:19) .
Revised: t ) [Equation] ∂log ( t ) (cid:19) [Equation]

Original: Speciﬁcally, conditional DRF (CDRF) µ ( t, X ) =
Revised: Speciﬁcally, conditional DRF (CDRF) µ ( t, X ) =

Original: E [ Y | T
Revised: E [ Y | T

Original: = t, X ] characterizes the treatment effect at individual levels and thereby leading to a CATE estimator as θ ( t, X ) = µ ( t, X ) − µ (0 , X ) , which is the target estimand hereafter.
Revised: = t, X ] characterizes the treatment effect at individual levels and thereby leading to a CATE estimator as θ ( t, X ) = µ ( t, X ) − µ (0 , X ) , which is the target estimand hereafter.

Original: 3. K ERNEL R EGRESSION AND D OUBLE /D EBIASED M ACHINE L EARNINGFor
Revised: 3. K ERNEL R EGRESSION AND D OUBLE /D EBIASED M ACHINE L EARNING

Original:  non-parametric modeling, kernel regression (Fan & Gijbels, 2018) works with solid theoretical foundations.
Revised: For non-parametric modeling, kernel regression (Fan & Gijbels, 2018) works with solid theoretical foundations.

Original: Speciﬁcally, given a non-parametric model y = g ( x ) + (cid:15) with data ( X i , Y i ) , kernel regression with linear kernel smoothers gives the estimator [Equation] g [Equation]
Revised: Speciﬁcally, given a non-parametric model y = g ( x ) + (cid:15) with data ( X i , Y i ) , kernel regression with linear kernel smoothers gives the estimator ˆ g ( x ) =

Original: x

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: K h ( x , X i ) is a scaled kernel density with h being the bandwidth for smoothness.
Revised: K h ( x , X i ) is a scaled kernel density with h being the bandwidth for smoothness.

Original: Typical choices of kernel include Uniform, Epanechnikov, Biweight, Triweight and Gaussian.
Revised: Typical choices of kernel include Uniform, Epanechnikov, Biweight, Triweight and Gaussian.

Original: The kernel-based DML estimator proposed in (Colangelo & Lee, 2020) performs a 2-stage DRF estimation maintaining asymptotically normal.
Revised: The kernel-based DML estimator proposed in (Colangelo & Lee, 2020) performs a 2-stage DRF estimation maintaining asymptotically normal.

Original: The DML estimator for the DRF µ ( t ) is proposed as follows.
Revised: The DML estimator for the DRF µ ( t ) is proposed as follows.

Original: First, we estimate the CDRF µ ( t, X ) with ˆ µ ( t, X ) and the GPS π ( T = t | X ) with ˆ π ( t | X ) .
Revised: First, we estimate the CDRF µ ( t, X ) with ˆ µ ( t, X ) and the GPS π ( T = t | X ) with ˆ π ( t | X ) .

Original: Then, with the nuisance functions plugged in, kernel-based double/debiased estimator gives
Revised: Then, with the nuisance functions plugged in, kernel-based double/debiased estimator gives

Original: [Equation]
Revised: [Equation]

Original:  G ENERALIZED C AUSAL F OREST
Revised:  G ENERALIZED C AUSAL F OREST

Original: In this section, we formally present the details of our algorithm, GCF.
Revised: In this section, we formally present the details of our algorithm, GCF.

Original: Building on CF, GCF relaxes partially linear assumption by adapting non-parametric DRF to the splitting criterion that can be estimated using kernel-based DML estimator described in the previous section.
Revised: Building on CF, GCF relaxes partially linear assumption by adapting non-parametric DRF to the splitting criterion that can be estimated using kernel-based DML estimator described in the previous section.

Original: In what follows, we give it to a workﬂow of GCF at both the training stage and prediction stage, followed by the elaborations on the details of splitting criterion, estimations, and practical tweaks for implementation.
Revised: In what follows, we give it to a workﬂow of GCF at both the training stage and prediction stage, followed by the elaborations on the details of splitting criterion, estimations, and practical tweaks for implementation.

Original: Algorithm1:
Revised: Algorithm

Original: Input Data set Ω =
Revised: Input Data set O =

Original: ( X i , T i , Y i ) , i = 1 , . . . , n ; number of trees B ; number of features sampled for growing a tree mtry ; the minimum sample size on each leaf node min.node.size ; honesty fraction α ; tolerance τ ; positivity threshold ζ ;
Revised: ( X i , T i , Y i ) , i = 1 , . . . , n ; number of trees B ; number of features sampled for growing a tree mtry ; the minimum sample size on each leaf node min.node.size ; honesty fraction α ; tolerance τ ; positivity threshold ζ ;

Original: Trainings begin
Revised: Trainings begin

Original: Data Ω 1 and data Ω 2 by a ( α , 1 − α ) split rule from Ω for honesty;
Revised: Data O 1 and data O 2 by a ( α , 1 − α ) split rule applied to O for honesty;

Original: Pre-train outcome regression model ˆ µ and treatment density estimation ˆ π on sample Ω 1 ; b ←− 1 ; while b ≤ B do
Revised: Pre-train outcome regression model ˆ µ and treatment density estimation ˆ π on sample Ω 1 ; b ←− 1 ; while b ≤ B do

Original: Randomly sample a feature set of size mtry X S from X ; for The stopping rule is not satisﬁed do
Revised: Randomly sample a feature set of size mtry X S from X ; for The stopping rule is not satisﬁed do

Original: Identify each parent node P ;
Revised: Identify each parent node P ;

Original: Compute the splitting criterion ∆( · ) (3) 
Revised: Compute the splitting criterion ∆( · ) (3) over the samples in O 1 with X S ;

Revised: Grow the tree T b by splitting at the parent node P according to the ∆( · ) ; end

Revised: Assign samples in O 2 to leaf nodes based on T b and let b ←− b + 1 ; end endOutput Causal forest with B trees by recursive partitioning on X ; node assignments for samples in O 2 ;

Original: Predictions begin
Revised: Predictions begin

Original: [Equation]
Revised: CDRF estimation ˆ µ b ( t, x ) by local weighted average over the outcomes of samples in Ωthat falls into L b ( x ) as (cid:80) ni

Original: [Equation] [Equation] [Equation]
Revised: =1 1 X i ∈L b ( x ) · 1 T i =

Original: [Equation] [Equation]
Revised: t · Y i |L b ( x ) | ; CATE estimation ˆ θ b ( t, x ) = ˆ µ b ( t, x ) − ˆ µ b (

Original: [Equation]
Revised: [Table]

Original:  We introduce the partial derivative of DRF, PDRF at treatment t with a slightly abuse use of notations
Revised: end We introduce the partial derivative of DRF, PDRF at treatment t with a slightly abuse use of notations

Original: [Equation]
Revised: [Equation]

Original: Intuitively, it tells about how µ ( · ) varies with treatments and characterizes the ”slope” along t .
Revised: Intuitively, it tells about how µ ( · ) varies with treatments and characterizes the ”slope” along t .

Original: Different leaf nodes on a tree shall have different slopes as the heterogeneity of covariates X .
Revised: Different leaf nodes on a tree shall have different slopes as the heterogeneity of covariates X .

Original: Let the number of trees be B and then GCF grows trees T b , b = 1 , . . .
Revised: Let the number of trees be B and then GCF grows trees T b , b = 1 , . . .

Original: , B , by repeating tree-growing process B times with bootstrapping.
Revised: , B , by repeating tree-growing process B times with bootstrapping.

Original: At the training stage, with PDRF we construct the tree T b by recursive partitioning based on a splitting criterion ∆( c 1 , c 2 ) .
Revised: At the training stage, with PDRF we construct the tree T b by recursive partitioning based on a splitting criterion ∆( c 1 , c 2 ) .

Original: It is proportional to the discrepancy between the PDRF in the left and right child nodes, to maximize the heterogeneity.
Revised: It is proportional to the discrepancy between the PDRF in the left and right child nodes, to maximize the heterogeneity.

Original: We elaborate on this splitting criterion in the next part.
Revised: We elaborate on this splitting criterion in the next part.

Original: Our algorithm is implemented on Apache Spark for large-scale data processing and the mechanism of the tree-growing process is different from that in CF.
Revised: Our algorithm is implemented on Apache Spark for large-scale data processing and the mechanism of the tree-growing process is different from that in CF.

Original: To elucidate on that, the data is stored at the master machine and trees are copied to each branch machine.
Revised: To elucidate on that, the data is stored at the master machine and trees are copied to each branch machine.

Original: 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.
Revised: 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.

Original: The updated tree will be copied to each branch machine again and we repeat the process until the tree stops splitting.
Revised: The updated tree will be copied to each branch machine again and we repeat the process until the tree stops splitting.

Original: This distributed computation and clustered storage leverage the computational power of multiple machines for speeding up the training process.
Revised: This distributed computation and clustered storage leverage the computational power of multiple machines for speeding up the training process.

Original: We refer readers to the appendix for more information on GCF’s Spark implementation.
Revised: We refer readers to the appendix for more information on GCF’s Spark implementation.

Original: The splits or the tree-growing process terminate when various pre-speciﬁed stopping criteria are met, for example, the sample size of child nodes is smaller than the parameter min.node.size or the distance D between PDRF is smaller than min.info.gain τ .
Revised: The splits or the tree-growing process terminate when various pre-speciﬁed stopping criteria are met, for example, the sample size of child nodes is smaller than the parameter min.node.size or the distance D between PDRF is smaller than min.info.gain τ .

Original: For prediction, we also follow the honesty principle proposed in (Athey et al., 2019) by partitioning the training samples into two parts where one is for constructing the tree and the other is for estimating CATE on the leaf node.
Revised: For prediction, we also follow the honesty principle proposed in (Athey et al., 2019) by partitioning the training samples into two parts where one is for constructing the tree and the other is for estimating CATE on the leaf node.

Original: That is to say that each training data can either be utilized to estimate CATE or contribute to growing a tree.
Revised: That is to say that each training data can either be utilized to estimate CATE or contribute to growing a tree.

Original: At the prediction stage, the leaf node of given samples x on each tree T b is denoted by L b ( x ) .
Revised: At the prediction stage, the leaf node of given samples x on each tree T b is denoted by L b ( x ) .

Original: The estimation of CDRF ˆ µ b ( t, x ) on L b ( x ) takes a local weighted average on training samples in L b ( x ) and ˆ θ b ( t, x )
Revised: The estimation of CDRF ˆ µ b ( t, x ) on L b ( x ) takes a local weighted average on training samples in L b ( x ) , which is given by (cid:80) ni

Original:  =
Revised: =1 1 X i ∈L b ( x ) · 1 T i = t · Y i |L b ( x ) | and ˆ θ b ( t, x ) =

Original: ˆ µ b ( t, x ) − ˆ µ b (0 , x ) .
Revised: ˆ µ b ( t, x ) − ˆ µ b (0 , x ) .

Original: The ﬁnal CATE estimation for X is the average of L b ( x ) over B trees, as ˆ θ ( t, x ) = 1 B (cid:80)
Revised: The ﬁnal CATE estimation for X is the average of L b ( x ) over B trees, as ˆ θ ( t, x ) = 1 B (cid:80)

Original: Bb =1 ˆ θ b ( t, x ) .
Revised: Bb =1 ˆ θ b ( t, x ) .

Original: 4. S PLITTING C RITERION
Revised: 4. S PLITTING C RITERION

Original: Given a parent node P and training samples with covariates X ω , our splitting criterion ∆( c 1 , c 2 ) is proposed as 
Revised: Given a parent node P and training samples with covariates X ω , our splitting criterion ∆( c 1 , c 2 ) is proposed as [Equation]

Original: [Equation]
Revised: [Equation]

Original: where ˆΦ c 1 , ˆΦ c 2 are the PDRF estimators, respectively.
Revised: where ˆΦ c 1 , ˆΦ c 2 are the PDRF estimators, respectively.

Original: The sample sizes of parent node, left child node and right child node are n P , n c 1 and n c 2 , respectively.
Revised: The sample sizes of parent node, left child node and right child node are n P , n c 1 and n c 2 , respectively.

Original: The ratio n c 1 n c 2 n
Revised: The ratio n c 1 n c 2 n

Original: P in ∆( c 1 , c 2 ) is to balance the sample sizes between the two child nodes.
Revised: P in ∆( c 1 , c 2 ) is to balance the sample sizes between the two child nodes.

Original: Distance metric D measures the distance between ˆΦ c 1 ( t, X ω ) and ˆΦ c 2 ( t, X ω ) in the functional space of PDRF, which is to be maximized in the splitting so as to optimize the heterogeneity.
Revised: Distance metric D measures the distance between ˆΦ c 1 ( t, X ω ) and ˆΦ c 2 ( t, X ω ) in the functional space of PDRF, which is to be maximized in the splitting so as to optimize the heterogeneity.

Original: Some commonly used metrics are L 1 , L 2 and L ∞ as in (Dette et al., 2018)
Revised: Some commonly used metrics are L 1 , L 2 and L ∞ as in (Dette et al., 2018)

Original: D 1 = (cid:82) t | ˆΦ c 1 ( t, X ω ) − ˆΦ c 2 ( t, X ω ) | dt, D 2 = (cid:82) t | ˆΦ c 1 ( t, X ω ) − ˆΦ c 2 ( t, X ω ) | 2 dt, D ∞ =
Revised: D 1 = (cid:82) t | ˆΦ c 1 ( t, X ω ) − ˆΦ c 2 ( t, X ω ) | dt, D 2 = (cid:82) t | ˆΦ c 1 ( t, X ω ) − ˆΦ c 2 ( t, X ω ) | 2 dt, D ∞ =

Original: max t | ˆΦ c 1 ( t, X ω ) − ˆΦ c 2 ( t, X ω ) | .
Revised: max t | ˆΦ c 1 ( t, X ω ) − ˆΦ c 2 ( t, X ω ) | .

Original: We next show how to estimate PDRF Φ c 1 and Φ c 2 using kernel-based DML estimator (Colangelo & Lee, 2020).
Revised: We next show how to estimate PDRF Φ c 1 and Φ c 2 using kernel-based DML estimator (Colangelo & Lee, 2020).

Original: 4. PDRF E STIMATION
Revised: 4. PDRF E STIMATION

Original: The estimation for DRF is a precondition for the estimation of PDRF.
Revised: The estimation for DRF is a precondition for the estimation of PDRF.

Original: In this paper, we point out kernel-based DML estimator in (Colangelo & Lee, 2020) can be adapted to the DRF estimation in our GCF and guides the tree splitting.
Revised: In this paper, we point out kernel-based DML estimator in (Colangelo & Lee, 2020) can be adapted to the DRF estimation in our GCF and guides the tree splitting.

Original: Recall that, the DML estimation for DRF is given by
Revised: Recall that, the DML estimation for DRF is given by

Original: [Equation]
Revised: [Equation]

Original: where ˆ µ ( t, X i ) is the pretrained estimator for µ ( t, X ) at X i ,
Revised: where ˆ µ ( t, X i ) is the pretrained estimator for µ ( t, X ) at X i ,

Original: ˆ π
Revised: ˆ π

Original: ( t | X i ) is the pretrained estimator for
Revised: ( t | X i ) is the pretrained estimator for

Original: GPS at X i and C denotes the child node.
Revised: GPS at X i and C denotes the child node.

Original: When adapted to our framework, we additionally derive the following estimator for CDRF by innovatively combine kernel regression with the DML estimator.
Revised: When adapted to our framework, we additionally derive the following estimator for CDRF by innovatively combine kernel regression with the DML estimator.

Original: CDRF estimator The estimator is given by
Revised: CDRF estimator The estimator is given by

Original: [Equation]
Revised: [Equation]

Original: PDRF estimator The PDRF estimator is given by ˆΦ( t ) =
Revised: PDRF estimator The PDRF estimator is given by ˆΦ( t ) =

Original: ∂
Revised: ∂

Original: ˜ µ ( t ) ∂t and ˆΦ( t, x ) =
Revised: ˜ µ ( t ) ∂t and ˆΦ( t, x ) =

Original: ∂
Revised: ∂

Original: ˜ µ ( t, x ) ∂t .
Revised: ˜ µ ( t, x ) ∂t .

Original: In estimating PDRF, the smoothness condition of DRF plays an important role.
Revised: In estimating PDRF, the smoothness condition of DRF plays an important role.

Original: For smooth DRF, we can explicitly derive the closed-form PDRF.
Revised: For smooth DRF, we can explicitly derive the closed-form PDRF.

Original: For smooth Gaussian kernels, Φ( t ) is
Revised: For smooth Gaussian kernels, Φ( t ) is

Original: [Equation]
Revised: [Equation]

Original: However, for general DRF, the PDRF may be discontinuous or even ill-deﬁned.
Revised: However, for general DRF, the PDRF may be discontinuous or even ill-deﬁned.

Original: For example, nondifferentiable kernel functions like uniform, epanechnikov, biweight and triweight, Φ can only be estimated via numerical approximations of Φ( t ) =
Revised: For example, nondifferentiable kernel functions like uniform, epanechnikov, biweight and triweight, Φ can only be estimated via numerical approximations of Φ( t ) =

Original: E [ µ ( t + δ ) − µ ( t )] δ 
Revised: E [ µ ( t + δ ) − µ ( t )] δ .

Original: [Equation]

Original: 4. P RACTICAL C ONSIDERATIONS
Revised: 4. P RACTICAL C ONSIDERATIONS

Original: 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 h deals with the trade off between bias and variance.
Revised: 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 h deals with the trade off between bias and variance.

Original: Usually, a small h avoids a large variance while a large h reduces the bias.
Revised: Usually, a small h avoids a large variance while a large h reduces the bias.

Original: The empirical ways of choosing an optimal h are Cross Validation and Rule-of-thumb (Dehnad, 1987)
Revised: The empirical ways of choosing an optimal h are Cross Validation and Rule-of-thumb (Dehnad, 1987)

Original: When utilizing kernel densities for weighting, we often suffer from the boundary bias if without normalization.
Revised: When utilizing kernel densities for weighting, we often suffer from the boundary bias if without normalization.

Original: To this end, we normalize estimators with the cumulative density function of range [ T min , T max ] of treatments.
Revised: To this end, we normalize estimators with the cumulative density function of range [ T min , T max ] of treatments.

Original: More speciﬁcally, the estimators are divided by (cid:82) T max T min K h
Revised: More speciﬁcally, the estimators are divided by (cid:82) T max T min K h

Original: ( T i − t ) dt .
Revised: ( T i − t ) dt .

Original: Regarding the positivity assumption in practice, we use a hyperparameter ζ to regularize for the boundary of σ ( π ) that is the variance of GPS.
Revised: Regarding the positivity assumption in practice, we use a hyperparameter ζ to regularize for the boundary of σ ( π ) that is the variance of GPS.

Original: This is also a guarantee on the weak assumption as stated in Assumption 4.
Revised: This is also a guarantee on the weak assumption as stated in Assumption 4.

Original: Formally, our proposed splitting criterion will be a regularized one as
Revised: Formally, our proposed splitting criterion will be a regularized one as

Original: [Equation]
Revised: [Equation]

Original: where ˆ π is the estimated GPS.
Revised: where ˆ π is the estimated GPS.

Original: With this additional term, the splitting will tend to maximize the variance of GPS and hence taking the variety of treatments across samples into account. 
Revised: With this additional term, the splitting will tend to maximize the variance of GPS and hence taking the variety of treatments across samples into account. 4. A SYMPTOTIC P ROPERTY

Revised: Let Assumptions 1-3 in (Colangelo & Lee, 2020) hold and thusfar the convergence of kernel-based

Original: E XPERIMENT
Revised: 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 to say that we can utilize a subsample s of order √ n to achieve a converged kernel DML estimator. Then s satisﬁes that s/n → 0 and s → ∞ which is stated as Speciﬁcation 1 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.

Revised: Throughout our algorithm, we also follow the honesty principle.

Revised: Then by Theorem 3 (Athey et al., 2019), our ﬁnal estimator is asymptotically unbiased.

Revised: This shows the doubly robustness of our estimator by combining kernel DML and Causal Forest under slightly stronger assumptions on the score function.

Revised:  E XPERIMENT

Original: In this section, we validate our proposed algorithm 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)
Revised: 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)

Original: GCF is ﬂexible yet effective compared to SOTA which provides enough evidence for its practical use.
Revised: GCF is ﬂexible yet effective compared to SOTA which provides enough evidence for its practical use.

Original: 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.
Revised: 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.

Original: 5. E VALUATION
Revised: 5. E VALUATION

Original: In what follows we introduce the commonly used metrics for model evaluation on both synthetic data and real-world data sets, named as model errors and Qini score and Qini curve, respectively.
Revised: In what follows we introduce the commonly used metrics for model evaluation on both synthetic data and real-world data sets, named as model errors and Qini score and Qini curve, respectively.

Original: For real-world data sets, the model errors are not available since the counterfactual outcomes in the potential outcomes cannot be observed.
Revised: For real-world data sets, the model errors are not available since the counterfactual outcomes in the potential outcomes cannot be observed.

Original: Synthetic Data
Revised: Synthetic Data

Original: We adopt PEHE and RMSE as the measure for model precision.
Revised: We adopt PEHE and RMSE as the measure for model precision.

Original: Formally, let n be the number of treatment groups, ˆ θ ti and θ ti be the predicted and true treatment effect of i th sample at treatment t , ˆ θ t = (cid:80) ti =
Revised: Formally, let n be the number of treatment groups, ˆ θ ti and θ ti be the predicted and true treatment effect of i th sample at treatment t , ˆ θ t = (cid:80) ti =

Original: 1 ˆ y ti − ˆ y 

Revised: 1 ˆ y ti − ˆ y

Original: i , ˆ y 

Revised: i , ˆ y

Original: i be the reference outcome that is pre-determined.
Revised: i be the reference outcome that is pre-determined.

Original: PEHE,RMSE,and ADRF are [Equation] | [Equation]
Revised: PEHE,RMSE,and ADRF are (cid:80) ni =1 ( | ˆ θ t − θ t | ))

Revised: n

Original: [Equation]
Revised: [Equation]

Original: Real-world data sets We refer readers to (Gutierrez & G´erardy, 2017) for  formal deﬁnition of Qini Score and Qini Curve.
Revised: Real-world data sets We refer readers to (Gutierrez & G´erardy, 2017) for the formal deﬁnition of Qini Score and Qini Curve.

Original: In practice, a larger Qini Score indicates  model has better performance in distinguish HTE.
Revised: In practice, a larger Qini Score indicates that model has better performance in identifying HTE.

Original: 5. M ODEL S ETUPS
Revised: 5. M ODEL S ETUPS

Original: The baselines for GCF include RF, CF, and Kennedy (Kennedy et al., 2017).
Revised: The baselines for GCF include RF, CF, and Kennedy (Kennedy et al., 2017).

Original: Both RF and CF use num.trees equals to 500, min.node.size to 50.
Revised: Both RF and CF use num.trees equals to 500, min.node.size to 50.

Original: For Kennedy (Kennedy et al., 2017), we use the code of function cts.eff from R package npcausal available in (ehkennedy, 2017).
Revised: For Kennedy (Kennedy et al., 2017), we use the code of function cts.eff from R package npcausal available in (ehkennedy, 2017).

Original: SuperLearner library is set to SL.ranger and SL.glm .
Revised: SuperLearner library is set to SL.ranger and SL.glm .

Original: We made small tweaks to SL.ranger to make share same hyperparaters with RF and CF for comparability.
Revised: We made small tweaks to SL.ranger to make share same hyperparaters with RF and CF for comparability.

Original: GCF We implement the proposed GCF on Apache Spark Mllib (Meng et al., 2016) with kernelbased DML estimator, denoted by GCF .
Revised: GCF We implement the proposed GCF on Apache Spark MLlib (Meng et al., 2016) with kernelbased DML estimator, denoted by GCF.

Original: For the implementation of GCF, we ﬁx the distance metric
Revised: For the implementation of GCF, we ﬁx the distance metric

Original: D to be d 2 out of simplicity.
Revised: D to be d 2 out of simplicity.

Original: The number of trees, the minimum node size and other hyperparameters are set to match those in RF and CF.
Revised: The number of trees, the minimum node size and other hyperparameters are set to match those in RF and CF.

Original: RF is a forest-ensemble method without controlling confoundedness.
Revised: RF is a forest-ensemble method without controlling confoundedness.

Original: CF makes the partially linear assumption on DRF.
Revised: CF makes the partially linear assumption on DRF.

Original: Kennedy is a non-parametric doubly robust algorithm but uses a normal splitting criterion.
Revised: Kennedy is a non-parametric doubly robust algorithm but uses a normal splitting criterion.

Original: The comparisons inclusively validate our algorithm for both non-parametric DRF estimation and specially designed splitting criterion.
Revised: The comparisons inclusively validate our algorithm for both non-parametric DRF estimation and specially designed splitting criterion.

Original: 5. S IMULATION
Revised: 5. S IMULATION

Original: Here we show a comprehensive simulation study on synthetic data.
Revised: Here we show a comprehensive simulation study on synthetic data.

Original: First, we elaborate on the data generating process.
Revised: First, we elaborate on the data generating process.

Original: It is followed by the details of model implementations.
Revised: It is followed by the details of model implementations.

Original: Finally we illustrate the numerical results that consistently show the superior performance of our GCF.
Revised: Finally we illustrate the numerical results that consistently show the superior performance of our GCF.

Original: 5.3. D ATA G ENERATING P ROCESS
Revised: 5.3. D ATA G ENERATING P ROCESS

Original: Recall that the covariate matrix X =
Revised: Let n denote the number of samples and p =

Original: ( X ji ) =
Revised: p X + p U

Original:  ( X 1 , . . .
Revised: + p Z be the dimension of covariates. The covariate matrix is ( X ji ) j =1 ,...,p X i =1 ,...,n = ( X 1 , . . .

Original: X n ) ∈ R n × p  .
Revised: X n ) ∈ R n × p X .

Original: Let n denotes the number of observations and p be the dimension of covariates.

Original: Data generating process (DGP) is
Revised: Data generating process (DGP) is

Original: [Equation]
Revised: [Equation]

Original: where φ is the sigmoid function and Ψ is the pdf of Beta distribution.
Revised: where φ is the sigmoid function and Ψ is the pdf of Beta distribution.

Original: Here we set DRF µ as polynomial(poly), exponential(Exp), and sinusoidal functions(Sinus).
Revised: Here we set DRF µ to be polynomial(Poly), exponential(Exp), and sinusoidal functions(Sinus).

Original: The above DGP gives us a total of 6 data sets by  the combination of DRF µ ( t ) and  covariate Σ, speciﬁed by p and DRF .
Revised: The above DGP gives us a total of 6 datasets by taking the possible combinations of 3 DRFs µ ( t ) and 2 setups of covariate Σ .

Original: Parameter (cid:15) , ν are noises following [ Unif ( − 1 , 1)] p .
Revised: Noises (cid:15) , ν follow Unif ( − 1 , 1) .

Original: Covariates X and coefﬁcients
Revised: Covariates X and coefﬁcients

Original: β follow N (0 , I p  ) .
Revised: β follow N (0 , I p X ) .

Original: We also allow sparsity in covariates by randomly set some of the coefﬁcients to 0.
Revised: We also allow sparsity in covariates by randomly setting some of the coefﬁcients to 0.

Original: Following this rule, we generate 100 rounds for each setup.

Original: Testing data sets are generated with randomly assigned treatments to allow unbiased evaluation.
Revised: Testing data sets are generated with randomly assigned treatments to allow unbiased evaluation.

Original: 5.3. S IMULATION R ESULTS
Revised: 5.3. S IMULATION R ESULTS

Original: Since some models output the ﬁnal prediction of ˆ Y ( t ) and some output the treatment effect ˆ θ ( t ) , we deliberately compute the pseudo outcome ˜ Y ( t ) =
Revised: Since some models output the ﬁnal prediction of ˆ Y ( t ) and some output the treatment effect ˆ θ ( t ) , we deliberately compute the pseudo outcome ˜ Y ( t ) =

Original: Y ( t ) − Y (0) as the label in the purpose of the comparability across baselines.
Revised: Y ( t ) − Y (0) as the label in the purpose of the comparability across baselines.

Original: The simulation results  are summarized in Table ?? .
Revised: The simulation results pf 3 datasets are summarized in Table 1 and those of the other 3 datasets are in Appendix.

Original: Here PEHE and RMSE are averaged and standard deviations of the metrics across simulation runs are also reported.
Revised: Here PEHE and RMSE are averaged and standard deviations of the metrics across simulation runs are also reported.

Original: Overall, GCF signiﬁcantly outperforms baseline methods, which again validates the superiority of the newly proposed algorithm.
Revised: Overall, GCF signiﬁcantly outperforms baseline methods, which again validates the superiority of the newly proposed algorithm.

Original: More speciﬁcally, GCF exhibits the smallest biases across multiple DGP setups which we defer details to the supplementary.
Revised: More speciﬁcally, GCF exhibits the smallest biases across multiple DGP setups which we defer details to the supplementary.

Original: Additionally, we compare the ADRF curve given by different models to the ground truth.
Revised: Additionally, we compare the ADRF curve given by different models to the ground truth.

Original: An example of ADRF under a DGP speciﬁcation is shown in Fig.
Revised: An example of ADRF under a DGP speciﬁcation is shown in Fig.

Original: [Table]
Revised: [Table]

Original: 5. R EAL - WORLD DATA SETS
Revised: 5. R EAL - WORLD DATA SETS

Original: Our method is also implemented on real-world data sets  sourced from a major ride-hailing company.
Revised: Our method is also implemented on real-world data sets with size 10,000,000 which sourced from a major ride-hailing company.

Original: A discount that randomly sampled (one of 6 different values { d 0 , d 1 , d 2 , d 3 , d 4 , d 5 : d 0 < d 1 < . . .
Revised: A discount that randomly sampled (one of 6 different values { d 0 , d 1 , d 2 , d 3 , d 4 , d 5 : d 0 < d 1 < . . .

Original: < d 5 } ) is assigned to all trips in a given origin-destination-time tuple.
Revised: < d 5 } ) is assigned to all trips in a given origin-destinationtime tuple.

Original: The effect of discount on the trip complete rate is of primary interests.
Revised: The effect of discount on the trip complete rate is of primary interests.

Original: We apply GCF to this data set and evaluate the performance of models by calculating Qini Scores.
Revised: We apply GCF to this data set and evaluate the performance of models by calculating Qini Scores.

Original: [Table]
Revised: [Table]

Original: The Qini scores of different models are summarized in Table 2.
Revised: The Qini scores of different models are summarized in Table 2.

Original: The performance of GCF is superior to the company’s current models as GCF has the highest Qini scores across all levels of discounts.
Revised: The performance of GCF is superior than the company’s current models as GCF has the highest Qini scores across all levels of discounts.

Original: We further create an incentive policy based on GCF output and deploy it to the online environment.
Revised: We further create an incentive policy based on GCF output and deploy it for online purpose.

Original: Online A/B testing shows a signiﬁcant lift in KPI for GCF-driven policy.
Revised: Online A/B testing shows a signiﬁcant lift in KPI for GCF-driven policy.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we propose a novel forest-based non-parametric algorithm to address the problem of
Revised: In this paper, we propose a novel forest-based non-parametric algorithm to address the problem of

Original: HTE estimation with continuous treatments.
Revised: HTE estimation with continuous treatments.

Original: Under the situation that a fully non-parametric and localized ML-based algorithm has not been proposed yet, we extend Causal Forest with a DRFbased splitting criterion computed by the distance in the functional space of PDRF.
Revised: Under the situation that a fully non-parametric and localized ML-based algorithm has not been proposed yet, we extend Causal Forest with a DRFbased splitting criterion computed by the distance in the functional space of PDRF.

Original: To estimate this
Revised: To estimate this

Original: DRF, We use the kernel-based DML estimator to guarantee double robustness that mitigates model misspeciﬁcations.
Revised: DRF, We use the kernel-based DML estimator to guarantee double robustness that mitigates model misspeciﬁcations.

Original: We implement GCF on Spark to leverage the computational efﬁciency and test it on both synthetic data and real-world data sets and compare it with the prior art approaches.
Revised: We implement GCF on Spark to leverage the computational efﬁciency and test it on both synthetic data and real-world data sets and compare it with the prior art approaches.

Original: The numerical resultsconsistently demonstrate that our method signiﬁcantly outperform these methods.
Revised: The numerical results demonstrate that our method signiﬁcantly outperform competing methods.

Original:  A PPENDIX
Revised:  A PPENDIX

Original: The appendix runs as follows.
Revised: The appendix runs as follows.

Original: First, we list all the notations used in the appendix.
Revised: First, we list all the notations used in the appendix.

Original: 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.
Revised: 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.

Original: Subsequently, we brieﬂy outline GCF’s Spark framework, which enables large-scale instances processing and faster model training.
Revised: Subsequently, we brieﬂy outline GCF’s Spark framework, which enables large-scale instances processing and faster model training.

Original: Finally, we demonstrate how to apply our method to an online schema and achieve superior performance as its another validation.
Revised: Finally, we demonstrate how to apply our method to an online schema and achieve superior performance as its another validation.

Original: 7. N OTATIONS
Revised: 7. N OTATIONS

Original: We restate the notations that are consistent with the main paper.
Revised: We restate the notations that are consistent with the main paper.

Original: Following the potential outcome framework in (Neyman, 1923; Rubin, 1974), we let T be the continuous treatments, X =
Revised: Following the potential outcome framework in (Neyman, 1923; Rubin, 1974), we let T be the continuous treatments, X =

Original: ( X j ) pj =1 be the p X -dim confounder, U be the p U -dim outcome-speciﬁc adjustment variable, Z be the p Z -dim treatment-speciﬁc adjustment variable, and Y be the observed outcome.
Revised: ( X j ) pj =1 be the p X -dim confounder, U be the p U -dim outcome-speciﬁc adjustment variable, Z be the p Z -dim treatment-speciﬁc adjustment variable, and Y be the observed outcome.

Original: The potential outcomes under treatment t is Y ( t ) .
Revised: The potential outcomes under treatment t is Y ( t ) .

Original: The population Ω =
Revised: The population Ω =

Original: ( Σ =
Revised: ( Σ =

Original: ( X , U , Z ) , Y, T )
Revised: ( X , U , Z ) , Y, T )

Original: ∈ R p X
Revised: ∈ R p X

Original: + p U
Revised: + p U

Original: + p Z +1 satisﬁes
Revised: + p Z +1 satisﬁes

Original: Y
Revised: Y

Original: [Equation]
Revised: [Equation]

Original: g ( T, X , U ) + (cid:15)
Revised: g ( T, X , U ) + (cid:15)

Original: [Equation]
Revised: [Equation]

Original: ν where (cid:15), ν are noises of zero mean, and g : R p X + p U
Revised: ν where (cid:15), ν are noises of zero mean, and g : R p X + p U

Original: × R → R and f : R p X + p Z → R .
Revised: × R → R and f : R p X + p Z → R .

Original: { ( Σ i , Y i , T i ) , i = 1 , . . . , n } are i.i.d.
Revised: { ( Σ i , Y i , T i ) , i = 1 , . . . , n } are i.i.d.

Original: samples drawn from the population Ω .
Revised: samples drawn from the population Ω .

Original: Then the covariate matrix Σ =
Revised: Then the covariate matrix Σ =

Original: (Σ ji ) 1 ≤ j ≤ p 1 ≤ i ≤ n = ( Σ 1 , . . . , Σ n ) ∈ R n × p where p =
Revised: (Σ ji ) 1 ≤ j ≤ p 1 ≤ i ≤ n = ( Σ 1 , . . . , Σ n ) ∈ R n × p where p =

Original: p X + p U
Revised: p X + p U

Original: + p Z .
Revised: + p Z .

Original: The Generalized
Revised: The Generalized

Original: Propensity Score is π ( T = t | X ) , which is the probability density for a unit receiving treatment t given the covariate X .
Revised: Propensity Score is π ( T = t | X ) , which is the probability density for a unit receiving treatment t given the covariate X .

Original: 7. D ATA G ENERATING P ROCESS
Revised: 7. D ATA G ENERATING P ROCESS

Original: Recall that the covariate matrix Σ
Revised: Recall that the covariate matrix Σ

Original: =
Revised: =

Original: ( X ji , U ji , Z ji ) =
Revised: ( X ji , U ji , Z ji ) =

Original: ( X 1 , . . .
Revised: ( X 1 , . . .

Original: , X n , U 1 , . . .
Revised: , X n , U 1 , . . .

Original: , U n , Z 1 , . .
Revised: , U n , Z 1 , . .

Original: , Z n ) ∈
Revised: , Z n ) ∈

Original: R n × ( pX + pU + pZ ) and treatment T ∈ R where n denotes the number of observations and p is the dimension of covariates.
Revised: R n × ( pX + pU + pZ ) and treatment T ∈ R where n denotes the number of observations and p is the dimension of covariates.

Original: Data generating process (DGP) works as
Revised: Data generating process (DGP) works as

Original: [Equation]
Revised: [Equation]

Original: where φ is the sigmoid function and Ψ is the pdf of Beta distribution with shape parameters set toand 3.
Revised: where φ is the sigmoid function and Ψ is the pdf of Beta distribution with shape parameters set toand 3.

Original: Here we set DRF µ as polinomial(Poly), exponential(Exp), and sinusoidal functions(Sinus).
Revised: Here we set DRF µ as polinomial(Poly), exponential(Exp), and sinusoidal functions(Sinus).

Original: [Equation]
Revised: [Equation]

Original: The above DGP gives us a variety of data sets by taking the combination of DRF µ ( t ) and covariate Σ that are speciﬁed by n , p and DRF .
Revised: The above DGP gives us a variety of data sets by taking the combination of DRF µ ( t ) and covariate Σ that are speciﬁed by n , p and DRF .

Original: Parameter (cid:15) , ν are noises following N (0 , 1) .
Revised: Parameter (cid:15) , ν are noises following N (0 , 1) .

Original: Covariates
Revised: Covariates

Original: X, U, Z follows N (0 , I ) and coefﬁcient vector β follow [ Unif ( − 1 , 1)] .
Revised: X, U, Z follows N (0 , I ) and coefﬁcient vector β follow [ Unif ( − 1 , 1)] .

Original: Following this rule, we generate 100 rounds.
Revised: Following this rule, we generate 100 rounds.

Original: Meanwhile, in test data, we randomly assigned the treatments to make sure unbiased evaluations.
Revised: Meanwhile, in test data, we randomly assigned the treatments to make sure unbiased evaluations.

Original: 7. S IMULATION RESULTS
Revised: 7. S IMULATION RESULTS

Original: In addition to the results provided in the main paper, here we provide the inclusive results of our simulation study, which covers all 6 data sets from the Data Generating process.
Revised: In addition to the results provided in the main paper, here we provide the inclusive results of our simulation study, which covers all 6 data sets from the Data Generating process.

Original: The results are summarized in Table 3.
Revised: The results are summarized in Table 3.

Original: Compared to CF (Athey et al., 2019) that is the benchmark of our generalization, our GCF outperforms it with signiﬁcantly smaller errors.
Revised: Compared to CF (Athey et al., 2019) that is the benchmark of our generalization, our GCF outperforms it with signiﬁcantly smaller errors.

Original: With the baseine ﬁxed to be Kennedy (Kennedy et al., 2017), our algorithm almost enjoys smaller errors except one dataset
Revised: With the baseine ﬁxed to be Kennedy (Kennedy et al., 2017), our algorithm almost enjoys smaller errors except one dataset

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: where DRF is sin.
Revised: where DRF is sin.

Original: Overall, these experimental evidence demonstrates that our GCF is superior than the algorithms for continuous treatments.
Revised: Overall, these experimental evidence demonstrates that our GCF is superior than the algorithms for continuous treatments.

Original: Meanwhile, ADRF curves are also inclusively presented here as validations of our proposed GCF.
Revised: Meanwhile, ADRF curves are also inclusively presented here as validations of our proposed GCF.

Original: Figure 3 − 8 are ADRF curves of baseline methods and GCF on different synthetic data sets.
Revised: Figure 3 − 8 are ADRF curves of baseline methods and GCF on different synthetic data sets.

Original: Across all the data sets, our proposed GCF performs the best by being the closest to the ground truth.
Revised: Across all the data sets, our proposed GCF performs the best by being the closest to the ground truth.

Original: 7. S PARK I MPLEMENTATION
Revised: 7. S PARK I MPLEMENTATION

Original: Apache Spark (Meng et al., 2016) has the power of large-scale data processing and provides APIs of any machine learning algorithms.
Revised: Apache Spark (Meng et al., 2016) has the power of large-scale data processing and provides APIs of any machine learning algorithms.

Original: Consequently, we build our proposed algorithm on Spark to do parallel computing and distributed model training that leverage the resources of multiple machines simultaneously.
Revised: Consequently, we build our proposed algorithm on Spark to do parallel computing and distributed model training that leverage the resources of multiple machines simultaneously.

Original: The workﬂow of Spark with GCF is shown in Figure 9, which include integrated data and distributed computation.
Revised: The workﬂow of Spark with GCF is shown in Figure 9, which include integrated data and distributed computation.

Original: More speciﬁcally, with the parallel structure of spark, the tree-growing process runs and the process of GCF is depicted in Figure 10, which is signiﬁcantly different from that in CF (Athey et al., 2019) by distributing tasks for computations
Revised: More speciﬁcally, with the parallel structure of spark, the tree-growing process runs and the process of GCF is depicted in Figure 10, which is signiﬁcantly different from that in CF (Athey et al., 2019) by distributing tasks for computations

Original: 7. O NLINE D EPLOYMENT AND E XPERIMENTS
Revised: 7. O NLINE D EPLOYMENT AND E XPERIMENTS

Original: The deployment of models in the online marketplace is twofold as illustrated in Figure 11.
Revised: The deployment of models in the online marketplace is twofold as illustrated in Figure 11.

Original: On the one hand, the ofﬂine model that is trained on historical data provides HTE estimation that decisionmakers need to customize incentives at the origin-destination-time level.
Revised: On the one hand, the ofﬂine model that is trained on historical data provides HTE estimation that decisionmakers need to customize incentives at the origin-destination-time level.

Original: On the other hand, the outcome in response to the customized incentives contributes to the data prepared for the subsequent model training.
Revised: On the other hand, the outcome in response to the customized incentives contributes to the data prepared for the subsequent model training.

Original: The following is an example of an online experiment with the GCF-driven policy.
Revised: The following is an example of an online experiment with the GCF-driven policy.

Original: We conduct online A/B testings with the model on 2 equal-size non-overlapping groups, i.e. the cohort.
Revised: We conduct online A/B testings with the model on 2 equal-size non-overlapping groups, i.e. the cohort.

Original: The evaluation metric of the online experiment is the ﬁnish order (FO) increment.
Revised: The evaluation metric of the online experiment is the ﬁnish order (FO) increment.

Original: The numerical results of GCF and CF are shown in Figure 12.
Revised: The numerical results of GCF and CF are shown in Figure 12.

Original: GCF improves on CF by 15.1 % , beneﬁting from the relaxation of the assumptions on the model speciﬁcation.
Revised: GCF improves on CF by 15.1 % , beneﬁting from the relaxation of the assumptions on the model speciﬁcation.

Original: 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.
Revised: 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.