Doubly Robust Proximal Causal Learning for Continuous Treatments
Authors: Yong Wu
Yanwei Fu
Shouyan Wang
Xinwei Sun
Venue: ICLR 2024 poster (2024)
Keywords: causal effect
continuous treatment
kernel function
doubly robust
Primary area: causal reasoning
TL;DR: We propose a kernel-based estimator that can well handle continuous treatments for proximal causal learning.

Abstract: Proximal causal learning is a powerful framework for identifying the causal effect under the existence of unmeasured confounders. Within this framework, the doubly robust (DR) estimator was derived and has shown its effectiveness in estimation, especially when the model assumption is violated. However, the current form of the DR estimator is restricted to binary treatments, while the treatments can be continuous in many real-world applications. The primary obstacle to continuous treatments resides in the delta function present in the original DR estimator, making it infeasible in causal effect estimation and introducing a heavy computational burden in nuisance function estimation. To address these challenges, we propose a kernel-based DR estimator that can well handle continuous treatments for proximal causal learning. Equipped with its smoothness, we show that its oracle form is a consistent approximation of the influence function. Further, we propose a new approach to efficiently solve the nuisance functions. We then provide a comprehensive convergence analysis in terms of the mean square error. We demonstrate the utility of our estimator on synthetic datasets and real-world applications.

Published as a conference paper at ICLR 2024
DOUBLY ROBUST PROXIMAL CAUSAL LEARNING FOR CONTINUOUS TREATMENTS
Yong Wu
Yanwei Fu? Shouyan Wang! Xinwei Sun
"Istitute of Science and Technology for Brain-Inspired Intelligence, Fudan University
School of Data Science, Fudan University ZZhangjiang Fudan International Innovation Center *Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence (Fudan University)
'MOE Frontiers Center for Brain Science, Fudan University
"Shanghai Engineering Research Center of AI & Robotics, Fudan University
"Engineering Research Center of Al & Robotics, Ministry of Education, Fudan University
ABSTRACT
Proximal causal learning is powerful framework for identifying the causal effect under the existence of unmeasured confounders. Within this framework, the dou- bly robust (DR) estimator was derived and has shown its effectiveness in estima- tion, especially when the model assumption is violated. However, the current form of the DR estimator is restricted to binary treatments, while the treatments continuous in many real-world applications. The primary obstacle to continuous treatments resides in the delta function present in the original DR estimator, mak- ing it infeasible in causal effect estimation and introducing heavy computational burden in nuisance function estimation. To address these challenges, we propose a kernel-based DR estimator that can well handle continuous treatments for prox- imal causal learning. Equipped with its smoothness, we show that its oracle form is consistent approximation of the influence function. Further, we propose a new approach to efficiently solve the nuisance functions. We then provide a compre- hensive convergence analysis in terms of the mean square error. We demonstrate the utility of our estimator on synthetic datasets and real-world application
be
INTRODUCTION
The causal effect estimation is significant issue in many fields such as social sciences Heedström & Ylikoski 2010), economics (Varian 2016), and medicine (Yazdani Boerwinkle 2015). critical challenge in causal inference is non-compliance to randomness due to the presence of unobserved confounders, which can induce biases in the estimation.
One approach to address this challenge is the proximal causal learning (PCL) framework (Miao) jet al., 2018a; Tehetgen et al. 2020 Cui et al. 2023), which offers an opportunity to learn about causal effects where ignorability condition fails. This framework employs two proxies a treatment- inducing proxy and an mutcome-inducing proxy to identify the causal effect by estimating the bridge/nuisance functions. Particularly, Cui al. (2023) derived the doubly robust estimator within the PCL. framework, which combines the estimator obtained from the treatment bridge function and the estimator obtained from the outcome bridge function. The doubly robust estimator has been widely used in causal effect estimation (Bang & Robins 2005), as is able to tolerate violations of model assumptions of bridge functions.
However, current doubly robust estimators (Cui et al. 2023) within the proximal causal framework mainly focus on binary treatments, whereas the treatments can be continuous in many real-world scenarios, including social science, biology, and economics. For example, in therapy studies, we are not only interested in estimating the effect of receiving the drug but also the effectiveness of the drug dose. Another example comes from the data (Donohue III & Leviti (000)) that focused on policy- making, where one wishes to estimate the effect legalized abortion on the crime rate.
"Corresponding author
Code is available httppp////ttttbb.ccc////e/ee///e wwwountiuuuusuioo reetttttttrrrretttttteeaoaaamet.

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Previous work causal effect for continuous treatments has focused primarily the unconfound- edness assumption (Kallus & Zhou 2018 Colangelo & Lee 2020). However, extending them within the proximal causal farmework encounters several key challenges. Firstly, the Proximal Inverse Probability Weighting (PIPW) part in the original doubly robust (DR) estimator relies on a delta function centered around the treatment value being analyzed, rendering it impractical for empirically estimating causal effects with continuous treatments, Secondly, deriving the influence function will involve dealing with the Gateaux derivative of bridge functions, which particularly intricate due to its implicit nature. Lastly, the existing estimation process of bridge functions requires running an optimization for each new treatment, rendering it computationally inefficient for practical applica- tions. In light of these formidable challenges, our contribution lies in addressing the open question of deriving the DR estimator for continuous treatments within the proximal causal framework.
To address these challenges, we propose a kernel-based method that can well handle continuous treatments for PCL. Specifically, we incorporate the kernel function into the PIPW estimator, as a smooth approximation to causal effect. We then derive the DR estimator and show its consistency for a broad family of kernel functions. Equipped with smoothness, we show that such a DR estima- tor coincides with the influence function. To overcome the computational issue in nuisance function estimation, we propose to estimate the propensity score and incorporate it into a min-max optimiza- tion problem, which is sufficient to estimate the nuisance functions for all treatments. We show that estimator enjoys the O(n -4/5) convergence rate in mcan squared error (MSE). We demonstrate the utility and efficiency on synthetic data and the policy-making (Donohue III & Levitt 200I
Contributions. To summarize, contributions are:
1. We propose a kernel-based DR estimator that is provable to be consistent for continuous treat- ments effect within the proximal causal framework.
2. We efficiently solve bridge functions for all treatments with only single optimization.
3. We present the convergence analysis of our estimator in terms of MSE.
4. We demonstrated the utility of our estimator on two synthetic data and real data.
2 BACKGROUND
Proximal Causal Learning. The proximal causal learning (PCL) can be dated back to Kuroki & Pearl (2014), which established the identification of causal effects the presence of unobserved con- founders under linear models. Then Miao
Than et al. (201 8a b) and its extensions (Shi et al., 2020 Tchet- gen et al. 2020) proposed to leverage two proxy variables for causal identification by estimating the outcome bridge function. Building upon this foundation, Cui et al. (2023) introduced treatment bridge function and incorporated it into the Proximal Inverse Probability Weighting (PIPW) estima- tor. Besides, under binary treatments, they derived the Proximal Doubly Robust (PDR) estimator via influence functions. However, continuous treatments pose a challenge as the treatment effect is not pathwise differentiable with respect to them, preventing the derivation of a DR estimator. In this paper, we employ the kernel method that is provable to be consistent in treatment effect estimation. We further show that the kernel-based DR estimator be derived from influence functions.
Causal inference for Continuous Treatments. The most common approaches for estimating continuous treatment effects are regression-basee models (Imbens 2004 Hill (01I), generalized
propensity score-based models (Imbens) 2000 Hirano & Imbens 2004; Imai & Van Dyk 2004),
and entropy balance-based methods (Hainmueller [2012] & Ratkovic Tübbickkk 202T) Furthermore, (Knnedy et al. (2017); Kallus & Zhou (2018) and Colangelo & Lee (2020) extended the DR estimation to continuous treatments by combining regression based models and the gen- eralized propensity score-based models. However, it remains open to derive the DR estimator for continuous treatments within the proximal causal framework. In this paper, we fill in this blank with new -rrnnll-aseddaee DR estimator that is provable to derive from influence function.
Nuisance Parameters Estimation. In proximal causal learning, one should estimate nuisance pa- rameters to obtain the causal effect. Many methods have been proposed for this goal (Tchetgen et al.) 2020 Singh 2020 Xu et al., 2021 Kompa et al.] 2022), but they primarily focus on the estimation of the outcome bridge function. Recently, Kallus et al. (2021); Ghassami al. (2022) y have provided non non-maraettric estimates of treatment bridge function, but they are restricted to binary treatments. When it comes to continuous treatments, existing methods can be computationally inefficient since
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it has to resolve optimization problem for each treatment. In this paper, propose method that can efficiently solve bridge functions for all treatments with only a single optimization.
PROXIMAL CAUSAL INFERENCE
Problem setup. We consider estimating the Average Causal Effect (ACE) of a continuous treatment on an outcome Y: [a)[, where (a) for any E supp(A) denotes the potential outcome when the treatment A is received. We respectively denote X and U observed covariates and unobserved confounders. To estimate B[Y(a)], we assume the following consistency assumptions that are widely adopted in causal inference (Peters et al.) 2017p:
Assumption 3.1 (Consistency and Positivity). We assume (i) Y(A) almost surely (a.s.); and (ii) 0 < p(A = alU = X z) < 1 a.s.
Assumption 3.2 (Latent ignorability). We assume (a) 1 A
Assump. 3.2 means that the strong igmorability condition may fail due to the presence of un- observed confounder U. To account for such a confounding bias, the proximal causal learning incorporates a treatment- innducccnn- proxy and an outcome-inducing proxy W. As illustrated in Fig. I these proxies should satisfy the fol- lowing conditional independence:
Progy vazable
Assumption 3.3 (Conditional Independence of Proxies). The pratment-inducing proxy Z and the outcome-inducing inducing proxy W satisfy the fol- lowing conditional independence: (i) Y Z A, X; and (ii) W 1 (A, 2) U.X.
Figure I: Illustration of causal DAG proximal
causal learning, where Z, W are proxy variables.
Equipped with such conditional independence, previous work by Miao et al. (2018)); Cui et al. 2023 demonstrated that we can express the causal effect, denoted as B(a). as follows:
E [Y (a)] EE (a, W, X]] IE [L(A aa, Z, X))], where hg and do are two nuisance/bridge functions such that the following equations hold:
R₂ (ho: y) B[Y - ho,, W, X) A, Z, X] = 0,
(2)
(10: p) := E [uo A, Z, X) 1/p(A//: X))A, W,, X] 0.
To ensure the existence and uniqueness of solutions to the above equations, we additionally assume that (Miao et al. 2018a Tchetgen et al. 2020 Cui al. 2023):
Assumption 3.4. Let v denote any square-integren funttrrll For any (a, x), we have
1. (Completeness for outcome bridge functions). 0 iff v(U) = 0 almost surely. 2. (Completeness for treatment bridge functions). E[p(W)) 2; iff (U) = almost surely.
We assume that Ely(U))W, 0 and
a, x] Elu(Z))
We assume that E(v(U) 0 and
Under assump. 3.4 we can solve hu and via several optimization approaches derived from con- ditional moment equations, including two-stage penalized regression (Singh 2020 Mastourí al.)
1 Xu et al. 2021), maximum moment restriction (Zhang et al. [2200] Muandet et al. 2020a). and minimax optimization (Dikkala et al., 2020) Muandet et al. 2020b; Kallus et al. 2021). solved ho, 9o.
With
estimate E{())) (a) via:
1 n Σ ho (a,, wi, 4;), or [a)]
1 Σ I(a; = a) a)qo(a, 2,, xx)y.. n i=1
E, Y (a)]
Furthermore, Cui al. (2023) proposes a doubly robust estimator improve robustness against misspecification of bridge functions.
(a)) = TT[T(A A a a) (a, 7, X, (Y -ho(a, W, X)) + ho W, X)),
(4)
= - 24) ho w., x:)) + (a, We x:))
(5)
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Although this proximal learning method can efficiently estimate (a)] for binary treatments, it suffers from several problems when comes to continuous treatments. First, for any a E supp(A), it almost surely holds that there does not exist any sample that satisfies a; for i = I,...,n making Eq. | infeasible. Besides, it is challenging to derive the influence function for continuous treatments it involves the derivative computation for implicit functions hg and ạo- Lastly, to estimate 9o. previous methods suffered from a large computational cost since they had to re-run the optimization algorithm for each new treatment, making inapplicable in real-world applications.
To resolve these problems for continuous treatment, we first introduce a kernel-based method in Sec. 4 which can estimate (a) in a feasible way. Then in Sec. 5 we introduce a new opti- mization algorithm that can estimate ho, qu for all treatments with single optimization algorithm. Finally, we present the theoretical results in Sec. 6
4 PROXIMAL CONTINUOUS ESTIMATION
In this section, introduce kernel-based doubly robust estimator for B(a) (a) with continuous treatments. We first present the estimator form in Sec. S.1] followed by Sec. 4.2 to show that such estimator well approximate the influence function for ((a).
4.1 KERNEL-BANE BASSD PROXIMAL ESTIMATION
As mentioned above, the main challenge for continuous treatments lies in the estimation infcasibility caused by the indicator function in the proximal inverse probahility weighted estimator (PIPW) with go: ((a) n2 E I(a; a)qo(e (a, 2i, T:))i. To resolve this problem, we note that the indicator function can be viewed as a Dirac delta function 5a (a,)) The average of this Dirac delta function over n samples 1 Ex= (a,) approximates the marginal probability P(a) (Doucet al. 2009).
n which equals to Ô when A is continuous.
To address this problem, we integrate the kernel function K(A a) that can alleviate the unit concentration of the Dirac delta function. We can then rewrite the PIPW estimator as follows, dubbed as Proximal Kernel Inverse Probability Weighted (PKIPW) estimator:
B(a)
Σ A=1
Know (a: -a)4o 24, 2;)0;
(6)
where haw >> is the bandwidth such that Knn.. (a; (0;-00 huw
K D=-0
The kernel function
K K (A-o) a) that has been widely adopted in density estimation, assigns a non-zero weight each sample, thus making it feasible to estimate 3(a). To demonstrate its validity, we next show that can approximate (a) well. This result requires that the kernel function K is bounded differentiable, as formally stated in the following.
Assumption 4.1. The second-order symmetric kernel function K (-) is bounded differentiable, i.e., k(()dd = f uk(u)du O, (2( u =²)(())du < oo. We define Di (("((K) (k)))
Assump. 4. adherrs to the conventional norms within the domain of nonparametric kernel estima- tion and maintains its validity across widely adopted kernel functions, including but not limited to the Epanechnikov and Gaussian kernels. Under assump. 4. we have the following theorem:
Theorem 4.2. Under assump. 4.1 suppose 3(a) = E (IA A a)qu(a, Z, X)Y] is continuous and bounded uniformly respect to a, then we have
(a) = E[[(A = a) a)4 X.X) lim E [ hm (A a) X,X)), how
Remark 4.3. The kernel function has been widely used in machine learning applications (Kallus & Zhou 2018 Kallus & Uehara 2020; Colangelo & Lee. 2020: Klosin 2021). Different from these works, we are the first to integrate them into the proximal estimation to handle continuous treatments.
Remark 4.4. The choice of bandwidth how is trade-off between bias and variance. When how is small, the kernel estimator has less bias as shown Thm. [4.2] however, will increase the variance. In Sec. 6, show that the optimal rate for low is O(n 11/9), which leads to the MSE converges rate of O( 4/5j for our kernel-based doubly robust estimator.
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Similar to Eq. we can therefore derive the Proximal Kernel Doubly Robust (PKDR) estimator
a
B(a)
Σ
K
(yx ho(a, 24)) go 2i, 2ra) ha W2, mn)).
nhbw
how
i=1
Similar to Thm. 4.2 we can also show that this estimator is unbiased as hbw + 0. In the subsequent section, we show that this estimator in Eq. 7 can also be derived from the smooth approximation of the influence function of ((a).
4.2 INFLUENCE FUNCTION UNDER CONTINUOUS TREATMENTS
In this section, employ the method of Gateaux derivative (Carone et al. 2018 Ichimura & Newey 2022) to derive the influence function of {(a). (For our non-regular parameters, we borrow the terminology "influence function" in estimating regular parameter. See Hampel Ichimura & Newey 2022), for example.) Specifically, we denote x as the distribution function for any variable X. and rewrite 3(a) {(;; 1P where Po denotes the true distribution for O == (A, Z, W, X, Y)) Besides, we consider the special submodel Tp (1 - e)T + <Po where Pr (-) maps a point o to a distribution of O, i.e., 1 (o) for a fixed o denotes the distribution of O that approximates to a point mass at o. Different types of Fhe (o) lead to different forms of Gateaux derivative. In our paper, we choose the distribution Palt (o) whose corresponding probability density function (pdf) Po Nt- (o) = Kew (0 - (0 ())() > how)., which has littoww -0 Po hbw (o) = limme -+0 Know ()-
We can then calculate the limit of the Gateaux derivative (Ichimura & Newey 2022) of the functional
3(a; (peh)w) ) with respect to deviation E/o 1 The following theorem shows that our kernel-
based doubly robust estimator corresponds to the influence function:
Theorem 4.5. Under a nonpurametric model, the limit of the Gateaux derivative is
lim how
① de
B(a;pehow)
B(a; pettow)
-Y ho(a, W, X)) (a X) lim Krow (A - a) + ho (a, W. - ((a)
c=0
hiw -0
Remark 4.6. For binary treatments, the DR estimator with the indicator function in Eq. corresponds
to the efficient influence function, derived within the non- non-parametric framework (Cui et al. 2023).
Different from previous works [Colangelo & Lee 22200), deriving the influence function within the proximal causal framework is much more challenging it involves the Gateau derivatives for nui- sance functions ho, that have implicit functional forms. By employing our estimator, even when the unconfoundedness assumption from Colangelo & Lee (2020) is not satisfied, we can still effec- tively obtain causal effects.
5 NUISANCE FUNCTION ESTIMATION
In this section, we propose to solve ha, qu from integral equations Eq. for continuous treatments. We first introduce the estimation of 4o- Previous methods (Kallus et al. 2021 Ghassami et al. 2022) solved go (a, Z, X) by running an optimization algorithm for each a 0, 1. However, it is computa- tionally infeasible for continuous treatments. Please see Appx. D.2 for detailed comparison. Instead of running an optimization for each a, we would like to estimate qo A, Z, X) with a single optimiza- tion algorithm. To achieve this goal, propose two-stage estimation algorithm. We first estimate the policy function p( A [w, 2) and plug into Eq. To efficiently solve qo. we note that it is equivalent to minimize the residual mean squared error denoted as (g p) = E[ (q. (e.)))', According to the lemma shown below, such a mean squared error can be reformulated into a maximization-ste -syle optimization, thereby converting into a min-max optimization problem.
Lemma 5.1. Denote [[ Il2, [x)] For any parameter Am > O, have
Cg(gp)) sup E [m([, W, X) (,o(A, - 1/p(A|W, X))] Am || W, W)I),
meM
where M is the space of continuous functions over (A, X)..
We leave the proof in Appx. D Motivated by Lemma. 5.1 can solve qo via the following min-
max optimization:
min max can.m. m) Σ
(g(as,
(g(a), 21, 23)
p(aslw,, m(a;, Wi, Am [m |m 12,00
1
Q MEM
(8)
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where Am Am/mm//2. is called stabilizer with |m : E m² ma W/ 21). We parameterize
q and m as reproducing kernel Hilbert space (RKHS) with kernel function to solve the min-max problem. We derive their closed solutions in the Appendix [F] Besides, we can also use Generative Adversarial Networks (Goodfellow et al. 2014) to solve this problem.
Estimating the policy function p(A al To optimize Eq. 8 should first estimate
p(a/w, z).. Several methods can be used for this estimation, such as the kernel density estimation and
normalizing flows (Chen 2017 Bishop 1994 Ambrogioni et al. 2017; Sohn al. 2015; Rezende et al. 2016). In this paper, we employ the kernel density function (Chen) 2017) that has been shown to be effective in low-dimension scenarios. When the dimension of XW.X) is high, we employ the conditional normalizing flows (CNFs), which have been shown to be universal density approximator (Durkan et al. 2019) and thus can be applied to complex scenarios.
& Mohamed,
Nuisance function hg- Since the estimation of ho does not involve indicator functions, we can apply many off-the-shelf optimization approaches derived from conditional moment equations, such as two-stage penalized regression (Singh, 2020; Mastouri al. 2021] Xu et al. [021), maximum moment restriction (Zhang et al. 2020; Muandet et al.) 2020a), and minimax optimization (Dikkala et al. 2020; Muandet et al. (020b). To align well with yo- here we choose to estimate ho via the following min-max optimization problem that has been derived in Kallus et al. (202I)
min max " (h, g) == n kEH gEG
Eo(a,, 2,, 24) (v h(a;, W/, x.) >
where H and G respectively denote the bridge functional class and the critic functional class.
6 THEORETICAL RESULTS
In this section, we provide convergence analysis of Eq. 8 991 for nuisance functions ho, qu, well as for the causal effect ((a) with the PKDR estimator in Eq.
We first provide convergence analysis for go. while the result for hg is similar and left the Appx. B Different from previous works Dikkala et al. (2020); Ghassami et al. (2022), our analysis encounters
significant challenge arising from the estimation error inherent in the propensity score function. By addressing this challenge, our result can effectively account for this estimation error.
Formally speaking, consider the projected residual mean squared error (RMSE) E(proi,, {o)3]. where (-) == E [-A,, X.. Before presenting our results, we first introduce the as- sumption regarding the critic functional class in M, which has been similarly made in Dikkala et al. (2000) Ghassami et al. (2022): Qi et al. (203).
oo and p is uniformly bounded; (2) (Symmetric) M is a symmetric class, i.e, if m E M, then -m E M: (3) (Star-shaped) M is star-shaped class, that is for each function m in the class, am for any a E [0, 1] also belongs to the class; (4) (Realizability) qo E Q: (5) (Closedness)
Assumption 6.1. (I) (Boundness) |
Drm proj, %)) E M.
Under assumption 6.1 we have the following convergence result in terms of || ( 9o 1o)ll
Theorem 6.2. Let 8% respectively be the upper bound on the Rademacher complexity of M. For
any n E (0, 1). define 89 = 5% + <.0 /ogtes//) for some constants cg. ci; then under assump. 6.1 we have with probability 1 - that
|rooo (Ā 1-rooll22 O
80 80'\\8 12
+ Am +
1
1
1 1 P
P stands for plappl//pppaooppa ((alw,aa x).
P
Remark 6.3. Inspired by Chen & Pouzo 2012):
Dikkala et al. (2020): Kallus et al. 2021 we can up to measure of ill-posedness denoted as 40 2ol12,
obtain the same upper bound for the RMSE l
Tq == sup ||q 40 //lll///lroii((((
yo)ll2 < xc.
The bound mentioned above comprises components. The first part pertains to the estimation of q. while the second part concerns the estimation of 1/p. The first part is mainly occupied by the Rademacher complexity 5g. which can attain O(n 1/44 if we parameterize M as bounded met- ric entropy such Holder balls, Sobolev balls, and RKHSs. For the second part, can also
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achieve 0/4= for 1/pp 1/2l|2 under some conditions (Chernozhukov et al. 2022 Klosin
2021 Colangelo & Lee 2020).
Now we are ready to present the convergence result for B(a) within the proximal causal framework.
Theorem 6.4. Under assump. 3.1 3.4 and 4.1 suppose I| - hl22 = o(1), llq - all = o(1) and || hlllll 4ll2 = (((nhbw) 1/2), nhe = O(1), nhow - oc. ho w, w, p(a, 2 2/w, x) and p(a, w|2, w12,,) x) are twice continuously differentiable wrt well as hu, qu, h, 9 are uniformly bounded. Then for any a, we have the following for the bias and variance of the PKDR estimator given Eq. 7
K2 (K) (KKK))BB(iliiiii +o((n/bb)) ) Varl3(a) 922(K) (V + +o()),
Biaas =
=
2
where Elqu(a, 2) L ho W, x) BAp(a, W 2,X)+ + a)qo(a, a, (Y ha(a, W, W)))
8² 0.42 ha W. X)I, V =[[AA
Remark 6.5. The smoothness condition can hold for broad family of distributions and be thus similarly made for kernel-based methods (Kallus & Zhou 2018 Kallus & Ucbara (020)_ According to Thm. 6.2 we have I holl2 = OO and |1@ 22 = O(n-1/4 thus can satisfy the consistency condition required as long as how = (1). Besides, we show in Thm. E.9 in Appx. E.5 that this estimator -
From Thm. 6.4 we know that the optimal bandwidth is /tw = (1/5)) in terms of MES that converges at the rate of Onn1//5. Note that this rate is slower than the optimal rate O(n=)), which is reasonable sacrifice to handle continuous treatment within the proximal causal framework and agrees with existing studies (Kennedy et al. 2017
Colangelo & Lee 2020).
EXPERIMENTS
In this section, we evaluate the effectiveness of our method using two sets of synthetic data one in low-dimensional context and the other in a high-dimensional context as well as the legalized abortion and crime dataset (Donohue III & Leviti 2001). In Appx. G we conduct experiments on more benchmark datasets, including time-series forecasting.
Compared baselines. We compare our method with the following baselines that use only ho for es- timation, i.e., 3(a) = 1 Ex ho(a, W4, ): (i) Proximal Outcome Regression (POR) that solved Eq. [9 for estimation; ti) PMMR (Mastouri et al.) I that employed the Maximal Moment Re- striction (MMR) framework to estimate the bridge function via kernel learning; iii) KPV (Mastouri et al. (021) that used two-stage kernel regression; iv) DFPV (Xu et al., (021)) that used deep neu- ral networks to model high-dimensional nonlinear relationships between proxies and outcomes; v) MINMAX (Dikkala al. 2020)) that used Generate adversarial networks to solve Eq. 9 vi) NMMR Kompa et al. 2022) that introduced data-adaptive kernel functions derived from neural networks.
For our method, we implement the Inverse probability weighting (IPW) estimator PKIPW that uses qo for estimation via Eq. [6 and the doubly robust estimator PKDR that used both the nuisance function hg and go to estimate causal effects through Eq. 7 For simplicity, we only present the result of PKDR that uses POR to estimate hg-
Implementation Details. In the PKIPW and PKDR estimators, we choose the second-order Epanechnikov kernel, with bandwidth how = cō AN 1/5 with estimated std ã and the hyperpa- rameter e > O. In our paper, we vary e over the range (0.5, 1, 1,1, ,.0} and report the optimal in terms of cMSF. To estimate nuisance functions, we parameterize Q and M (resp., H and g) via RKHS for go (resp., ho), where we use Gaussian kernels with the bandwidth parameters be- ing initialized using the median distance hcuristic. For policy estimation, we employ the KDE in the now-dimensional synthetic dataset and the real-world data, while opting for CNFs in the high- dimensional synthetic dataset. We leave more details about hyperparameters in the Appx. H
Evaluation metrics. We report the causal Mean Squared Error (cMSE) across 100 equally spaced points in the range of supp (A): cMSE == 100 D (E E Here, we respectively take sApp(A) := 11, 1, 2], [0, 1], [0, 2] in low-dimensional synthetic data, high-dimensional synthetic data, and real-world data. The truth [[|""] is derived through Monte Carlo simulations comprising 10,000 replicates of data generation for each a.
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Sample 1030
Sample size 2000
Sample
N0W
Deatmaat
Tea
Figure 2: ATE comparison of different methods across various methods on three datasets; Left: ATE comparison using 1000 samples in the first experiment: Middle: ATE comparison using 2000 samples in the second experiment; Right: ATE comparison for the abortion and crime dataset.
7.1 SYNTHETIC STUDY
We consider two distinct scenarios. The first scenario demonstrates the effectiveness of the kernel method in the context of the doubly robust estimator under model misspecification, while the second scenario evaluates the utility in high-dimensional settings. For both scenarios, we report the mean cMSE of each method across 20 times.
7.1.1 DOUBLY ROBUSTNESS STUDY
Data generation. We follow the generative process in Mastouri et al. (0021) and leave details in the Appx. H. Similar to Kang & Schafer (2007); Cui et al. (202)), , we consider four scenarios where either or both confounding bridge functions are misspecified by considering a model using a transformation of observed variables:
Scenario 1. We follow Mastouri et al. 2021 to generate data;
Scenario 2. The outcome confounding bridge function is misspecified with W |W|1/2 + 1;
Scenario 3. The treatment confounding bridge function is misspecified with 2 = 12|1/2 + 1;
Scenario 4. Both confounding bridge functions are mis-specified.
Table cMSE of all methods on two synthetic data and the real-world data.
Datasct
Size
PMMR
KPV
DFPV
MINMAX
NMMR
POR
PKIPW
PKDR
500
0. 1..600
0.37
0.30
0.20
0.20
0.10
0.11
Scenario
1000
g,14
0.21
0.26
0.10
0.25
0.16_.010
0,11
0.08 na
500
2.02
3.50
1.03
7.48
4.72
3.47=0.08
0.16g0
0.16
Scenario
Doubly Robust
1000
3.32 Looot
3. 10:19
0.97
5.27 1.27
6.71
3.48
0.20
Scenario 3
500
0.15
1.29
0.28
0.33
0.38
0.20_..11
2,38
0.20
1000
0.14
0.20 2.20
0.30
0.27
0.47gaar
0.22-0.14
2. 2.15ga.0
0.19g010
3.32 Luos
3.51
1.00.ozz
6.69
5.000
3.46_
1
3.38
Scenario
1000
3.20
3.46
1 1.02=0.3
5.1040.00
5.65
3.44
1.874001
3.56
High Dimension
1000
0.74g0.00
1.04
0.22go.11
0.12
0.14
0.31_ao.
0,200
0.08
2000
0.69 su.0
0.24
0.07
0. 1,05
0.30
0.19
0. 0.00
Abort. & Crim
1500
0.02
0.03
0.07
0.05
0.01
0.01 =0.00
0.04
0.0220.00
Results. We present the mean and the standard deviation (std) of cMSE over 20 times across four scenarios, as depicted in Fig. 2 and Tab. [ For each scenario, we consider two sample sizes, 500 and 1,000. In the first scenario, on h. For scenarios with misspecification, the PKIPW estimator and the baselines with only ho respectively perform well in scenario 2 and scenario 3. Notably, the PKDR can constantly perform well in these scenarios, due to its doubly robustness against model mis-specifiicatins. In scenario 4 where both models of hy and qu misspecified, all methods suffer from large error. Besides, we can see that the PKIPW method has a large variance in scenario 4, where both estimations of the policy function and go be inaccurate due mis-specifications (Robins al. [0077] [iang al.) 2022). It is worth mentioning that compared to others, DFPV exhibits minimal errors in scenario This could be attributed their approach of individually fitting each variable's 's kernel function using different neural networks, thereby enhancing flexibility in their models.
PKDR is comparable and even better than the estimator based
Sensitivity Analysis. According Thm. 6.4 how is the trade-off between bias and variance. To show this, we report the eMSE in how Aă 115 varies in (0.5, 1.0, 1.5, 1.,,... 4.0). As (i.e.,
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hbw) increases, the cMSE first decreases, then rises, and reaches its optimum at 1.5, which is consistent with the optimal value derived in Kallus et al. (2021).
PRDB Coumple s2e 5015
PRDS mple 500g
10
Cuust
Cousa
13
bandwid
bandw
Figure 3: Sensitive analysis of e in how Cō An in PKIPW (left) and PKDR (right) estimators.
7.1.2 HIGH DIMENSIONAL STUDY
Data generation. We follow Colangelo & Lee (2020): Singh (2020) to generate data, in which we
set dim(X) 100, dim (Z) 10, and dim
10. Specifically, set X (0, E) with
= for i * The outcome Y is generated from Y a + 1.2.4 + 1.2(XX) 8r W B + AX1 + 0.25U, where 8z, S exhibit quadratic decay, i.e., [3,1] --- More details can be found in the Appx. H
E E IR 10000000 has E₁ = for E [dim( [ and Enj
report the mean and std of cMSE over 20 times with sample sizes set to 1.000 and 2,000, as depicted in Fig. 2 and Tab. [T] As shown, we find that the ATE curve fitted by PKDR estimator is closest to the real curve, and its cMSE is also the lowest. This result suggests the robustness of our methods against high-dimensional covariates.
Results.
7.2 LEGALIZED ABORTION AND CRIME
We obtain the data from Donohue III & Levitt (2001) >; Mastouri et al. (2021) that explores the rela- tionship between legalized abortion and crime. In this study, we take the treatment as the effective abortion rate, the outcome variable Y as the murder rate, the ieaauunn-innuuiig proxy Z as the generosity towards families with dependent children, and the outcome-inducin proxies W as beer consumption per capita, log-prisoner population per capita, and concealed weapons laws. We follow the protocol Woody et al.. (2020) to preprocess data, We take the remaining variables as the unob- served confounding variables U. Following Mastouri FollowingMMastouri
al. (2021) the ground-trun value of B(a) is taken from the generative model fitted to the data.
Results. The results are presented in Fig. and Tab. It evident that all three methods effectively estimate 3(a), which suggests the utility our method in real-world scenarios. However, when falls within the range of [1.5, 2), deviations become apparent in the fitted curve. We attribute these deviations to an inadequate sample size as Fig. 2 It's worth noting that the DFPV method employing Neural Networks (NN) exhibits higher variances. This suggests potential numerical instability in certain experiments, a phenomenon in line with observations made in Kompa et al. (2022).
8 CONCLUSION
In this paper, propose kernel-based doubly robust estimator for continuous treatments within the proximal causal framework, where we replace the conventional indicator function with a kernel function. Additionally, we propose more efficient approach to estimating the nuisance function qo by estimating the policy function and incorporating it into min-max optimization. Our analysis reveals that the MSE converges at a rate of On (4-4) when we select the optimal bandwidth to balance bias and variance. We demonstrate the utility of our PKDR estimator in synthetic well as the legalized abortion and crime dataset.
Limitation and future works. Our estimator is required estimate the policy function, which may lead to large variance especially when the policy function is mis-specified. Potential solutions include the variance reduction method including the stabilized IPW estimator, whose estimation forms and theoretical analysis will be explored in the future.

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9 ACKNOWLEDGMENTS
This work was supported by the National Key Research and Development Program of China (No. 2022YFC2405 100); STI 2030-Major Projects (No. 20222D0205300;; Shanghai Municipal Sci- ence and Technology Major Project No.22211H222XX1033 the State Key Program of National Natural Science Foundation of China under Grant No. 12331009. The computations in this research were performed using the CFFF platform of Fudan University.
REFERENCES
Luca Ambrogioni, Umut Güçlü, Marcel AJ van Gerven, and Eric Maris. The kernel mixture net- work: A nonparametric method for conditional density estimation of continuous random vari- ables. arXiv preprint arXiv: 1705.07111, 2017.
Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, (44)9662,973, 2005.
Christopher M Bishop. Mixture density networks. 1994.
Marco Carone, Alexander R Luedtke, and Mark van der Laan. Toward computerized efficient estimation in infinite-dimensione models. Journal of the American Statistical Association, 2018.
Marine Carrasco, Jean-Pierre Florens, and Eric Renault. Linear inverse problems in structural econo- metrics estimation based on spectral decomposition and regularization. Handbook of economet- rics, 6:5633-5757 2007.
Xiaohong Chen and Demian Pouzo. Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica, 80(1):277-321, 2012. 2012.
Yen-Chi Chen. A tutorial on kernel density estimation and recent advances. Biostatistics Epi- demiology, 1(1):161-187, 2017.
Victor Chernozhukov, Whitney K Newey, and Rahul Singh. Automatic debiased machine learning of causal and structural effects. Econometrica, 90(3):66-1002,, 2022.
Kyle Colangelo and Ying-Ying Lee. Double debiased machine leamning nonparametric inference with continuous treatments, arXiv preprint arXiv:2004. 2000000000.003 03036, 2020.
Yifan Cui, Hongming Pu, Xu Shi, Wang Miao, and Eric Tchetgen Tchetgen. Semiparametric prox- imal causal inference. Journal of the American Statistical Association, pp. 1-12, 2023.
Nishanth Dikkala, Greg Lewis, Lester Mackey, and Vasilis Syrgkanis. Minimax estimation of con- ditional moment models. Advances in Neural Information Processing Systems, 33: 12248-2222, 2020.
Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. arXiv preprint /605v /60.00803,, 2016.
John J Donohue III and Steven D Levitt. The impact of legalized abortion on crime. The Quarterly Journal of Economics, 116(2):379-42) 420, 2001.
Arnaud Doucet, Adam M Johansen, et al. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704):3, 2009.
Conor Durkan, Artur Bekasov, lain Murray, and George Papamakarios. Neural spline flows. Ad- vances in neural information processing systems, 32, 2019.
Dylan J Foster and Vasilis Syrgkanis. Orthogonal statistical learning. arXiv:1901.09036 2019.
arXiv preprint
AmirEmad Ghassami, Andrew Ying, Ilya Shpitser, and Eric Tchetgen Tchetgen. Minimax kernel machine learning for class of doubly robust functionals with application to proximal causal inference. In International Conference on Artificial Intelligence and Statistics, pp. 7210-7239.
PMLR, 2022.
10

---

Published as a conference paper at ICLR 2024
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farkes Sherjil Ozair,
Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Advances in neural information processing systems, 27, 2014.
Jens Hainmueller. Entropy balancing for causal effects: A multivariate reweighting method to pro-
duce balanced samples in observational studies. Political analysis, 20(1):25-46, 2012.
Peter Hedström and Petri Ylikoski. Causal mechanisms in the social sciences. Annual review of sociology, 36:49-67, 2010.
Jennifer I. Hill. Bayesian nonparametric modeling for causal inference. Journal of Compulational and Graphical Statistics, 20(1):21-2240, 2011.
Keisuke Hirano and Guido W Imbens. The propensity score with continuous treatments. Applied Bayesian modeling and causal inference from incomplete-dar data perspectives, 226164:73 84, 2004.
Jie Kate Hu, Dafne Zorzetto, and Francesca Dominici. A bayesian nonparametric method adjust for unmeasured confounding with negative controls. arXiv preprint arXiv:2309.026311 2023.
Hidehiko Ichimura and Whitney K Newey, The influence function of semiparametric estimators.
Quantitative Economics, 13(1))29-61, 2022.
Kosuke Imai and Mare Ratkovic. Covariate balancing propensity score. Journal of the Royal Statis- lical Society Series B: Stutistical Methodology, 76() ((1):2433223) 2014.
Kosuke Imai and David A Van Dyk. Causal inference with general treatment regimes: Generalizing the propensity score. Journal of the American Statistical Association, 99(467))854-86,, 2004.
Guido W Imbens. The role of the propensity score in estimating dose-response functions.
Biometrika, 87(3)700--70, 2000.
Guido W Imbens. Nonparametric estimation of average treatment effects under exogeneity: A
review. Review of Economics and statistics, 86(1):4-3 2004.
Kuanhao Jiang, Rajarshi Mukherjee, Subhabrata Sen, and Pragya Sur. A new central limit theorem for the augmented ipw estimator: Variance inflation, cross-fit covariance and beyond. arXiv preprint arXiv:2205.101900 10198, 2022.
Nathan Kallus and Masatoshi Uehara. Doubly robust off-policy value and gradient estimation for deterministic policies. Advances in Neural Information Processing Systems, 33:10420-10430, 2020.
Nathan Kallus and Angela Zhou Policy evaluation and optimization with continuous treatments. In International conference on artificial intelligence and statistics, pp. 1243-1251. PMLR, 2018.
Nathan Kallus, Xiaojie Mao, and Masatoshi Uehara. Causal inference under unmeasured confound- ing with negative controls: A minimax learning approach. arXiv preprint arXiv:2103.14029 2021.
Joseph DY Kang and Joseph L Schafer. Demystifying double robustness: A comparison of alterna- tive strategies for estimating population mean from incomplete data. 2007.
Edward H Kennedy, Zongming Ma, Matthew D McHugh, and Dylan S Small. non-paraettric meth- ods for doubly robust estimation of continuous treatment effects. Journal of the Royal Statistical Suciety Series B: Statistical Methodology, 79(4): 2017.
Sylvia Klosin. Automatic double machine learning for continuous treatment effects. arXiv preprint arXiv:2104. 10334, 2021.
Benjamin Kompa, David Bellamy, Tom Kolokotrones, Andrew Beam, et al. Deep learning meth- ods for proximal inference via maximum moment restriction. Advances in Neural Information Processing Systems, 35:1 1 189-1 1201, 2022.
Manabu Kuroki and Judea Pearl. Measurement bias and effect restoration in causal inference. Biometrika, 101(2):423-433 2014.
11

---

Published as a conference paper at ICLR 2024
Afsanch Mastouri, Yuchen Zhu, Limor Gultchin, Anna Korba, Ricardo Silva, Matt Kusner, Arthur Gretton, and Krikamol Muandet. Proximal causal learning with kernels: Two-stage estimation and moment restriction. In International conference on machine learning, pp. 7512-75225 PMLR, 2021.
Wang Miao, Zhi Geng, and Eric J Tccetten Tchetgen. Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika, 105(4):987-993, 2018a.
Wang Miao, Xu Shi, and Eric Tchetgen Tchetgen. A confounding bridge approach for double negative control inference on causal effects. arXiv preprint arXiv: 04945, 2018b.
Krikamol Muandet, Wittawat Jitkrittum, and Jonas Kübler. Kernel conditional moment test via maximum moment restriction. In Conference on Uncertainty in Artificial Intelligence, pp. 41-50.
PMLR, 2020a.
Krikamol Muandet, Arash Mehrjou, Si Kai Lee, and Anant Raj. Dual instrumental variable regres- sion. Advances in Neural Information Processing Systems, 33:2710-2721, 2020b.
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.
Zhengling Qi, Rui Miao, and Xiaoke Zhang. Proximal learning for individualized treatment regimes under unmeasured confounding. Journal of the American Statistical Association, pp. 1-14, 2023.
Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Interna- tional conference on machine learning, pp. 1530-1538. PMLR, 2015.
James Robins, Mariela Sued, Quanhong Lei-Gomez, and Andrea Romnitzky. Comment: Perfor- mance of double-robust estimators when" inverse probability" weights are highly variable. Sta- tistical Science, 22(4):544-559 2007.
Bernhard Schölkopf, Ralf Herbrich, and Alex J Smola. A generalized representer theorem. In International conference on computational learning theory, pp. 416-426. 426. Springer, 2001.
Xu Shi, Wang Miao, Jennifer Nelson, and Eric J Tchetgen Tchetgen. Multiply robust causal infer- ence with double-negative control adjustment for categorical unmeasured confounding. Journal of the Royal Statistical Society Series B: Statistical Methodology. 92(2):521-544 2020.
Rahul Singh. Kernel methods for unobserved confounding: Negative controls, proxies, and instru-
ments. arXiv preprint arXiv:20/2. 10315, 2020.
Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. Advances in neural information processing systems, 28, 2015.
Eric Tehetgen Tchetgen, Andrew Ying. Yifan Cui, Xu Shi, and Wang Miao. An introduction to proximal causal learning. arXiv preprint arXiv:2009. 2020.
Stefan Tübbicke. Entropy balancing for continuous treatments. Journal of Econometric Methods, 11(1):71-89 2021.
Hal R Varian. Causal inference in economics and marketing. Proceedings of the National Academy of Sciences, 113(27):310--315, 2016.
Martin J Wainwright. High-ddiennsioal statistics: A non-asymptotic viewpoint, volume 48. Cam-
bridge university press, 2019.
Spencer Woody, Carlos M Carvalho, P Richard Hahn, and Jared S Murray. Estimating heteroge- neous effects of continuous exposures using bayesian tree ensembles: revisiting the impact of abortion rates on crime. arXiv preprint arXin:2007.098845 2020.
Liyuan Xu, Heishiro Kanagawa, and Arthur Gretton. Deep proxy causal learning and its application to confounded bandit policy evaluation, Advances in Neural Information Processing Systems, 34: 2021.
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Published as a conference paper at ICLR 2024
Afsanch Yazdani and Eric Boerwinkle. Causal inference in the age of decision medicine. Journal of data mining in genomics & proteomics. 6(1), 2015.
Andrew Ying, Wang Miao, Xu Shi, and Eric Tchetgen Tchetgen. Proximal causal inference for complex longitudinal studies. Journal of the Royal Statistical Society Series B: Statistical Method- ology, 83(3):684-704 2023.
Rui Zhang, Masaaki Imaizumi, Bernhard Schölkopf, and Krikamol Muandet. Maximum moment restriction for instrumental variable regression. arXiv preprint arXiv:2010.07684, 3(4), 2020.
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APPENDIX
Preliminaries
15
A.1 Notation
15
A.2 Critical Radius
16
Related Works
17
Regularity condition
18
D Estimating nuisance function
19
D.I. Proof of Lemma 5.1
19
D.2 Comparison of existing methods
20
Proofs and Derivation
22
Proof of Theorem
22
Proof of Theorem
22
Proof of Theeorm 6.2
24
Proof of Theorem 6.4
29
E.5 Consistency of the Estimator
35
Computation Details
38
Additional Experiments
40
G.
Experiments with Different Data Generating Process
40
G.2
Experiments for Time Series Data
41
Rate
H
Experiments
43
H.1
Data Generating Process in the Low-Dimensional Synthetic Experiment
43
H.2
Data Generating Process in the High-Dimensional Synthetic Experiment
43
H.3
Legalized Abortion and Crime
43
H.4
Hyperparameters Selection
44
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A PRELIMINARIES
A.I NOTATION
In this section, we will define some notations used throughout the proof in the appendix. Moreover, we will introduce other notations in the corresponding subsection.
Table 2: Table of Notations
Notation Z, W
Meaning
Treamenn-innuucnng proxy and outcome-inducing proxy Covariates and unobserved confounders
X,U
(a)
The potential outcome with A =
do(a)
do(A a), intervention variable As with value of ag
ATE at point a obtained by Eq. Average causal effect at point
1 or
B(a)
B(a)
Estimate of B(a)
ho = ho(a, x)
The nuisance/bridge function that satisfies Eq.
40 ho(a, 2,
The nuisance/bridge function that satisfies Eq. 3
K-))
Kernel function
(2)
f fu²k(u)du
265 (K)
Second moment of K, i.e.
Khow
Kernel function with bw, i.e. Know (u) h k(/ hb//)))) The distribution approach a point mass at O at how +
Pow (-)
pell (-)
The special submodel, i.e. pehe- (1 e) + slphe
(-)-)
The true distribution
B(; ((c/)))
A statistical functional that map the distribution pprtow to real
Q H (resp. M,
The bridge functional class (resp. The dual functional class) Projection operator, i.e. E [-|A, WX
(-)
R, (6:9)
The empirical Rademacher complexity the function class G
R., (6:
The population Rademacher complexity of the function class G
8n
The upper bound on the Rademacher complexity of the function class
(E, G, 11b0)
The size of the smallest empirical cover of G
(c,, 9. II)
The empirical metric entropy of g
pg 4h
Moment restriction, defined by Eq. 14
T 1
Moment restriction with regularization. defined by Eq. 12
T T
The empirical version of 4y Ph
Am n.
The empirical version of d. 2
E[-, P[-
The expectation and the probability distribution of a random variable
|| 12
VS (x)) di (x)
||0|||
Σ, |ed|
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A.2 CRITICAL RADIUS
The bound we provide is based on the critical radii of the function classes involved, as described in Sec. 66 These critical radius defined in terms of the empirical localized Rademacher critical radius, which characterizes the critical radius of a function class up to constant factor. Specifically, the empirical Rademacher complexity and population Rademacher complexity of function class G V + [-1, 1] is defined as follows:
(Ō; = B(a)%
sup @EG.
Gg (v:)
R,; = ((
Gg (va)
MEG.
where {u:} are i.idd samples from some distribution D on V and {e are i.i.d. Rademacher random variables taking values equiprobably {-1,1}. The empirical critical radius defined as any solution 6, to (6; g) < 8 and the critical radius of function class G is the smallest solution 5 to the inequality R. (6; 9) < 6². Proposition 14.1 of Wainwright (019 shows that w.p. 1 - S.
6n = 100(, +
log(1/{) n
Thus we can choose 6, based on the empirical critical radius 8, Moreover, Dikkala al. (2020) suggests using the covering number to obtain an upper bound on the empirical Rademacher com- plexity and thus the critical radius. We denote with (e, G, II- as the size the smallest empirical- cover of G. The empirical metric entropy of is defined as [e, G, || II) log(N G, || []). An
empirical d-slice of defined as G3 = {g E 9 ||g||2,n I 8). Then the empirical critical radius of G is upper bounded by any solution to the inequality:
H(c, Gs.
II)
8²
de S
(10)
82/8
20
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B RELATED WORKS
Regarding the proximal causal inference framework, there is a growing literature on using proxy variables for causal inference from observational data, due to its ability account for unmeasured U through two confounding proxy variables: a treatment-inducing proxy Z and an outcome-inducing proxy W, which are respectively independent to the outcome and the treatment.
The proximal causal learning (PCL) can be dated back to Kuroki & Pearl (2014), which established the identification of causal effects in the presence of unobserved confounders under linear mod-
els. Then Miao et al. (201&b) and its extensions (Shi et al. 2020 Tchetgen et al. 2020) proposed to leverage two proxy variables (Z, W) for causal identification by estimating the outcome bridge function (Eq. 2. Building upon this foundation, Cui et al. (2023); Ying et al. (2023) introduced treatment bridge function 3) and incorporated it into the Proximal Inverse Probability Weight- ing (PIPW) estimator. Besides, under binary treatments, they derived the Proximal Doubly Robust (PDR) estimator via influence functions. However, their methods cannot handle continuous treat- ments, whereas the treatments can be continuous in many real-world scenarios, including social science, biology, and economics. The main challenge for continuous treatments lies in the estima- tion infeasibility caused by the indicator function in two estimators. To the best of our knowledge, we are the first to generalize the PIPW estimator and PDR estimator to the continuous case by replacing the conventional indicator function with a kernel function. The advantage of using the kernel function is that we do not need to discretize the continuous data, but use the kernel function to incorporate local information about the similar treatments. Besides, we derive the corresponding influence function, a process that involves handling the Gateaux derivative of bridge functions.
Most existing work focuses on how estimate the outcome bridge function. Singh (2020) and Mastouri et al. (202T) propose to use a two-stage kernel estimator for the outcome bridge function (KPV), and Xu al. (202T) further improved upon this with an adaptive features derived from neural networks (DFPV). Besides, an alternative approach based maximum moment restriction (MMR) uses single-stage estimators of the bridge function. The masterpiece in this regard that
Mastouri al. (202T) extends the MMR framework to the proximal setting through the use of kernel functions (PMMR). Kompa et al. 2002)) introducex data-adaptive kernel functions derived from neural networks. Another traditional method is transform the conditional moment equation into an unconditional moment equation, and then solve a minimax optimization problem (Dikkala al. 2020 Ghassami et al. 2022 Qi et al. 2023) (MINIMAX, POR). However, these methods that estimate only the outcome bridge function, rather than also estimating the treatment bridge function, which would permit us construct doubly robust estimator. For the treatment bridge function, (Kallus et al. 2021 Ghassami et al., 2022) solved (a.a Z, X)) by running an optimization computationally infeasible for continuous treatments.
algorithm for each Instead of running optimization for each a, we would like to estimate (A, Z, X) with a single optimization algorithm. To achieve this goal, we propose a two-stage estimation algorithm. We first estimate the policy function and then convert the conditional moments into equivalent forms, which can lead min-max optimization problem.
0. However,
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C REGULARITY CONDITION
One conventional approach to studying their solutions is through singular value decompasition, as discussed by Carrasco et al. (2007). We first introduce the singular value decomposition of the operator:
Given Hilbert spaces Hi and H2, a compact operator K Hi + H2 and its adjoint operator K' H2 + H1. there exists singular system (Any Ph: Vn )=) of K with nonzero singular values {An} and orthogonal sequences (Pn H1} and {ea E H2) such that
Kin, Xn K' K'u,, An
By means of singular value decomposition, Picard's theorem characterizes the conditions for the existence of solutions of the corresponding Fredholm integral equations of the first type. We apply Picard's & theorem to the setting of proxy variables.
Let (P(x)) denote the space of all square integrable functions of æ with respect a cu- mulative distribution function P(x), which is Hilbert space with inner product (91,92) For brevity, we replace Waj and Zi with W and Z below. Let Ta. denote the operator: (P((a( x)} - (P( x)}, Taah E(h(W) z, a, x| and let (Aa,z,n, Pa,z, oa ) denote singular value decomposition of Ta.z- Also let To denote the operator: L2 xP)
9 (x))))))
(2 x)), Ty,29 Elg(Z)/u; a, æ. and let (Na Pa,a, d, denote singular value decomposition of a. We assume the following regularity conditions:
Assumption C.1. (Regularity conditions)
1. (Existence of compact operator Ta and To
/
alll oo;
2. (Existence of solutions)
1' < oo;
3. (Eigenvalue structure of compact operator Ta,x)
Na KE da) oo;
4. (Existence of solutions)
f f (tllw, watt x) < oo;
5. (Eigenvalue structure of compact operator T Tr.s)
Σ 2n (alw, () Va..,)) ao.
Theorem C.2 (Cui et al. (2023). functions hu a, 2)) such that
Under Assumption C.I 1,2,3) and Assumption 3.4 1), there exist
E Z. = A, A,2,,
almost surely.
Theorem C.3 (Cui et al. (2023), ). Under Assumption C. 1,4,5) and Assumption 3.4 there exist functions q(2, a, 2) such that
E
W. =a,
p(A al W.
almost surely.
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D ESTIMATING NUISANCE FUNCTION
We next return to how to solve the nuisance function q. For simplicity, define
1 w, (
(g) E
A, A,
Notice that (g. being zero almost surely is actually conditional moment equation, which is equivalent to finding such that the following residual mean squared error (RMSE) is minimized for all q:
min Lg(p:)) = E (R₂ (g.p))²
(11)
D.1 PROOF OF LEMMA
In the following steps, we will utilize the technique of minimax estimation. To begin with, we will introduce Interchangeability:
Definition D.1 (Fenchel duality). Let R X R. be a proper, convex, and lower semi- continuous loss function for any value in its first argument and f == E (y. convex conjugate of by ((y,)) :) which is also proper, convex, and lower semi-continuous w.E.t the second argument. Then, (v( = max, fwv Lvv - (u))- The maximum is achieved at v E de (u), or equivalently E al(v).
Theorem D.2 (Intechaangeabilitt). Let be random variable Qt and, for any Ω. the function f(;ww) : + (-00, 0o) is proper and upper semi-continuous concave function. Then,
Ew max (u, wER
max [ulf(u(w),,)), a(-IEL(()))
where ((2) = (u(-) : + R} is the entire space of functions defined on the support Q.
Lemma 5.1. Denote || [[]2(X)]. For any parameter Am > O, we have
Lg(g:p) = sup E [A, W, (A, X. - Am |/ W, W.))
mEM
where M the space of continuous functions over (A, W,
Proof. Notice that the squared loss function (g(v) = 4x (v v)2, we have (u) = wy + Au?. Then by Fenchel duality, have
(v) 41
(y 0)² max {vu - ( ueR
max ueR
(vu - uy Au?) = max
{(v - y)u Xua
wER
Let = P(A W.X)
A, W. x - E (A, (A, Z.X) A.W. a m, we have
A,W, - E l (A, 2, A,W.X1)
4Am
PPA W, X)
= max mER
[A) Z, X)
A,W, E
1
A, 1.,, x)) m Am
(PA W,X)
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Therefore, applying the interchangeability. we have
1
1
A, W. X 1)
27
Lg( = 4Am E
(A W) (((,,,)))
4Am
m Am mm? }
=4 E
max MER
E (A, Z, X)
A. W x]
E
A, A.W.X
(AA W, X)
m Amm? }
=4 E
max mER
Fzz
9 (A,,
A, W. X
(A W,
1 ( W.
=4xm max E
(A,,
1,, X
m (A, W) >mm (A, W. X)
MEM
=4Am max E
((A,2,X) P(A WX
m (A,W,X) Am A,, WA)
MEM
We define
2>-h (p) = (1 p (A )) m (AA,, - Am 11m21/ (12)
As long the dual function class M is expressive enough such that - Rg (g:P) EM, have
2xm
pg(p:)
max Tr..Am Am (g. m:;
mE.M
can express Eq. M in an alternative form:
min max $ Am (g, p)p.. gEQ MEM 4
(13)
We denote the empirical version
- Σ (2i)
an (q, m:
(a;, Wi, 244) Am lImll
p (a; w;, 2;))
Furthermore, for simplicity, define Py(9, m) the non regularized version of o>
l, m) E [1 p (A W, W,X)
1
(A, WA)
(14)
In fact, due the fact that need to estimate the density function p. consider the following min-max optimization problem:
min max T. (q, m; p)
(15)
Q mEM
Similarly, for the nuisance function h, we consider the following min- min-max optimization problem:
min max "h n. (h, = Σ (ys - h (Wi, a4, x;)) g (as, 24, 2,, x.) - Ag 11all2..
(16)
NEW 969
D.2 COMPARISON OF EXISTING METHODS
Closely related is the estimation of the general treatments proposed by Kallus al. 2222 They hope to estimate the generalized average causal effect (GACE):
[ Y(a)m(a X)du(a)
J= E
where (a X) is contrast function. Based on the idea, they propose the method for solve the muisance function q
min max E, [A (,) (Tm)(WW, X)] A. |mml2 meM
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where (Tm) 2²) ff m(a, w, ())((())du(())
However, consider the continuous treatments E supp(A), then in this case, T(a X) = 1(A a).
Correspondingly, the conditional moment equation for the action bridge function q is equivalent to
min max [T (A a) ((,, Z, Xm W,,W, W, - (W, X)] Am m m
4EΩ mEM
As mentioned above, the main challenge for continuous treatments lies in the estimation infeasibility caused by the indicator function. Therefore we cannot estimate the nuisance function q when the treatment is continuous.
The second paper to solve 9 is from Ghassami et al. (202)). They estimate the causal effects of binary treatments and illustrate how the double robustness property of these influence functions can be used to formulate estimating equations for the nuisance functions. Specifically,
min max E, [[-I(A - a))(aa) X) + 13 m (W, X) - .n² (W, X)] R MEM
Then functions ((a 2, I(a (2) x>) + I((a 1)41 (2.z). Note that since the indicator func- tion appears in their optimization equation, we still cannot solve the case of continuous treatments.
Instead of running optimization for each a, would like estimate go A, Z, X) with single optimization algorithm. To achieve this goal, we propose a two-stage estimation algorithm. We first estimate the policy function and then convert the conditional moments into equivalent forms, which can lead to a min-max optimization problem.
1 =
((a; JT. ((as//us,z)) m(ai, Wis (t:) Am
min max QEQ MEM TI
Since the treatment is continuous, we have estimate the policy function before solving the moment equation. However, when the treatment is binary, we can transform the moment equation so that the policy function disappears. This the cost of estimating the causal effects of continuous treatments.
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E PROOFS AND DERIVATION
E. PROOF OF THEOREM
Theorem 4.2. Under assump.
4.1 suppose 3(a) E[I(A a)qo(a, X)Y] is continuous and
bounded uniformly respect to a, then we have
E (a)) [[A A = a)yo (a, 2, x
lim E [ (A a) Z, X)Y]
Nbw
Proof. By definition we have
[[T(A ( ao) Z, X 1 yo(ag, 2, x)PP (ag, 2,
/ Sas (a) vqu(a, 2, x)P (a, 2, x)
/ dao (a) 1 2, (a, 2, 2, du(z, 27 y)
[ 8av (a) da (8a. 3) = 3 (an)
where the last equation uses the properties of the Dirac function. Similarly for the kernel function, also have
KK B3 Kin (a - 4 (a) da a=howw+an
K(S) (how + ag) ds
As 3 is continous and bounded uniformly, this integral is dominated by CK(s). Moreover, be- cause 3 is continuous, the integral converges point-wise to (as)) Applying the dominated convergence theorem and Assumption 4.1 kernel function integral is 1.) yields
lim K(s)8 (hows + an) ds / K(s) (0o) ds 3 (ao)
lim Kro B)
how 00
how -0 Js
which finally shows
B = lim E [Kin (A 1 a))o(, XYY]
Now
E.2 PROOF OF THEOREM 4.5
We denote Fx as the distribution function for any variable X, and rewrite 3(a) as B(a; P8) where FO denotes the true distribution for O == (A, Z, W, X, Y). Besides, we consider the special sub- model Pohi.- = (1 (1 +ePhow where Pri (-) maps point for any o denotes the distribution of that approach point mass at o as how + O.
distribution of O, i.e., Iph (o)
Theorem 4.5. Under a nonparametric model, the limit of the Gateaux derivative is
lim ② Now DD OE B(a; polmo
= (Y - ho(a, W, X)) (a X,X) lim
Kin (A a) + ho (a, W, X) - B(a)
Niw -00
Proof. Similar to B(a) rewritten as {(a; Pg), we can rewrite ho (a, w. as ho (a, w, v, po PAwx For simplicity, we omit the subscript of the distribution. Please identify according to context.
0 3(a; (petew))
/ bo (a, w, xr; (pshew)) dipshow (w, xx)
05
0e
e=0
i
ho (a, w. 2, pehsow)) ) dFD (w,x)
(1)
② p phnn (w, x)) 1=p du (w,
ho (a, w.
05
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Since Pell (1 --) + ero have
i
Tpshow
Po.
Ōe
l==
For term (II)
III) / ho (a, w, 2) p (w, x)) du (w,
Oe
f ho (a, w. (pm (w. - pd (w,x)) dp (w, x)
how -0 ho (a, W, X) (a)
For term (I)
(I)
Os ho (a, w, pshb ) le-o dIP0 (w,z) x)
pd (a, w)
Ōe ho (a. w; pehbo
le-o p (a w)
dec (w,
/ 0. ho (a, wo,w,a x;
e== pd (a, w, w. x) / go (a, 2, x)) pd (z a, w. x) duzdp (w, x)
p: pehbw)
Oe
/ qu (a, 2, de ho (a, w, 2; pethw l pd (1,y a,2, pd (a, (w,, (w, 2,2)
And by Eq. 2 we have
1 / (y ho (a, w. x; pehww))))) )) pe (y, w a, a, (y, w)
de
==0
Then
| ① ho (a, w. x;; pehne ) Il- pe (w.y y a, 2, x x) (w, y)
= / [y- - to (a, 2))
② p (w, y, a. 2, x)) -= du (w,y)
nd (a, x)
ff [y ho (a, w., x)] pd (w, y, a, 2, x, De ge (a, 2, x) )===0=0000 dp (w, y)
-
(n (a, 2)
where we use the equation
i eb (w,y a, x)
① p thr (w, y. 2,
pd (w, y. a, 2. 2 p (a, 2) c=
0e p
c=
0 (a, 2.
(n0)2 (a, 2,
==0
Substituting (I), obtain
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# pf (w,Y, a, a, 2, ==0 du (w, (w,
(I) / 90 (a, 20 (y ho (a, w, x)) dd (a, (0,2,31 2,
Te
pd (a,
pe (0) De of (a, 2, 2) (0 (w, x 2,3)
- f 20 (a, 2) ho (a, pd (a, 2,
(a,
/ 20 (a, (y ho w)) (a, x) p (0) -0--p" -00 du (w, v. 2,
p 2,,
f 40 (a. x) (y ho (a, w. x)) (a, 2,
(0) (n (a, - pd (a, 2, 2))
dp (w, 2,9)
(p0)² (a, x.
40 (a, Z, X) (Y - ho (a, w. x))
lim Pa Nnw
how (a)
+ $f g0 (a, 2, (y ho (a, w, x)) pd (w. a. p 20) 2,
New +0
4o (a, Z, XY ho (a, W. lim how
PA how (a)
where the last line is because of Eq.
The corresponding probability density function (pdf) PO our kernel density, and we thus have lim Aswe (0) two terms, we get
Now (o) Khom (0 ())(pp(()) how is
listl//emm -0 K Kinn (0 o). Combining the
② how 00 0e
lim
B(a; pehww)
= -Y - -olaa, W, (a 2, X),
lim Khow (A - a) + he (a, W, X) - 5(a)
how -a
E.3 PROOF OF THEOREM 6.2
To prove Theorem 6.2 we first give some Lemma.
Lemma E.1 (Theorem 14.1 in Wainwright (2019). Given star-shaped, b-uniformly bounded func-
tion class F. let 8n be any positive solution of the inequality R, S /b. Then for any 2 8n,.
we have
t
111f112, 11 S / 11
YfEF,
2
with probability least - cie ccce-ceett comt" If in addition no? 2 2log (4 (1/8,)) /c2, then we have that
111112,. - 1 S coò..
VfEF,
with probability at least & exp(- /62))
Lemma F.2 (Lemma 11 in Foster & Syrgkanis 2019)). Consider a function class F, with suP ol < 6, and pick any f E F. Also, consider a loss function é IR x y + E which
L-Lipschitz its first argument with respect to the la norm. Let 62 4d 4d log(2ogn) be any
solution to the inequalities:
82
Rs (ō:star (F -f)) S
where star (F f)=f)=m a for - E [0, 1). Then for any 2 6n and some universal
constants C1, 02 > 0, with probubility 1- oie cie -ennt /6? it holds that [( [e (f(())))) E,, - for any E F. Iffurthermore, the loss function é is linear in J. iee. ((fff')))),))) (z),)) {(f()))))) (z)( and f(af(z), y) = af(f(x)) v), then the lower bound on 8% is not required. If the outcome f of constrained ERM satisfies that with the same probability,
(E] v)l
((e((F
< 18Ldt {llf f
+ +t)
E ([(((())))))
< 18Ldt {||
+
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Lemma E.3. Let some a, b, d 2 0 be given, and suppose that
ax² S bx + d,
for some X O. Then, we have
b b+ v ad
X <
Proof. Since b and d are both positive, the quadratic ax² bX d must have positive and negative root. Therefore, this quadratic is negative and only if X less than the positive root; that is, we have
ax² - bx - -500 0
b vb2
+ 4ad
X
2a
X b+ v ad
Proposition E.4. Let 8% respectively be the upper bound on the Rademacher complexity of M. For lugles//) for some constants cg c; then under assump. 6.1
any v E (0, 1), define 89 == 8% + cj v
n
we have with probability 17 that
1
Am
2C2 + +c0 (872
sup mEM
9 Am (go. m: p) S
+
+
Am
p
2
2
Am
Proof. To relate |m 2 and 11ml22. by Lemma E.1 it holds with probability at least n that
| |1ml]2 < 2
|m| 1
(89)2
Vm E EM,
(17)
2
as long as we choose i equal to 89 = 8% + co / Iog(ea/1)
where 6% an upper bound on the
critical radius of M. On the other hand, to relate Φ.-dm m; D (g. m: p) - Am || 2
to its empirical version, we apply Lemma E.2 to E(a1,a2) = 02 aj = m(A, W, W)
(A, Z, X) b[|[. that is Ci-Lipschitz with respect aj by noting (A, X) in [-C1, Ci] with some constants C1 = 1/ll + [|Q]
A(WX
/e (a1, 02) l 02)) s C₁ la- - -
Therefore, we have that there exists positive constant such that, with probability at least 1 - 7,
@ (go, p) L (4o, m; P) s cCi 80 |mll2 + (
Vm EM
(18)
Thus, we can further deduce that, for some absolute constants e > O, with probability at least 1 - 17,
sup A. (go, m, = sup MEM
( (4o, m; - Am || ||mm122n
2.n
mEM
{ { (qu, m) + c01 f ||
+ (89)2)
Am |l [ml22.0)
sup
mEM
(2)
Am
Am
4 (go. m: p) + cCi |m|]]2 2 + (8))2)
|| 2 +
(89)2
sup MEM
2
2
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where (1) is derived from Eq.
18 and (2) is derived from Eq.
17
We cam further bound the right-
hand side of the above inequality as
Am
sup T. (q,,m; S MEM
sup meM
o (go, m:
|ml]2
Am
Am
||mlll + (89)2)
|| 1
(89)2
sup mEM
4
2
Am
eCG
sup MEM
by (40, m; +
+
+ + (89)2
2
Am
Am
+ sup mE.M
$ (20, m; D) - F, m.po:;;p))
4
sup Me.M
E
1
m
Am
mmml
+
Am
e09 + +o0s }
(89)2
P
p
4
2
Am
IA
1
Am
Am
eC
} (80)2
sup
1l
11
11mll2
+
+ c0i
mEM
P
4
2
Am
IVB
Am
2C7
+
+
c0i
(87)2
2
Am
where (1) from the fact suP 2 [allmll2 - bllml13) < a² a²/4b /4b for any b > 0, (2) holds from the fact that D (go, pp = 0 and (3) holds from Cauchy's 's inequality.
Theorem 6.2. Let 8% respectively be the upper bound the Rademacher complexity of M. For
any n E (0, 1), define 8 == 6% + cgv /logle//)) for some constants cg ci; then under assump. 6.1
we have with probability 1 - that
V X2, + Am +1 1 1 -11)) p stands for o())w.x)).
|iroo, (Ā 10)ll2 O 88
Proof. First note that,
sup Am m; = sup mEM
nh (q. m;;))) - -b" (go. m: p) + Φ (go: m; P) - Am mmml
4
mEM
d (q. m: P) - Dy (go. m: P) - 2Am l|mmll2..)
sup meM
+ inf { (b (go. m; + Am 11mll2.n }
mcM
By the symmetry of M, we have
inf mEM { Φ6 (go. p) + Am [1 = inf
Th (go, -m;; p) + Am |1
-MEM
inf -MEM
{
-." (qu, m: p) Am || 12.2..1
{ (b (go. m; p) - Am |
sup
-MEM
= sup am..m (40, m; p)
-meM
sup Dnn.Am (go, m. p) mEM
In the sequel, we upper and lower bound term (+) respectively.
(i). Upper bound of term (x).
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By definition of the estimator á and the assumption qu E Q, we have sup mEM
"n (q. (q.m:)) - sup Tu..Amm (go, m: D) mEM
We have
(+) - sup
Dn mm (g. m: m) + sup gm (qo, m;
mEM
mEM
Am
209 Am
+ cCs (0o)))
2 sup aa.dm (qo, m: p) 2 2 MEM
+
+
Am
p
2
(ii). Lower bound of term (+).
We now invoke Lemma E.2
with E(a1, a2), a1 and 02 q- that Cy-Lipschizz with respect
to a₁ by noting (A, 2, X)
40(A, 2, ) is in Cc2 C2] with some constants C22 = 2||Q
|l(a), 02) f(ai, 02) I C2l01 - d'I.
Therefore, we have that there exists positive constant such that, with probability at least 1 - 17.
{ Φ (g. m: D (go. m; p) g (g. m: p) b (4u. m: p) - cC2(88 |mmll2 + (69)²) I 2Am proja (q 20) M.
(19)
Now we are ready to prove the lower bound on term (+). Since my and Star-shaped, we have ma M
2
{ m; D (40, m); - 22m [|ml12,07]
(+) = sup
mEM
Φ q. 2
mg
:p)
qu.
myg
:))
Am
m
2
2
2 g
q.
myg
2
:
2
qo
my
2
:
c0'²
8
mg
2
+
+88)2)
Am
lmm
112 +
(89)2
2
where the last line holds from the Eq. 17 and
19
For the term (o), have
(0) = $, (. my
2 :p) 7.
+
mg
2
:)
$
i
my
2
g
q0:
mgg
2
:P
mg
mg
+
$ 4
90+
P
+
go:
2
2
2 (g. my
1
my
1
mg
my.
2 :p
E
E
4
q.
p
P
2
p
2
where we used b (go. me: p)
O. Moreover, since we have
E ((A,,
p (A W A,W,
=
- 40 A,W,
= projj (q - (g))
and my 2x.,, proj, (q 40), we have
1
1 ④ my :1)
= E
q (Z, A,X,
1
my (A, W, x)]
2
P W,X))
1
= E
my (A, X, IE q (,)
( W, A, W.
E.
(proi, (q - q0))²]
=mm 11mm112 =
41m
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Therefore, obtain the lower bound of (+):
(+) 2 Am m cC2 fou
myg 2 ||
(09))
Am
3
mg l1
(80)²
2
2
(iii). Combining upper bound and lower bound of term (+).
Now we are ready to combine the upper bound and lower bound of (x).
Am |m
c02 { 8"
mg
+ [87)2)
Am
3
mg
l2
)89)
2
2
2
2
1
1
2
Am
c² C²
+ +c01 (88)22
S
+
Am
P
2
2
Am
This give a quadratic inequality on ||m 112- i.e.
8c2
Am ||m 2
20C209 1|m
+
5m +
+ 8cC1 + 1cOs (87)2)
5o
Am
Am
(B)
(C)
By Lemma R.3
we have that
B+ V 2 47m C
(B+ [ VanCC
1
|| 12
<
2)m
Am
Applying the definition of A and B, we conclude that, with probability at least
20C284
8c
+ ScC1 + 4cC2 } (80)2
| 112 5
+
Am
+
5Am
Am
Am
Am
p
Am
2c0²
ScC' + 4c02
80" +
2
1
1
+ /
15
89 +
Am
Am
X2
Am
p
11
80C1 + 4c02
402 (2C + C3)
2
1
5+
+
89
Am
Xa
Am
P
1
1
1
84
Am
X2
Am
P
where the third line holds from the vā + vb S /2(a + b). According Eq. 11
we have
Coy V IE (proi, (a (0)))²]
| 122, E (mg) ?|
2Am l1m
we can bound the Projected RMSE by
|rroi, 40)ll2 = VCa (x:p) = 2Am || 1222 S 89 V X2, +
Am + +1+
P
For bridge function h, we also the similar theorem. We first give some assumption.
Assumption E.5. (1) (Boundness) [[g [|[]|pg < oo and y uniformly bounded; (2) (Symmetric) G is symmetric class, i.e, if E G, then -g E G; (3) (Star-shaped) G is star-shaped class, that is for each
function g in the class, og for any a E j0, 1] also belongs to the class; (4) (Realizability) by H; (5) (Closedness) 2x, proja ho) E H.
I
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Theorem E.6. Let 8% respectively be the upper bound on the Rademacher complexity of G. For any
7 E (0, 1), define 8 == 8/; + cb V Ingte3/1) /m) for some constants c), ct; then under assump. E.5 we
have with probability 1) that
proin (h- Sonoin(hhhooll 0 Vi Ag +1
13
The proof of the Thm. E.6 is detailed in Kallus et al. 2021
E.4 PROOF OF THEOREM 6.4
We prove the bias and variance of the PKDR estimator given Eq. 7 respectively. The proof method comes from Colangelo & Lee (2020); Kallus & Uchara (020).
Theorem E.7. Under assump. 3.1 3.4 and 4.1 suppose 1| Mll2 = o(1). l1 - 2l12 = o(1)
and [ 11]l2ll 4lI2 = of (nhhm)) /2), nhg = O(1), nhwo + oc, ho(a, w. w,, p(a, w. x))
and p(a, wlz, are twice continuously differentiable wrt a, well as ho- 9o, h are uniformly bounded. Then for any o, we have the following for the bias of the PKDR estimator given Eq. 7
E [B(0)] -3(a)=
1it
K2 (²(KB + )
where Eago(a, X) 2 ho(a, W, X) p(a, (,X))xp( W Z, X) + 82 hola, W, WX]]]
Proof. We calculate the expectation of the estimator for a single data point. For simplicity, we treat this data point as random variable. We have
E ((a)) - B(a) = E [ (()) - -(aa)
E Knew (A- - a) {(r - h (a, W,,x) (a, 2, x)
(I)
E Noo (A- a) ((Y - ho (a, W, X)) 90 (a. X)X
(II)
h (a, W,X) - ho (a, W. XXX)
(I))
+ E [Khm (A (A- a) (Y ho (a, W. 90 (a, X)
(IV)
For the (IV) term, we first have
E Kn (A a) 90 (a, Z, X) (Y - ho (a, W, WX)
=E Know (A - a) 90 (a, Z, X) B[ hy (a, W, X)) | A, 7, X]]
=B Kmn (A a) 90 (a, Z, X) E ((ho (A, W, X) ho (a, W, X))
A, A,
EE lgo (a, X) E [Kn.. (A - a) (ho (A, W, X) ho (a, W, W))
X,]
go (a, 2.X) f Knn (a' a) (ho (a', w, x) - ho (a, w. x) p (a', w x) du (a', w)
EE
90 (a, x) / K (u) (ho (a + low w. x) ho (a, w,, w),) p (a + h U, w 2) due (u, w) =E
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where the last line holds from a' hbw + Consider Taylor expansion of ho (a, w, x) and p (a, w 2, 2) around A = a:
p (low u + a, w 2, 2, p (a, w 2, hbw U
0A p (a, w 2, + O (liw))
ho (a + hbw U, w, hu (a, w, x) how 2
0A hu (a, 2w 2²)
(how u) 2
0²
T.. ho (a, wx O (h
Then, we can compute the conditional expectation by integrating the approximation to the density term by term. Here, Ky (K) represents the jth kernel moment, defined as Ky (K) u² (((K(u)d (u)du. It's important to note that for symmetric kernel, the odd-order moments integrate to O. Therefore, we have
E KNow (A- 20 (a, X) - ho (a, W,X)]
=B 90 (a, Z, x) / K (u) (ho (a + Now n) ho (a, w. æ:) p (a hbw w 22 du (u, w)
θ
8
1 / d2
ho (a, WX ]]
hbw N2 (2( (K)E
qo (a, Z, x)
ho (a, X)
0A P (a, (a,
X, 2 0.12
aA
+ o (liww)
For the (III term, we have
(II) + (III)
E Knon (A- a) (1, 20 (a, X)) (ao X) W, ((a,
(20)
+E
Kho (A- a) (a, X. 40 (o, X,)) (o W2X))l
(21)
+E Knom (A a) 90 (a, X, (ao W) - h(,, - (a,
(a, W. - h (a, W, x))
(22)
We will explain in turn that the above three convergence rates are ((nh -1/2), o(1) x 0
and o(1) x Oh respectively. From now on, we prove Eq. 21 is o(1) x Oh,,,
K (A - a) (a, X) - qo (a, Z, X)) (Y - ho (a, W, X))
E KNow (A a) (a, Z) qu (a, Z, X)) E [Y ho (a, W. X)) A, A,]
E [K (A a) (a, Z,X) - qu (a, 2, x)) E (/hp (A, W, - ho (a, W,X)) | [A]
E [K (A- a) (a, 20 (a, Z, X)) (ho (A, W, WX ho (a, W, X))
E [(ğ (a, X) - 90 (a, ( M. (A a) (ho (A, W.X) - ho (a, W,X)) Z,
E [( (a, X) 90 (a, X) {O(kb)))
(3) (1) x Oh
where (1) is derived from Eq. 2 (3) is derived from assumption || - -9l22 = =(1) and (2) is because
Nrme (A- a) (hu (A, W, X) - ho (a, W,X) X,
/ Kim (a' aa) a) (ho (a', w. 2) ho (a, w. x)) p (a', w 2, du (a', w)
-f K (1) (ho (a + hbw u, w, x) ho (a, w. x)) p + how w 2,2)) du (u, w)
= / K (u)
a ho (a, w, x) + O (m) (p (a, w x,z) + (hb du (u, w)
Nbw u 0A
(N
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Next, we prove Eq. 22 is o(1) (.
BlK (A - a) 9o (a, x) (ho (a, X) h W, - (he (a, - h (a, Wx))
=B( (a, W X) - h (a, W. X) | [ how (A - a) 20 (a, 2, xx W.
We consider
E New (A - a) qu (a, X) 1 W
[ (Khoo (a' - a) 90 2,, 1) (al, ,2 w. x)) du (a', 2)
-f Kinw (a' - a0 (a, p (d', dp (a', 2)
f K (u) qu 20, hbw u, w. du (u,
/ K (u) 40 20, (a, w. + hbw 1
①
ho (a, w. + 0 (l) du (u, z) -
DA
= -f to (a, 2, p (a, w, dy (z) - + 0
where we use the first-order Taylor expansion of p (a, = w, x):
p uh a, x) = p (aa, 2 w, x) + hbw u ho (a, w.) + O (himw)
0A
Therefore, we have
(no (a, -a (a,,,,,,)) x)) (f ao (a, (a, 2 w, x) du (x) - (k(w))
dp (a, w,X) (fo (a ( (z) - 1)]
h(a (no (a
+ (1) x 0 (w
where (1) is derived from assumption ||h - h1l2
11hl1l12 = o(1). We assert that the first expression
(ho (a, - (a, wx)) (ʃ qu (a, 2, 2)) p(, w, du (2) -2) 1)]
/ - h (a, w. 22) 90 (a, 2, x) (a, w. du (2, w.
- p (w, x)) du (w, x)
(0, (a, w.
(1) h (a,
(w,x) h (a, w. x) (w, x) du (w, x)
f h (a,w, u. x) (a, 2, (a, w. dje (z, w,
/ h (a, x)
p pla, w.
du (w, x) = 0
p w,xx
where we used the following property for (1)
3 (a) = to (a, w, x) (w, x) du (w, x)
ho (a, w. (a, (a, 2, w, 2) du (z, w. w).
By assumption | 11|22l/ 4|12 = (((nhw))-11) 1/2), we have Eq. 20 is (n(nbw))))l -1/)). Combining these terms we get
(III) (((hb))) - 1/2) o(1) x (hh + o0) x ((hhm) of (nnhbw)) 11/2)
where we use nh 0(1). Therefore.
12 (K)B + (((nlo()))))//)))) 1/2),
E (a)) B(a) 2
where B Elgo(a, 7, X2 2o W, X) pap(a, W 7, X)
- Sozhola, W, X)]]-
82
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Theorem E.8. Under assump. 3.1
3.4 and 4.1
suppose ||h hl[2 o(1), l1 2l12 o(1) and nhe O(1), nhow + sc, (a w, p(a, 2 2|w, x) and
|| h1l22l 412 (((how)) 1/2)
p(a, w w/2, x) are twice continuously differentiable wrt as well as hos qu- h, 9 are uniformly bounded. Then for any a, have the following for the variance of the PKDR estimator given Eq.
(2(KK n//w
Var[$(a)]
(V +o()),
where V = E[[(A = a)go(a, 2, XY - hola, W, {
Proof. For convenience, we let
m (o; h. q) = Kin. (A- a) (Y - h (a, (a,
(23)
(o; h. q) m (o: h q) + h (a, W.
(24)
We first cross-fitting which allows exchange the order of summation and variance. More specifically, we split the data randomly into two halves O₁ and O2. Then we have
E
En
(o:4.
(o: h
E
En
Φ
(z h, (orh,))
102
Φ hr 2)
)) 102
n
E
We will omit this step later, please identify it according to the context. According to the definition of variance, we have
Var Bn @ (o 0) (a))
Var (EE ( (o: ho: [0)l 33 (a)) + Var En ob (oz ho, 2o) Φb
(II)
En o (o; hu, qu) (o:h.
+2 V Var (En iØ (o; ho qu)l 3 (a)) Var
(III)
We use cross-fitting and by Eq. 24
(I) Var (m (o; ho, go) + ho (a. W,X) (a))
(25)
We first consider Var (m (o; ho, qu)). Since
Var (m (o; ho. 9o)) =
E m (o: ho, 90)²
(E Im (o; ho, (0)))))
< E m (o: Nu: qu)
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only consider the second moment of term in the estimator
E
m (o; ho, -0)²
(K (A-o) (Y ho (a. W, qu (a, Z, x))
E
E
40 (a, Z, x
Kinn (A a)² (V ho (a. W, x))² z, x
9o (a. 2,)) /
1 K
a' (d'-a) (y ho (a, w. x²) p (a', v, x) du (a', y, w)
E
how
20 (a, x)x /
E
K (u))² (y - ho (a, w. x)) (a + whow v.w 2, du (u, Us w)
hbw
(a, 2)x² /
1
=E
K (u)² (y ha wx)) (p (a, N. w 2, + (kemw) du (u, Vo, w)
hbw
[[2(KK) + (Hbbw)
hbw
(26)
where V = E [[A - a) 4o (a, (Y - -ho (a, W, x))"
Because ho is bound, we have Var (ho (a, X) - 3 (a)) is bound. Therefore, substituting the above equation to Eq. 25 we have
(I) Var (m (o; ho, 20) + ho (a, W. - -3 (a))
1
Var (m (o; hos (o)) + Var (ho (a, W, - 3 (a))
n
n
/aa (m ho, Var (ho (a, W) - 3 (a))
C +
1/2 /2/CC)/ 1/2
(9c(K)V o (how))
+
(V + (bbb))
nhbb
n
n
nhow
(92(K)V + (1im))
nhow
For the term of (II) we have
(II) -Var
En ① (o: ho: (o:
④ (o: 90) 0 - (o:h. ④) 27
(B
E
(27)
(0 ho 90) 1 (o:h. 0)) )
E
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Similar to Kallus & Uehara 2020),
according the definition of Eq. 24 and decomposition of
Eq.
Eq.
22 we have
④ (o; ho, 40) - -0 (0:0,,11) ))
(1)
Kno (A - a)² (7 (a, 2, x) - to (a, (x) ( (a, W. h (a, W, xi) |
+ KNu (A - (1 2, -0 (a, 2, x))²(x - (a, W, x))²
E Knew (A (4, ( W - !
+
+ E (b, - ho (h (x)) 27 +A
(2)
Num (A - a)² (i(a, - 20 (a, 2, x))² (io (a, W. h (a, W. x)) x))
SE
+ Kno (A - a) (1 2,X) 20 (a, 2, x))² 1
- ho (a, W, x)))
Kmo (A a)² 90 (a, x,X (ho W (a,
+
(h - ho (a, W, x)x))² + +0(1)
+
(3) bbw max
ho (alla, - h (a, W. x)
| E (a, Z. qu (a, Z, x) + 0(1)
E
=o(how)
where (I) is the square expansion of Eq. 20 and A is sum of cross terms, (2) derived from h, q, h, q uniformly bounded, (3) uses the same approach as Eq. 26
Therefore, substituting the above equation to Eq. 27 we have
(II) Var (En (o (o; ho, go) @ (o: f h)) ((1hbm)))))
(28)
For the term of (III), we only need to substitute (1) and (II) to (III)
(III) Var (E,, job (o; ho, qu)] - (a)) Var (nn o (o; ho, 20) - (o: i))
1/2 o (n (112// 1112 1/2
nhow (V + o (hbow) }}
bw
Therefore, combining the three terms (I), (II) and (III), we get
Var En ④ (o:h. (a))
1/2
1 n/on
(V + o (how)) o (n (1-1//1 Now
-1/2) + o o (m-²)
{L2 (k) V + [bbbw) +2
n/tto
(Ko(k)V +1))
nhow
Theorem 6.4. Under assump, 3.1 3.4 and 4.1
suppose llf - loll2 o(1), |lq - a/l2 o(1) and
|| Allall 4/12 o((nhw)) 1-12
1/2) nh
O(1), nhow oc, ho(a, w. w)) p(a, 2/ z/w, x) and
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p(a, w|2, x:) are twice continuously differentiable wrt well as bu- qu. h, 9 are uniformly bounded.
Then for any a, we have the following for the bias and variance of the PKDR estimator given Eq. 7
1i
()) (V + o(1)),
Bias (Ba))) =
= -
-3(a)
K2 (2) + (nhhtw))) )11/2,, Var = -hho
2
8² where B 3= Elquu(a, Z.X) A4 (a X) pla, W Z.X)+ + 0A22 ho X)I,,,, V = [[ a)yo(a, Z, X) ho(a, W, ))
Proof. By Theorem. E.7 and E.8 we completed the proof. If we want to optimize the
tradeoff of the asymptotic mean squared error, we choose the optimal bandwidth how such that neither term dominates the other.
MSE ( (a) 3 (a)) Bias + Variance
how
(n (K) B) + n//bw
(K) +
4
nhbo
Optimizing the leading terms of the asymptotic MSE with respect to the bandwidth how:
i MSE (x (K) B) hi
22 (K) 0
nh
Therefore, we can select the optimal bandwidth is how = 1/n- in terms of the mean squared error (MSE) that converges at the rate of On )/5).
E.5 CONSISTENCY OF THE ESTIMATOR
Theorem E.9. Under assump. 3.7 3.4 and 4.1 suppose bola, w. w. and pla, w w(z,x) are twice con- tinuously differentiable wrt as well as has qo: h 9 are uniformly bounded. Then, for some universal constants and o2, with probability 1 - f). the PKDR estimator given Eq.
7 Zrrrrr is bounded by:
log (02/1)
(ci-go)l: B(a) - -(a) = [TA = a) (ho in) [lrroj (4 - -00 20)12tt11
K2 (K) R-o (w
2
where R E ( x) [2 ho (a, X) Sap (a, W x) + (o (a. W, wxx)
Proof. From the relationship between causal effect and nuisance function, we have (a) -
E [T(A a) (a, Z, X) (Y ha (a, W, X)) + ho (a, W, x)] - M(a)
E [I = 90 g0 (Y ho) + ho] - E [(A = a) a (Y -h) +
(T)
E (I a) Know (A (A-a)) 4 (Y [i]]
(II)
+ Knuw (A- a) (Y -h) +h] 3(a)
(1I1)
where hg ho (a, W, ,0 go (a, 7, x), h h (a, W, X) and g = & (a, 2, X).
We can bound each term as follows. For the first term, we have
(I) = E [(A a) foo (Y hoo) - q - -)}}] - E hno in]
= [A = a) Y (40 - 7) + [(A = a) {qh- - qoho }} - E ho - in]
@ | a) ho (g0 4) + E [[A a) phh 9oho }|
-E [[(A a)go (ho - -h)-)
= E [(AA = a) (20 i) Inu - -i)l 2 || (A a) (hu kn]l_
||proj (6 20)1l2
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where (2) is derived from Cauchy's 's inequality and (1) is derived from
B (A a) Y (g0 - 4)]
E [I (A a) (go 9) E Y A, Z]]
= E [I (A a) (go () E [No (A, IA,
E (A a) (90 q) hol
E ho - 11] = =f (ho- h)
(a, w. x))
du du((w (u. .r) = / hho - h) (a, w. x.)
p (alw, x))
[20 (a, Z, X) A, W,W, X] du((e,
E [(A - a) [no fi)(0) g0
For the second term, we have
E
K hom (A (A-a) a) ( 2, x, (
Khow (A-a) Z.X) E (x- |A, 2, 2,x]]
E
hiw (A- a) (a, x,x) (oo (A, W - ( wxx)]
E
E
i (a, Z, X)E E Kno (A- a) A, h (a,
x)) (a, |z, |2,x]
@ (a, z, x) / K (a' aa) ho (a', w. x) -h (a, w, (a,
x)) p (w, a (2, x2 du (w, at)
+ how u, w, x) h (a, w, 2)) p (w, a + how x)) dp (w, u) ( Z.) / (u) (ho (a (ho
=E
where the last line holds from a' how + Consider Taylor expansion of ho (a, w. x) and p (a, w 2, around A = a:
p (how u + w z, x) - (a, w 2, 2,x) = how 2
DA p (a, w 2, 2. + O (him))
ho (a + huw w) ho (a, w. how 1
0A to (a, w,,zz)
(N u)² 2
0A ho (a, w. w. + 0 (1)
02
2
Therefore, we have
(x)
ho (a, w. - h (a, w. p (aa 2,,x)
+ ho (a, w, - -h (a, w., Now
(a, w 2,
OA
④ 0A
+ hbw 21
ha (a, w. wx) p w 2,
(hbb u)²
82
0A2 1i (a, (,) p (a, w 22,x)
2
how 2
hn (a, w. w.x)
/mw DA (a, w 2,
0A
(Mow u2
② ho (a, (a, m,,))) how a n (a, w 2, (1i 0.2²
82
2
Then, we can compute the conditional expectation by integrating the approximation to the density
term by term. Here, Kt (K) represents the jth kernel moment, defined as Kș (K) I u² K (u)du. It's
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important to note that for symmetric kernel, the odd-order moments integrate 0. Therefore, have
Knn (A- (a-a)a(aa Z,X) (a,
E
0 (a, x, / K ((u) (u) (ho (a Nbw w) (a, wx (a how x) du (2,, w) E
① ho (a, w)
1
0²
hi Ni B
(a,, x)
0A a W Z,X) +
ho (a, W, x)X)
0A
2
0.²2
+ E a) (a, Z,
(a, (hiwe)
Therefore, we obtain
(II) = E [I( = =a)- ~N (A - aA (Y- 1i]]
2 n2 KRR + (hi
where = ==E(a(a, Z,
2 ho (a, W.X) x) P (a. W Z.X)
ho (a, W. xx) )
The third terms are upper-bounded by Bernstein inequality. This concludes
B(a) - )(a)) & I( a) (ho 1 11. |roj (4 2o) 12 te
log (02/n)
1h
K2 (K) KR- (Now)
2
Remark E.10. According to Thm. 6.2 and E.6 we have l|A - holle = O(n 1/44 and | qu =
O(n 0n1-1/4) Therefore the order of the estimation error is controlled by how- From Thm. 6.4 we know that the optimal bandwidth hbww = O(n 1/55 Onn5// in terms of estimator error that converges at the rate O(n 2/5). Note that this rate is slower than the optimal rate O(n (11),, which is reasonable sacrifice to handle continuous treatment within the proximal causal framework and agrees with existing studies (Kennedy et al. 2017 Colangelo & Lee 2020).
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F COMPUTATION DETAILS
We consider the nuisance/bridge function class Q or H. and the dual/critic functional class M or are the RKHS class. The inner maximization Eq. and 9 may no longer have closed- form solutions with the RKHS norm constraints. Similar to Dikkala et al. (2020), we consider the following optimization problem
M (g(as, 21,0s) p(ailux, m2) m(as, wi, wi) - Am ||vall2,n - 7m m + llqll
min max
4E mEM 7
1 Σ (y) (wi as, as, az)) )) (ai, 2,, Ag 19l1 To Ilol n 1ill gEg
min
Proposition F.1. Suppose M and G are RKHS spaces with kernel KM and Kg equipped with the canonical RKHS norm, then for any q, h, we have
Am
T" Am |ml12 -i [1 mEM
ehi KM,
KM, + Via
Hngnn
4'
I'm
Am 1
-1
Un KMn
KM.n + +ymdd
KM tn
4mm
Im n
Xg
max 47 >g 11912,m 79 1llll g6g
Wh, Kgnn
Kann + rigg/
Uh,n
Arg
7ig
wh. 1//
Xg
Kg + 7/9
Kg 4h
4
"i n
where KM.a = (K Ws 24, aj. Wj, xz 1) = Kgnn = (Kgla;, 24, Ti Oj. Bgz 23 22)) the empirical kernel matrix and 19.n ( (1 2i, palwwa.)) Di Uhn (2 (y: - h (a;, (1,,,4;,000 ()))
Proof. By the generalized representer theorem of [Schölkopt al. (200T), implies that an optimal solution of the constrained problem takes the form
m(a, w. x) = Σ a KM Km Wi, Iz, a, w,
We denote KM,a- (Km(a;, W1 2i, Wy F.y) 1))) the empirical kernel matrix. And have [| = a KM, f(;) = e, MM, and ||ml|2, La K K a. Thus the penalized prob- lem is equivalent to the finite dimensional maximization problem:
(Am KM. + +mmdd
Am
max 2o 0E 9.n. KMnaa - T
K KM.aa
where an = (+ (((a,, 2,, x;) has closed form optimizer of:
1 p(aclernnas))
))== i=1"
By taking the first order condition, the latter
1
Am 1
KM. +
27m
Ym n
and optimal value of:
1/ KM,
Am Ym
KMan +1)
2g
1b 12/2 M.n
(mm 1 Kma +1)
K
1/2
M.n Mgn
4vmm Similarly, we have
4mm
Ym
Ag 1 Kgnn +1)
-1
max $ >g 1la11... - 11al13 = gEg
whn Kgn
4hhn
47g
"i n
wT K112
Ag Kg +1)
-1
1/22
Kgnn Pha
47g
"g n
where vhe.. ( (yi (as, Wi, ax))))
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If further Q and H. are RKHS, obtain the closed form solution about the outer maximization, by solving
arg min 15
MM,
Am 1
K KM, (
+7)
Vynn + 4am llall
qEQ qi
Y' n
Ag Kgn +1)
h
min 2 kgnn kEH
Shnn M7 l1bb1
79
Again by the representation theorem, we have
h (.) = Σ d₂ (la,, x.), (()) Σ
Byk (laz, 22, 23)
where
â = (KH.n GnKHH + 4 H. KH.nGaWt
B M, + Im en)) K Ke p(a, luws p(a; xz)
for Gb.
ggn
1 Kg + +1) To n
K1/2 and M, K
1 n
KMM + 1)
K
K1/2
G.n
7m
There are several tuning parameters in the estimation of ho and 40- We accept the tricks and recom- mendation defaults by Dikkala et al. (2020). The following parameters will used to determine.
Ag (n)
(29)
7g
n.4
s > (n)
M 7g (s, (s
(30)
79
For Am and Tm, we also choose parameters like this.
7m
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G ADDITIONAL EXPERIMENTS
Sample size 1000
Sample size 1000
3.5
2.5
-1
ElY(do(A)]
-2
1.5
0.0
5.6
5.8
6.0
6.2
6.4
6.6
6.8
7.0
4.0
4.2
4,4
4.6
4.3
5.0
5.2
5.4
Treatment
Treatment
Sample size
1000
Sample size
1000
4-
E[Y]do(A)
2-
ELY(d(((A)
POR
1-
KPV
MIKIMAX
NKMR
5.6
5.8
6.0
6.2
6.4
6.6
6.8
7.0
-2.0
-15
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Figure 4: ATE comparison of different methods across various methods on three different data- generating mechanisms; Left Top: ATE comparison using 1000 samples in the Scenario 1; Right Top: ATE comparison using 1000 samples in the Scenario 2: Left Bottom: ATE comparison using 1000 samples in the Scenario 3; Right Bottom: ATE comparison using 1000 samples in the Times series data.
In this section, we consider three more synthetic settings introduced in Hu et al. (023). as well the times-series setting introduced in Miao al. (2018b) that satisfies the proximal causality framework. Similar to Tab. [ our methods are comparable or better than others.
Implementation Details. In the PKIPW and PKDR estimators, we choose the second-order Epanechnikov kernel, with bandwidth fow = cō AN
an 1/5 with estimated std & and the hyperpa- rameter e 1.5. For policy estimation, we employ the KDE in the two datasets. The rest of the implementation details are consistent with the experiments in the text.
Evaluation metrics. We report the causal Mean Squared Error (cMSE) across 100 equally spaced points the range supp cMSE == 100 Ei (Y ) Here, we respectively take sApp(AA := [5.5, 7], [4, 5.5], [5.5, 7], 2, in three synthetic data and the times series data. The truth [[ is derived through Monte Carlo simulations comprising 10,000 replicates of data generation for each
G.1 EXPERIMENTS WITH DIFFERENT DATA GENERATING PROCESS
Data generation. We consider three different data gotta-gnnerating mechanisms (Hu et al. 2023). Under each of scenarios, simulate the continuous unmeasured confounder U following normal distri- bution with mean and variance 0.2, denoted by U ~ N1,,..2), Similarly, we simulate two type of proxy variables WIU N 2 0.2) and ZUU N(-1+ 1.5 U, 0.2). The three scenarios vary according to the data generation process based on the models for the outcome Y|A, U, and for the treatment AU.
Scenario 1. We assume that the true distribution of the outcome Y is parabola, i.c., the outcome is a second-order regression function of the treatment:
Y|A,U NN-10 10 2.2 (A 6) 4 U, 0.2)
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and AIU (2.5 + 4. 0.2).
Scenario 2. We assume that the true distribution of the outcome Y has a sigmoidal shape:
YAA ~ N(15 + sign(A (A- 5) VIA - 5| + 1.7 U, 0.005,,
where sign is the sign function such that it is equal when 2 and otherwise. We assume
AUU ~ (1 + 4 U, 0.2).
Scenario 3. We assume that the true distribution of the outcome Y monotonically increasing with non-linear relationship with both variables A and U:
YIA,U U N(-2 e0--2.--1440 -14((1)) + -
e. 002)..
We assume A U ~ (2.5 + 4 U,0.2)
Results. We report the mean and the standard deviation (std) of cMSE over 20 times across four scenarios, as depicted in Fig. 4 and Tab. []] For each scenario, we take n = 1, 000. We can see that PKIPW method suffers from large errors in scenarios and 2 while performing well scenario 3. where the treatment-inducing proxy Z is misspecified. However, the PKDR method still performs well due to its doubly robust. In addition, we find that the MINIMAX method does not perform well because it requires more samples to fit the neural network.
Table 3: cMSE of all methods three different data-generating mechanisms.
Dataset
Size
PMMR
KPV
DFPV
MINIMAX
NMMR
POR
PKIPW
PKDR
Scenario
1000
05
0.25
0.50
1.45
0.17
0.16
0.16 2.29
Fl al. 2023
Scenario
1000
0.16g0002
0.16
0.22ga.7
0.88
0.05
0.08
0.15 o6
0.07
0.10g000
0.10.a.02
0.28
0.45
0.2
0.22
0.09 oa
0.21
500
0.11
0.13g0.12
0.20
0.21
0.18g0.12
0.10
0.21
0.09
Time Series
1000
0. 0.
0.08
0. 0.16
0.22
0.18
0.12
0.20
0.12
G.2 EXPERIMENTS FOR TIME SERIES DATA
Data generation. We follow Miao (0188) generate data.
U, -U,-++++0 22 V; 0.6U, + E
= + 1.5V, + r/U; + E3s:
Y = 0.5 + 0.7A, + 1.5% + 0.90; Esi. &1i. E2i, C3i, ~ NO0, 1),
where Ug is a stationary autoregressive process with autocorrelation coefficient ç, and 1/ controls the magnitude of confounding. Here, we let 0.8, n 0.5. For proximal causal approach, use W, = Y,-1 and Z, = A4+1 as two types of proxy variables and do not need auxiliary data.
Results. We report the mean and the standard deviation (std) of cMSE over 20 times across four scenarios, as depicted in Fig. 4 and Tab. 3 For each scenario, we consider two sample sizes, n = 500 and n 1, 000. As shown in Fig. 4 our PKIPW and PKDR accurately estimate the causal effect across all treatment values, making its overall cMSE comparable or better than other baselines. This result suggests the effectiveness of our methods for different scenarios.
G.3 RATE
Due to the error introduced in kernel approximation, this is reasonable sacrifice to handle contin- uous treatment within the proximal causal framework. Besides, according to Ichimura & Newey (022)), since the estimand is non-regular, therefore it may not enjoy the properties of /n--consistent and asymptotically normality. Such flexible kernel function approximation will make a non- negligible contribution to the limiting behavior of the estimator, preventing asymptotic normality and root-n consistency.
We conducted empirical numerical verification using Scenario I from the initial synthesis experi- ment outlined in Appendix G As per the synthesis mechanism, we can easily obtain the density function
f(A(W) = v0.4m
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We compute the empirical estimator error with sample sizes {200, 400, 600, 800, 1, 0000 and com- As we speculated before, the convergence rate difficult to
pare the estimator error Figure 5 reach n 1/2
n7-3/2
0.14
nn-2/9
0.30
0.00
0.04
Sample s:e
Figure 5: Empirical estimator error of the Scenario I in Appendix G
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Published as a conference paper at ICLR 2024
H EXPERIMENTS
In this section, we present the data generation process of experiments and the detailed settings of hyper-parameters.
H.1 DATA GENERATING PROCESS THE LOW-DIMENSIONAL SYNTHETIC EXPERIMENT
We describe the data generating mechanism in the Synthetic- Data Experiment. The generative pro- cess from Mastouri al. (2021). Since the original data generates E [0, 1] and the overall trend is flat, we modily the structural equation of Y make it easier to distinguish.
= UU1 U2]; U ~ Uniform [-1, 2]
U1 Uniform [0, 10 - U2 S 1]
W = WW, W2] = [U1 + Uniform [1, 1.11UU22 + e1]
Z [1, 72] [Us E2, U2 Uniform 11,
A U2 +
3 (2 (0.3U1 0.30 + 0.2) 1.5.4) E4
where (6:)==1 ~ 1).
H.2
DATA GENERATING PROCESS IN THE MIGH--DMENSIONAL SYNTHETIC EXPERIMENT
For X Rdim(X), Z (2)m((2) W Rdim(W) and (A, R, first generate the unobserved noise:
(ehE (),,)), 1~~ Uniform[-1, dj)mm()) Ui Uniform[ 11, dim(()
Next, we generate the following data structure
For unobserved confounders U, have
U= + 63.
Uw 63
For two types of proxies Z and W, we have
= V= + 0.25 1dm)
W = Uw + 0.25 Un ddmm(W)
where 1, E TRP is the vector of ones of length p.
For covariates X, we have
1 11(li - for i * j.
X ~ N(O, where E E pddmm (xlim(X)xxiii xxdim(X)xdim(x)) En 1 and Σij
2
For treatment A, we have
= A(3X 3r + +3 {2) 0.25 Uw.
where 8, E (X) and 3_ E d2) are quadratically decaying coefficients, e.g. [8x]; =j-?
A is the truncated logistic link function A(t) (0.9 - 0.1), , - + 0.1.
exp(t)
For outcome Y, we have
Y ATE (A) + 1.2(X 3r BT)) AX, + 0.25 U.,
where B,, E [d)m(W) are quadratically decaying coefficients, e.g. [Sw]; = =j-
Follow Colangelo Lee 2020). we use the quadratic design, 0ATE (a) = a² + 1.2a.
H.3 LEGALIZED ABORTION AND CRIME
In the Abortion and Criminality dataset, described in the reference Woody al. (2020), the key variables are as follows:
Treatment Variable A: Effective abortion rate;
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Outcome variable Y: Murder rate;
Treatment-inducing proxy Z: Generosity towards families with dependent children;
sutcome-inducing proxy W: Beer consumption per capita, log- reg-prisoner population per capita, and concealed weapons laws.
We take the remaining variables the unobserved confounding variables U. Following Mastouri
et al. (2021). the ground-truth value of ((a) is taken from the generative model fitted to the data.
The dataset is available at https::/githhue http://www..cooocccceeee///eeeee wha.eeceeennneneeeeeeeeeeeeeeeee htap://///a//a////////ooo//////e proxiiess /ata/aaaaaaa/// htaaaaaaanaaaaaaaaaaaaaaaaaaaaaa
tree/ma n
sim _d_no_x
H.4 HYPERPARAMETERS SELECTION
In our numerical studies, RKHSs 9, M, Q are equipped with Gaussian kernels
52) exp(2lla; x2 x2
The median heuristic bandwidth parameter -
median [|=:; zÿ 1 hijeer for indices subset C (I....,n). n). For the regularization coefficient, we automatically select it according to Eq. 29
and 30
For KDE, we also choose the Gaussian kernel. For bandwidth, we employ three fold cross- validation, where the bandwidth chosen 20 values uniformly distributed in logarithmic space between 10-0.1 and 10 raised to 10
For CNFs, recommend using the package probaforms, where the prior distributions is multivari- ate normal distribution.
Table 4: CNFs-block
Layer
Configuration
Input( A,, W,, X₂) FC(in-dim, 128), ReLU
2
3
FC(128, 64), ReLU FC(64, 32)
3
We stack four CNFs-block and finally solve the density function.
Table 5: Hyperparameters for CNFs.
Hyperparameter
Learning rate
le-4
Epochs
500
Batch
512
Reight_decay le-4
For the KPV method, used the Gaussian kernel where the bandwidth is determined by the median trick. We select the regularizers A A2 = 0.005.
For the PMMR method, used the Gaussian kernel where the bandwidth is determined by the median trick. We select the regularizers A = X2 = 0.1.
For the DFPV, optimize the model using Adam with learning rate 0.001, ;1 0.9, ; 0.999 and 10-8, Regularizers \ X2 are both set to 0.1 as a result of the tuning procedure.
For the MINIMAX, we used learner and adversary networks where learner 12= le-4,learner Ir=le-4, adversary 12 = 5e-3 and adversary Ir 5e-4.
For the NMMR, optimize the model using Adam with learning =3e-3, decay=3e-6 and epoch 10000.
44


Reviews:
[{"review_id": "pCHsU5aw48", "reviewer": "Reviewer_jsr3", "summary": "This work studies two very important settings in causal inference: continuous treatment & unmeasured confounding. This work introduces a new kernel-based DR estimator designed for continuous treatments, and it presents an efficient approach to solving nuisance functions and demonstrates the estimator's effectiveness through synthetic and real-world data.", "questions": "See weakness.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Clear writing & solid theoretical results", "weaknesses": "The reviewer is not an expert in deep learning for causal inference but would assume there are working targeting the studies setting: causal effect estimation for continuous treatment under potential missing confounders. \n\nThe reviewer understands the page limitation and would recommend a detailed comparison to existing literature in the appendix --- how is this method novel and why this novel kernel modification is necessary to handle the pitfalls in the current literature?\n\nThe reviewer would like to raise the score once there are more literature survey on that direction and added numerical experiments: comparing with **SOTA** DL method on **benchmark** datasets in the revision.\n\nA minor comment: will it be better to call section 2 \"background\" instead of \"related works\"? And it is not clear to the reviewer why \"in this paper\" is highlighted in the second paragraph of section 2...", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7FvOxTDgCO", "reviewer": "Reviewer_sZEk", "summary": "This paper introduces a method for proximal causal inference when dealing with continuous treatments. The proposed method is doubly robust in terms of estimating nuisances.", "questions": "1. I believe that the statement of Theorem 4.5 is incorrect. More precisely, the influence function of $B(a)$ does not exist, meaning that $B(a)$ is not pathwise differentiable. However, $B(a; P^{\\epsilon,h_{bw}})$ is pathwise differentiable, indicating that the influence function does exist. Therefore, to accurately state this, the term \"lim\" should be removed and the statement should be made with \"for any $h_{bw} > 0$\". \n2. I understand that $q_0$ represents the maximum value for the equation in Lemma 5.1. However, I am unsure about the connection between Lemma 5.1 and Equation (8). In other words, why do we need to minimize the empirical quantity for the equation in Lemma 5.1?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. This paper is technically sound and strong. \n2. The experimental studies are well done. A sufficient amount of empirical evidence for the proposed method is provided.", "weaknesses": "1. I believe that the statement of Theorem 4.5 is incorrect. More precisely, the influence function of $B(a)$ does not exist, meaning that $B(a)$ is not pathwise differentiable. However, $B(a; P^{\\epsilon,h_{bw}})$ is pathwise differentiable, indicating that the influence function does exist. Therefore, to accurately state this, the term \"lim\" should be removed and the statement should be made with \"for any $h_{bw} > 0$\".\n2. A practical guide is needed to solve the optimization problem in Equations (8,9) and apply these methods in practice.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZT0rrJo9Vr", "reviewer": "Reviewer_eqxp", "summary": "The paper presents a kernel-based doubly robust estimator designed for proximal causal learning, particularly adept at handling continuous treatments. The authors demonstrate that this estimator is a consistent approximation of the influence function. They also introduce an innovative method for efficiently resolving nuisance functions. Additionally, the paper includes an in-depth convergence analysis in relation to the mean square error.\n\nThe paper proposes a novel and intriguing problem within the proximal causal learning framework.  Despite the motivation not being particularly compelling, the problem itself remains highly intriguing and of considerable importance.", "questions": "The authors have well clarified their setting and their problem is also novel. I only have several comments below.\n\n* It will be more appropriate to call Assumption 3.2 the \"latent ignorability\" instead of \"conditional randomization\". \n* Although one single optimization reduces computational burden, but asymptotical normality of causal effect estimate may not hold using the proposed kernel estimation method. So please make the motivation clear about why this paper does not focus on the inference issue.\n* Why does the projected residual mean squared error make sense? Does it mean \\hat{q} is consistent for q0?\n* this paper seems to consider the high-dimensional setting in experiments, what about theoretical properties?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is well-structured and flows naturally.\n* This paper proposes an intriguing research topic within proximal causal inference work.\n* Theoretical guarantee is also provided.", "weaknesses": "* The inference issue seems to be ignored without the analysis of asymptotical distribution for causal effect. \n\n* The paper’s rationale for the calculation of the influence function appears to be lacking, as it directly selects a specific submodel. It might be more appropriate to refer to the function derived in this paper as the efficient influence function, given that the previous doubly robust estimator for binary treatment is efficient.\n\n* The empirical coverage probability is not given. MSE only contains both bias and variance terms, which may not display the true statistic estimation acuraccy and precision.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)