Generalization Bounds for Estimating Causal Effects of Continuous Treatments
Authors: Xin Wang
Shengfei Lyu
Xingyu Wu
Tianhao Wu
Huanhuan Chen
Venue: NeurIPS 2022 Accept (2022)
Keywords: dose-response
causal inference
selection bias
causal effect
continuous treatment

Abstract: We focus on estimating causal effects of continuous treatments (e.g., dosage in medicine), also known as dose-response function. Existing methods in causal inference for continuous treatments using neural networks are effective and to some extent reduce selection bias, which is introduced by non-randomized treatments among individuals and might lead to covariate imbalance and thus unreliable inference. To theoretically support the alleviation of selection bias in the setting of continuous treatments, we exploit the re-weighting schema and the Integral Probability Metric (IPM) distance to derive an upper bound on the counterfactual loss of estimating the average dose-response function (ADRF), and herein the IPM distance builds a bridge from a source (factual) domain to an infinite number of target (counterfactual) domains. We provide a discretized approximation of the IPM distance with a theoretical guarantee in the practical implementation. Based on the theoretical analyses, we also propose a novel algorithm, called Average Dose- response estiMatIon via re-weighTing schema (ADMIT). ADMIT simultaneously learns a re-weighting network, which aims to alleviate the selection bias, and an inference network, which makes factual and counterfactual estimations. In addition, the effectiveness of ADMIT is empirically demonstrated in both synthetic and semi-synthetic experiments by outperforming the existing benchmarks.

Generalization Bounds for Estimating Causal Effects of Continuous Treatments
Xin Wang
Shengfei Lyu
Xingyu Wu
Tianhao Wu
Huanhuan Chen"
University Science and Technology of China
{wz520, saintfe, xingyuwu, wutianhao8888)@mm @mail. ust..uut.. cn hchenduste.. edu.cc
Abstract
We focus on estimating causal effects of continuous treatments (e.g., dosage in medicine), also known as dose-response function. Existing methods in causal inference for continuous treatments using neural networks effective and some extent reduce selection bias, which is introduced by non-randomized treatments among individuals and might lead to covariate imbalance and thus unreliable inference. To theoretically support the alleviation of selection bias in the setting of continuous treatments, we exploit the re-weighting schema and the Integral Probability Metric (IPM) distance to derive an upper bound on the counterfactual loss of estimating the average dose-response function (ADRF), and herein IPM distance builds bridge from a source (factual) domain to infinite number of target (counterfactual) domains. We provide discretized approximation the IPM distance with theoretical guarantee in the practical implementation. Based on the theoretical analyses, we also propose a novel algorithm, called Average Dose- response estiMatlon via re-weigh schema (ADMIT). ADMIT simultancously learns a re-weigtting network, which aims to alleviate the selection bias, and inference network, which makes factual and counterfactual estimations. In addition, the effectiveness of ADMIT is empirically demonstrated in both synthetic and semi-synthetic experiments by outperforming the existing benchmarks.
Introduction
Causal inference, which estimates causal effects from observational data, is fundamental problem in identifying causal reactions between actions/treatments and effects. It facilitates the docisiionmmainnn process wide variety of domains, such as economics [I]. public policy (2), medicine [3] and advertising [ b. Specifically, causal inference aims to estimate the causal effects of taking different treatments (e.g., whether take medication or not), where neural network-basee methods
have made breakthrough in the binary setting 7 8 9 10]. In real-world applications, e.g. medical experiment, treatments usually cannot be simplified as binary treatments since the dosage is critical factor for the effect of the medication. Compared with causal inference for binary treatments, causal inference for continuous treatments, also known as dose-response estimation, deals with an infinite number of treatments, resulting in an infinite number of counterfactuals, far more than one counterfactual in the binary setting.
The success of neural networks on causal inference for binary treatments [T] 12 13 can be attributed
part the alleviation of selection bias, which leads unreliable counterfactual inference since the covariates of subpopulations receiving different treatments are unbalanced due to non-randomized treatments among them. Specifically, the two distributions of the treated and control groups can
"Corresponding author
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Factual
Loss
DX,,
Weight
yt
Inference Network
Pxr CXjO
weighting
ADRF
IPM
Bound
Network
Loss
yt
PxT (35) s- [0. 11, sit
Countrractual
yt
Loss
The framework of estimating ADRF using a re- re-weighting schema and IPM. The counter- factual loss cannot be computed since counterfactual outcomes are not unobserved. In the figure. unobservable variables and incomputable results are indicated by dotted rounded boxes and infeasible computations are indicated by dashed arrows. We learn sample weights using a re-weighting network to make the distributions of the subpopulation receiving treatment & and another subpopulation receiving treatment E [0, 1], s F be closer in the IPM distance, i.e., more similar, which results in alleviating selection bias in the continuous treatment setting. We also learn inference network to make factual and counterfactual estimations. Then we exploit re-weighted factual loss and the IPM distance to bound the counterfactual loss. Finally, we combine the factual loss and the bound of the counterfactual loss as the ADRF loss, which minimized to simultaneously learn the re- ee-eeighting network and the inference network.
Figure 1:
In the continuous setting, since the control groups are potentially infinite, these standard methods for alleviating selection bias in binary treatments cannot be easily extended to mitigate selection bias among a theoretically infinite number of subpopulations.
be balanced using Integral Probability Metric (IPM) [|| or re-weighting schemes 112
13
Currently, Schwab et al. [4 propose the Dose Response Network (DRNet) that extends "two heads", which are two outcomes a neural network for making factual and counterfactual estimations in the binary setting, to "multiple heads" for continuous dosage by discretizing the dosage into multiple intervals. Several regularization schemes in DRNet, such as distribution matching 77 propensity dropout [15] and matching on balancing scores [7], are exploited to handle selection bias. However, these schemes are heuristic and not provide theoretical evidence back up their intuition.
To theoretically support the alleviation selection bias, this paper derives an upper bound on the counterfactual loss during estimation of the average dose-response function (ADRF). The key points of the derivation lie in the re-weighted factual loss obtained from observational data and an IPM term that measures the distance between the factual (re-weighted) and counterfactual distributions, as illustrated in Figure [ The infinite number of counterfactual distributions is the main challenge since the distance cannot be calculated with finite samples in practice. In the practical implementation, we provide a discretized approximation of the IPM term and prove that the difference between its approximate and true values converges zero in probability under the assumption that the distributions of subpopulations receiving different treatments shift smoothly. On the basis of derived upper bound and its implementation with theoretical guarantee, we propose novel algorithm called Average Dose-response estiMatlon via re-weighTing schema (ADMIT). ADMIT jointly learns inference network and re-weighting neural network to reduce selection bias while making factual and counterfactual estimations. The effectiveness ADMIT is empirically confirmed by outperforming the existing benchmarks on both synthetic and semi-synthetic experiments in the continuous treatment setting.
Our contributions in this paper are 4-fold: (1) to the best of our knowledge, this is the first study that provides generalization bound for estimating ADRF, alleviating selection bias; (2) we provide discretized approximation of the IPM distance with theoretical guarantee; (3) we propose an algorithm, ADMIT, capable of reducing selection bias while making factual and counterfactual estimations; (4) conduct both synthetic and semi-synthetic experiments in the continuous treatment

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setting, empirically demonstrating the effectiveness of ADMIT with superiority over the existing benchmarks.
2
Related Work
Causal inference for continuous treatments is broadly studied through statistical methods [8 19
21. These methods are often based upon the generalized propensity score (GPS) 122], or the entropy balancing based method [23] that learns the unit weights by incorporating information about known sample moments to achieve covariate balance, e.g., entropy balancing for continuous treatments
(EBCT) [24]. Adaptation of neural networks for causal inference in the continuous treatments setting has attracted the attention of research community recently [14 25 26)). Schwab et al. [4 present the
DRNet to estimate counterfactuals for continuous treatments, which is an extension of the network architecture proposed in [1]]. Specifically, DRNet divides continuous treatment into several successive intervals, and trains one separate head for each interval. Bica et al. [26] point out that
DRNet might be inflexible since it cannot determine the divided intervals dynamically and propose SCIGAN that generates counterfactual outcomes for continuous treatments based on a generative adversarial network (GAN) framework. Meanwhile, for the issue that DRNet produces discontinuous ADRF due to interval discretization of continuous treatments, Varying Coefficient Network (VCNet) 25 lets the weights of the prediction head be continuous functions of the treatment to preserve the continuity of ADRF Moreover, Nie et al. (25] obtain a doubly robust estimator of ADRF by generalizing the targeted regularization proposed in [0),, and provide theoretical guarantees for its asympiotic correctness. Unlike VCNet, our work is derived from the generalization error of reducing the selection bias when estimating ADRE. To balance the covariates among infinite subpopulations, we learn re-samplnng weights that reduce the IPM distance between observed and counterfactual groups. As such, we derive an upper bound on the estimated counterfactual error and demonstrate experimentally the proposed algorithm ADMIT based on the derived upper bound outperforms GPS, EBCT, DRNet, SCIGAN and VCNet.
See Appendix A for discussion research related to causal inference for binary or continuous treatments. Appendix A also introduces the relationships between causal inference for continuous treatments and domain adaptation [27 22 29
3 Problem Formulation
Let X' denote the d-dimensional space of covariates and ) the outcome space. We refer to T as continuous treatment random variable in an interval T = [a, b). Without loss of generality, we set [0, in this study. Suppose have an i.i.d. sample of units, indexed by 1,2, ... n. Following Rubin's 's potential outcome framework [30), we postulate the existence of potential outcomes, Y for t E T. For each unit i we observe the treatment f;. and the covariate æ,, The observed outcome (denoted as Y2) is defined to be the potential outcome (denoted as Y corresponding to the received treatment t, i.., Y, = Y
Given an input covariate and a treatment I, we refer to sa(;,.) as the individual dose-response function, which defined as:
(tt, .)) := E X = æz)
(1)
Our goal is to derive an estimator of the average dose-responnse function (ADRF):
nx(t) = EY
(2)
By the law of iterated expectation, n(()= = Elu(t, X). In this study, the estimator of ADRF p(() is based on estimating the individual dose-response function pll, æ)).
The following assumptions have been made to ensure that ADRF identifiable from observational data.
Assumption 1. (Unconffoundedness) Conditional on the covariates, X, the potential outcomes, Y!, are independent of the treatment assignment, T,
[ E [0, 1}} LTX.
This assumption is also known as no unmeasured confounding and ensures that the individual dose-
response function is identifiable from observational data, i.e., j((,, x) E = m]] = E X T t].
3

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Assumption 2. (Overlap) There exists some constant C > 0, Vo E X and Vt E T,
(1 2
This assumption is the condition that every point in the space of covariates has non-zero conditional probability density of receiving any treatment.
Theoretical Analyses and Algorithm
This section firstly presents theorem that the error of estimating ADRF is bounded by the marginal loss comprising the factual and counterfactual losses. The counterfactual loss cannot be calculated from observational data since counterfactual outcomes the ground truth are unobserved. In the meanwhile, the counterfactual inference might be unreliable due to selection bias. We then derive theoretically optimal weights in alleviating selection bias from the perspective of importance sampling. Although the optimal weights are theoretically guaranteed, they are impractical to learn due to their instability. Under similar re- re-weighing schema, theoretical guarantee provided to guide the learning of adaptive weights that mitigate the impact of selection bias. We refer the reader to Appendix B for detailed proofs.
4.1 Error of estimating ADRF
Let ji()) be an estimator for put), and we are interested in estimators with a small expected mean squared error (EMSE) in estimating ADRF,
EMSE(μ, A) = ((((TT) (T))
(3)
Since u(t) cannot be observed from observational data, this study uses hypotheses fo E H. to estimate the individual dose-response function p(t, x) first. For case expression, we write f(t, x) as f(()). After that, we combine these hypotheses to estimate p(t).
(tt) (X).
(4)
We bound the error EMSE(H, pi) by the loss of hypotheses ft. for T. with respect to the corresponding potential outcomes. Let L y x y + R4 be loss function. The expected loss for the treatment and covariate pair (t, x) is defined as:
1 (x) == Ex ,.())) æ².
(5)
Theorem 1. Let T. be the squared loss function, iee. I(y.y') (y y/)22. For hypotheses f of individual dose-response function p(t,)) with marginal loss c(fi) = (X)), there exists a constant mmin 2 0, such that,
EMSE(#. p)) Erle(f)) -miin-
(6)
Though u()) cannot be observed from observational data, Theorem implies that its expected error can be bounded by the marginal loss c(ft), which are related the prediction loss concerning potential outcomes, i.e., (x).. However, the model cannot obtain marginal loss due to the inability to observe the counterfactuals B1]. A re-weighting schema to address this issue is presented next.
4.2 Re-weighting via importance sampling
The fundamental problem of estimating ADRF is that it is impossible to observe all potential outcomes for any unit in data. The potential outcome Y" could only be observed in the subpopulation that received treatment f,, i.e., T f. We refer to Px (2 (æ=(t) as the distribution of T t subpopulation, and px (æ) as the distribution of population. The two distributions, Px yr (x(t) and px (æ), are not equal since X and are not independent. Their difference leads to the margin loss e(ft) not being equal the conditional loss of hypothesis ft, (felT s), which is defined
e(fiT 3) = (X) s).

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The conditional loss e(fiTT s) is the expected loss with respect to the distribution of the subpop- ulation that received treatment s, i.e., T s. and the marginal loss e(ff) is the expected loss with respect to the distribution of the population. When the treatments l. and 8 are equal, the conditional loss <(fa]T = s) called faceefactaall loss; otherwise, it is called counterfuctual loss. The counterfactual loss is infeasible since the potential outcome Y could only be observed in the subpopulation who receive the treatment t. A minimizer ft of {(ft) is required to minimize the error of estimating ADRF according to Theorem L which may be different from minimizer of eft:TT' t) since Px(r(()) () + px (z). Importance sampling is introduced to estimate e(ft) in this work, which aims to estimate Ex= [[())]] of particular distribution g(a) while only samples generated from a different distribution g(x) are available. The basic idea is rewrite the expectation as:
g(x)
Ex-gta) [h(x)] = Ewrg(a))
h(xz)
(8)
g(x))
where g(m) is called sampling weight. To estimate e(fs) based on the factual loss e(felT' = t), a q(a)
weight w(x) is utilized for each unit to define re-weighted factual loss:
Eu (f( = = [x)r(x((X) (X) t).
(9)
By Equation 8 the re-weighted factual loss is equal to the marginal loss when w(x) w² wx)) where
px
Pr(t)
w" (x)
(10)
(x
Prix
(f)x))
The term Prix (t|zz) is the conditional density of receiving treatment given the covariate æ, which is also known as the Generalized Propensity Score (GPS) [22 32). In binary treatment cases, the propensity score could balance the distributions of covariates between the treated and control groups [33] Hirano and Imbens (32] stress that the GPS has similar balancing property in
continuous treatment cases. Fong al. 20 propose the covariate balancing generalized propensity score methodology by defining a stabilized weight based on the GPS. Interestingly, the weight defined in [20) is exactly the same as w* (x) derived from the sample weight via importance sampling.
The weight w² (x:) based on the GPS is theoretically optimal in expectation [22] but may be impracti- cal to obtain. On the one hand, the approaches based the propensity score may be unstable in the estimations 34 355. What's worse, this issue may be amplified in continuous treatment cases since the conditional density difficult to estimate accurately [31]. On the other hand, it could be proved
g(m) that a large sampling weight q(ar) may result in high variance the estimations, which could lead to a poor finite sample performance. Therefore, the following subsection focuses on learning adaptive weights balance observed covariates across subpopulations that received different treatments and
giving a generalization bound of the marginal loss.
4.3 Generalization bound of the marginal loss
The imbalance of covariates across subpopulations that received different treatments caused by selection bias is closely connected with covariate shift in domain adaptation. This study alleviates this imbalance using an IPM distance, which is similar to the distribution distance metrics in domain adaptation, and gives a generalization bound of the marginal loss when using adaptive weights in Equation
The IPM is a class of metrics that measure the distance between two probability distributions, which is defined as:
IPMg(p. suppeg If -())))(a)) qxx))
(11)
where G is function family of functions g : X' - R. For simplicity, we abbreviate px (xx(((=)))) and Px(((2) (ol) (x) as Pt and p!" respectively. We first state Lemma bounding the difference between the re-weighted factual loss, (f. t), and the counterfactual loss, c(felT s), where + t.
Lemma 1. Let G be a family of functions 1 : X' - R. Assume the per-unit expected loss function T(f.8") E G for all f,f' E H. Then for any E [0, 1] and + we have:
efi s) S ew (fel T t) + IPMg(Pe,P"))
(12)
5

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Based on Lemma 1 we could give generalization bound of the marginal loss through the re-weithted factual loss as follows.
Theorem 2. Let 1PM = | P.)). The following inequality holds under the conditions of Lemma
<(Si) S ( = =0) () + 1PI
(13)
Theorem 2 shows that the marginal loss is upper bound by the re-weighted factual loss and an IPM distance. The first term could be estimated with observed samples. The second term is defined based on the distance between the two probability distributions of the subpopulation receiving treatment t and another subpopulation. Choosing a family of norm-l reproducing kernel Hilbert space (RKHS) functions leads to IPM being the Maximum Mean Discrepancy (MMD) metric [36]. When involving
discrete domains, there may have samples {xp, ... 2 ] ~ p and {z), .mh ~ an estimator
of the MMD distance between p and q could be obtained as follows:
MMD = n
n
2 VITE
n m
k( mr)
ΣΣ
ka) (14) f=1
m
m2
1=1 j=1
j=1
where i denotes a differentiable kernel, e.g., the Gaussian RBF kernel. A certain number of samples needed to estimate the MMD shown in Equation [4 When the treatment is discrete and finite, all the choices of the treatment can be observed in the samples and thus one can estimate the MMD from these samples. However, when the treatment is continuous, the samples that received some treatment /. may be unavailable since only a finite number of samples are observed while the choice of is infinite. In other words, the IPM term in Theorem 2 involves continuously shifting domains (for each E [0, ID. which cannot be estimated empirically. To overcome this challenge, we bound the difference between the IPM and its discretization under the assumption that the probability distributions of subpopulations that received different treatments shift smoothly.
Assumption 3. Let pt, and Pt, denote the conditional probability densities of subpopulations that received treatment 11 and 12, respectively. We assume that there is a constant a such that the following
inequality holds for all t1, l2 [0, 1]:
(pM. Pt.)) S a atlt1 - t2l.
(15)
could be proved that the minimum a that meets the conditions of Assumption
3 is
[0 (lims-o) JPMppppuP according to the triangle incquality for the IPM. Intuitively, a in Assumption
3 iidicates the maximum rate of the probability distribution shift of subpopulations, and it is easy to find a constant that satisfy the condition since the IPM term is bounded when the outcome of the hypothesis ft is finite, e.g., survival years (outcome) after taking some medicine (treatment). Therefore, Assumption iss practical and feasible in real-world applications. When we set a continuous treatment t. it is generally assumed that the conditional probability of t given smoothly shift (26). For instance, the difference in the probability of taking similar dose given an age is small if the dose of particular medicine depends on the age of the patient. Therefore, the age distributions of groups taking similar doses vary smoothly, or in other words, it holds most of the time that a is relatively small.
Based on Assumption 3 the following Lemma states that the IPMmaz corresponding to continuously shifting domain (for each E [0, 1) could be approximated as an IPM term based on discrete domains (for each t E {tz,.... ).
Lemma 2. Suppose we have i.idd. sample of units, and the ith unit received a trealment p(t). This study assumes Assumption hholds for a constant a. Then we have,
IPM S <maxiell.... .n) (TTr:d,.....I((PMPPPPP.P.. +0 n/
(16)
The RHS of Equation 16 bounds the worst case of IPM by discrete set (IPM (pr. Pf")
[1]} since all ag are enlarged to a during the proof of Lemma 2 where 0g = IPM(P..P.)
Therefore, the difference between the IPM and its discretization will be generally smaller than Lemma 2 displays. following Lemma states that point &; could be replaced by its neighbourhood [t,, t + 6].
ma t-t;
To empirically estimate IPM (Pt. )!",, the

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Algorithm/ ADMIT: Average dose-response estimation via re-weighting schema
Input:
Observed samples D { 9;, Yi .....,n,, loss function L(.,-) function family G
for IPM, batch size na, neighbourhood size 8. Initialized parameters: 0 = [0s, Bh Aw)
while f converged do
2: Sample mini-batch {( $i, Bt.). ).... (Kin, Stn, Vi. }} from D
3: Calculate the weight of each unit: Wiy = w(8ig ((11,)))
4:
for - to L = [1/0] do
5:
for = to and do IPMg = IPMg((ws, Riy ba E(( 1)526): foen Jas, E((R- 128.86)
6:
end for 7:
8: IPM
(I.....) A (IPMg (Pak,
9:
end for
10:
Calculate the gradients of the IPM term:
91 = Aa. I E, TPM Amaz
O2 = Ae.. Σ, IPM
11:
Calculate the gradients of the re-weighted factual loss:
93 Do,, ni
Ej Way L(h(si, ((xr, Viy)
94 = Ao. Σ, Wi, L(h(si, Φ( )). Vig)
12:
Update parameters: 10s. 0, [w] + [0+ 1g92 + g4), 8% -193. Ow 7911
end while
Output: Learned representation function (b.), outcome function h(,,)), re-weighting function wi.;))
Let Pas (xIr(alt E s + 5)) (0 < 5 < 1) denote the conditional density of covariates when t E ss 6). Then the following inequality holds under Assumption
Lemma 3.
(P., p(") S 1PMg(Ps..Pg P.) + ad.
(17)
Following Lemma and Lemma 3 we bound the marginal loss as follows.
Theorem 3. Suppose we have n i.i.d. sample of units, and the ith unit received treatment t.- Let IPMAmaz mazie .n) (IPMg(Pss, p.")} This study assumes Assumption holds for constant Then. for neighbourhood size O < we have,
(ft) S ( 1) + IPM Amaa + Op Vn
ad.
(18)
Contrary to the second term in Theorem IPM A Smar could be estimated empirically, since may get samples that received treatment E [., t + 8) by choosing an appropriate neighbourhood size
To minimize the error in estimating the ADRF, Theorem and Theorem 3 suggest that we should minimize the upper bound Equation (18) with respect to the re-weighted conditional loss and the IPM term, which leads us to Algorithm I detailed in the following subsection.
4.4 Proposed algorithm
Our algorithm, ADMIT, proposed based on the theoretical results derived in the previous sub- sections. As illustrated in Figure 2 neural networks to build the model, which is mainly composed of three components: a representation network ((x), ; a re-weighting network w11,, ((x)), and inference network hit,, d(x:))
This work firstly learns treatment-agnostee representation influence of treatment is different from that of æ (II. which might be lost during training if Φ(x) is high-dimensional, To address this issue, the varying coefficient model [37 25 is adopted to build ltt,, ((x)) and wtt, (())). Take the former an example, hit, d(x)) is defined as:
((x) of æ using all data. The
h(t, (())) NNe.(o(@(z)),
(19)

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Inference network
WL(Ps,
(x(t)
X
Z
(1)
IPM (p..pp
Representation network
Re-weighting network
Figure 2: Model architecture of the proposed ADMIT.
where NNo.(t) (t) is neural network with parameter 0n (t). Thus, the treatment t determines the nonlinear functions in (N)
Based on the theoretical analyses aforementioned, ADMIT aims to seek a re-weighting function w and inference function h minimize the error EMSE (t) of estimating the ADRF, using the following objective:
71 E (wll ((x:)) ((((4,)6(4))) Vi) + maxje(1, () (PPM
ming
(20)
where A 10. B 0,). To effectively calculate the IPM, the interval [0. 1] is divided into [1/5] intervals confirm the neighbourhood of the ith unit, and details of the calculation presented in lines 4 to 9 of Algorithm The model is trained using stochastic gradient descent to minimize the objective. Finally, an estimator of ADRF is built
(()) Σ hi,, ((x:)). 2=1
(21)
5 Experiments
difficult to get the ground truth treatment effect on real-world datasets due to the inability to observe the counterfactuals. To deal with this, existing literature [14 25 26 often synthetic/semi- synthetic data for meaningful evaluation. We use one synthetic dataset and two semi-synthetic datasets, News [14 7 and TCGA (38), demonstrate the effectiveness of ADMIT. Details of these datasets and implementation can be found in Appendix C.
5.1 Experimental setup
Synthetic data generation. We generate covariates æ Unif(0, 1) 185, and the assigned treat- ments and their corresponding outcomes are generated as follows:
(1
N (1l, 0.5) where
sin(max Ø1, F2, 13)) + max(ax, 21, + sin(0.5rz3)( + exp(1r - 0.5xz))
#1 = 10
+ (xr + xx)²
+ x3 + 2sin(xa) + 2rs- - 6.5,
y ~ (u2,, 0.5) where μ22 cos(2:r(t (t max(#) 28) 1+2n3
sin(x)))..
Semi-synthetic data generation. We obtain covariates from two real-world datasets, News 14 and TCGA [38]. To obtain the assigned treatments and their corresponding outcomes, we generate set parameters, v; = un:l|ee || and i =,,2, 2, 3, where u, sampled from normal distribution (O, 1), then

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Method
Simulation
TCGA
News
DRNet
0.209 + 0.0127
0.216 0.0281
0.274 0.0194
SCIGAN
0.638 + 0.0467
0.301 + 0.0204
0.707 + 0.0074
VCNet
0.129 + 0.0186
0.139 0.0107
0.201 + 0.0193
VCNe++EBCT
0.162 = 0.0256
0.126 = 0.0234
0.196 = 0.0123
GPS
0.179 + 0.0016
0.196 = 0.0011
0.176 0.0049
ADMIT
0.081 + 0.0048
0.071 + 0.0074
0.106 + 0.0135
Table 1: Comparison of ADMIT with (non) neural network- -based baselines. Reported performance (VEMSE) of average dose-respons, estimation synthetic and two semi-synthetic datasets. Metrics are reported as Mean Std.
~BBeta 3) where 3 = 7-1 d" +
and d= -yand
w. æ 20
vT y ~ (w,, 0.5) where y= 40t((00.5)) sin( 2(max((22)))) exp(( 0.3)))+10uf æ).
As discussed in [26]. larger y will result in higher selection bias. Without specific instruction, we 7 2 in experiments.
Baselines and metries. For neural network baselines, we compare ADMIT with DRNet [4], VCNet
(25 and SCIGAN (26). With the same settings in [25). we apply targeted regularization on both DRNet and VCNet, and add a conditional density estimation head for DRNet. For statistical baselines, compare ADMIT with GPS (32) and entropy balancing for continuous treatments (EBCT) [24]. When evaluating EBCT, we adopt the sample weights learned by EBCT, and take VCNet as its inference network estimate the ADRE For metrics, Expected Mean Squared Error (EMSE) p(t.))²
on the dataset, where EMSE
5.2 Benchmarks comparison
Table shows that the proposed ADMIT more accurate in estimating the average dose-response in both synthetic and semi- synthetic datasets. The dimensions of co- variates in TCGA and News are around 4000 and 3000, respectively. This result may in- dicate that ADMIT is well adapted to the nigh-dimensiooal data. Moreover, using the sample weights obtained by pre-processing the data with EBCT also enhances the per- formance on the two semi-synthetic datasets compared with VCNet. This result demon- strates the sample weights learned by EBCT could alleviate the selection bias to some extent. However, the performance boost is smaller than ADMIT, which indicates the superiority of the learned sample weights guided by the derived bound in this study. Additionally, the performance of ADMIT may benefit from the design where ADMIT jointly learns the re-weighing network and the inference network in optimization by minimizing the derived upper bound. This one-step approach may be preferable to two-step approaches, i.e., EBCT, which is demonstrated by Zhang et al. [39
SCIDAN
VONLt
ADMIT
EEMEE
selection bias y
Figure 3: Comparison of four models in terms of their / EMSE for varying selection bias on TCGA.
"The implementation of ADMIT is available at https//gittub.co//wwwx//////

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5.3 Selection bias
As indicated in [26], the variance of the Beta distribution decreases with the increase of y in the semi-synthetic data generation, resulting in higher selection bias. We compare the performance of ADMIT with VCNet. SCIGAN and DRNet on TCGA when varying choices of from (no bias) to 8 (high bias) to assess the robustness of these methods in different levels of selection bias. We found that ADMIT shows consistent performance and outperforms the three baselines across the entire range of evaluated selection bias. This result demonstrates the effectiveness of ADMIT in empirically mitigating the impact of selection bias.
Conclusion
In this paper, propose novel re- re-weighting schema to mitigate the impact of selection bias in causal inference with continuous treatments. Under this schema, this paper provides and proves generalization error bound on the estimation of ADRF based on IPM distance between the observed (re- reightet and counterfactual distributions, with theoretical evidence on learning sample weights that alleviate the unreliable counterfactual inference problem caused by selection bias. In the practical implementation, we provide a discretized approximation of the IPM distance with a theoretical guarantee, leading algorithm ADMIT that jointly learns re-weighting and inference networks.
7 Acknowledgement
This research is supported by National Key R&D Program of China (No. 2021ZD01 1 11700), National Nature Science Foundation of China (No. 62137002. 62176245), Key Research and Development Program of Anhui Province (No. 202104a05020011, 1), Key Science and Technology Special Project of Anhui Province (No. 2021 0320000000000),, Fundamental Research Funds for the Central Universities. We thank the anonymous reviewers for their constructive comments that help improve the manuscript.
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For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [No]
(c) Did you discuss any potential negative societal impacts of your work? [No]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix
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(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix B.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either the supplemental material or as URL)? [Yes] See Appendix
C.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix C.
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] See Section
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix C.
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13


Reviews:
[{"review_id": "ypUAHmAHziN", "reviewer": "Reviewer_XSmM", "summary": "This paper derives an error bound for treatment effect estimation in the continuous treatment (dosage) regime. The key insight is to relate this error to the bias introduced by distributional shift in treatment assignment using IPM distances between distributions. The authors further provide an algorithm inspired by the theoretical upper bound as well as empirical validation for the proposed method.\n\n", "questions": "Extracted from the above section:\n\n* Lemma 1 seems somewhat restrictive given that the loss function has to be in the family of functions that defines the IPM. What are the implications for the squared loss?\n* EMSE is a population quantity. What are the finite sample guarantees (if any)?\n* How do you actually calculate the IPM gradients and how does it contribute to the computational complexity? \n* How do the sample and computational complexities depend on the choice of $\\delta$?\n*  What were the computational bottlenecks of ADMIT? How does the GPU usage compare across methods?\n* Given the minor DGP modifications, have you tuned the parameters of the baselines (e.g. VCNet) to the new DGP?", "limitations": "The authors have not addressed the computational limitations of their algorithm besides the fact that $\\simeq 3000$ GPUs were used.", "rating": 3, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is by-and-large an extension of [1] to continuous treatments domains by leveraging a discretized version of the IPM metric used in [1], as well as importance sampling with learned weights ([2], [3]). The idea itself is a good addition to the causal inference literature, but I have reservations regarding the execution, both from a technical and quality of writing standpoint. \n\nOverall structure:\n* There is not enough contextualization with previous work. For example, [1], [2], [3] are only briefly mentioned and their connection with the current work is not made obvious. The entire *Related Work* section lacks clarity. \n* The introduction is also confusing, especially for those unfamiliar with the advances in this specific area of causal inference. The different paragraphs don't seem connected and it's unclear what the state of the art that this paper is improving upon actually is. \n* Section 4.4 and the Algorithm Box are too underdeveloped. It it unclear what the different quantities ($\\phi, h, w$) until you read the text and the text doesn't have enough explanations about how to execute the different steps of the algorithm (e.g. how to computer the IMP gradient or how to choose $\\delta$ appropriately). The appendix is very sparse and doesn't contain additional information that could answer this question.\n\nTechnical weaknesses:\n* Theorem 1 also holds for $\\sigma_{min}=0$, for example when $Y^t=f_t(x)$, i.e. the counterfactual outcome under treatment $t$ depends solely on observed features.\n* Lemma 1 seems somewhat restrictive given that the loss function has to be in the family of functions that defines the IPM. What are the implications for the squared loss?\n* EMSE is a population quantity. What are the finite sample guarantees (if any)?\n* How do you actually calculate the IPM gradients and how does it contribute to the computational complexity? How do the sample and computational complexities depend on the choice of $\\delta$?\n* Concerns about reproducibility: the simulations were run on 3080 GPUs according to the appendix. What were the computational bottlenecks? How does the GPU usage compare across methods?\n* The experimental results are encouraging, but there are some issues with the performance of the benchmarks. For example, the DGP is similar to the one in the VCNet paper [4], but their performance were more along $\\simeq 0.15$, rather than the $\\simeq 0.19$ found in this paper. And if the discrepancy comes from the minor DGP modifications, have you tuned the parameters of the VCNet to the new DGP?\n\nOverall, I don't think this work is ready for publication yet. \n\n[1] Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: Generalization bounds and algorithms.\n\n[2] Negar Hassanpour and Russell Greiner. Counterfactual regression with importance sampling weights. \n\n[3] Fredrik D Johansson, Uri Shalit, Nathan Kallus, and David Sontag. Generalization bounds and\n360 representation learning for estimation of potential outcomes and causal effects. \n\n[4] Lizhen Nie, Mao Ye, Dan Nicolae, et al. VCNet and functional targeted regularization for learning causal effects of continuous treatments.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "diL2WbwHVVw", "reviewer": "Reviewer_WSLo", "summary": "This paper estimates the causal effects of continuous treatments. To balance the covariates among infinite subpopulations, they learn re-sampling weights that reduce the IPM distance between observed and counterfactual groups. Thus, they derive an upper bound on the estimated counterfactual error and demonstrate experimentally the proposed algorithm ADMIT based on the derived upper bound outperforms GPS, EBCT, DRNet, SCIGAN, and VCNet.", "questions": "This work provides a discretized approximation of the IPM term.  Since there are limited samples while the number of domains is infinite, to overcome this challenge, they bound the difference between the IPM and its discretization under the assumption 3 that the probability distributions of subpopulations that received different treatments shift smoothly. However, how could this model handle a substantial shift and give a continuous and complete dose-response curve?", "limitations": "The experimental analysis is relatively insufficient for the continuous treatments. ", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. provide a theoretical guarantee to the causal effect estimation of continuous treatments.\n2. give a comprehensive summary of the current studies on the continuous treatment effect estimation\n\nWeaknesses:\n1. lack of innovation. The proposed framework is mostly based on previous works such as GPS, CFR, and DRNet.\n2. limited contribution. although the author provides the generalization bounds for estimating causal effects of continuous treatment, the theoretical proofs are the extension of CFR [Shalit, Uri, Fredrik D. Johansson, and David Sontag. \"Estimating individual treatment effect: generalization bounds and algorithms.\" International Conference on Machine Learning. PMLR, 2017.]", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "_Hb8Sp_0g1A", "reviewer": "Reviewer_TaAx", "summary": "The authors tackle the problem of estimating causal effects under continuous treatments by proposing a novel framework to estimate the average dose response function (ADRF). The core idea of their framework is to minimize a bound on the ADRF loss (instead of minimizing the error in estimating the ADRF directly). The ADRF loss consists of both a factual loss (exact) and an upper bound on the counterfactual loss – which is derived via a re-weighting schema  and the use of an Integral Probability Metric (IPM) that measures the distance between the factual and counterfactual distributions. \n\nThe authors provide a practical implementation of this theoretical bound under a smoothness assumption on the shift between covariate distributions in different subpopulations. They call their algorithm ADMIT, which they test in synthetic and semi-synthetic settings.\n", "questions": "It would be nice for the authors to give more intuition for how strong assumption 3 is? Seeing as this is a core assumption for the approximation of the IPM term, the authors need to provide a little bit more discussion surrounding it.\n\nMinor typos/spelling errors: \n\nLine 127: $\\hat{u}(t)$ should be $\\hat{\\mu}(t)$\n\nLine 173: What’s worse is that \n\nLine 16/Line 119: A transition word other than “besides” may be more appropriate \n", "limitations": "\nNo, there was not a significant discussion of the limitations of the approach, which is a weakness of this paper. See above for suggestions to improve this (i.e. assumption 3 discussion, more thorough empirical analyses). \n", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\nStrengths: \n\n[quality + significance] - The authors provide a clear and well-motivated development of their upper bound on the ADRF loss. Their connection of the theory to practical implementation was also particularly well done.\n\nWeaknesses: \n\n[quality] - There are a few improvements that could be made to the empirical results section of the paper. Increasing the thoroughness of the empirical section is the only major weakness in my mind. For one, it would be useful to provide another view of the results by e.g. providing the actual dose response curve (i.e. response vs dosage for the estimation and the ground truth). This would help the reader get a better idea as to which regions of the treatment may be better (in terms of estimation) vs not. Secondly, it would be interesting to see how the method performs in the setting where some of the confounders were unobserved (or simulated to be unobserved, i.e. you conceal them). Although an initial assumption is one of unconfoundedness, characterizing the failure modes of the method is generally useful. At a high level, the authors need to do a better job at discussing the limitations and failure modes of their approach.\n\n[clarity] (minor) - There are a few areas where there are some grammatical and spelling errors. See below for a few examples (not exhaustive). \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "1KZWUEoak6w", "reviewer": "Reviewer_hMRr", "summary": "This work derives a new generalization bound for estimating causal effects of continuous treatments. The new bound is based on the idea of generalized propensity score reweighting, and is obtained via a sequence of inequalities, translating the original marginal treatment effect which is hard to estimate to a new quantity that is easier to estimate. The generalization bound has been used as the objective function for training a deep neural network, whose architecture has been adapted from existing works. Experimental results show state-of-the-art performance of the proposed method (objective function).", "questions": "About soundness and contribution (primary):\nThe bound in Equation (18) is obtained by chaining a sequence of bounds. It seems to me the bounds are loose, for example, Equation (17). So I am curious why it can still outperform the other methods by large gap in the experiments section. And is it because of the use of reweighting schema (GPS) + neural network (a more flexible model), or is it because of using the generalization bounds? It is important to understand which part contributes in the improvement, and to give more explanations on the tightness of the bounds.\n\nAbout originality:\nAre there any literature in causal inference deriving generalization bounds? Not necessarily of a similar form to the one in this work. If so, the authors may want to compare your bound to theirs, and cite them in the paper.\n\n\n", "limitations": "The authors do not discuss this.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Originality: \nThe  proposed bound in this work seems novel to me. One question about related literature on generalization bounds is raised in the next section.\n\nQuality and clarify:\nThis work is overall of high quality and is very clear in terms of presentation. The only concern I have is about the soundness of the bounds, and is listed in the next section.\n\nSignificance:\nJudging from the experimental results, the improvements seem significant. However, it is not entirely clear to me why the method can perform so well. The bound seems loose and it is interesting to understand why they can still perform well. A better understanding is needed in order for the method to be more trustworthy.\n\nSome minor remarks:\n1. in line 127, \\hat u is a typo.\n2. in line 198, \"the sume of the re-weighted factual loss\" -- should it be average?\n3. more details are needed for line 201 to line 207;  think the MMD is defined for both continuous and discrete domains, while the authors seems to argue that it cannot be obtained for continuous domains — am I understanding correctly?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept