Causal Effect Inference for Structured Treatments
Authors: Jean Kaddour
Yuchen Zhu
Qi Liu
Matt Kusner
Ricardo Silva
Venue: NeurIPS 2021 Poster (2021)
Keywords: Causal Inference
Causal Effects

Abstract: We address the estimation of conditional average treatment effects (CATEs) for structured treatments (e.g., graphs, images, texts). Given a weak condition on the effect, we propose the generalized Robinson decomposition, which (i) isolates the causal estimand (reducing regularization bias), (ii) allows one to plug in arbitrary models for learning, and (iii) possesses a quasi-oracle convergence guarantee under mild assumptions. In experiments with small-world and molecular graphs we demonstrate that our approach outperforms prior work in CATE estimation.

Causal Effect Inference for Structured Treatments
Jean Kaddour" Centre for Artificial Intelligence University College London
Yuchen Zhu Centre for Artificial Intelligence University College London
Qi Liu
Matt J. Kusner Centre for Artificial Intelligence University College London
Department Computer Science University of Oxford
Ricardo Silva
Department of Statistical Science University College London
Abstract
We address the estimation of conditional average treatment effects (CATEs) for structured treatments (e.g., graphs, images, texts). Given weak condition the effect, we propose the generalized Robinson decomposition, which (i) isolates the causal estimand (reducing regularization bias), (ii) allows models for learning, and (iii) possesses quasi-oracle convergence guarantee under mild assumptions. In experiments with small-world and molecular graphs we demonstrate that our approach outperforms prior work in CATE estimation.
plug in arbitrary
Introduction
Estimating feature-level causal effects, so-called conditional average treatment effects (CATEs), from observational data a fundamental problem across many domains. Examples include understanding the effects of non-pharmaceutical interventions the transmission of COVID-19 in specific region [12], how school meal programs impact child health [13], and the effects of chemotherapy drugs on cancer patients [52]. Supervised learning methods face two challenges in such settings: (i) missing interventions, the fact that we only observe one treatment for each individual means models must extrapolate new treatments without access to ground truth, and (ii) confounding factors that affect both treatment assignment and the outcome means that extrapolation from observation to intervention requires assumptions. Many approaches have been proposed overcome these issues [1, 2, 3, 4, 5, 6, 7, 9, 10, 15, 18, 19, 21, 22, 23, 25, 27, 29, 33, 39, 41, 42, 45, 52, 56, 57, 60, 64, 67].
naturally structured. For instance, drug is commonly represented by its molecular structure (graph), the nutritional content of a meal as a food label (text), and geographic regions affected by new policy map (image). Taking this structure into account provide several advantages: (i) higher data-efficiency, (ii) capability to work with many treatments, and (iii) generalizing to unseen treatments during test time. However, the vast majority of prior work operates on either binary or continuous scalar treatments (structured treatments are rarely considered, a notable exception to this trend is Harada & Kashima [16] which we describe in Section 2).
In many cases, treatments
To estimate CATEs with structured interventions, our contributions include:
Generalized Robinson decomposition (GRD): A generalization of the Robinson decomposition [47] to treatments that can be vectorized as a continuous embedding. This GRD reveals learnable
"Corresoondence to k0ddkaadd
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Causal graph
Observed data
CATE
X
X
t
y
Medical history
do(pw) do(xtt) do
(
olocrved
unoloctvedd
Drug T
Health
Y
Figure 1: Illustration of CATE estimation with structured treatments (e.g., molecular graphs). Left: Problem setup with features X, treatment T, and outcome Y. Center: Observations the estimator has access to, typically containing only one outcome per individual. Right: The CATE is the difference between the expected outcomes given a fixed individual and pair of treatments.
pseudo-outcome target that isolates the causal component of the observed signal by partialling out confounding associations. Further, allows one to learn the nuisance and target functions using any supervised learning method, thus extending recent work plug-in estimators [42, 29].
Quasi-oracle convergence guarantee: result that shows that given access to estimators of certain nuisance functions, as long as the estimates converge at an O -1/4) rate, the target estimator for the CATE achieves the same error bounds as oracle who has ground-truth knowledge of both nuisance components, the propensity features, and conditional mean outcome.
Structured Intervention Networks (SIN): A practical algorithm using GRD, representation learning, and alternating gradient descent. Our PyTorch [43] implementation is online."
Evaluation metries designed for structured treatments. Since previous evaluation protocols of CATE estimators have mostly focused on binary scalar-continuous treatment settings, we believe that our proposed evaluation metrics can be useful for comparing future work.
Experimental results with graph treatments in which SIN outperforms previous approaches.
2 Related Work
Closest to our work is GraphITE [16], a method that learns representations of graph interventions for CATE estimation. They propose minimize prediction loss plus regularization that aims to control for confounding based on the Hilbert-Schmin Independence Criterion (HSIC) [14]. This technique suffers from two drawbacks: (i) the HSIC requires multiplication kernel matrices and scales quadratically in the batch size: (ii) selecting the HSIC kernel straightforward, as ground-truth CATES are never observed, and empirical loss does not bound CATE estimation error [1]. We discuss other related work not on structured treatments in Appendix A.
parameter is not
3 Preliminaries
3.1 Conditional Average Treatment Effects (CATEs)
Imagine a dataset where each example (x,, t;, yi) E 'D represents a hospital patient's medical history record x,, prescribed drug treatment t,, and health outcome yı. as illustrated in Figure (Center). Further, we wish to understand how changing the treatment changes a patient's health outcome. The CATE, (t', t;, x.). describes the expected change in outcome for individuals with history x,, when treatment is replaced by t', depicted in Figure 1 (Right). In real-world scenarios, only observe one outcome for each patient at one treatment level. Further, the patient's pre-treatment health conditions x; influence both the doctor's treatment prescription and outcome, thereby confounding the effect of the treatment on the outcome.
Formally, we have the dataset D= (xx, tú vi) sampled from joint distribution p (X, T, Y). where Y (X, T) + depicted in Figure (Left), We define the causal effect of fixing

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treatment variable T T to a value on outcome variable Y E IR using the do op-oraat [44] as [Y do = t)]. Crucially, this estimate differs from the conditional expectation [Y T T t] in that it describes the effect of an external entity intervening on T by fixing it to a value t (removing the edge X + T). We further condition on pre-treatment covariates X define the conditional causal estimand E [Y X = x, do(T t)] The conditional average treatment effect (CATE) is the difference between expected outcomes at different treatment values t, t' for given covariates X.
,t,,,,
A E Y X = x, do - t')
E
X = x, do = t)
(I)
Pipe (x)
pi (x)
where f4 (x) is defined as the expected outcome for covariate vector x under treatment t.
Because we do not observe both treatments t, t' for single covariate x, we need to make assumptions that allow us to identify the CATE from observational data.
Assumption 1. (Unnconfuundedness) There are no confounders of the effect between and Y beyond X. Therefore, Pr (Y s x, do(t)) Pr (Y
x.t), for all (x, v)
Assumption 2. (Overlap) It holds that 0 < x) < f.r all (x, t).
Assumption 2 means that all sub-populationss have some probability of receiving any value of treatment (otherwise, some r(t', x.x) may be undefined or impossible estimate.) These assumptions allow us to estimate the causal quantity r(t', t.x) through statistical estimands:
T (t', t, He (x) - μ (x) = Y X = x, T = t - X-xT=t)
(2)
While one model (x) with regression models, such approaches suffer from bias [9, 26, 29] due to two factors: (i) associations between X and T, due to confounding, makes it hard to identify the distinct contributions of X and T on Y, and (ii) regularization for predictive performance effect estimation. Mitigating these biases relies on exposing and removing nuisance components. This transforms the optimization into (regularized) regression problem that isolates the causal effect.
harm
3.2 Robinson Decomposition
One way to formulate such nuisance components is via the Robinson decomposition [47]. Originally reformulation of the CATE for binary treatments, it used by the R-learner [42] to construct plug-in estimator. The R-learner exploits the decomposition by partialling out the confounding of X on T and Y. It also isolates the CATE, thereby removing regularization bias.
Let the treatment variable be
(0, 1) and the outcome model p (y x, t) parameterized
Y = fXX 7) + Mo X
T
76
X E;
where we define error term such that E [E
x, t
E
0, and Tu (x) (1, x)).
Define the propensity score [48]
(x)
A p (T'
x)
and the conditional mean outcome as
m (x)
Y x = .lo (x) + (x) To (x).
(4)
From model (3) and the previous definitions, it follows that
Y (X) (I (T- (x)) To (x) +E,
(5)
allowing us to define the estimator
1
1J 7 (x))) (1(+++(())))))))
(.) arg min
n
where 0: y: (xs) and ū, e I, - (x:) are pseudo-data points defined through estimated nuisance
functions m)()) 2(-), which can be learned separately with any supervised learning algorithm.
3

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4 The Generalized Robinson Decomposition
Our goal is to estimate the CATE 7(t', t, x) for structured interventions t', (c.g., graphs, images, text) while accounting for the confounding of X on T and Y. Inspired by the Robinson decomposition, which has enabled flexible CATE estimation for binary treatments [6. 9, 33, 42], we propose the Generalized Robinson Decomposition from which we extract a pseudo-outcome that targets the causal effect. We demonstrate the usefulness of this decomposition from both a theoretical view (quasi-oracle convergence rate in Section 4.2) and practical view (Structured Intervention Networks in Section 5). For details on its motivation and derivation, we refer the reader to Appendix B.
4.1 Generalizing the Robinson Decomposition
To generalize the Robinson decomposition to structured treatments, we introduce two concepts: (a) we assume that the causal effect is product effect: the outcome function f* (X, T) can be written as an inner product of two separate functionals, one over the covariates and one over the treatment, and (b) propensity features, which partial out the effects from the covariates on the treatment features. Similar techniques have been previously shown to add to the robustness of estimation [9, 42].
Assumption 3. (Product effect) We consider the following partial parameterization of p(y x, t),
Y= (X) h (T) +
where g : X + h. R and Els
x, = E [ x = for all (x, t) EXX..
This assumption mild, as we can formally justify its universality. The following asserts that provided we allow the dimensionality of g and h to grow, we may approximate any arbitrary bounded continuous functions in (X' x T) where x T is compact.
Proposition 1. (Universality of product effect) Let HxxT be Reproducing Kernel Hilbert Space (RKHS) on the set X x T with universal kernel k. For any > 0, and any I E HXxT, there is a E such that there exist two d-dimensional vector fields g : X - 12d and h T - Rd, where |lf - gg" h(llx((Pxx)) S 6. (Proof in Appendix C)
This assumption allows us to simplify the expression of the CATE for treatments t', given x,
T (t', = g (x) (t') h (t))
= Define propensity features e (x) E [h (T) x] and m (x) A < (x) ch (x).
Following the same steps as in Section 3.2, the Generalized Robinson Decomposition for eq. (7) is
Y m (X) 9 (X)
h (T) - e" (x) +e.
(9)
Given nuisance estimates (a.)),z(()) (.), we can use this decomposition to derive an optimization problem for (-)) (note 2th (-) implicitly depends on h(.), ), we address this dependence Section 5).
((),(())) arg min 9.h
1 Σ
Y,
m (xx)) ( - (b(T))-e* (xxi)) ( +A((1)
n 1=1
(10)
4.2
Quasi-oracle error bound of Generalized Robinson Decomposition
We establish the main theoretical result of our paper: quasi-oracle convergence guarantee for the Generalized Robinson Decomposition under finite-basis representation of the outcome function. This result is analogous to the R-learner for binary CATES [42]: when the true e (-),,, (-) are unknown, and we only have access to the estimators ē ((-), m (-), then as long as the estimates 1/4 converge at -111 rate, the estimator 5h (-) achieves the same error bounds as an oracle who has ground-truth knowledge of these two nuisance components.

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More formally, provided the nuisance estimators m-)) and (-) converge at an O 1114/4 rate, our
CATE estimator will converge at On भाोनग rate for arbitrarily small p O, recovering the parametric convergence rate for when the true m(-) and e-) are provided oracle quantities.
Our analysis assumes that the outcome Y X = x, T = t] can be written as a linear combina- tion of fixed basis functions. By Proposition as long as we have enough basis functions, this representation is flexible enough to capture the true outcome function.
Assumption 4. Let (xX) Rda, ((T) Rd be fixed, known orthonormal basis features on X E & T RR,,, respectively. The true outcome function (x, t) KE|Y X = T t| can be written as (x, t) = a (x)@* (x() for some (unknown) matrix of coefficients θ*
Note that by setting g= a To and h B, we recover eq. (7). Additionally, we will need overlap in the basis features ,x
Assumption (Overlap features). The marginal distribution features P Pa(X)x((T) is positive, xx(x x ((T).
i.e. supp[Pa(x)xx))
Assumption is typically weaker than requiring overlap in X and T, i.e., when da d3 < dx de.
With further technical assumptions specified in Appendix F, we establish the following theorem. Theorem 2. Let O' denote the representer of the true outcome function. Suppose Assumptions 5, 6, and 4 hold. Moreover, suppose that the propensity estimate 2h is uniformly consistent,
sup XEX
[|e((x) (x) - ex(x]]l 7p
(11)
and the La errors converge at rate
(x) m² [X)
TE 112 (x) e² (X) [3] = O(a)))
(12)
for some sequence on 0, where (ax) is such that on O( with > 1. Further, we define the regret as the excess risk
R ) & (@..) (on)) L(0)), L(
E {x- m² (x))) a (X) (B ( c² (x) (x))
(13)
Suppose that obtain On via a penalized basis function regression variant of the Generalized Robinson Decomposition, with a properly chosen penalty A, (1@mll2) (specified in the proof). Then, θ satisfies the regret bound: R (0m) = Ö(r2) with In n शां. for arbitrarily small >> 0.
5 Structured Intervention Networks
We introduce Structured Intervention Networks (SIN), a two-stage training algorithm for neural networks, which enables flexibility in learning complex causal relationships, and scalability to large data-sets. This implementation of GRD strikes a balance between theory and practice: while we assumed fixed basis-functions in Section 4.2, in practice, often need to learn the feature maps from data. We leave the convergence analysis of this representation learning setting for future work.
5.1 Training Algorithm
We propose to simultancously learn feature maps ğ (X). h (T) using alternating gradient descent, so that they can adapt to each other. remaining challenge that learning 2h (X) is now entangled with learning h (T). While the R-learner is based on the idea of cross-fittigg where at each data point we pick estimates of the nuisances that do not that data point, introduce pragmatic representation learning approach for (g. h) that does not use cross-fitting"
We could in principle use cross- fittinii for 2h although the loop between fitting alternating with 2t would break the overall independence between ě, (X) and data point i. While it is possible that cross- fitting for 2" is still beneficial in this case, for simplicity and for computational savings, did not implement it.
5

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a SIN Training.
b Pseudocode in a Py PyTorch-like style.
Input: Stage 1 data D1 == {x,, y/4 j=
Initialize submodels and optimizers m, e, 8, h MLP(...), MLP(...), MLP(...), GNN
Stage data D2 == {x,, t; Mi )=) Step sizes Ao, An: Ag, Ax- Number of update steps K. Mini-batch sizes B1, B2.
opt, e_opt, g_opt, h_opt Adam params Adam(m.params()) (), m_1r), adam(e.params() e_lr),
I: Initialize parameters: 0, 0, 20,
Stage
2: while not converged do
Stage
for batch in train_loader:
3: Sample mini-batch {(x,, yt yb) b=t
X, batch.X, X, batch.Y p..pp..zeo.graee F.mse_loss (m(X), Y) (.bckwardd))) m_opt.sste((
4: Evaluate m (0)
5: Update 0 < 0 Xo Vo)(()
6: end while
Stage
7:
while not converged do
Stage 2
for batch in arain_loader.
8: Sample mini-batch {xx, to, }b)
x, T, Y batch. X, batch. I, batch.
for range (num (ntm_uuuuttttt update_steppe):
9:
Evaluate Jg.h (eb, Φ) Jeh (1)
g_opt.zero_gee ggrdd l.opt.zero, gradggrrr
10:
for k 1 to K do
11:
Update Φ t Φ X a. g, (2)
F. F.mse_losae ([g (( e(X))) sum
(-1), (e)=(())))bccuuerrr)
12:
Update 1/b + 1/- - Xx V 2 g (4b,
g_opt.step()
13:
end for
b_ htopt.step(
14:
Update 7) 77 in V Nn (n)
F. P.mse_loss loose__losse (e(X), h(T)) backvard
zero_grad()
15:
end while
e_ept..step(
Figure 2: The two-stage algorithm for training SIN.
We learn surrogate models for the mean outcome and propensity features (x and 24 (X) with parameters 0 E de 17 R8,,, as well as feature maps for covariates and treatments Fe (X), h (T), parameterized by & E Rdv, &f Rde We denote regularizers by A Figure 2 summarizes the algorithm. As the mean outcome model mg (X) does not depend on the other components, we learn it separately in Stage 1. In Stage 2, we alternate between learning 1b, Φ n7
Stage 1: Learn parameters 0 of the mean outcome model ma (X) based on the objective
Jm (0)
(ye mo (x:)) 2 + (0)
(14)
which relies only on covariates and outcome data D1 : {xx, M1) I=r
Stage 2:
Learn parameters 2b Φ for the covariates and treatments feature maps Pyp. (X), hos (T), well as parameters n for the propensity features e (X).
2 mo (x:) + Fx (xx) (Tis (t:) - 2h (x:)) (xx))))
1
Jg (b)
V1
A (v)) + A
n
(15)
This loss hinges on eh (X), which needs to be learned by
Jen (n) Σ Tho (t:) 2t (xi) An))
(16)
note again the dependence on h (T). While it may be tempting learn w Φb and n jointly, they have fundamentally different objectives (et) (X) defined an estimate of the expectation E (T) xx). Therefore, we employ an alternating optimization procedure, where we take & e (1...., K} optimization steps for th & towards Jy.h (zh, Φ) and one step for learning 7). We observe that setting K 1, i.c. updating vb & more frequently than 7,, stabilizes the training process.

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SW (In-Sample)
SW (Out-Sample)
TCGA (In-Sample)
TCGA (Out-Sample)
UPE
100
1000
40
20
20
10
Number of trestmennts K
Number of treatmentt K
Number of treatmenta K
Number treatmenta K
SIN
GraphITE
GNN
CAT
Zero
Figure 3: UPEHE K for K E (2,...,10), 103.
5.2 Advantages of SIN
We conclude by describing the beneficial properties of SIN, particularly in finite-sample regimes:
Targeted regularization: Regularizing Te (X), Thpp (T) in eq. (15) after partialing out con- founding is type of targeted regularization of the isolated causal effect. In contrast, outcome estimation methods can suffer from regularizationindduss effect estimate away from zero in the service of trying to improve predictive performance [29].
confounding, c.g., regularizing the
2. Propensity features: Learning propensity features can help us to (i) partial out parts of X that cause the treatment but not the outcome, and (ii) dispose unnecessary components of T.
3. Data-efficiency: In contrast to methods that split the data into disjoint models for each treatment group (known as T-learerss for binary treatments [8, 10]), sharing causal effect parameters between all covariates regardless of their assigned treatment increases data-efficiency.
4. Partial data: In settings without access both the treatment assignment and the outcome but only access to one of them, one can leverage that data to improve the (nuisance) estimator further, e.g., when patient's recovery is observed one year after drug was administered [33].
Experiments
Here we evaluate how CATE estimation with our proposed model SIN compares with prior methods.
6.1 Experimental Setup
Datasets. To be able to compute CATE estimation error w.r.t. ground truth, we design two causal models: simpler synthetic model with small-world graph treatments and a more complex model with real-world molecular graph treatments and gene expression covariates. The Small-Worls World (SW) simulation contains 1,000 uniformly sampled covariates and 200 randomly generated Watts-Strogatz small-world graphs [61] as treatments. The Cancer Genomic Atlas (TCGA) simulation uses 9,659 gene expression measurements of cancer patients for covariates [62] and 10.000 sampled molecules from the QM9 dataset [46] as treatments. Appendix D details the data- eetta-generatigg schemes.
Baselines. We compare method to (1) Zero, sanity-check baseline that consistently predicts zero treatment effect and equals the mean squared treatment effect (poorly regularized models may perform worse than that due confounding), (2) CAT, categorical treatment variable model using one-hot encoded treatment indicator vectors, (3) GNN, a model that first encodes treatments with GNN and then concatenates treatment and individual features for regression,
(4) GraphITE [16], CATE estimation method designed for graph treatments (more details in Section 2). GNN and CAT reflect the performance of standard regression models. The contrast between these two provides insight into whether the additional graph structure of the treatment improves CATE estimation. To deal with unseen treatments during CATE evaluation, we map such the most similar ones seen during training based on their Fuclidean distance in the embedding space of the GNN baseline.
Graph models. For small-world networks, we use k-dimensional GNNs [38], as to distinguish graphs they take higher-order structures into account. To model molecular graphs, we use Relational Graph Convolutional Networks [50], where the nodes are atoms and each edge type corresponds to specific bond type. We use the implementations of Py Torch Geometric [11].
7

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Table 1: Error of CATE estimation for all methods, measured by WPEHE@6. Results are averaged
over 10 trials, + denotes std. eтror (each trial samples treatment assignment matrix W).
SW
TCGA
Method
In-sample
Out-sample
In-sample
Out-sample
Zero
56.26 + 8.12
53.77 + 8.93
26.63 + 7.55
17.94 + 4.86
CAT
51.75 + 8.85
49.76 + 9.73
155.88 + 52.82
146.62 + 42.32
GNN
37.10 + 6.84
36.74 + 7.42
30.67 + 8.29
27.57 + 7.95
GraphITE
34.81 + 6.70
35.91 + 8.07
30.31 + 8.96
27.48 + 8.95
SIN
23.00 4.56
23.19 + 5.56
10.98
3.45
8.15 1.46
Evaluation metrics. We extend the expected Precision in Estimation of Heterogeneous Effect (PEHE) commonly used in binary treatment settings [19] to arbitrary pairs of treatments (t.t') as follows. We denote the Unweighted PEHE (UPEHE) and the Weighted PEHE (WPEHE) as
& Sfx (F(',tt (t', t.x)) p p( p (t' xx) dx,
CUPERECWPHITT)
(17)
where the weighted version gives less importance to treatment pairs that are less likely; to account for the fact that such pairs will have higher estimation errors. In fact, as the reliability of estimated effects decreases by how likely they in the observational study, we evaluate all methods U/WPEHE truncated to the top K treatments, which we call U/WPEHE@ K. To compute this, for each x, we rank all treatments by their propensity (t x) (given by the causal model) in descending order. We take the top K treatments and compute the U/WPEHE for all KK treatment pairs.
In-sample vs. out-sample. A common benchmark for causal inference methods is the in- in-sample task, which we include here for completeness: estimating CATEs for covariate values x found in the training set. This task is still non-trivial, as the outcome of only one treatment is observed during training" 4. In contrast, and arguably of more relevance to decision making, the goal of the out-sample task is to estimate CATEs for completely unseen covariate realizations x'.
Hyper-paramette tuning. To ensure fair comparison, we perform syper-parameter optimization with random search for all models on held-out data and select the best hyper-parameters over 10 runs.
Propensity. We define the propensity (or treatment selection bias) p (T x) softmax (xWTX) X), where W E TR. Tixd Vi,j Wi, ~ u [0, 1] is random matrix (sampled then fixed for each run). Recall [7] is the number of available treatments and let d be the dimen- sionality of the covariates. Here the bias strength k is a temperature parameter that determines the flatness of the propensity (the lower the flatter, i.e., x corresponds to the uniform distribution).
6.2 Comparison of Performances different K Treatments
Figure shows the @PEHE@ K of all methods for K E {2...., 10). of all methods in Table 1. Unless stated otherwise, we report results for bias strengths 10 and K 0.1 in the SW and TCGA datasets, respectively across 10 random trials.
10). We also report the WPEHE@6
The results indicate that the relative performance of each method, for both the in-sample and out- sample estimation tasks, is consistent. Further, they suggest that, overall, the performance of SIN is best due to a better isolation of the causal effect from the observed data compared to other methods. The performance difference between CAT and GNN across all results indicate that accounting for graph information significantly improves the estimates. We observe from the SW experiments that GraphITE [16] performs slightly better than GNN, while is nearly the same as GNN on TCGA.
Surprisingly, the results of the TCGA experiments with low bias strength ς = 0.1 expose that all models but SIN fail isolate causal effects better than the Zero baseline. These results confirm that confounding effects of X on Y combined with moderate causal effects can cause severe regularization bias for black-box regression models, while SIN partials these out from the outcome by mh (X). We include additional results on convergence and larger values of K in Appendix E.
"The original motivation comes from Fisherian designs where the only source of randomness is on the treatment assignment [20]. Our motivation is simpler: rule out the extra variability from different covariates, highlighting the difference between methods due to different loss functions and less due smoothing abilities.
8

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SW (In-Sample)
SW (Out-Sample)
TCGA (In-Sample)
TCGA (Out-Sample)
6il
60-
WPC
100
100
40
40
25
26-
20-
20-
20
15
0.3
Blas
Buss
Blas
Bias
SIN
GraphITE
GNN
CAT
Zero
Figure 4: WPEHE@6 over increasing bias strength
6.3 Comparison of Robustness to different Bias Strengths
strong selection bias (i.e. large k) the observed data makes CATE estimation more difficult, as it becomes unlikely to see certain treatments t T for particular covariates Here, we assess each model's robustness to varying levels of selection bias, determined by к, across random seeds. In Figure 4, we see that SIN outperforms the baselines across the entire range of considered biases. Interestingly. SIN performs competitively even in a case with no selection bias (x=00, which corresponds randomized experiment). Importantly, all performances (SW) or to increase (TCGA) with increasing biases. Notably, the poor performance of CAT suddenly improves datasets with high bias. We believe this is because, in high bias regimes, distinct treatments overall, which allows the CAT model to approach the performance of GNN.
to either stagnate
fewer
7 Limitations, Future Work and Potential Negative Societal Impacts
Limitations and future work. Firstly, in some real-life domains, Assumption I (Unconfounded- ness) can be too strong, as there may exist hidden confounders. There are two common strategies to deal with them: utilizing instrumental variables [17, 58, 63] proxy variables [35, 59]. Devel- oping new approaches for structured interventions such settings is a promising future direction. Secondly, SIN is based neural networks; however, neural network initialization impact final estimates. To obtain consistency guarantees, GRD can be combined with kernel methods [35, 58].
Potential negative societal impacts. Because causal inference methods make recommendations about interventions to apply in real-world settings, misapplying them have negative real- wolld impact. crucial to thoroughly test these methods on realistic simulations and alter aspects of them to understand how violations of assumptions impact estimation. We have aimed to provide comprehensive evaluation of structured treatment methods by showing how estimation degrades as less likely treatments considered (Figure and treatment bias increases (Figure 4).
Conclusion
The main contributions of this paper two-fold: (i) the generalized Robinson decomposition that yields a pseudo-outcome targeting the causal effect while possessing a quasi-oracle convergence guarantee under mild assumptions, and (ii) Structured Intervention Networks, practical algorithm using representation learning that outperforms prior approaches in experiments with graph treatments.
Acknowledgements
We thank Antonin Schrab, David Watson, Jakob Zeitler, Limor Gultchin, Mare Deisenroth and Shonosuke Harada for useful discussions and constructive feedback on the paper. JK and YZ acknowledge support by the Engineering and Physical Sciences Research Council with grant number 5P/S021566/11 This work was partially supported by ONR grant number N62909- 19-1-2999 to RS. We thank the Alan Turing Institute for the provision of Azure cloud computing resources.
References
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14


Reviews:
[{"review_id": "yyRprYHOYrX", "reviewer": "Reviewer_rvHB", "summary": "This paper studies the estimation of conditional average treatment effect (CATE) with multiple treatments that are graph structured. The proposed method extends the R-learner (Nie and Wager, 2020), which is designed for binary treatment, by learning a representation that reduces the dimension of the treatments. The representation is parametrized by a graph neural network that takes the graph structure into account. Unlike the standard R-learner that learns the nuisance components $E[Y|X]$ and $E[T|X]$ separately from the CATE, the proposed method only estimates $E[Y|X]$ separately while learns the representation $h(T)$, the propensity features $E[h(T)|X]$, and CATE jointly via gradient-based methods. ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The idea of combining R-learner and representation learning to reduce the dimension of complex treatments is quite interesting. The method can be useful in many applications that involve graph-structured treatments, such as those in two-sided markets. The paper is well-written overall and the technical arguments are sound. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "(1) I would suggest stating the product effect model (9) directly without describing the general version (8), which looks odd to me. In fact, R-learner, or Robinson's method, is designed for the product effect model which breaks the response function $f(x, t)$ into the target parameter (i.e., CATE in this case) and two nuisance functions. It is shown that (e.g., Nie and Wager, 2020) R-learner yields a faster rate for the target parameter if only the two nuisance functions are estimated at an $O(n^{-1/4})$ rate. Without this structure, there is no reason to estimate $f$ through (8); instead, one can directly estimate it by minimizing $(1/N)\\sum_{i=1}^{N}(y_i - f(x_i, t_i))^2$. \n\n(2) The overlap assumption (Assumption 2) is arguably too strong when the treatment is complex. Under the model (9), is it fine to just assume overlap on $p(h(t)|x)$?\n\n(3) Algorithm a (GIN training) updates $\\eta$ only once for every $K$ updates of $\\phi$ and $\\eta$. Is there any specific reason for this asymmetry?\n\n(4) It would be nice to provide a slightly more detailed description on how to parametrize the treatment via a graph neural network. \n\n(5) An advantage of cross-fitting is to reduce bias and enable valid inference (e.g., valid pointwise confidence intervals). Is there any approach for inference with the proposed method? ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "imEL0MoamJR", "reviewer": "Reviewer_fdib", "summary": "The paper proposed a CATE estimation method for graph-structured treatment by Robinson Decomposition. It is an extension of Robinson Decomposition, and the experiment shows the effectiveness of GIN on two small world datasets. The writing is good, but some contents are missing, so that the paper is not that easy to read. My detailed comments and concerns are described in the following.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "1. My biggest concern is about m(x). m(x) is related to the  e^h(x), according to the definition. However, in the training algorithm, the estimated function m(x) is fixed after Stage 1, only with the observational x and y. The design of training mismatch Eq(11), and e^h(x) is always updating when training the model. How can m(x) be fixed in the first place?\n\n2. how to mitigate dependency between X and T towards unbiased treatment effect estimation? It is not clear in the manuscript.\n\n3. GIN seems to work well on mitigating the selection bias according to the experiments. Which component of GIN contributes to controlling it? It is also not clear in the paper.\n\n4. GIN uses GNN to handle the graph treatment, i.e., developing the function h(T). However, its detailed mapping process is missing. For example, how to get the readout output of the graph treatment?  \n\n5. the paper aims to estimate the CATE, so how to select the covariate values x in the training set? or use all values shown in the training set?\n\n6. Under the setting of the GIN model, the identification of causal effect is another concern.\n\n7. The number of baseline methods is a bit limited. Authors can use more treatment effect estimation models and expand them to the graph-treatment setting. For example, using the CFR model to obtain covariates' representation and use GNN to map the treatment into low-dimensional space, then concatenate them to predict potential outcomes.\n\n8. BTW, as far as I know, GIN usually means \"Graph Isomorphism Network\", which is published in ICLR 2019.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "None", "review_text": "", "extra_scores": {"ethical_concerns": "None", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4 hours"}, "extra_text": {}}, {"review_id": "aGxHDBq2bH-", "reviewer": "Reviewer_1AJd", "summary": "This paper proposes graph intervention network (GIN) to estimate conditional average causal effects when treatments are graph-structured. GIN builds on an extension of traditional Robinson decomposition to arbitrary treatment types. In particular, GIN factorizes the outcome Y into an inner product of two terms $g(X)^\\top h(T)$. During training, GIN first estimates a mean outcome model $m(X)$ and then estimate $g,h$ using the estimated $m(X)$ function. The method is evaluated on two datasets with synthetic outcome functions.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall, the proposed method is an interesting algorithmic solution to graph-structured treatment effect estimation. The method is a novel approach and the assumption on product effect is reasonable. My major concern of the paper are the following:\n\n1. Assumption on overlap. When treatment itself is graph-structured or high-dimensional, it is hard to satisfy the overlap assumption in general. There are many unseen combinations of covariates, treatments, and outcomes $(x,t,y)$. Therefore, for many treatments $t$, the empirical estimate of $p(t|x)=0$. I don't think this paper has sufficiently addressed this issue. \n\n2. The TCGA experimental setup is too synthetic. The outcome function is purely simulated (a predefined function). In addition, molecules from the QM9 datasets are not really drug like and they are typically not qualified as chemotherapy compounds. Using drug like molecules from the ChEMBL database is a more realistic setup.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The paper sufficiently addressed its limitations and societal impacts.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1 hour"}, "extra_text": {}}, {"review_id": "SaUjrkEEWXL", "reviewer": "Reviewer_5P78", "summary": "This paper proposed a CATE estimation approach when the treatment variable $T$ is a graph. Under a separable assumption (A3) between confounders $X$ and the treatment $T$ in the outcome structural equation ($Y$), Robinson's decomposition of CATE for binary $T$ can be applied. The performance of the authors' proposal was evaluated in both synthetic experiments and real data analysis.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The authors set out to address a very important problem in causal inference, which is methodology development for non-Euclidean treatment variables. In particular, the authors considered graph-valued treatment $T$. However, from my own perspective, there are several major limitations in this work.\n\n(1) The type of intervention on graph-valued treatment is not spelled out clearly in the paper (or at least in the introduction section).\n\nAs a referee, when I saw the title of this paper, the immediate question I had in mind is what type of intervention on graphs the authors were considering. But until Section 4, I finally realized that the intervention is simply changing one graph $\\mathcal{G}$ to another $\\mathcal{G}'$. I think this should be made much clearer at the beginning of the paper.\n\n(2) As a follow-up comment of (1), I would think it will be more interesting to consider problems like the direct effect of changing one set of vertices on a graph and the indirect effect of changing this set of vertices on the overall network structure, further on to the outcome. Such effect decomposition might be more interesting to applied researchers. For example, there might be targeted cancer therapy that can inhibit the expression of one particular gene. Then it is very interesting to understand whether the effect of such a drug is manifested through its target gene or through the entire gene regulatory networks. Directly comparing the change of the network may not be entirely meaningful to applied researchers.\n\n(3) Kennedy (2020) proposed a novel CATE estimation method based on the efficient influence function. I am wondering if this or related method can also be used in the graph-valued treatment setup. \n\nSome of the following minor limitations might have been pointed out by the authors themselves. If so, I will not describe in more detail:\n(1) Lack of theoretical guarantees. But personally, I do not think this is a very important limitation for a really novel idea.\n(2) In Section 4.4, with partial data, I thought the best one can do is to produce bounds on the treatment effect. But maybe I am wrong.\n\nReferences:\nKennedy EH. Optimal doubly robust estimation of heterogeneous causal effects. 2020", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes. The authors listed a few limitations and what can be done in future work and wrote a quite thorough paragraph on the potential negative societal impact of applying causal inference methods in real life.", "review_text": "", "extra_scores": {"ethical_concerns": "No ethical concerns.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "Eesu9_rHdVd", "reviewer": "Reviewer_kq9J", "summary": "The paper considers the problem of estimating conditional average treatment effect when the treatment is a graph. The paper generalizes the Robinson decomposition, adapts the R-learner to this graph-treatment setting and introduces a two-stage algorithm. The performance of the proposed algorithm is compared to a few benchmarks through numerical experiments. ", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: The proposed method is original. The paper generalizes the Robinson decomposition and adapts the R-learner in a very interesting and novel way. The two-stage training algorithm is different from the cross-fitting method considered in R-learner. \n\nQuality: As far as I checked, all claims are well supported. The experimental results are explained very clearly. One question I have is on cross-fitting. My impression is that cross-fitting is usually very helpful (sometimes even necessary) in developing theories. In practice, it is not the most data efficient way, since it uses only part of the data to train things. I’m wondering in the experiments, how much improvement in accuracy owes to not doing cross-fitting? Do you think there are specific settings where not doing cross-fitting would make the method fail? Would theories still go through without cross-fitting? I understand there is no guarantee of convergence so far, but I think a slightly deeper discussion on these would be very helpful.\n\nClarity: The paper is well written and I enjoyed reading it. \n\nSignificance: The problem this paper considers is very important. The paper also provides a strong solution to the problem. The proposed algorithm appears to perform very good empirically. Indeed, I believe the proposed method works for any non-binary treatment variables. It seems that both the Robinson decomposition and the two-stage algorithm can be generalized to other types of treatment, e.g., real numbers, vectors (potentially high dimensional), or categorical variables. I wonder whether this is true and how much the algorithm needs to be modified to adapt to other types of variables. If this can be discussed in depth, I think this can make the conclusions of the paper even stronger. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have adequately addressed the limitations and potential negative societal impact of their work. They have discussed limitations including lack of a convergence guarantee and potential hidden confounders. ", "review_text": "", "extra_scores": {"ethical_concerns": "NA", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The proposed method extends the R-learner (Nie and Wager, 2020) to deal with multiple treatments that are graph-structured. Dealing with non-Euclidean treatments is a very important problem that is understudied in the literature. The proposed two-stage algorithm is an extension of the Robinson decomposition that several reviewers found promising (kq9J, rvHB). An important stepping stone of the approach is to generalize the propensity scores to propensity features. Reviewers rvHB and 1AJd criticized the overlap assumption, which is very strong if the treatment is graph-structured. In the discussion, it has been clarified that the overlap assumption is only required for the features maps of X and T. In addition, the authors provided a sketch for a quasi-oracle regret bound. Reviewers kq9J, rvHB and fdib agree that the paper is well-written. Reviewer fdib pointed to some issues in the presentation. Overall, the authors have addressed the main issues in their responses and the theoretical results have been strengthened during the review process.
Decision: Accept (Poster)