Estimating Treatment Effects via Orthogonal Regularization
Authors: Tobias Hatt
Stefan Feuerriegel
Venue:  (2021)
Keywords: Treatment Effects
Regularization
Neural Networks

Abstract: Decision-making often requires accurate estimation of causal effects from observational data. This is challenging as outcomes of alternative decisions are not observed and have to be estimated. Previous methods estimate outcomes based on unconfoundedness but neglect any constraints that unconfoundedness imposes on the outcomes. In this paper, we propose a novel regularization framework in which we formalize unconfoundedness as an orthogonality constraint. We provide theoretical guarantees that this yields an asymptotically normal estimator for the average causal effect. Compared to other estimators, its asymptotic variance is strictly smaller. Based on our regularization framework, we develop deep orthogonal networks for unconfounded treatments (DONUT) which learn outcomes that are orthogonal to the treatment assignment. Using a variety of benchmark datasets for causal inference, we demonstrate that DONUT outperforms the state-of-the-art substantially.

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ESTIMATING TREATMENT EFFECTS VIA ORTHOGONAL REGULARIZATION
Anonymous authors Paper under double-blind review
ABSTRACT
Decision-makinn often requires accurate estimation of causal effects from obser- vational data. This challenging as outcomes of alternative decisions are not observed and have to be estimated. Previous methods estimate outcomes based on unconfoundedness but neglect any constraints that unconfoundedness imposes on the outcomes. In this paper, we propose novel regularization framework in which we formalize unconfoundedness as an orthogonality constraint. We provide theoretical guarantees that this yields an asymptotically normal estimator for the av- erage causal effect. Compared to other estimators, its asymptotic variance is strictly smaller. Based on our regularization framework, we develop deep orthogonal networks for unconfounded treatments (DONUT) which learn outcomes that are orthogonal the treatment assignment. Using variety of benchmark datasets for causal inference, we demonstrate that DONUT outperforms the state-of-the-art substantially.
INTRODUCTION
Estimating the causal effect of intervention (i.e., treatment effect) is integral for individual
decision making in many domains such as marketing (Brodersen et al. [0155] Hatt& & Feuerriegel 2020), economies (Heckman et al. 1997), and epidemiology (Robins et al. 2000). For instance, in order to control an epidemic, it is relevant for public decision-makers to estimate the causal effect of school-closures (intervention) on the infection rate (outcome).
The causal effect of an intervention can be estimated in two ways: randomized control trials (RCTs) and observational studies. RCTs are widely recognized as the gold standard for estimating causal effects, yet conducting RCTs is often infeasible (Robins et al. 2000). For instance, randomly allocating different policy interventions during an epidemic might be unethical and impractical. Unlike RCTs, observational studies adopt observed data to infer causal effects. For this, covariates must be collected that contain all confounders (i. e., variables that affect both treatment and outcome). This is becoming increasingly common due ease of access to rich data. In this paper, we estimate the average causal effect treatment from observational data.
In order to estimate the causal effect of a treatment, the outcome of alternative treatment has to be estimated. However, this is challenging, since we do not know what the outcome would have been if another treatment had been applied. Existing methods for estimating treatment effects use the treatment assignment as feature and train regression models to estimate the outcomes (Funk) al. 2 Kallus 201 2b). Methods based nearest neighbors and matching are adopted to find similar subjects (Ho et al. 2007 Crump et al.) 2008 Kallus 2017a 2020). Tree and forest-based methods (Wager& & Athey 2018) estimate the treatment effect at the leaf node and train many weak learners to build expressive ensemble models. Gaussian process- based methods provide uncertainty approaches re-weight the outcomes using weights based on covariate and treatment data (Kallus) 2018).. For instance, Fong et al. (2018); Yiu & Su (2018) seek weights such that the treatment assignment is unassociated with the covariates. However, they do not require the treatment assignment to be unassociated with the potential outcomes. Doubly robust methods combine a model for the outcomes and model for the treatment propensity in manner that is robust to misspecification (Funk et al.) 2011 Benkeser et al. 2017; Chernozhukov et al. 2018). Recently, deep learning has been successful for this task due its strong predictive performance and ability to learn representations of the data
quantification (Alaa & van der Schaar 2017 Ray & Szabo, 2019). Weighting-

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(e. g.. Johansson et al. 2016 Louizos al. 2017| Shalit al. 2017 Yao et al. 2018; Yoon et al. 2018 Shi et al. 2019). To ensure identifiability of the causal effect, state-of-the methods for estimating treatment effects based on unconfoundedness (i. all confounders measured, and thus included in the covariates). Hence, unconfoundedness assumed for identifiability, yet, during estimation of the model parameters, any implications the unobserved outcomes that arise from unconfoundedness have been neglected.
Contribution In this paper, (i) we introduce a regularization framework that exploits unconfound- edness. To this end, we formalize unconfoundedness as an orthogonality constraint. This constraint used during estimation of the model parameters to ensure that the outcomes are orthogonal to the treatment assignment. We prove sufficient conditions under which this yields an asymptotically normal estimator for the average causal effect. Compared to other estimators, its asymptotic variance is strictly smaller. (ii) Based on our regularization framework, we develop deep orthogonal networks for unconfounded treatments (DONUT) for estimating average causal effects. DONUT leverages the predictive capabilities of neural networks to learn outcomes that are orthogonal to the treatment assignment. Using variety benchmark datasets for causal inference, we demonstrate that DONUT outperforms the state-of cice-of-hhe-rrr the- substantially.
2 PROBLEM SETUP
Our objective is to estimate the average treatment effect (ATE) of a binary treatment from obser- vational data. For this, we build upon the Neyman- Rubin potential outcomes framework (Rubin) 2005). Consider population where every subject is described by the d-dimensional covariates X, Rd. Each subject is assigned a treatment T; E {0, 1). The random variable Yi(1) corresponds to the outcome under treatment, i. e., T; 1, whereas (0) corresponds
the outcome under no treatment, e., Ti = O. These two random variables, (1),,, ((()) E R. are known as the potential outcomes. Due the fundamental problem of causal inference, only one of the potential outcomes observed, but never both. The observed outcome is denoted by Y,.
Our aim is to estimate the average treatment effect
tb (1) Y(0)-
The following standard assumptions are sufficient for identifiability of the causal effect (Imbens & Rubin 2015;: consistency (i.e., E [0, Y ((),, T poxitivity (i. e., Vz Rd < P(7 X x) < 1); and unconfoundedness. Unconfoundednes, assumes that all confounders are measured and, hence, conditioning on them blocks all backdoor paths. This is equivalent to assuming that the potential outcomes Y (1) and Y (0) are independent of the assigned treatment T given the covariates X, i.e.,
(()), 1 T X.
(2)
Based on this, the ATE is equal to
e = E|E X, T = 1| E| X, T 0J].
(3)
Our task is to estimate the function f(x, = E| X = x, T for all Rd and t E {0, 10,1} based observational data D {(X,, T,, Y;)
ORTHOGONAL REGULARIZATION FOR ESTIMATING TREATMENT EFFECTS
The key idea of our regularization framework is to exploit the implications the outcomes. that result from unconfoundedness. For this, we formalize unconfoundedness as an orthogonality constraint. This orthogonality constraint ensures that the outcomes are orthogonal to the treatment assignment. We later introduce a specific variant our regularization framework based on neural networks, which yields DONUT.
'Code available at wwwggittubbb... http://atgtttb..b.////oooooooooo httpub/...oommoooooooo//ooottttt (anonymized for peer-review).
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3.1 Under unconfoundedness, the outcomes are independent of the assigned treatment given the covariates, e.e.
UNCONFOUNDEDNESS ORTHOGONALITY CONSTRAINT
(O)) T J.
(4)
Using the inner product (V, = 1 E V/2 W, we formalize the following orthogonality con- straint:
a
(Y - 1, = F(x,
T (Xx)) = 0,
(5)
fort {0, 1}. and where f t) E| x= T= and (x))=
z The function
и is the propensity score This is a necessary condition for unconfoundedness, since unconfound- edness requires the covariance between Y (t) and T given X to be zero due to the independence of the outcomes and the treatment assignment. Hence, since the inner product in (5) is the empiri- cal covariance between (t) and T given X, unconfoundedness requires the (centered) outcomes to be orthogonal to the (centered) treatment assignment with respect to the above inner product An appropriate method for estimating outcomes based on unconfoundedness should ensure that the orthogonality constraint holds. Although existing methods are based on unconfoundedness, the orthogonality of the outcomes to the treatment assignment is ignored during estimation of the model parameters. As remedy, we propose a regularization framework that accommodates the orthogonality constraint the estimation procedure.
3.2 PROPOSED REGULARIZATION FRAMEWORK
In our regularization framework, the orthogonality constraint in is included in the estimation procedure as follows. Let H, S R x [0,1) + IK and H, S [R [0, 1]] be function classes for and æ, and E IR model parameter. Then, the objective is to find the solution to the optimization problem
inf (,,, x
CFu(f. nT; D) ARoR(f E),
where CFL is the so-called factual loss (see Section B.2.1 between the estimated models and observed (factual) data. The term Pon is our new orthogonal regularization that enforces the orthogonality constraint using the model parameter (see Section B.2.2p. The variable R is hyperparameter controlling the strength of regularization. We describe both components of (6 i. e., factual loss and orthogonal regularization, in the following.
3.2.1 FACTUAL LOss LPL
Similar to previous work (c. g.. Shi et al. 2019), the outcome model f and propensity score model π are estimated using the observed data and the factual loss given by
n n Σ (f(X,, Ti) Y,)² + (rossEnnroppy(((X)....
1
CF
(7)
where a E IR, is a hyperparameter weighting the terms of the factual loss. We note the following about the estimation of the outcome model in (7). In the first term of the factual loss, f is fitted to the observed outcomes and, thus, no information about the unobserved outcomes is used. As consequence, the model learns about the observed outcomes, but not about the unobserved outcomes. -fate-of-thhear methods exclusively rely on the factual loss to estimate the outcomes (e. Shallit al. 2017). This is based on the assumptions that similar subjects have similar outcomes (Yao et al. 2018) and that for every subject in the data, there is similar subject in the data with the opposite treatment. However, this unlikely in presence of selection bias, i. e., when treatment and control groups differ systemically. This pitfall is addressed by orthogonal regularization in the following.
Note that the propensity score is defined as T2x) = P (T - X x)) Rosenbaum & Rubin 1983
which is equivalent to E[T' X æ) for binary treatments.
2 "Note that we use notion orthogonality notion of Neyman orthogonality in (Nie & Wager 2017 2 Chernozhukov et al. 20118). which requires the Gâteau derivative of a debiased score function to vanish at the true parameter. In contrast, our orthogonality constraint exploits unconfoundedness ensure that the outcomes orthogonal (w.r.t. the inner product in (5) to the treatment assignment.
respect to inner product in 5). This differrnt from the
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3.2.2 ORTHOGONAL REGULARIZATION ΩOR
We now specify the orthogonal regularization term ΩoRR that ensures that the orthogonality constraint in 55) is satisfied. We first state SlR and then explain each of its parts. The orthogonal regularization term is given by
Do = Σ -
(8)
with
Y7(0)) Y; 10 Ti 3²= 1
f(,, 1) - ()
(9)
and
+ tX,,, ((T for E (0.1),
(10)
where the pscudo outcome (0) and the perturbation function f are required to learn outcomes that are orthogonal to the treatment assignment, as described the following.
Pseudo outcome. The untreated outcome of subject be expressed Y₁(0) = YY; (X,) T; where (X) = f(X. 1) - f(X, 0) is the true treatment effect at X. Hence, if we had access to the would also have access the untreated outcome (0) even if we did f We use the average treatment effect of the current model fit, i. i.e., v5" in (9), as proxy
treatment effect (X).
observe it
for (X,)). This creates a pseudo outcome. Y," (0) = Y 36 T. Under sufficient conditions. this converges to the true outcome (see Section 4
Perturbation function. order ensure that the orthogonality constraint is satisfied, e.. (Y f(X, t). π (X)) = O, we use perturbation function f Simply adding the inner product as regularization term is not sufficient, since this does not guarantee that the orthogonality constraint holds at solution of (6). A similar approach is used in targeted minimum loss estimation (e. g., van der Laan & Rose 2011) and, more general, dual problems in optimization (Zalinescu) 2002). We extend the function I to perturbation function / using the model parameter as in (TO). As result, solving the optimization problem in (6) forces the outcome estimates f to satisfy the orthogonality constraint. Mathematically, this can be seen by taking the partial derivative of (6) and setting it to zero, e.e
0 (C AAOO+ de
= Σ -(x,)) -(()),,, (1I)
xX)) = 0. is analytically sufficient to regularize only the and, hence, (Y*(0) -
untreated outcome Whenever one of the outcomes is orthogonal to the treatment assignment, the other outcome is also orthogonal. A proof of this is provided in Appendix A
Then, the outcomes are estimated using fx, t) and, therefore, the estimate for the ATE is T EE D== f=((.,,) (xx, 0). which coincides with sh* since the perturbation terms in (1 cancel.
3.3 DEEP ORTHOGONAL NETWORKS FOR UNCONFOUNDED TREATMENTS
Our regularization framework works with any outcome model f and propensity score model æ that estimated through loss function. The regularization framework ensures that the outcomes are estimated such that they are orthogonal to the treatment assignment. We introduce specific variant of our regularization framework based on feedforward neural networks, which yields deep orthogonal networks for unconfounded treatments (DONUT). Neural networks present a suitable model class due their strong predictive performance. The architecture of DONUT is set as follows. For the outcome model f, we use the basic architecture of TARNet (Saaltt et al. 2017). TARNet uses deep feedforward neural network to produce a representation layer, followed by two 2-layer neural networks to predict each of the potential outcomes from the shared representation. For the propensity
"This can be by distinction of cases: if 7; 1, then (())) Y. v(X.). and if T. 0. then (i(0) Yi. 'It is straightforward to extend the orthogonal regularization term to incorporate both outcomes by adding the same term for the treated outcome Y(1), including the pscudo outcome Y*((1) and the perturhation function
f" (ax 1) to enforce orthogonality. We demonstrate this in Appendix A
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score model иτ, use logistic regression (Rosenbaum & Rubin 1983). The choice of feedforward
neural networks is particularly strengthened by the theoretical discussion in Section
4 THEORETICAL GUARANTEES
We prove sufficient conditions under which our regularization framework yields an asymptotically norma O estimator v for the true ATE vh. Asymptotic normality is partieularly favorable for an estimator, as such estimators converge to the true ATE at a rate 1/vn. We further show that, compared to other estimators, our estimator has strictly smaller asymptotic variance.
Theorem 1. (Asymptotic Normality.) Suppose
1. The estimator i = (f. A) for the outcome and propensity score model converges some + = = in the sense that | - 7i = (1(),, where either f f or ग л (or both)
corresponds to the true function.
2. The treatment effect homogeneous, i. e., Vx E Rd dd: +(xx) vh.
3. The estimators ff and it take values in P-Donsker classes, e., Hj, Hr E CL
Then & is asymptotically normal, i.e.
vn (2b - w) 3 SN((,
a²
(12)
(ar(T x)]
where x² is the variance of the outcome.
The proof is provided in Appendix C We make the following remarks about the result and its conditions. Condition I requires that either the model for the outcomes f or the model for the propensity score π (or both) is correctly specified, but not necessarily both. This means that our estimator is doubly robust, since it is consistent under correct specification of either f or æ. In particular, neural networks converge at a fast enough rate to invoke the doubly robustness condition (Farrell et al.) 2018). Condition 2 can be easily relaxed to any specification of z/s as long as it has finitely many parameters and given that the appropriate identification criteria hold. This changes the orthogonal regularization in accordingly; see Appendix D for details. However, the advantage of Condition 2 is that it explains why the asymptotic variance of & is smaller compared to other estimators. In particular, the difference in asymptotic variance to some estimators (e. g., the inverse probability weighted estimator) can be sizable if the propensity score is close zero or one. Often, this difference is not offset by weaker restrictions imposed by heterogencous treatment effects (c. g.. Vanstcelandt Joffe 2014). Condition captures large class of functions of estimators for and from which we can choose. An overview of F-Donsker classes is given in Mikosch et al. (199)). In partieular, the class of feedforward neural networks is P-Donsker class, since Lipschitz parametric functions are P--onsker functions, and any Lipschitz transformation of IP-Donsker function is again a F-Donsker function. For mathematical details, see Appendix Together with the convergence rate for doubly robustness, this provides theoretical justification for the use of neural networks in our regularization framework, and therefore in DONUT.
Similarity to partially linear regression. We find interesting similarity between estimator á and the estimator obtained by partially linear regression (PLR) (see (1.5) in Chernozhukov et al. (2018)) ). The analytical expression of & yield by regularization framework in 8 is given in 311 in Appendix [ This is similar to the estimator obtained by PLR after partialling the effect of X from T' (see (1.5) in Chernozhukov et al. (2018)). However, the PLR separately estimates the nuisance functions and स and then plugs them into the estimator. Our approach does not use the analytical expression as plug-in estimator, but the estimator arises directly from solving the
We refer to Appendix B for a brief discussion on asymptotically normal estimators. A class F of measurable functions on a probability space (2,A, P) is called P-Donnkkr class if, for GP Vn(P,, -P), IF), the empirical process [Gf E F}.21 converges weakly to a P-Brownian bridge G... Donsker classes include parametric classes, but also many other classes, including infinite-dimensional classes, g., smooth functions and hounded monotone functions. See (Mikosch 1997) for more details.
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optimization problem in 66) via (8. The difference becomes apparent in the experiments (Section where we include the PLR as baseline and find that the performance of DONUT is superior.
Comparison to other estimators. We compare the asymptotic behavior of our estimator v to other regular estimators (e. g.. Fu.k et al. (20TT); Nie & Wager (2017). We find that, under the conditions of Theorem [ there does not exist a regular estimator that achieves strictly smaller asymptotic variance than our estimator. The reason for this is follows. We proof in Appendix C that our regularization framework yields an asymptotically normal estimator for the ATE. In particular, we proof that our estimator is efficient (see Appendix Čl the efficient influence function), and therefore it achieves the efficiency bound. As consequence, among all regular estimators, our estimator achieves the smallest asymptotic variance. In Appendix [G] we make an example in which we compare the inverse probability weighted estimator to our estimator. We show that the difference in asymptotic variance becomes particularly pronounced in presence of selection bias (i. e., when treatment and control groups differ systemically).
To summarize, we prove sufficient conditions under which our estimator & is asymptotically normal. Hence, our proposed estimator converges to the true ATE da at a fast rate. Compared to other regular estimators, its asymptotic variance is strictly smaller, since we prove that estimator is efficient. Since feedforward neural networks converge at fast enough rate to invoke doubly robustness and belong a IP-Donsker class, these results hold true when using feedforward neural networks in our regularization framework, which yields DONUT.
EXPERIMENTS
Our proposed DONUT is evaluated against state-of-the-art baselines, where find that its ATE estimation is superior (Section[5.2) Its benefits appear especially for problems subject to selection bias, which is confirmed as part of simulation study (Section 5.3 In the latter, the theoretical guarantees from Section are confirmed.
5.1 SETUP
Evaluating methods for estimating causal effects is challenging as we rarely have access to ground truth causal effects. Established procedures for evaluation of such methods rely on eemi-synthetic data, which reflect the real world. Our experimental setup follows established procedure regarding datasets, baselines, and performance metrics (e. g.. Johansoooon al. 2016 Shalit et al.) 2017).
Datasets. We evaluate all methods across four benchmark datasets for causal inference: IHDP (e. g..
Johansson et al. 2016). Twins (c. g.. Yoon et al. 2018), ACIC 2018 (c. g.,Shi et al. 2019). and Jobs (c. Ve.g..,Shal /hhalt et al. (017). The first three datasets are semi-synthetic, while the last originated from RCT. Details on IHDP, Twins, ACIC 2018, and Jobs are provided in Appendix
Training details. DONUT is trained using the regularization framework in (6). where both the outcome model and the propensity score model are trained jointly using stochastic gradient descent with momentum. The hidden layer size is 200 for the representation layers and 100 for the outcome layers similar to Shalit et al. (017)): : Shi et al. (2019). The hyperparameter a in the factual loss 7 is set to 1 and À in the orthogonal regularization 88 determined by hyperparameter optimization over {10*}2 k For IHDP, we follow established practice (c. g., Shalit et al. 2017) and average over 1,000 realizations of the outcomes with 63/27/10 train/validationttes spints Following (Shalit et al. (2017): Yoon et al. (2018); ; Shi et al. 2019), we average over 100 different saunn/valiatiio/teen splits Twins and Jobs, and over 10 splits for each dataset for ACIC, all with ratios 56/24/20.
Baselines. We compare DONUT against 17 state-of-ther art methods for estimating treatment effects, organized in the following groups: (i) Regression methods: Linear regression with treatment as covariate (OLS/LRR-)). separate linear regressors for each treatment (OLS/LR--2), and balancing
linear regression (BLR) (Johansson et al. 2016); (ii) Matching methods: k-nearest neighbor (k-NN)
(Crump et al.) 2008); (iii) Tree methods: Bayesian additive regression trees (BART) (Chipman et al.] 2012). random forest (R-Forest) (Breiman 2001). and causal forest (C-Forest) (Wager &
Athey 2018) (iv) Gaussian process methods: Causal multi-task Gaussian process (CMGP) (Alaa
& van der Schaar 2017) and debiased Gaussian process (D-GP) (Ray & Szaboj 2019); (v) Neural
network methods: Balancing neural network (BNN) (Johansson al. 2016), arenotcnt-agnostic

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Table I: Results for estimating average treatment effects IHDP, Twins, and Jobs. Lower hetter.
Results
Datusets (Mean + Std)
IHDP (EATE)
TWINS (EXTE)
JOBS (EATT)
METHOD
IN-S.
Ours.
IN-S.
QUT-S.
IN-S.
QUT-S.
OLS/LR-1
.73 = .04
.94 = .06
.0038 + .025
.0069 + .0056
.01 + .00
.08 + .01
OLS/LR-2
.14 .01
.31 + .02
.0039 + .0225
.0070 + .0059
.01 + 001
.08 + .03
BLR
.72 + .04
.93 .05
.0057 .0036
.0334 + .0092
.01 + .01
.08 + .03
k-NN
.14 .01
.79 .05
.0028 + .0021
.0051 + .0039
.21 - .01
.13 1 .05
BART
.23 .01
.4 .2
.1206 + .0266
.1655 + .0234
.02 + .00
.08 + .0
R-FOREST
.73 .05
.66 = .06
.0099 + .0034
.0080 + .0051
.03 - .01
.09 + .04
C-FOREST
.18 .01
.40 = .03
.286 + .0035
.0335 + .0083
.03 + .01
.07 + .03
D-GP
14 .32
.17 .47
.046 + .0033
.0058 + .0043
.03 .1
.06 4 .05
CMGP
.1 .10
.13 .12
.0124 + .0051
.0143 + .0116
.06 + .06
.09 + .07
BNN
37 .03
4 .03
.0566 - .0032
.0003 + .0071
.04 # .01
.09 t .04
TARNET
26 -01 .01
.28 .01
.0088 + .0017
.0151 + .0018
.05 # .02
.11 1 .04
CFR-WA WASS
25 =01 .01
.7 .01
.0122 + .0016
.0244 + .0032
.04 .01
.09 + .03
GANITE
.43 135005 .05
.49 .05
.0088 + .0017
.0089 + .0075
.01 .01
.06 + .03
AIPWE
22 28
.0027 = .0033
.0488 + .0032
.03 0 .11
.10 - .8
TMLE
.1 .10
.33 32
.0034 .0200
.0533 + .0025
.02 02 01
.06 + .05
PLR
.63 - .09
1,32 - 31
.0133 + .235
0084 + .0354
.09 + 04
.10 + .08
DRAGONNET
114
.01
.20 - .05
.0022 .0031
0064 .0054
00 - .01
.06 .05
DONUT
.3 .01
.19
+ .02
0025 .0016
0033 .0026
.01 + .00
.06 + .05
representation network (TARNet) (Shalit et al. 2017). counterfactual regression with Wasserstein
distance (CFR-WASS) (Shalit al.
(017). generative adversarial networks (GANITE) Yoon et al.
2018), and Dragonnet (Shi et al. 2019): (vi) Plug-in estimators: Augmented inverse probability
weighted estimator (AIPWE) (Cao al. 2009), targeted maximum likelihood estimator (TMLE)
(van der Laan & Rubin 2006), and partially linear regression (PLR) (Chernozhukov et al. 2018 all
of them using the outcome and propensity score model of Dragonnet as nuisance functions.
Performance metrics. Following established procedure, we report the following metrics for each dataset. For IHDP and ACIC 2018, we use the absolute error in average treatment effect (Johansson al. 20166: <ATE = 1/_ Σ (f(x;; (x, - Σ (f (0fk;1))-0 (x,, 0)) For Twins, we use the absolute error observed average treatment effect ( Yoonn et al. 2018): сĄTE 1 E. (v)
y: (0)) - Σ (Mx (1) ÿ: (0)) For Jobs, all treated subjects were part of the original randomized sample E, and hence, the true average treatment effect can be computed on the treated by ATT [T|-1 ΣiET - |CnEl- Σiecne Vt. where C' is the control group. Similar to (Shalit et al. 2017
we then use the error: EATT = |ATT [T] "7-'EEr EiET (f(z, 1) f(zx,
5.2 RESULTS
IHDP, Twins, and Jobs have been used to evaluate many methods for estimating treatment effects. In Table 1. the results of the experiments IHDP, Twins, and Jobs are presented. Overall, DONUT achieves competitive performance across datasets. On IHDP, the CMGP baseline achieves slightly lower estimation eгrоr (mean: .11 (CMGP) vs. .13 (DONUT)). but the drawback an inferior standard deviation (std.: 11 (CMGP) vs. ,01 (DONUT)). However, as shown by previous work, the small sample size and the limited simulation settings of IHDP make it difficult to draw conclusions about methods (c. g., Yoon et al. 2018 Shi et al. 2019). In contrast, DONUT achieves superior performance on both Twins and Jobs, where the number of samples is much larger (Twins: =11,400; In particular, we point out the
Jobs: =3.212). On these datasets, DONUT is state-of-the-are difference to PI.R which uses the analyiteal expression of our estimator similar to Chernozhukov al. 2018) discussed in Section 4 but plug-in estimator. For reasonable comparison, use the outcome models of Dragonnet as nuisance functions. We find that DONUT achieves superior performance.
Among neural betserk-based methods (i.e., BNN, TARNet, CFR-WASS, GANITE, Drag- onnet, and DONUT), DONUT performs superior across all datasets. In compari- son to TARNet, which shares the basic architecture of DONUT (but without orthog- onal regularization), achieve substantial reduction in ATE estimation across
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all datasets (out-of-sample eπror reduction: 32.1 % 45.5% % on Jobs). This demonstrates the effectiveness of our regularization framework.
IHDP, 78.1 % on Twins, and
We further evaluate DONUT using ACIC 2018. This collection of datasets was introduced for eval- uating neural networks for estimating average treat- ment effects (Shi al. (019). We compare DONUT against the most competitive, and current state-of- the-art method, Dragomnet, In addition. we compare against TARNet (i. c., DONUT, but without orthogonal regularization). Table 2 presents the results of the experiments on ACIC 2018. The main observation that DONUT improves estima- tion relative to TARNet (DONUT without orthogo- nal regularization) and relative Dragonnet across a large collection of datasets. These results further
Table 2: Results for estimating average treatment effects on ACIC 2018. Comparison between Drag- onnet (most competitive baseline). TARNet (same architecture DONUT, but without orthogonal reg- ularization) and DONUT. Lower better.
Results
ACIC (EATE)
МETHOD
IN-S.
OUT-S.
TARNET
4.53 + 0.72
4.48 E 0.74
DRAGONNET
2.97 1.46
2.99 1.48
DONUT
1.22 + .33
1.26 + 0.29
confirm the benefit of including orthogonal regularization the estimation procedure.
5.3 SIMULATION STUDY ON SELECTION BIAS
To evaluate the robustness of DONUT with regards to selection bias (i. when treatment and control groups differ substantially), we generate synthetic data with varying selection bias according to a similar protocol as in Yao et al. (2018); Yoon et al. (2018). Details on the protocol are provided Appendix We compare DONUT against Dragonnet, which makes use the AIPWE to estimate the average treatment effect. This is particular interesting for the comparison with DONUT due to the results in Section 4 We showed that, compared any other estimator, our estimator obtains strictly smaller asymptotic variance, especially in pres- ence of selection bias. Figure | presents the
DONUT
Dragenict
on
mean and standard deviation of caTE for varying selection bias. We report two major insights: (i) Dragonnet is outperformed by DONUT across different levels of selection bias. As selection bias increases, the estimation error of DONUT remains stable. (ii) The standard deviation of both DONUT and Dragonnet increases as selec- tion bias increases. However, in line with our findings that DONUT has the smallest asymp- totic variance among all regular estimators, the standard deviation of DONUT remains consis- tently smaller than the standard deviation of Dragonnet. In addition, this difference becomes more pronounced the larger the selection bias. Hence, we observe empirical properties of our estimator that coincide with the theory derived Section
210
Kullbook diveggncce
Comparison of DONUT and Dragonnet in presence of selection bias. Reported is the mean and standard deviation of eXTe synthetic data selection bias varied (measured by the Kullback-Lete Leibelr diver- gence). For cach level, both algorithms are run on 100 realizations of the datasets. Lower is better.
Figure 1:
CONCLUSION
Understanding causal effects is crucial for reliable decision-making. In this paper, we present regularization framework for estimating average causal effects. We formalize unconfoundedness as an orthogonality constraint that is used to learn outcomes that are orthogonal to the treatment assignment. We prove theoretical guarantees that our regularization framework yields an asymptotically normal estimator. Based on this, we develop DONUT, which leverages the predictive capabilities of neural networks to estimate average causal effects. Experiments datasets show that, in most cases, DONUT outperforms the state-of-the- substantially. This work provides an interesting avenue for future research causal inference. We hypothesize that most existing models can be improved by
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incorporating orthogonal regularization. We leave the derivation of unifying theory for future work. revised version of this work can be found in Hatt & Feuerriegel (2021).

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A SUFFICIENCY OF REGULARIZING THE UNTREATED OUTCOME
We show that it is sufficient regularize (0),, since orthogonality of Y (0) implies orthogonality of (1) T given X. Suppose Y (0) satisfies the orthogonality constraint, i. e..
-
Σ - f(x, -X())) 0.
(14)
Then,
((X)
(Y1(1) f(x, 1)) -xx))
(15)
Σ (X((1) E| X, 1))(1) (xx))
(16)
Σ (Y((1) EI X - x(x))
(17)
Σ
+ (X)+ - -XX + (X.) (1- T:) (.))(TT +(x)))
(18)
Σ (% (X,) - EYY -xx,) X.)(T, -x(x))
(19)
Σ (V((0) ( X.)( - ((x))
(20)
Σ (V/(0) B[Y X,, T = =0)(T; x(X.))
(21)
1 Σ (Y7(0) f(x, O) (Xx))
(22)
(0
T(X) O.
(23)
using that Y 1), T (X)) is zero and, therefore, Y(1) is orthogonal to T given X. Nevertheless, it is straightforward to adapt the orthogonal regularizer (8) to accommodate both outcomes. Consider the following adapted orthogonal regularizer
and
= Y ( in (18) and (20). Hence, (Y
(, = Σ -0)
-
(24)
n
with
(2(()) = Y, - +0 Tú,
YT(1) = T:+ 1 (1 TT), 1' = 1(x,1)- (25)
and
0), f(X,, + +e[; T(:)),
(26)
= 1) 1) + e(T) x(x)))
(27)
where e is additional model parameters. Similar the result in Section 3.2.2., this yields that
(0) - (X, 0), T T(X)) and (1*(1) (X, (X)) are zero, when the partial derivative of 224 t. is zero.
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B ASYMPTOTICALLY NORMAL ESTIMATORS
In this section, we give a brief discussion on asymptotically normal estimators. We refer to van der Vaart (1998) for an in-depth discussion. Suppose v is an estimator for the true parameter y. Then, e an asymptotically normal estimator if it satisfies
e - 6
Σ o(Z)) + op(1//n,
(28)
where oZZ is referred to as the influence function at Z (X Kandasamy et al. 2015
The influence function has mean The existence and uniqueness of the influence function follows by the Riesz representation theorem. Asymptotic normality is a favorable property of an estimator, since, by the central limit theorem,
and finite variance (i.e., E() O and Var( (Z)) 20).
Vn(Ø e)) & NO, Var(o(Z)))
(29)
Hence, an asymptotically normal estimator is asymptotically normal distributed and unbiased. More- over, the variance shrinks in proportion 1/v the sample size grows and is given by the variance of the influence function. As a consequence, the distribution of the estimator u converges weakly to dirac delta function centered around the true v/.
C PROOF OF THEOREM
In observational studies, the covariates X are often high-dimensional and limited knowledge about the nuisance functions (i. e., outcome and propensity score functions f and T) is available. In such a case, it reasonable to use flexible, data-adaptive methods for estimating the nuisance functions, e.g., neural networks. However, the complexity of these methods makes asymptotic analysis difficult, because the estimators used to construct (ff f) are not described by single tnite-dimensional parameter. Nevertheless, under some conditions, we can learn about the asymptotics of & using tools from empirical process theory.
First, we need introduce some notation. Throughout the proof, we will use P[f(Z)) f(=) to denote expectations of f(Z) for random variable Z (treating the function as fixed). Hence, P(f(Z)} is random if f is random (e. g., estimated from the sample). In contrast, E[f(Z)) is fixed non-random quantity, which averages over randomness in both Z and ff and thus will not equal P(f(Z) except when
fixed and non-random. Moreover, we let 72 Σ 82, denote the empirical measure such that sample averages can be written as 1 n +EE1 f(Z)) = J dz) =
P. (Z)). For clarity, we will denote the true nuisance functions and average treatment effect as Voo = (Jo, 7o) and voo respectively. We further denote the triplet (Y,T,X) X) by Z.
For f = fr, f.. and & a E (X, - fz (X;, 0) from
( the regularization framework yields
(x)) +(X)) -
1
Σ (Y, 1 TT. O)( *(X.)) 0,
(30)
i=1
by construction. Hence, we receive an analytical expression of the estimator of the average treatment effect by solving the above for v i.e.
Σn
#= (Y: - (Xx #Xx))
(31)
1 Σこ Ti(Ti Ax()))
"
Therefore, the estimator is given by & = P., (m(2: 2; fi), where
ml (Z:1) (Y S(X, 0) -Tx,)))(T(((x))
(32)
P(T(T T(X))
and 77 = (T. f) denotes the nuisance functions.
We recall the conditions in Theorem 1. Suppose the estimator n = (ff: #) converges to some fj = (J, #) in the sense that || (1f-m)) 7| o, where either / = So or 75 (or both) corresponds the
(1/rr) employs the usual stochastic order notation so that X,, (1/rr) means that r, X, 0 in prohahility.
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true nuisance function. Thus at least nuisance estimator needs to converge to the correct function, but one can be misspecified. Then, (mm(22 Z: n)} = (mmm( (2; } = Vo. from the straightforward to check fact that ( n, fo)) = (m Z; no. f)} for any ā and f. Consider the decomposition
3 - t/u = P,, (m f)) - (m
= PP
P(mm (2; m(Z
If the estimators for the nuisance functions f take values in P--Donsker classes, then m mZ (i) also belongs to PP-Donsker class, since Lipschitz transformations of Donsker functions are again Donsker functions. The Donsker property together with the continuous mapping theorem yields that (P,,
P) {) Z; f asymptotically equivalent (P,, - -P)) {m((z ri)} up to (p(///T) error (see Mikosch ct al. 1997) for more details on P--Donsker classes). Therefore.
w/ (T,,, - (m (2: + (m((2))) m( + (p(//m).
(33)
is left to show that (m (2; n) - m(Z asymptotically negligible. This term equals
(T
(T- (X)) - fu(X,0))
P
(34)
P[T(T- =(x)))
PTT(T - +x(Xi)}
(1- (X))((XX - f(X,0)) (1- - P(T(T - =xo(x)}
P
Tx(
(35)
F[T(T - +(X))}
P
(36)
P[T(T - *(x))}
(1 -
(1 -X=,
P
(o(
(37)
([mo(X](155(X)
P[mo(X)( (x)))
To (X))))( ((x
P
(38)
- (1-())((-(x))
(X) - 1x - f(X,0))
P
(39)
((m(()(1--(x))
(1 (TXXX))) O))
P
(40)
(X) - (X),,, - f(X,0) xo(X(( ((X)) fxx,0)
P
(41)
(10o(X(1-a(x)))
(1 To
(10())(1=(())) (x.0)
P
(42)
where we use P(m(Z; 7)} (m ((mm(:mm)) mo)) in (34), iterated expectation in (35), and P
*(X))= (( X) (1 - =xx))} (likewise for To instead of R) in (37). In (39), and similarly in
(341), use that (X) thu and, therefore, fo(X, So(X, vig As result, by simplifying.
the above equals
(X) #(X) (X)o(X)(1 +X))) (fo(X,0)-fF) f))
(43)
Therefore, by the fact that To and & are bounded away from zero and one, along with the Cauchy- Schwarz inequality (i. e., P[fg}] < |[]]]|[]], we have that (up to a multiplicative constant) (P(m((;;) m (Z:1) } H is bounded above by
[X)o(X) - -x) f(x,
(44)
Thus, for example if स is based on a correctly specified parametric model (e. g., logistic regression), so that || Toll op (n). then we only need f to be consistent, i.c., |l - foll (1). to make the product term doubly robust estimator satisfies b tho (P,, [m no))) + (p(1/vn) and it is efficient with
fi) m Z: n = (p /z) asymptotically negligible. Then the
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influence function (Z 1), m(Z - vb Thus, this proves the asymptotic normality of v.. It remains show that the asymptotic variance of vi equals
o²
(45)
P[Var[T X)}
Note that in the paper, E[S(Z)] is used similarly to P(S(Z)) We introduced the notion P(S(Z)} here to make clear that only integrate over randomness in Z and not the function estimates f. As
seen in Appendix B the asymptotic variance of an estimator is the variance of its influence functions.
From above, we know that the efficient influence function of vb is of (ZZ; using that Pf m))
Z:;4; 16, 7) m(Z; v. Thus,
(by definition).
Var(o n)) P( n))]]
(46)
+P(mm(Z:)) - +b)
(47)
= =P((m(Z:n)* - +²
(48)
This, together with
P(U2] P(m((Zn))) P[Var(T X)] + x/2
(49)
where U = - f(X,T) o²) and P[T(T m(x))} IP Par X)] proves the statement.
D ORTHOGONAL REGULARIZATION UNDER LINEAR SPECIFICATION OF THE TREATMENT EFFECT
In Theorem I, condition 2 ensures the homogeneity of the treatment effect, e., (XX 1b. This can be easily relaxed to any specification i as long as it has finitely many parameters and given that the appropriate identification criteria hold (for linear specification, this is the non-singularity of the design matrix). We explain in this section how the orthogonal regularization in (8) changes when linear specification of the treatment effect is considered, i. e.. (xx) 0 xr, where E Rdd. Recall the orthogonal regularizer from 8
" DoR(F,T,E Σ (0) (),,
(50)
where the perturbation function fc remains unchanged. The pseudo outcome Y;* is now given
(7) Y,, (0 X,) T;
(51)
where 0 is given by the system of linear equations,
0 X; 1
Σ (T, -)
(52)
where 2)) is arbitrary function of the dimension of 6, which ensures that the system of linear equations possesses a unique solution. For instance, the case (xx)= = A x, the choice v(t,, 22, 2; could be made. The procedure goes as follows. First, the above system of linear equations is solved for 0. Second, the pseudo outcome (,) = Y; (0 X,)T, can be computed. Finally, this is plugged into the regularizer. The proof for asymptotic normality goes similar the one under homogeneous treatment effect.
E FEEDFORWARD NEURAL NETWORKS ARE IN P-DONSKER CLASS
We show that feedforward neural networks are a P-Donsker class. feedforward neural network
fa is defined by the recursion
(x) WTT,, sr )(x)= WT 2 d,
(53)
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for d E N21, matrices { (Wi) of appropriate dimensions, activation function 0, which applies element-wiie wiie,, and is the depth of the neural network.
fi is clearly in a P-Donsker class as it is Lipschitz parametric. The Donsker property is preserved under Lipschitz transformations. Hence, fz), for 2,..., d, in a I--Donsker class, if the activation function is Lipschitz transformation. This is the case for most common activation functions. We give two examples in the following.
The ReLU activation function, (R) max(0. (0. z), preserves the Donsker property, since constant function is clearly in a IP-Donsker class and taking the maximum of two PP-Donsker functions again P--Donsker function due the preservation under Lipschitz transformations. The same follows for the ELU activation function,
if > { I, aaee 1),
GEL
if <
where 0: 2 0, since it is Lipschitz transformation.
F ASYMPTOTICS OF & UNDER HETEROGENEOUS TREATMENT EFFECTS
Our estimator retains useful interpretation under heterogeneous treatment effects. In this case, & converges to
P[Var(T) (x){(X)
(54)
P(Var(T) XX)
where (X) = 1)
f(X,0) -(X,)) is the true treatment effect at X. This can be interpreted weighted average of treatment effects ((X). As consequence, most weight is given to
subpopulations with large Var(T X), e.., which are most informative about the treatment effect.
We consider the same notation in the proof in Appendix C and resume it. From the proof in Appendix C we know that
-ー = (P, - () Pfm((P m(Z + opp///n)),
(55)
since the estimator of the nuisance functions = (f. #) belong a IP-Donsker class, and therefore m(Zi (; ri) belongs to a P--onskker class. Similar to the proof in Appendix C it is left to investigate the term (2m((Z; f) m ( }. We show that this term is asympiotically not negligible when the treatment effect is heterogeneous. Again, by iterated expectations, this term equals
(To (X) -) 0)) PP
no (1) mo( fo(X, (()v((X(1 no (x)))
P
(56)
P(mo(X)(1 (Ix))
The first term can be bounded from above (up a multiplicative constant) by the Cauchy-Schwan inequality and, therefore, is asymptotically negligible if one of the nuisance functions is correctly
specified. Thus, only the second term remains. Using (X) fo(X, 1) fo(X, 0), the second term can be written as
P To -
(57)
P To (x) (X))))
(T)())) Since To(X) To(X)) = Var(T) X), this concludes the proof.
G COMPARISON TO INVERSE PROBABILITY WEIGHTED ESTIMATOR
We compare our estimator to the inverse probability weighted (IPW) estimator (c. g., Funk et al. 2011. The IPW estimator is popular estimator for treatment effects due to its improvement in efficiency and reduction in bias compared to unweighted estimators.
Under identical conditions as in Theorem 1 the IPW estimator, denoted as Spw is asymptotically normal with asymptotic variance x² E /Va((T) X)] (e. g.. Kennedy] 2016). We can compare the asymptotic behavior of our estimator & and the IPW estimator 2/PWw Both estimators are asymptotically unbiased, but with different asymptotic variance the following result shows.
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Corollary 1. Under the conditions in Theorem / than the asymptotic variance of thrrw
the asymptotic variance of a is strictly smaller
Proof. The statement follows by Jensen's inequality.
The difference between the asymptotic variances becomes particularly pronounced in presence of selection bias, i. c., when the treatment group is systematically different from the control group.
Corollary 2. If the propensity score x (x)) is close to 0 or I for some x E R2, then the asymptotic variance of siew can lake arbitrary large values.
Proof. For some small > 0. let x(X) E (e,,, e).. Then, by Bhatia-Davis inequality and
since is binary, Var(TT X) = (1 x(X)). Hence, Var(T X) I c, and, therefore. B[1/Var() (TL/Var(T)) x)] 1/c, which proves the statement.
H DETAILED DESCRIPTION OF THE DATASETS
H.1 IHDP
Hill (Hill 2011) introduced semi-synthetic dataset created from the Infant Healthand Development
Program (IHDP). This dataset is based on a randomized experiment that examines the effect of home future cognitive scores. The dataset consists of 747 children (t 1: 139, I = 0: 608) with 25 covariates. Similar to Shalit et al. (2017). we use 1,000 realizations from setting in the NPCI package (Dorie 2016).
visits by specialists
H.2 TWINS
This dataset is made up of all births in the USA between 1989 and 1991 (Almond et al., 2005).
Only twins are considered among these births. Treatment (i. T = is defined as being the heavier twin (and T 0 as being the lighter twin). The outcome defined 1-year mortality. There are 30 covariates available for each pair of twins that relate to the parents, pregnancy and birth: marital status; race; residence; number of previous births; pregnancy risk factors; quality of care during pregnancy; and number of gestation weeks prior to birth. Only twins that weigh less than 2 kg and have no missing covariates (list-wise deletion) are taken into account. This creates complete dataset (without missing data) | The final cohort consists of 11,400 twin pairs, whose mortality rate is 17.7 % for lighter twins and 16.1 % for heavier ones. this setting, we observed both T 0 (lighter twin) and T treatment effect in this dataset is known. In order to simulate an observational study, one of the twins is selectively observed based information using the covariates (which leads to selection bias) as follows: T x ~ Bern(Sinn (einSimmoid((u x + n)) where T ~ ((-001 0.1 30001 and n ~ (0, 0.1).
(heavier twin) for each pair of twins; Therefore, the true
H.3 JOBS
Jobs studied in Landn (996) consists of randomized data based on the National Supported Work program and -andomnddd observational study data. A (random) subset of randomized data is used to evaluate the algorithms. The dataset consists of 722 randomized samples (I" = 1: 297, T = 0: 425) and 2,490 non-randomized samples (T 1: 0, T 0: 2,490), all with covariates.
H.4 ACIC 2018
ACIC 2018 is a collection of semi-synthetic datasets derived from the linked birth and infant death data (LBIDD) (MacDorman & Atkinso 1998), and was developed for the 22018 Atlantic Causal Inference Conference competition (ACIC) (Shimoni et al., 2018). The simulation includes 63 different data generation processes with sample size from 1,000 to 50,000. Each dataset is realization from a separate distribution, which itself randomly drawn in accordance with the settings of the data generation process. Similar to Shi et al. (201). we randomly pick 3 dataset [00 of size either 5k or
We provide the complete dataset at g thub. com http///nnnmmss///// /dontt (anonymized for peer-review) pererrrreeeww 10We provide the list of unique dataset identification numbers at g giggttthhh...o. com www.//aoonoosus//oott.o. (anonymized for peer-review) for reproducibility.
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10k for each of the 63 data generating process settings and exclude all datasets with indication of strong selection bias. This yields a total number of 97 datasets.
I DATA GENERATING PROCESS FOR SIMULATION STUDY
For the simulation study, we follow a similar protocol as in Yao et al. (2018); Yoon et al. (2018). We generate 2,500 untreated samples from [0 0.5 ΣΣ and 5,000 treated samples from (11,, 0.5 x ΣΣT) where E (O0, 1))0x10). Varying d1 yields different levels selection bias, which is measured by the Kullback-Leible divergence. The larger the Kullback-Leibler divergence, the greater the distributional distance between treatment and control group, and, thus, the larger (w + n), where
the selection bias. The outcome is generated as Y X x. T
~ ((((1,,1) 10x1), and n ~ N(0, 0.1).
19


Reviews:
[{"review_id": "tc7uaEcMQkY", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 7, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThe present paper introduces a new approach, deep orthogonal networks for unconfounded treatments (DONUT), that allows to estimate (average) treatment effects exploiting an orthogonality property implied by the classical unconfoundedness assumption. The authors propose a regularization framework based on the orthogonality constraint and prove that a resulting estimator is doubly robust, asymptotically normal and with efficient variance. They supply multiple simulations to demonstrate their theoretical claims and to show state-of-the-art performance of their estimator.\n\nRecommendation:\nClear accept. In summary, I am convinced that the this paper would be a valuable addition to this year's conference. It considers a novel approach to improve average treatment effect estimation on observational data using combining classical causal inference assumptions and predictive power of deep learning.\n\nStrong points:\n - The authors propose a new methodology that seems theoretically solid and that has an implementable estimator for ATE estimation, exploiting a necessary condition implied by a standard causal inference assumption.\n - The article is well written and easy to read.\n\nWeak points:\n - The code for their simulations is not accessible (broken/incorrect url?).\n - A discussion about the impact of the hyperparameters, especially the orthogonality regularization parameter $\\lambda$, would give more insight into the importance of the contribution of the regularization term.\n\nQuestions/Issues:\n - The provided url to access the anonymized code did not work (at least for me), would it be possible to fix this or to provide the complete code as supplementary zip file?\n - How sensitive are the results to the hyperparameter choices, especially the $\\lambda$ parameter?\n - The orthogonality constraint is implied by the unconfoundedness assumption, but it is not a sufficient condition for unconfoundedness. Have the authors studied the behaviour of their method and its performance in (simulated) cases where unconfoundedness (4) does not hold but the orthogonality constraint (5) holds?\n - The authors mention the R-learner (Nie \\& Wager, 2017) which uses the notion of Neyman orthogonality and use the R decomposition to propose an estimator of treatment effects. Would it be possible to add this method to the list of compared methods?\n - Since the authors theoretically compare their estimator to the IPW estimator, it would be interesting to add this to the experiments to confirm the theoretical results.\n - For the ACIC datasets, how does DONUT compare to BART, which is known to perform well on these data (Dorie et al., 2018)?\n\nMinor comments (that did not impact the score):\n - p. 2/3: equations (2) and (4) are the same.\n - p. 5: yield by our $>>$ yielded by our\n\nReferences:\n - Dorie V, Hill J, Shalit U, Scott M, Cervone D. Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competition. Statistical Science. 2019;34(1):43--68.\n - Nie X, Wager S. Quasi-oracle estimation of heterogeneous treatment effects. arXiv preprint arXiv:1712.04912, 2017.\n\n===================\n\nPost Rebuttal Update:\n\nAfter the discussions and reading the other reviews, I lower my score by one point. I could not find the changes to the manuscript announced by the authors during the discussion, especially the additional intermediate results necessary for the theorem's proof pointed out by reviewer 1.", "extra_scores": {}, "extra_text": {}}, {"review_id": "igTbZru8MpX", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Overall, I found the main idea of this paper very interesting and the experimental results promising; however, there were several major and minor technical issues with the work that need to be resolved.\n\n--- Major comments ---\n\n1. There appear to be at least two major technical issues:\n\n    a. A substantial portion of the work is based on the author's assertion that $Y_i(0) = Y_i - \\psi(X_i)T_i$ which is not, in general, true. We can see this with a simple counter-example. Let $Y(0)$ be a binary variable. Then $\\psi(X) = E[Y(1)|X] - E[Y(0)|X]$ will be some value in $[-1,1]$, but if $\\psi$ is any value other than $1$, $0$, or $-1$, then $Y_i - \\psi(X_i)T_i$ will be non-binary and thus not equal to $Y_i(0)$. This assertion only holds if $X$ and $T$ uniquely determine $Y$, which is not generally the case. Thus, it remains to be shown that enforcing equation (8) is equivalent to enforcing equation (5).\n\n    b. In the first condition of Theorem 1, the authors assume that either $\\hat{f}$ or $\\hat{\\pi}$ is consistent. If $\\hat{f}$ and $\\hat{\\pi}$ were estimated separately and used as plug-in estimators, as in Chernozhukov et al. (2018), this would be a reasonable assumption, as it would be up to the user to show that their nuisance parameter estimates are consistent. However, $\\hat{f}$ and $\\hat{\\pi}$ are estimated using Equation (6) and thus it is up to the authors to show that solving (6) gives consistent estimates of $\\hat{f}$ or $\\hat{\\pi}$ under correct model specification. This very well may be the case, but it cannot simply be assumed.\n\n2. I found the introduction very hard to follow. In particular, it is never really made clear what the motivation for the work is. Extrapolating from the experiments, it appears that the goal is to derive an estimator with lower variance than existing estimators, but that is not stated in the intro. I would recommend restructuring to something like: CI from observational data is important because XYZ. It is desirable that estimators for the ATE have the lowest possible variance. We propose a new estimator that, empirically, has lower variance than a variety of state-of-the-art causal estimation methods. We do this by translating the exchangeability assumption into an explicit constraint on the estimator objective. This constraint reduces variance because ABC. Additionally, lit review portion of the intro reads as a random list of methods with no clear connection between them or to the proposed method. Some specific issues are:\n\n    a. It is incorrect that \"the outcome of an alternative treatment has to be estimated\". In particular, IPW methods do not do this. In fact, methods based on estimating the conditional expected outcome do not do this either since, as stated above, the expected outcome is not equal to the outcome. \n\n    b. \"to find similar subjects\" --> \"to find similar subjects who received different treatments\"\n\n    c. \"seek weights such that the treatment assignment is unassociated with the covariates\" --> \"reweights the data so that treatment assignment is unassociated with the covariates in the reweighted distribution\"\n\n    e. \"However, they do not require the...\": None of the methods described here require this, but they do require that they be conditionally independent. Also, potential outcomes have not yet been defined.\n\n3. Page 2, Contribution section, \"Compared to other estimators, its asymptotic variance is strictly smaller.\": It is my understanding that TMLE is also semi-parametrically efficient, so I believe this statement is incorrect as are similar statements in the \"Comparison to other estimators\" section. Further, both Chernozhukov et al. (2018) and the work on CV-TMLE show $\\sqrt{N}$-consistency without relying the Donsker class assumptions.\n\n--- Minor comments ---\n\n1. In equations (4) and (5), the authors jump from a constraint on the true distribution to a constraint on the empirical distribution. I recommend adding a statement indicating this.\n\n2. What score was used to select the best $\\lambda$ on the validation set? \n\n3. RHS of Equation (11): $\\epsilon$ --> $\\hat{\\epsilon}$ or drop the \"evaluated at\" bar.", "extra_scores": {}, "extra_text": {}}, {"review_id": "YvVHV4BaVLn", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a novel regularization term for designing loss functions to estimate outcome and propensity score models, where the end goal is to estimate ATE.  The regularizer is derived from the assumption of conditional independence of potential outcomes and treatment given covariates (i.e. the no hidden confounding assumption).  The authors observe that this assumption implies that residuals of potential outcomes and treatments are orthogonal.  The authors derive a loss function which yields this orthogonality at the optimum.\n\nThe Theorem 1 shows asymptotic normality and a double robust property. The authors also perform extensive empirical comparisons to other causal inference methods on 4 standard benchmarks, as well as on the ACIC challenge. The method performs admirably: it is competitive on IHDP, and in a statistical tie with the best-performing methods on Twins and Jobs.\n\nUnconfoundedness is an assumption that ensures that an estimator of the form of equation (3) is unbiased for a causal effect. Without the assumption, an estimator may ascribe a causal effect to the treatment that really comes from a common cause.  To use it to inform a penalty on the estimator is an unusual step, as it seems unrelated to the question of efficiency.  However, the results are compelling.\n\nThe first question that arises is: have the authors presented a clear ablation analysis, similar to the TARNet vs CFR analysis of Shalit et al.?  The authors state that they use the TARNet architectures for the outcome models.  In that case, can we interpret DONUT vs TARNet results in Table 1 as an ablation analysis on the $\\Omega_{OR}$ regularizer? The numbers for TARNet in this submission are identical to those in Shalit et al., so perhaps not all experimental variables have been exactly matched such that we can consider it a true ablation analysis?\n\nI have some technical questions that I believe the authors should address:\n*    Moving from equation (31) to (32) seems to require omitting any randomness in the denominator.  Why is this justified?\n*    Since $\\hat f = f^{\\epsilon}$ in equation 31, can we ever expect that $\\bar f$ is equal to $f_0$?\n*   What are the numbers in table 2?  The authors state they evaluate on 97 models from the ACIC challenge. Is table 2 presenting an average of ATE estimation errors?\n*   In describing the Pseudo Outcomes, the authors state that $\\psi(x)$ is the true treatment effect at X.  However, it is the expected treatment effect, and hence can’t be used to exactly impute an unobserved outcome, which may have some noise in general.  This also affects the argument in Appendix A (steps 18 and 20), where this imputation is performed.  Are the authors assuming throughout that the treatment effect is deterministic given $X$?  In Theorem 1, a constant treatment effect is assumed, but this seems to be an assumption to make the theory go, and the method is designed to handle heterogenous treatment effects.  If such an assumption is at play in the entire paper, it should be presented earlier.\n*   Can the authors address how to construct confidence intervals, which are generally always reported for ATE estimation?  Can one plug in quantities for equation 12 from in-sample estimates?  Can the authors compare widths and coverage properties of confidence intervals constructed in this way?\n*    The authors are noncommittal about regularizing orthogonality of both $Y(0)$ and $Y(1)$: appendix A indicates it’s not necessary if plugging the true outcome and propensity score model: however, in practice, does it make a difference?\n*    A small complaint about terminology: the authors state in 3.1 that “the inner product in (5) is the empirical covariance between $Y(t)$ and $T$ given $X$.” However, equation (5) is not a function of X so this statement cannot exactly hold.  The orthogonality constraint requires some removal of conditioning, which the authors should address.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "8MtcuFJeHU9", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a regularized framework for estimating the average treatment effect. They assume unconfoudedness and show that it implies a specific orthogonality constraint. The main idea is to use this orthogonality constraint during estimation of the model parameters as a regularizer. On the theoretical side, the authors provide sufficient conditions under which the regularization yields an asymptotically normal estimator for the average causal effect. Based on the regularization framework, an estimator for average causal effect via feedforward neural nets is developed.\n\n- The motivation for the regularization is not clear. It is not clarified in the paper that why the proposed regularization should in fact improve the estimation bias and variance. It is only shown that the resulting estimator is asymptotically normal (the authors also show efficiency but it is under the strong assumption 2 in the theorem), but it is not clear that why this regularization would be particularly helpful.\n\n- Page 4, the authors say \"if we had access to the treatment effect \\psi(X)=f(X,1)-f(X,0), we would also have access to the untreated outcome Y(0) even if we did not observe it. This does not seem to be true. (same for the footnote) f(X,0) is a function of X and if we take its expected value, under positivity, consistency and ignorability, it gives us the expected value of Y(0) not the random variable itself. \n\n- Then \\psi(X) is replaced by its empirical mean and the authors say \"Under sufficient conditions, this converges to the true outcome (see Section 4)\". This does not seem to be what is proven in Theorem 1.\n\n- How can we show that equation 11 implies equation 5, which was the original goal?\n\n- Assumption 2 in Theorem 1 is very strong and I suggest removing it from the manuscript.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a regularization term that enforces the orthogonality between (i) a residual between the observed outcome and its estimator and (ii) the treatment and propensity score. The method empirically performs competitively. However, there seems to exist a gap between the proposed method and the assumptions made to provide theoretical guarantees (e.g., R3, R2). R4 was also concerned about the issue and adjusted his/her score accordingly. Even though the authors provide a detailed discussion on most of the reviewers' concerns, some of the problems remain unresolved. Further, unlike other papers submitted to ICLR, the authors did not actually update the paper such that we could check whether the revisions were adequately made. As such, I believe this paper is not quite ready for publication in its current form.

Decision: Reject