Estimating Treatment Effects via Orthogonal Regularization

1 INTRODUCTION . Estimating the causal effect of an intervention ( i. e. , treatment effect ) is integral for individual decision making in many domains such as marketing ( Brodersen et al. , 2015 ; Hatt & Feuerriegel , 2020 ) , economics ( 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 to ease of access to rich data . In this paper , we estimate the average causal effect of a treatment from observational data . In order to estimate the causal effect of a treatment , the outcome of an 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 a feature and train regression models to estimate the outcomes ( Funk et al. , 2011 ; Kallus , 2017b ) . Methods based on 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 quantification ( Alaa & van der Schaar , 2017 ; Ray & Szabo , 2019 ) . Weighting-based 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 a model for the treatment propensity in a 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 to its strong predictive performance and ability to learn representations of the data ( e. g. , Johansson et al. , 2016 ; Louizos et al. , 2017 ; Shalit et al. , 2017 ; Yao et al. , 2018 ; Yoon et al. , 2018 ; Shi et al. , 2019 ) . To ensure identifiability of the causal effect , state-of-the-art methods for estimating treatment effects are based on unconfoundedness ( i. e. , all confounders are measured , and thus included in the covariates ) . Hence , unconfoundedness is assumed for identifiability , yet , during estimation of the model parameters , any implications on the unobserved outcomes that arise from unconfoundedness have been neglected . Contribution.1 In this paper , ( i ) we introduce a regularization framework that exploits unconfoundedness . To this end , we formalize unconfoundedness as an orthogonality constraint . This constraint is 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 a variety of benchmark datasets for causal inference , we demonstrate that DONUT outperforms the state-of-the-art substantially . 2 PROBLEM SETUP . Our objective is to estimate the average treatment effect ( ATE ) of a binary treatment from observational data . For this , we build upon the Neyman-Rubin potential outcomes framework ( Rubin , 2005 ) . Consider a population where every subject i is described by the d-dimensional covariates Xi ∈ Rd . Each subject is assigned a treatment Ti ∈ { 0 , 1 } . The random variable Yi ( 1 ) corresponds to the outcome under treatment , i. e. , Ti = 1 , whereas Yi ( 0 ) corresponds to the outcome under no treatment , i. e. , Ti = 0 . These two random variables , Yi ( 1 ) , Yi ( 0 ) ∈ R , are known as the potential outcomes . Due to the fundamental problem of causal inference , only one of the potential outcomes is observed , but never both . The observed outcome is denoted by Yi . Our aim is to estimate the average treatment effect ψ = E [ Y ( 1 ) − Y ( 0 ) ] . ( 1 ) The following standard assumptions are sufficient for identifiability of the causal effect ( Imbens & Rubin , 2015 ) : consistency ( i. e. , ∀t ∈ { 0 , 1 } : Y = Y ( t ) , if T = t ) ; positivity ( i. e. , ∀x ∈ Rd : 0 < P ( T = 1 | X = x ) < 1 ) ; and unconfoundedness . Unconfoundedness 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. , Y ( 1 ) , Y ( 0 ) ⊥ T | X . ( 2 ) Based on this , the ATE is equal to ψ = E [ E [ Y | X , T = 1 ] − E [ Y | X , T = 0 ] ] . ( 3 ) Our task is to estimate the function f ( x , t ) = E [ Y | X = x , T = t ] for all x ∈ Rd and t ∈ { 0 , 1 } based on observational data D = { ( Xi , Ti , Yi ) } ni=1 . 3 ORTHOGONAL REGULARIZATION FOR ESTIMATING TREATMENT EFFECTS . The key idea of our regularization framework is to exploit the implications on 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 of our regularization framework based on neural networks , which yields DONUT . 1Code available at github.com/anonymous/donut ( anonymized for peer-review ) . 3.1 UNCONFOUNDEDNESS AS ORTHOGONALITY CONSTRAINT . Under unconfoundedness , the outcomes are independent of the assigned treatment given the covariates , i. e. , Y ( 1 ) , Y ( 0 ) ⊥ T | X . ( 4 ) Using the inner product 〈V , W 〉 = 1n ∑n i=1 ViWi , we formalize the following orthogonality constraint : 〈Y ( t ) − f ( X , t ) , T − π ( X ) 〉 = 1 n n∑ i=1 ( Yi ( t ) − f ( Xi , t ) ) ( Ti − π ( Xi ) ) = 0 , ( 5 ) for t ∈ { 0 , 1 } , and where f ( x , t ) = E [ Y | X = x , T = t ] and π ( x ) = E [ T | X = x ] . The function π is the propensity score.2 This is a necessary condition for unconfoundedness , since unconfoundedness 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 empirical covariance between Y ( 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.3 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 a remedy , we propose a regularization framework that accommodates the orthogonality constraint in the estimation procedure . 3.2 PROPOSED REGULARIZATION FRAMEWORK . In our regularization framework , the orthogonality constraint in ( 5 ) is included in the estimation procedure as follows . LetHf ⊆ [ Rd × { 0 , 1 } → R ] andHπ ⊆ [ Rd → [ 0 , 1 ] ] be function classes for f and π , and ∈ R a model parameter . Then , the objective is to find the solution to the optimization problem inf ( f , π , ) ∈Hf×Hπ×R LFL ( f , π ; D ) + λΩOR ( f , π , ; D ) , ( 6 ) where LFL is the so-called factual loss ( see Section 3.2.1 ) between the estimated models and observed ( factual ) data . The term ΩOR is our new orthogonal regularization that enforces the orthogonality constraint using the model parameter ( see Section 3.2.2 ) . The variable λ ∈ R+ is a 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 LFL Similar to previous work ( e. 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 LFL ( f , π ; D ) = 1 n n∑ i=1 ( f ( Xi , Ti ) − Yi ) 2 + αCrossEntropy ( π ( Xi ) , Ti ) , ( 7 ) where α ∈ R+ is a hyperparameter weighting the terms of the factual loss . We note the following about the estimation of the outcome model f 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 a consequence , the model learns about the observed outcomes , but not about the unobserved outcomes . State-of-the-art methods exclusively rely on the factual loss to estimate the outcomes ( e. g. Shalit et 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 a similar subject in the data with the opposite treatment . However , this is unlikely in presence of selection bias , i. e. , when treatment and control groups differ systemically . This pitfall is addressed by our orthogonal regularization in the following . 2Note that the propensity score is defined as π ( x ) = P ( T = 1 | X = x ) ( Rosenbaum & Rubin , 1983 ) , which is equivalent to E [ T | X = x ] for binary treatments . 3Note that we use the notion orthogonality with respect to the inner product in ( 5 ) . This is different from the notion of Neyman orthogonality in ( Nie & Wager , 2017 ; Chernozhukov et al. , 2018 ) , which requires the Gâteau derivative of a debiased score function to vanish at the true parameter . In contrast , our orthogonality constraint exploits unconfoundedness to ensure that the outcomes are orthogonal ( w.r.t . the inner product in ( 5 ) ) to the treatment assignment .

Summary: 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.