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 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.