On Inductive Biases for Heterogeneous Treatment Effect Estimation
Authors: Alicia Curth
Mihaela van der Schaar
Venue: NeurIPS 2021 Spotlight (2021)
Keywords: heterogeneous treatment effects
potential outcomes
causal inference
counterfactual estimation

Abstract: We investigate how to exploit structural similarities of an individual's potential outcomes (POs) under different treatments to obtain better estimates of conditional average treatment effects in finite samples. Especially when it is unknown whether a treatment has an effect at all, it is natural to hypothesize that the POs are similar -- yet, some existing strategies for treatment effect estimation employ regularization schemes that implicitly encourage heterogeneity even when it does not exist and fail to fully make use of shared structure. In this paper, we investigate and compare three end-to-end learning strategies to overcome this problem -- based on regularization, reparametrization and a flexible multi-task architecture -- each encoding inductive bias favoring shared behavior across POs. To build understanding of their relative strengths, we implement all strategies using neural networks and conduct a wide range of semi-synthetic experiments. We observe that all three approaches can lead to substantial improvements upon numerous baselines and gain insight into performance differences across various experimental settings. 

On Inductive Biases for Heterogeneous Treatment Effect Estimation
Alicia Curth University of Cambridge
Mihaela van der Schaar University of Cambridge
amc2530cam ac.
University of California. Los Angeles The Alan Turing Institute mv4720cam ac uk
Abstract
We investigate how to exploit structural similarities of an individual's potential outcomes (POs) under different treatments to obtain better estimates of conditional average treatment effects finite samples. Especially when is unknown whether a treatment has an effect at all, it is natural to hypothesize that the POs are similar yet, some existing strategies for treatment effect estimation employ regularization schemes that implicitly encourage heterogeneity even when it does not exist and fail to fully make use of shared structure. In this paper, we investigate and compare three end-to-end learning strategies to overcome this problem based regularization, reparametrization and a flexible multi-task architecture - cach encoding inductive bias favoring shared behavior across POs. To build understanding of their relative strengths, we implement all strategies using neural networks and conduct a wide range of semi-synthetic experiments. We observe that all three approaches lead to substantial improvements upon numerous baselines and gain insight into performance differences across various experimental settings.
Introduction
The advent of fields such as personalized medicine has led to rapid growth of the machine learning (ML) literature on heterogeneous treatment effect estimation in recent years To further advance the understanding of how to incorporate insights from other areas of ML into treatment effect estimation, we revisit the well-established problem of estimating the conditional average treatment effect (CATE) of binary treatment within the potential outcomes (PO) framework [9. The PO framework allows conceptualize the problem as estimating the expected difference between an individual's 's expected 'potential' outcome with and without treatment, of which only one
7 g).
observed the factual world. This fundamental problem of causal inference [1O] leads to the consensus that CATE estimation is not 'just another' supervised learning problem 6
Under the standard assumption of ignorability- which precludes hidden confounding we consider two statistical features central to estimating CATE: (i) the presence of confounding and (ii) CATE being contrast between two PO functions, possibly exhibiting simpler structure than each PO separately. Much of the recent ML literature on CATE estimation has focused on the first feature, and treated confounding as a covariate shift problem. At this point, a range of sophisticated solutions exist which reduce the effect of confounding by balancing the covariate space K 4. importance weighting [1 12 end-to-end manner, however, has received little explicit attention so far and is what we focus on here.
or propensity drop-out [15]. How to exploit the second feature in an
We build on the intuition that the tasks in the CATE problem estimating the expected PO with and without treatment- are expected to be strongly related in practical applications (regardless of the
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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presence of confounding). In fact, under the common null hypothesis of no treatment effect, we expect them to be identical. Even when there is non-7ero treat- ment effect, we expect shared structure: In medicine, for example, one distinguishes between biomarkers that are prognostic of outcome regardless of treat- ment status - hence determining what is shared be- tween the POs- - and biomarkers that are predictive of treatment effectiveness determining heterogeneity [6 17]. The induced difference between the POs is often expected to be small relative to the complexity of the PO functions themselves 77]. which, as in Fig. could manifest in terms of CATE being a much simpler function than each PO.
Rasponn
Contro
Treated
Figure 1: Illustrative toy example: the treat- ment effect can be much simpler than each PO function separately
One possible way of exploiting this is to estimate CATE directly using recently proposed model- agnostic multi-stage estimation strategies 7 18 19]. However, these estimators output an estimate of CATE only and do not include an estimate of the untreated PO, which is often of independent interest in practical decision support problems- - e.g. to trade off outcomes at baseline with the effectiveness of high-risk treatments [20]. From this literature, we borrow the insight that explicitly targeting learning strategies towards CATE, instead of only the POs, can lead to better estimators. However, instead of relying
multi-stage strategies, we operate within end-to-end learning framework similar to 4 and recent extensions. Due to a focus on confounding, this line of work did not explicitly investigate how to exploit the similarity of the PO functions beyond them sharing jointly learned feature space. Combining the two lines of work, we investigate how to use end-to-end approaches output better estimates of both the POs and CATE by incorporating the assumption that the POs share much structure (which can result in a possibly simple CATE), as inductive bias.
We focus fundamental question that has received little explicit attention so far: How can we best exploit the structural similarities of the POs for CATE estimation? This question crucial even in randomized experiments, making its solutions orthogonal to any of the sophisticated strategies developed to handle confounding. Therefore, we investigate approaches exploiting the shared structure of the POs which can be applied to modify existing modeling strategies, and thereby aim to provide guidance for improving existing CATE estimators along a new dimension. As such, the goal of this paper is not promote the use of specific approach, method or architecture. Rather, we aim to build systematic understanding and greater intuition of the (dis)advantages of different approaches. This is crucial in the context of CATE estimation, where model selection is notoriously difficult due to the absence of ground truth treatment effects in practice. We focus on gaining insight into the effect of different approaches relying on the same underlying ML method and use neural networks (NNs) due to their flexibility and popularity in related work, yet (variants of) the approaches we consider are applicable many likelihood- or loss-based ML methods.
We investigate three approaches incorporating inductive biases for shared structure into the estimation of the POs: (1) soft approach, which relies regularization to encourage the PO functions to be similar and is hence easy to combine with existing methods, (2) a hard approach, which hardcodes an assumption the PO functions and (3) a flexible approach, in which we build on ideas from multi-tas learning to design new architecture for CATE estimation (FlexTENet), which adaptively learns what share between the PO functions. We implement instantiations of all approaches using NNs and evaluate their performance across wide range of semi-synthetic experiments, varying in the structural similarity of the PO functions. We empirically confirm that all approaches can improve upon baselines, including both end-to-end and multi-stage approaches, and present a number of insights into the relative strengths of each approach. We find that strategies significantly changing the model architecture hard and flexible approaches- usually lead to the largest improvements, with FEcxTENet performing best on average; yet even the simple soft approach often leads to notable performance increases an insight that can easily be incorporated into any existing method with treatment- specific parameters.
Contributions
similarity into the model specification by reparametrization of
2 Problem Definition and Key Challenges
Assume observe sample D = {(Y,, X,, W.)) with (Y,, W.) iid P. Here, Y, E y is continuous binary outcome of interest, X, E X C IRd vector of possible confounders (i.c.
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pre-treatment covariates) and W. E { is binary treatment, assigned according to propensity
score π (z) P(W = 1|X = z). Using the Neyman- Rubin potential outcomes (PO) framework (9) our main interest lies in the individualized treatment effect: the difference between the PO (1) if
treatment administered (W; = 1) and (:(0) if individual i not treated (W; = 0). However, only
one of the POs is observed as Y; = (1) + (1 (()) Therefore, we focus on estimating the conditional average treatment effect (CATE)
T(æ) = (1) - Y(O)XX = x]]
(1)
which is the expected treatment effect for an individual with covariate values X x. We operate under the standard identifying assumptions in the PO framework:
Assumption 1. |Consistency, unconfoundedness and overlap) Consistency: in is assigned treatment ws, we observe the associated potential outcome Y; Y;(w)). Unconfoundedness: there are no unobserved confounders, so that YO)), (() IL W|X. Overlap: treatment assignment is non-deterministic, i.e. 0 < ス (x) < Vx E E.
2.1 The Key Challenges of CATE Estimation
As the ability interpret a treatment effect estimate as causal ultimately relies on a of untestable assumptions, the unique difficulty in making causal claims lies in using domain expertise to argue whether a treatment effect is identifiable 211. Given identifiability, CATE estimation is purely statistical problem thus. if one is willing to rely on assumption [ CATE can be estimated using observed data. A simple strategy for doing so (also known as the T-learner 7 obtains regression estimates P. (2r) of Hw (z) = X = I, W = w). applying standard supervised learning methods using only observed data for which W = w, and finally sets (ax) = pu (Ir) - (z)...) Yet, this seemingly straightforward solution is oblivious to two statistical challenges of CATE estimation:
1. Confounding: If (2) is not constant, then the distribution of covariates in treatment and control groups differs. Such imbalance can be the result of confounders, which are variables that affect both treatment selection and outcomes, and can be problematic when the PO functions are fit on the factual data using empirical risk minimization (ERM) because each problem is solved with respect to the wrong empirical distribution namely X P-IIW = w) instead of X (-). While this problem is not unique to CATE estimation it is equivalent to the covariate shift problem encountered in e.g. domain adaptation [22 it is usually emphasized as one of its main difficulties and motivated the literature on balanced representation learning 3 4
2. CATE is the difference between two functions:
While supervised learning usually targets
single function, the goal of CATE estimation is to estimate the difference between two related functions most accurately- which may require different considerations than estimating each function separately,
To see this, consider the MSE of estimating CATE and let esg (f(x)) = Ex-P [x-pI[f(X)---) (f(x)
denote the MSE for an estimate f(x) of fx)) If we simply were to estimate (zx) as the difference between two separately learned functions, we would have that (up to constants) =x (x)) S (An (z))) (+)(
(z)
(f)o)))))((((())))))) X Rale,, +Rale The convergence rates Rale depend on e.g. smoothness or sparsity of the PO functions; a well-known example would be [23's nonparametric minimax rate. If we had oracle access to both POs and could regress (1)-Y(C Y (0) on X directly, we would have Egg (+()) S Rate,, The assumption that T(x) is often much simpler than each nad (x) separately (7] translates into Rate, < max Ratep... highlighting that targeting CATE directly could lead faster convergence. Similarly, any shared structure across the PO regression tasks could also be exploited to improve upon the simple additive bound above. These observations motivated much of this paper, as they have largely been neglected in the literature on md-to-end learning for CATE. They have, however, been the motivation for some model-agnostic multi-stage learners which we discuss next.
the used estimator and assumptions
3 Related Work
Direct and indirect meta-learners for CATE Recent literature has developed a number of model-
agnostic learning strategies for CATE estimation (also known as 'meta-learners" 7 7 Within this class, we consider an indirect estimator any strategy that uses observed data to obtain regression estimates of the PO functions (x)) and then sets F(x) = fun (a) - (a(z)) This includes [7]'s 's model-agnostic S- and T-leamerr additionally the majority of model-specific ML-based CATE estimators also follow
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an indirect strategy (including all NN-based estimators we discuss below). Conversely, we consider learners direct estimators if they target r(2) directly. As ( (0) is unobserved, multiple existing strategies construct pseudo-outcomes Y,, for which it holds that EE[Y, X = x] (2x) for some nuisance parameters vf(x)) which can be estimated from observational data. Different learners require estimation of different nuisance parameters, which often include propensity score T: and/or PO functions/ fou (xr). All direct estimation strategies that we are aware of - X-learner 7.. R R-learner [8] DR-learner [18), PW-learner and RA-learner [19] proceed in a two-stage manner: they first obtain plug-in nuisance parameter estimates fz()), and then estimate T (xx) by regressing Ya on X.
Under some conditions, these two-stage learners can attain the oracle rate Rate,. We give more detailed overview of all meta-learner strategies in appendix A.
NN-based CATE estimators Complementing model-agnostic strategies, many adaptations of spe- cific MI. methods for CATE estimation have been proposed recently. We build on NN-hased estimation strategies due to their flexibilityy and popularity in related work[] Much work on CATE estimation using NNs has focused on handling confounding, most prominently by learning shared and balanced feature representations for the two PO functions [3 4. Formally, this strategy entails jointly learning a shared feature map, and two PO-specific regression heads (fit using only the data of the corresponding treatment group) each parametrized by a NN. The output heads are then used for indirect estimation of CATE. Without further regularization, this leads to the TARNet specification, while CFRNet introduces a regularization term which encourages the network to learn representations that are balanced, i.c. have indistinguishable distributions across treatment groups [4]. Recent extensions investigated incorporating weighting strategies an additional remedy for confounding into this framework [2 13 14), considered targeting towards average treatment effects (27 28
or allowed for both shared and private feature spaces for the PO functions [9]
Relationship to multi-task learning The architectures proposed in 4 and extensions effectively take a multi-task learning (MTL) approach to PO estimation, relying on hard parameter sharing (29 in the first d, layers of the used network, and no sharing in the top dq layers. While sharing a feature space will lead to some shared behavior between the estimated PO functions, it does not allow to fully exploit underlying similarity c.g. if there are purely prognostic effects it might be better to share some information also between top layers. Below, we therefore investigate strategies to incorporate (additional) inductive bias for shared behavior into PO estimation. These are inspired by work in transfer learning (30), domain adaptation [1 32
and, in particular, MTL [3 B4 35), all allowing
for flexible modeling of shared and task-specific aspects of a problem. Note, however, that MTL and CATE estimation have distinct statistical target parameters and goals MTL is concerned with achieving a good average performance in prediction of outcomes across tasks, while the main target of CATE estimation is estimating the expected difference between outcomes- making it non-obvious a priori whether methods successful in the former will perform well in the latter problem.
4 Inductive Biases for CATE Estimation
In this section, we consider end-to-end approaches for incorporating the prior belief that (o( and u (x) will share much structure (which can imply that 7(x) simpler than pau (x)) as inductive bias. Here, we define inductive bias as the mechanism by which some hypothesis functions are preferred over others during learning [36]. Throughout and for the remainder of this paper, we focus on exploiting the expected similarity between the PO functions, and disregard the impact of confounding on ERM- with the understanding that existing strategies, such as balancing or weighting. are orthogonal solutions that could readily be applied to complement any strategy we discuss.
4.1 Implicit inductive biases in indirect learners
We begin by examining the nature of the inductive biases present in popular indirect learners: T- learner based on vanilla NNs (TNet) and TARNet. Recall that TARNet jointly learns a representation : X + S, parametrized by dense NN with d, layers and n, hidden units, and regression heads hmm S + V. each parametrized by dense NN with d, layers and nas hidden units. A TNet can be
As we discuss in Appendix A, other popular estimators, which we do not consider further as we are interested in the effect of different approaches relying on the same underlying ML method, are based on Generalized Random Forests 24.. Bayesian Additive Regression Trees 25). Gaussian Processes [26] and GANs [S)

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1)(
00 (1)
(2)
Ao(x)
D
10004
:4+
D
وa (x)
(1
(x)
f(x)
(I) Regularization for TNct (left) and TARNet (right)
(2) Reparametrization
(3) FlexTENet
Figure 2: The three approaches under investigation. Dark layers indicate parameters shared between
POs, light layers indicate private parameters. Green arrows indicate regularization encouraging parameters to be similar, red arrows indicate regularization that encourages orthogonalization.
seen special case of TARNet with ((x) sr i.e. no joint learning of feature space Below, will refer to the weights in hww and & as @.. and ②g respectively. During training, both TARNet and TNet use a loss function that takes the general form
DF + X Σ (e R(O)....
(2)
where CF = L hw, (o))))), R is regularizer (usually L2), which regularizes the com- plexity of each few separately, and we dropped the regularization term for the shared representation X R((@)) from the equation it is not of relevance for the following discussions.
While the PO functions can thus share jointly learned feature space, they are not encouraged to be similar beyond this. Instead, they are sepurately regularized to be simple function, leading to their difference- 7 (+) being highly instable and hence to an implicit inductive bias that encourages treatment effect heterogeneity priori (see also the discussions of this phenomenon in g 25 This
neither in line with scientific null hypothesis of no treatment effect (heterogeneity) nor the assumption that (41 (z) and (z) should be close due to the existence of prognostic effects. The instability of f FIr) can also be seen as a consequence of indirect learners not being well targeted
towards CATE. In fact, using this regularization scheme, it is not even possible to control the
complexity of CATE directly.
4.2 Explicit inductive biases for CATE estimation
We investigate three approaches modifying existing end-to-end learners to encourage shared structure in the POs and hence incorporate an inductive bias for simpler r(x). Ordered by ease of implementa- tion, we distinguish between (1) a soft approach relying on regularization, (2) hard approach based on reparametrization and (3) a flexible approach (FlexTENett which explicitly learns what to share between the POs. The architectures for each approach are depicted in Figure Note that although all approaches are targeted at estimating CATE, only (2) directly outputs an estimate of (xx).
Soft approach Regularization The most straightforward strategy to fixing the regularization- induced inductive bias towards heterogeneity discussed above would be simply change how the PO functions are regularized. Instead of regularizing them separately, one could regularize the difference between the weights in the output heads which ultimately determines Txx) coorespponding to an inductive bias towards small treatment effect heterogeneity Analogously to this leads to a loss
LF + X R(Or.) + A2R O%a)
(3)
Choosing A2 > A additionally reinforces the inductive bias towards simple T(x). We further discuss how to set hyperparameters such as X2 (which is shared by all considered approaches) in Appendix B.1. This regularization-bass based approach is attractive because it is extremely easy to implement, is directly applicable to any loss-based method with treatment -roup-specific parameters, has intuitive appeal and does not heavily constrain the functions the hypotheses are able to represent. At the same time, the latter point a downside of this approach, since this might also result in only marginal gains.
In our implementations, give similar capacity to the resulting PO estimators and in analogy with a 'no parameter sharing' strategy, we give each fow a TNet additional access d, layers with n, hidden units. "Note that, by convention, we regularize only the weights of NNs, and not biases (offsets), resulting in penalization only of non-constant treatment effects.
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Hard approach - Reparametrization Instead of regularizing the difference between the PO functions, we could also reparametrize our estimators, strategy that e.g. [37]ss LASSO for CATE estimation and [25 's Bayesian Causal Forest rely on. Instead of estimating (2) and in (x)
separately, we can build on the identity in (a)) = (o(x)) + ((x) and effectively estimate r(a:) directly
as an offset from (a)),, parametrized by NN h, (2)) with weights On,-- This leads to a loss
CF + A1 R(O,)) + )²R(O)..)
(4)
with ==== Ei= My,, hz( + wyhr (x:)) for continuous y, which seems analogous to (3) but parametrizes T(x) explicitly - giving the investigator control over the complexity of 7(x) directly- and hard-codes the assumption that the shared structure between the POs is additive. This approach may be at disadvantage if ur (z) simpler than (z(z), since one would then be better off using the reverse parametrization (o(a) fr (z) (xx). More generally, it is also possible that the relationship between (2) and μi (z) is not additive, e.g. if μ (a) is a simple transformation of (o(z) say a
multiplicative or logarithmic transformation (as is the case in the popular IHDP benchmark [I) such that different parametrization would lead to an easier learning problem.
Unfortunately, the parametrization leading to the easiest learning problem is usually not known in practice. Alternatively, one could thus rely on a strategy that can automatically and flexibly learn which information to (hard-)share between the PO functions. Inspired by architectures in MTL and domain adaptation that explicitly anticipate shared and private structure We therefore propose new architecture for treatment effect estimation, FlexTENe (Flexible Treatment Effect Network), as a final strategy. As depicted in Fig. 2 it has private (nhn (s)-spcccfi) subspaces- which ultimately determine r(x:) and shared subspace in each layer (including the output layer), allowing the model to automatically learn which information to share at each layer of the network. In principle, any MTL method could be adapted for PO estimation, yet, as we discuss in appendix B.2, we propose the Flex specification because it generalizes many existing strategies. Given its flexibility, we expect that such a general architecture should perform well on average an appealing feature given that model-selection is nontrivial in CATE problems.
Flexible approach- FlexTENet
31 32 35
We implement Flex TENe using a specification matching TARNet to allow for direct comparisons, and consider d layers, within which each private and shared subspace has ng.p and nA.s hidden units, respectively, where we let ng.p = n,k 2x, & E fr. h), for simplicity. For layer 1 >> 1, let m 1-1 and m denote the output dimensions of shared and private subspace of the previous layer, let 0g E
R xxm denote the weights in the shared subspace, while 0/___ R (m)) (mm--=mm +m/-1) "xmm,, denotes the
weights in each private subspace To discourage redundancy and encourage identification of private structure, we apply regularizers orthogonalize the shared and private subspaces. Like [32 35
we rely on 38 's orthogonal regularizer (O., O po: 0p,) = E (0 _0=1100 10, Pa, Pam
where ||: denotes the squared Frobenius norm. This leads to the following loss function
LF LF+
R(, Ppo Pp,)
(5)
where setting d2 > X adds inductive bias encouraging the shared space to be used first.
4.2.1 Underlying assumptions and theory
Which assumptions do these approaches encode? Motivated by real-world applications in which prognostic effects are often assumed stronger than predictive ones 7 25), our central assumption is 'there is much shared structure between no( and (1 (x)'. This assumption is purposefully abstract, allowing it to manifest different specific assumptions depending on the used ML method and and regularizer (R); c.g. in regression with LO- L0-penalty, r(x) would be assumed linear and sparse, while in our case, using NNs with L2-penalty, the uww (x) are implicitly assumed to be close in some function class, with smooth differences. We thus consider inductive biases for T(x) relative to the inductive bias in the original method used estimate the un (æ); we effectively investigate how to best target
"The difference in input dimensions arises as we only allow communication from shared subspaces to private subspaces and not the reverse; refer Appendix B.3 for pseudocode of FlexTENet forward pass.
'While both hard and soft approach are directly applicable to any loss-based method, our flexible approach is specific to NNs. When using different ML method, similar strategies could be constructed by adapting (i) existing MTL approaches or (ii) mgdel-agnostic approach similar to (31), creating PO-specific feature spaces.

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these biases to control ((x) explicitly. Further, there is conceptual difference between assumptions in soft & flexible and hard approach: only the latter assumes shared structure to be additive.
Why is shared structure inductive bias reasonable in this context? Shared structure is a reason- able assumption in many practical applications as usually one would expect at least some similarities between treated and untreated individuals: intuitively speaking, receiving a drug will most likely not change all biological processes related disease progression in a patient and attending a job training program is unlikely to neutralize all characteristies determining an individual's salary. In medicine, for example, this has led to the well-known distinction between prognostic and predictive (effect- (effect-modifyingg modifying) biomarkers [16] 17] in our context, the strength of such prognostic information would determine the degree of shared structure. Further, assuming shared structure is compatible with explicit assumptions made in recent theoretical work on CATE meta-leamnrss where CATE is assumed a simpler function than each of the POs
19 which implies shared structure (i.e. a shared baseline function) between the POs. Additionally, a related (but much stronger) assumption is made in papers considering the popular
[7
'partially linear regression model' analyzed in e.g. [39 40]; here all nonparametric ('complex') ) structure is shared between POs, while the treatment effect is assumed parametric (often constant).
Are there any theoretical expectations for CATE estimation performance? All theoretical results focusing on estimating the difference between two functions that we are aware of originate in the literature on (i) CATE estimation 77 19] or (ii) transfer learning via offset estimation [41] 42). They provide risk bounds for CATE estimation of the form Rate, + Rate Remoinders indicating that, if some remainder terms decay sufficiently fast, oracle rates for estimation of CATE can be attained. These results rely on two-stage estimation and stability conditions on the estimators, and as such are not directly applicable to our setting. Nonetheless, we hypothesize that our end-to-end learning strategies can match the performance of such estimators particularly in small sample regimes due to sharing of information between tasks (estimation of POs). It would therefore be an interesting next step to adapt theoretical results from MTL 43 44 to analyze end-to-end strategies for CATE estimation, yet we consider this non-trivial as (i) MTL is concerned with the average performance over tasks, and not the performance on estimating task differences, and (ii) the number of tasks in our context is small (T 2). We therefore defer theoretical analysis of our approaches to future research and focus on experimentally evaluating their performance below.
Experiments
5.1 Experimental setup
Simulation s As ground truth treatment effects are unobserved in practice, we use semi- synthetic setups based on real covariates and simulated (1)) To systematically gain insight into the relative strengths of different strategies, we consider a number of setups (A-D) comprising a total of 101 simulation settings. We provide brief descriptions below; refer to Appendix C for more detail. For setups A&B, we use the ACIC2016 covariates (n 4802, 55) of [45) but design our own response surfaces, allowing us to introduce our own 'experimental knobs' to enable structured evaluation of different approaches. We simulate response surfaces similar to [17 as
Y; Ez= 8,(1 W,, E
8y. - W X, + W; -X,,,,, (6)
where 3,. 75: 75: W. Bj,,, E 13 are sampled as Bernoulli random variables, and the probability of non-zero 8 331 f,1 is fixed throughout. In Setup A, we set all زا = W 0 and control the complexity of T(x) = Σ X, by varying the expected proportion p of non-zero yy; thus (o( is sparser than (z). Conversely, in setup we set all y, and instead induce treatment effect heterogeneity by sampling non-zero wi, wij with probability p. These cancel some prognostic effects for the treated, such that the complexity of 7 (xx) increases as the complexity of Pu (z) decreases; here μ4 (x)) is thus simpler than n4 (z). In both settings, knob p determines the complexity of r(x) through the number of predictive features; as p increases, μl (x) becomes less similar to fixed (o(x)) We randomly assign no E {200, 500, 2000} to control and {ni E {200, 500, 2000} ni < noo) to treatment, creating different levels of sample imbalance (without confounding). and use 500 units for testing.
For setups C&D, we use the IHDP benchmark (n = 747, d = 25), into which [ introduced confounding, imbalance (18% treated) and incomplete overlap. We use I''s original simulation as
'Code replicate all experiments is available httpp:///gtttt.....o////cccccaat
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nt 200
A: 0g 500
A: 2000
B: By 200
B: 500
B: 2000
1.c
Tet iepacins.
Baseline:
RMSE
TNet
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Poxtt
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A: D2 = 200
A: 01 = 500
A: 02 2000
B: 02 = 200
B: 02 500
B: n2 = 2000
Baseline: TARNet
RMSE
Fexts
RecTt
Figure 3: RMSE of CATE estimation by p for multiple n2 for setup A and B, using TNet (top) and TARNet (bottom) baseline, no 2000. In each graph, as p increases r(x: becomes more complex because fai (x) becomes less (more) sparse. Avg. across 10 runs, one standard error shaded.
setup C, consisting of simulated expected POs (x) = cxp( + (.5) and uf = - X,8, where is a randomly sampled sparse vector. Both response surfaces share the dependency on X albeit in different functional form; here the assumptions of additive similarity and r(x) being simpler than pu do not hold and there is very strong heterogeneity, We add setup D, in which these assumptions D
do hold, by slightly altering the response surfaces to 14g (z) (x) = no (x) and (x) =n0 (x) + uP (ar)
(i.e. an additive treatment effect). Here, we use the 90/10 train-test splits of [4]'s IHDP-100 benchmark to evaluate in- and out-of-sampy sampple performance. In Appendix E, we report additional results (102 additional settings) using the original ACIC2016 response surfaces and the real-world Twins dataset.
Models We compare the three approaches to T'Net and TARNet as baseline indirect estimators and [8's 's DR-learner, a direct two-stage estimator, which we implement using both TNet and TARNet in the first stage. We also considered RA- [9], R- [8] and X-learner [7]. but found them to perform worse than the DR- -R-learner on average (see appendix D). We perform further baseline comparisons, using DragonNet (27] and SNet [9 in appendix D. To ensure fair comparison across all models and all experiments, we fix hyperparameters across all models within each experimental study and ensure that each estimator/output function has access to the same number of hidden units across all model?" and effectively use each model "off-the-shelf". We discuss implementation further in Appendix C.
5.2 Results and Discussion
Below, we discuss our findings. We begin with setups A&B and evaluate performance on CATE estimation with imbalanced data (where no = 2000 unless stated otherwise). We perform ablations to gain insight into the relative effect of the different components of each strategy, consider the effect of smaller To and consider performance on PO estimation. Finally, we consider setups C&D and ACIC2016 & Twins (Appendix E) to consolidate our findings on well-established benchmark datasets. We highlight some aspects of the results here, and present additional findings in appendix (including varying no. further results on estimating the POs, analysing the weights of a trained TexT and additional baseline comparisons).
Comparison to indirect learners: All three approaches improve upon baseline indirect learners in CATE estimation with large no (setups A&B); regularization brings smallesi gains, reparamelrizatinn works well in selup A but not B, and FlexTENel performs best on average. Considering Fig. 3 we make a number of interesting observations. First, all three strategies consistently improve upon the respective baseline in almost all settings and largest improvements are made relative to TNet with n1 small. Additionally, the performance of the three approaches is closest when is small (r(x) sparse). Second, gains for regularization are most apparent with TNet for n1 < 500. Adding regularization to TARNet brings relatively smaller improvements because its baseline error is smaller. Third, for n1 S 500, FlexTENet and reparametrization perform equivalently in setups A but not in B, which is to be expected because the latter uses an impractical parametrization for B, while TexTeNet can freely adapt to the underlying problem structure. Fourth, comparing FlexTENet and TARNet, we observe that when we use only the Flex architecture without additional regularisation (A, = 0 and d1 = >2), their performance similar, while incorporating additional inductive bias, encouraging identification of predictive and prognostic effects, Ieads to substantial improvement of FlexTENet over TARNet across all setups. Finally, FlexTENet performs best on average and seems
Due different degrees of parameter sharing, TNet has more parameters than TARNet and FlexTENet.
8

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to have particular advantage for larger nı and p. where the latter indicates that FlexTENet is not only well-equipped to handle prognostic effects, but also treatment effect heterogeneity.
Comparison to multi-stage learners: Regularization rarely outperforms the DR-learner while FlexTEENet and reparametrization generally do. Unlike the other two approaches, regularization rarely outperforms the DR-learner, yet it often matches its performance (particularly in setup B), Both FlexTENet and reparametrization outperform the DR-learner, yet performance of reparametrization and DR-Iearner seem to converge as 7 increases, particularly in setup B. In Appendix D, we additionally investigate using the soft- and flexible approach as the first stage of DR- and X-learner.
Ablation study: All components contribute to better performance. We study ablations for setup A, R1 200 (Fig. 4 to gain insight into the relative effect of different components of each approach. We find that setting A < X2, i.e. regularizing CATE more heavily than the POs, indeed led to additional (albeit smaller) improvement. For FlexTENes orthogonal regularization alone adds more than setting λ < X2, yet together they lead to the greatest improvement.
Regullarization
Reparametrization
FlexTENet
RMSE
0.5
A A
Figure 4: RMSE of CATE estimation by p for ablations (n1 200, setup A). Avg. across 10 runs, one standard error shaded.
Effect of no: Improvements over baseline are less substantial for smaller ny- We investigate the effect of having smaller set of control units no in Fig. 5 While all strategies continue to outperform TNet, we observe that their gains are much smaller than for
no 500
Thlt
Thesanng
RMSE
OR (-
7o 2000 (Fig 3). Additionally, we observe that FlexTENet appears to perform somewhat less well for small p. indicating that it may need more training data than other approaches due its flexibility.
p
Figure 5: RMSE of CATE estimation by for rig = 200, 500 (n) 200, setup A). Avg. across 10 runs, one standard error shaded.
PO estimation: Most approaches also lead to im- provements in PO estimation. We consider whether the three approaches improved not only estimation of CATE but also of the POs separately in Fig. G For estimation of (x) at n1 = 500, we observe that almost all strategies improve upon TNet, but make two interesting observations: First, the per- formance gap between TNet and its regularized ver- sion is qualitatively smaller than the performance gap in estimating CATE (particularly in setup B), indicating that regularization could have larger impact on improving the CATE estimate than the PO estimate. Second, the reparametrization approach indeed unable to handle setup B where μn (z) becomes progressively simpler. Interestingly, even though it performs worse at estimating na (z) than
Setup A
Setup
RMSE
Peporers
ResTt
36-
04
0:00
0.5
0.50
0.75
8.00
Figure 6: RMSE of μu (x) estimation by P for
T1 = 500 for setup A and B. Averaged across 10 runs, one standard error shaded.
TNet for large p. Fig. still showed better performance at estimating rxx) in this case indicating that the reparametrization approach is better targeted towards CATE estimation than PO estimation.
Further benchmark results: Perfor-
Table 1: Normalized in- & out-of-sample RMSE of CATE, setup C & D. Avg. across 100 runs, standard emror in parentheses.
mance on IHDP, ACIC2016 and Twins se- tups reinforces findings and confirms ex- pectations. Even though the IHDP dataset smaller, subject to confounding and has limited overlap, performance across setups C Cand D [T largely confirms the find- ings previously discussed. The only major difference induced by setup in which ((x) is not simpler than each (xx sepa- rately and an additive parametrization does not lead to the easiest learning problem; as
i,
C, out
D, in
D, out
TNet
0.32 (.01) 0.34 (.01) 0.29 (.01) 0.29 (.01)
TNet + reg
0.30 (:01) 0.32 (.01) 0.26 (.01) 0.26 (.01)
DR (+TNet)
0.35 (.01) 0.37 (.01) 0.22 (.01) 0.22 (.01)
TARNet
0.29 (00)) 0.31 (.01) 0.22 (.01) 0.23 (.01)
TARNet reg
0.28 (.01) 0.31 (.01) 0.20 (.01) 0.01)
DR (+TARNet)
0.33 (.01) 0.34 (.01) 0.20 (.01) 0.20 (.01)
Reparam.
0.39 (.01) 0.40 (.01) 0.20 (.01) 0.20 (:01)
Flex TeNet
0.27 (.01) 0.29 (.01) 0.22 (.01) 0.23 (.01)
'Due to the exponential in (z))). RMSE varies by orders of magnitude across runs and seems unsuitable to assess relative performance. We report RMSE normalized by standard deviation of observed outcomes (see appendix for details and unnormalized results). results)
9

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expected both reparametrization and the DR-learner perform poorly in this scenario (yet methods using shared representations are nonetheless able to exploit the shared dependence of the POs on B). TARNet in combination with the soft approach and Flex show the best average performance across the two setups, where the weaker performance of FleXTENet on setup D is in line with previous findings for small no and simple T (æ)) The results on the ACIC2016 simulations and the real-world Twins dataset presented appendix E further confirm the relative performance of the different methods we observed throughout.
6 Conclusion
We found that altering the inductive biases in end-to-end CATE estimators can lead to performance increases and match, improve upon, the finite sample performance of multi-stage learners targeting CATE directly. We observed that all our approaches were particularly useful when only one treatment group had abundant samples, a situation arising naturally in practice when a new treatment is introduced. We found that strategies which change the model architecture - reparametrization and FlexTENer led to the largest improvements: The former seems be the best choice in smaller samples when treatment effects are additive and fo is simpler than μ1 (z)) while FlexTENer showed the best average performance due its flexibility and handled high heterogeneity well, but required more training data when r(z) is very simple. Additionally, we also found that the simple strategy of changing the regularizer leads to notable performance increases, an insight that could directly be incorporated into many existing methods with treatment-specifie parameters.
Here, limited our attention approaches. Therefore, an interesting next step would be to consolidate our findings by applying our approaches using different underlying methods. Finally, all approaches rely the ability to manipulate loss (or likelihood) functions, it would be interesting to consider how to equip methods that do not have casily manipulable loss functions (e.g. random forests) with similar inductive biases.
single ML method (NNs) to isolate the properties of different
Acknowledgments and Disclosure of Funding
We thank anonymous reviewers as well as members of the vanderschaar-lab for many insightful comment and suggestions. AC gratefully acknowledges funding from AstraZeneca.
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12


Reviews:
[{"review_id": "mCD8Tigpdyo", "reviewer": "Reviewer_yc6v", "summary": "This paper considers the problem of estimating the conditional average treatment effect (CATE) of a binary treatment with some potential outcomes of interest (either continuous or binary). Simply put, in, e.g., a medical scenario, CATE would measure the expected delta in administering some drug vs. not for a particular patient X. The main challenge of this problem is that Y(0) and Y(1), the outcomes given the value of the binary treatment, are never observed at the same time, so estimating CATE is not a standard supervised learning problem, where Y(1) - Y(0) can be observed for the same input X. One can estimate both Y(1) and Y(0) separately, but the errors add. \n\nThe key insight of this work is that the difference, $\\tau(x) = \\mathbb{E}[Y(1) - Y(0) \\mid X = x]$, is often of a much simpler form than either $\\mathbb{E}[Y(1) \\mid X = x]$ or $\\mathbb{E}[Y(0) \\mid X = x]$, due to inherent similarities in Y(1) and Y(0). To account for this, this paper investigates and compares several end-to-end learning strategies that encode inductive bias for shared/similar structure---namely, (1) regularization, (2) reparameterization, and (3) shared multi-task architectures.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "=== Strengths ===\n\n- The paper is very well written and provides extensive evaluation and discussion/comparison of past work. \n- In general, the problem that the paper focuses on is interesting and important.\n- It is nice to see *some* real world evaluation (Twins dataset), with the understanding that this is imperfect and hard to come by.\n- Empirically, the strategies proposed by the authors (in particular, FlexTENet) seem to show consistent improvements over competing baselines.\n- The proposed strategies are simple and easy to implement within most frameworks, and should be easy to adopt.\n\n=== Weaknesses ===\n\n- From a technical standpoint, the contributions of this paper are fairly limited. The inductive biases for shared/similar structure are, for the most part, well-known (at least in the multi-task learning community). This is not inherently bad---there remains value in the analysis. The wording of lines 61-70 is a bit confusing though, as shared structure inductive bias has already been a quite common \"dimension\" for improving MTL estimators (not just the architectures used). The use in CATE estimation specifically may be a bit less prevalent.\n\n- While it may be true that this shared structure inductive bias may be useful, it seems hard to systematically tune the regularization strengths, $\\lambda$. This is especially true in the real world, where development data will be mostly unavailable, and the true underlying setting (shared structure or not) will be unknown. In the worst case, it seems that strategies (1) and (3) with too high of a regularization could severely underestimate CATE (i.e., the models collapse together)? How should these weights be chosen?\n\n=== Justification for score ===\n\nOverall, the paper is well-written and provides an interesting survey of several reasonable inductive-bias inducing techniques for CATE estimation. The technical novelty of the paper, however, is fairly limited, and its experimental findings are not that surprising given what we know from MTL literature, which makes it borderline. That said, its presentation and analysis could still make a potential contribution to the treatment effect estimation community. \n\n=== Minor comments ===\n- It would be helpful to introduce the CATE estimation problem more formally (as done in Section 2) earlier in the introduction. This would help ground readers less familiar with PO frameworks.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I do not foresee any major limitations and/or societal impacts. How to choose regularization strengths would be the main implementation question/limitation.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "7"}, "extra_text": {}}, {"review_id": "V2Qd16n2n3l", "reviewer": "Reviewer_EVDi", "summary": "In this paper, the authors proposed methods to encourage structural similarities in the potential outcomes (POs) under different treatments when learning the conditional average treatment effects (CATEs). Specifically, three sets of different structure/model modifications were proposed on top of the existing TNet/TarNet model by utilizing regularizations or shared hidden units under different treatments. Proposed methods were evaluated on various simulated data. ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The proposed methods are structure modifications to existing methods for learning CATEs. The modifications are fairly straightforward, utilizing regularization or shared hidden units in neural networks. These techniques are commonly used in neural network models, so the contribution mainly lies in the application area for learning CATEs under the inductive bias assumption.\n\nThe proposed methods are valid and appropriate to address the research question. This paper is in general well organized. Extensive simulation studies were conducted. If the true POs share a common structure, the proposed methods are useful in improving the estimation for CATEs by borrowing strengths and encouraging a simpler model structure. My main concern is the soundness of the assumption and how the proposed models perform under violation of the assumptions. Also, how to test the assumption and whether people should proceed with this line of methods when there is no prior knowledge of POs need to be examined.\n\nAdditional comments and questions:\n\nIn the simulations, all settings assumed some common structure between the two POs. As a sensitivity analysis, I wonder how would the models perform if the two POs do not share any structure or parameter? In this circumstance, would the penalization parameter that encourages similarity to be chosen close to 0 by cross-validation? \n\nCan the authors talk about how would people check the assumptions that the POs are similar, and whether or not to encourage a similar structure between POs in learning CATEs when there is no prior knowledge?\n\nAlso, I wonder do the authors compare the accuracy in inferring the optimal treatments under different methods? MSEs of CATEs were reported. But sometimes a more meaningful question is whether the treatment recommendation direction is correct, and such result can be informative to viewers. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors should discuss more on the limitation and societal impact when the inductive bias for encouraging similarity between POs is not correct. How much would this impact the estimation and treatment recommendation should be evaluated. Potential remedies and proper guidance on whether this line of methods should be utilized should be discussed. ", "review_text": "", "extra_scores": {"ethical_concerns": "no", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "6RhVP_R8LBV", "reviewer": "Reviewer_9oJ5", "summary": "This paper contributes an investigation of the inductive biases of various neural network architectures in the context of conditional average treatment effect estimation. Specifically, the authors put forth the hypothesis that $\\mu_0(x) = E[Y|X=x, W=0]$ and  $\\mu_1(x) = E[Y|X=x, W=1]$ have similar structure and thus $\\tau(x) = \\mu_1(x) - \\mu_0(x)$ *can* be simpler than either $\\mu_w(x)$. \n\nThe authors investigate three approaches: (1) a two path (one per treatment) neural network, where the parameters $\\theta_w$ of both paths are regularized to be point-wise close, (2) reparametrization whereby they explicitly learn a $\\mu_0$ and a $\\tau$ and set $\\mu_1$ as $\\mu_0 + \\tau$, and (3) the authors' proposed approach (FlexTENet) where there are \"private\" functions $\\mu_w$ for each treatment $w$ and a \"shared\" subspace in each layer. ", "questions": "", "limitations": "", "rating": 8, "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": "Thanks to the authors on the hard work on this paper. It is obvious that they have put in time and thought to address the concerns of previous reviewers (ICML 2021).  The paper is ready for publication in my opinion, especially considering the wealth of additional experimental evidence in the appendix. CATE is hard, and having promising neural network architectures is important.\n\nThe work is largely clear and well-written. A few questions:\n\n(1) The section that actually describes FlexTENet starts on line 229. I found it lacked the details I was expecting. Some pseudocode or math in the appendix would be helpful. It's not clear exactly what the arrows mean in Figure 2.3. When 2 orange arrows go into the same block, does that mean the inputs are concatenated? That is, the private $\\theta$ are always being mixed with the shared $\\theta$?\n\n(2) A lot of this work is presented as being built on the idea that $\\tau$ may be easier to estimate than $\\mu_w$. However, I am unclear as to when this would actually be the case in practice. For example, if the true $\\mu_w$ was a decision tree, then $\\mu_1 - \\mu_0$ would be a difference of two decision trees, which is more complex than either one. From what I can tell, your toy example in Figure 1 is a lot of the justification of this idea. Can you justify it further? Perhaps using data from the twin study? Also, just because the experimental results are promising, does that mean you've found evidence of the hypothesis that $\\tau$ is simpler than $\\mu_w$?\n\n(3) In lines 401 and 402 you wrote \"The former seems to be the best choice ... when ... $\\mu_0$ is simpler than $\\mu_1$.\" Is it possible to fit 2 variants of the reparametrization model, where $h_0$ estimates $\\mu_0$ in one case and $h_1$ estimates $\\mu_1$ in the other case, and then choose the better one as part of the hyperparameter optimization? Would this approach significantly improve the reparametrization model model? After all, one or the other will be more complex in practice, although they could be of very similar complexity. Does the reparametrization model fail when $\\mu_0$ and $\\mu_1$ are of comparable complexity?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents a novel framing and approach to the problem of conditional average treatment effect estimation, and validates it on synthetic data and a small sample of natural data. Reviewers generally found it to be a technically-simple but interesting contribution, and raised only minor concerns.
Decision: Accept (Spotlight)