Estimating individual treatment effects under unobserved confounding using binary instruments
Authors: Dennis Frauen
Stefan Feuerriegel
Venue: NeurIPS 2022 Submitted (2022)
Keywords: Causal machine learning
treatment effect estimation
instrumental variables

Abstract: Estimating individual treatment effects (ITEs) from observational data is relevant in many fields such as personalized medicine. However, in practice, the treatment assignment is usually confounded by unobserved variables and thus introduces bias. A remedy to remove the bias is the use of instrumental variables (IVs). Such settings are widespread in medicine (e.g., trials where compliance is used as binary IV). In this paper, we propose a novel, multiple robust machine learning framework, called MRIV, for estimating ITEs using binary IVs and thus yield an unbiased ITE estimator. Different from previous work for binary IVs, our framework estimates the ITE directly via a pseudo outcome regression. (1) We provide a theoretical analysis where we show that our framework yields multiple robust convergence rates: our ITE estimator achieves fast convergence even if several nuisance estimators converge slowly. (2) We further show that our framework asymptotically outperforms state-of-the-art plug-in IV methods for ITE estimation. (3) We build upon our theoretical results and propose a tailored neural network architecture called MRIV-Net for ITE estimation using binary IVs. Across various computational experiments, we demonstrate empirically that our \modelname achieves state-of-the-art performance. To the best of our knowledge, our MRIV is the first multiple robust machine learning framework tailored to estimating ITEs in the binary IV setting. 

Estimating individual treatment effects under unobserved confounding using binary instruments
Anonymous Author(s)
Affiliation
Address
email
Abstract
Estimating individual treatment effects (ITEs) from observational data relevant in many fields such as personalized medicine. However, in practice, the treatment assignment is usually confounded by unobserved variables and thus introduces bias. A remedy to remove the bias is the use of instrumental variables (IVs). Such settings are widespread medicine (e.g.. trials where compliance is used as binary IV). In this paper, we propose novel, multiply robust machine learning framework, called MRIV, for estimating ITEs using binary IVs and thus yield an unbiased ITE estimator. Different from previous work for binary IVs, our framework estimates the ITE directly via a pseudo outcome regression. (1) We provide a theoretical analysis where show that framework yields multiply robust convergence rates: our ITE estimator achieves fast convergence even several nuisance esti- mators converge slowly. (2) We further show that our framework asymptotically outperforms state-of-the-art plug-in IV methods for ITE estimation. (3) We build upon theoretical results and propose tailored deep neural network architecture called MRIV-Net for ITE estimation using binary IVs. Across various compu- tational experiments, we demonstrate empirically that our MRIV-Net achieves state-of-the-ar performance. To the best of our knowledge, our MRIV is the first multiply robust machine learning framework tailored estimating ITEs in the binary IV setting.
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Introduction
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21 Individual treatment effects (ITEs) are relevant across many disciplines such as marketing [4] and personalized medicine (5T]. Knowledge about ITES provides insights into the heterogeneity of treatment effects, and thus help in potentially better treatment decisions.
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Many recent works that machine learning estimate ITEs based the assumption of 25 unconfoundedness [ confounders
36 42), practice, however, this assumption is often violated because reported in the data. Typical examples race, 27 income, gender, or the socioeconomic status patients, which are not stored in medical files. the confounding is sufficiently strong, standard methods for estimating ITEs suffer from confounding
it is common that
29 bias 31 which may lead to inferior treatment decisions.
To handle unobserved confounders, instrumental variables (IVs) can be leveraged to relax the assumption of unconfoundedness and still compute reliable ITE estimates. IV methods were originally developed in economics [48). but, only recently, there is growing interest in combining IV methods 33 with machine learning (see Sec. (3). Importantly, IV methods outperform classical ITE estimators if sufficient amount of confounding is not observed [I7]. We thus aim at estimating ITEs from observational data under unobserved confounding using IVs.
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Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

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36 In this paper. we consider the setting where a single binary instrument is available. This setting is widespread in personalized medicine (and other applications such as marketing or public policy) 38 [9]. In fact, the setting is encountered in essentially all observational or randomized studies with observed non-compliance [19.. As an example, consider a randomized controlled trial (RCT), where treatments are randomly assigned to patients and their outcomes are observed. Due to some potentially unobserved confounders (e.g., income, education), some patients refuse to take the treatment initially assigned to them. Here, the treatment assignment serves as a binary IV. Moreover, such RCTs have been widely used by public decision-make maakes,,,,,,s e.g., to analyze the effect of health insurance on health outcome (see the so-called Oregon health insurance experiment) [6] the effect of military service on lifetime earnings [2]
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We propose novel machine learning framework (called MRIV) for estimating ITEs using binary IVs. Our framework takes an initial ITE estimator and nuisance parameter estimators as input to perform a pseudo-outcome regression. Importantly, our framework uses a multiply robust parametrization of the efficient influence function as pseudo outcome.
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We provide a theoretical analysis, where we use tools from 22 to show that our framework achieves multiply robust convergence rate, i.e., our MRIV converges with a fast rate even if several nuisance 52 parameters converge slowly. We further show that, compared to existing plug-in IV methods, the performance of our framework is asymptotically superior. Finally, we leverage our framework and, on top of it, build tailored deep neural network called MRIV-Net.
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Differences to existing literature: Our framework is multiply robus i.e., it is consistent in the union of three different model specifications. This is different from existing methods for ITE estimation using IVs, which are only doubly robust (e.g., Syrgkanis et al. 40 plug-in estimators
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Contributions (1) We propose a novel, multiply robust machine learn- ing framework (called MRIV) to learn the ITE using the binary IV setting. To the best of our knowledge, ours is the first that is multiply robust, i.e., consistent in the union of three model specifications. For comparison,
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existing works for ITE estimation are only double robust 45
40 (2) We
U
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prove that MRIV achieves a multiply robust convergence rate. This is different to methods for IV settings which are only doubly robust, such
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as 40). We further show that our MRIV is asymptotically superior to ex- isting plug-in estimators. (3) We propose tailored deep neural network, called MRIV-Net, which builds upon our framework to estimate ITEs, We demonstrate that MRIV-Net achieves state-of-the-art performance.
Figure I: Underlying causal graph. The instru- ment Z has a direct influ- ence on the treatment A, but does not have a direct effect on the outcome Y. Note that we allow for un- observed confounders for both Z-A (dashed line) and A-Y (given by U). Our setting is general in that U can be correlated or uncorrelated with the observed confounders X.
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2 Problem setup
71 Data generating process: We observe data = (Zi, Zi, as y con- sisting of E N observations of the tuple (X, 2, A, Y). Here, X E X 73 are observed confounders, Z E {0, 1} is a binary instrument, A E {0, 1}
74 is a binary treatment, and Y IR is an outcome of interest. Furthermore,
75 we assume the existence of unobserved confounders U E Ut which affect both the treatment A and the outcome Y. The causal graph is shown in
77 Fig. T
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Applicability: Our proposed framework is widely applicable in practice, namely to all settings with the above data generating process. This includes both (I) observational data and (2) RCTs with non-compliance. For (1), observational data is commonly encountered in, e.g., personalized medicine. 81 Here, modeling treatments as binary variables is consistent with previous literature on causal effect estimation and standard in medical practice (33) For (2), our setting is further encountered in RCTs 83 when the instrument Z is a randomized treatment assignment but individuals do not comply with their treatment assignment. Such RCTs have been extensively used by public bocision-mmaert, e.g.,
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For a detailed introduction to multiple robustness and its importance in treatment effect estimation, we refer 44). Section 4.5. Codes are in the supplementary materials. https://anonymous.oo wwwp:yymuuuouooo.ccoeooooo wwapeenssiincce//MiiiWeeee--..e (Upon acceptance, we replace the link and point a public GitHub repository).
Codes are also available at

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85 to analyze the effect of health insurance on health outcome [16] or the effect of military service on lifetime earnings (
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We build upon the potential outcomes framework [34 for modeling causal effects. Let Y =) denote the potential outcome that would have been observed under À = and 7 = Following previous literature on IV estimation [45]. we impose the following standard IV assumptions on the
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data generating process.
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We assume: (1) Exclusion: za, = (()) for all a, E {0. 1), i.e., the instrument has no direct effect on the patient outcome; (2) Independence: Z T1 U X; (3) Relevance: Z J. A X, (iv) The model includes all A-Y confounder: (a)) 1
Assumption 1 (Standard IV assumptions 45
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(A, 2) (X, for all a E 10,
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Assumption is standard for IV methods and fulfilled in practical settings where IV methods
96 are applied 2
4 19 Note that Assumption [ does not prohibit the existence of unobserved Z- A confounders. On the contrary. it merely prohibits the existence of unobserved counfounders that affect all Z, A. and Y simultaneously, as it is standard in IV settings [47]. A practical and widespread example where Assumption I iss satisfied are randomized controlled trials (RCTs) with non-compliance 119. Here, the treatment assignment Z is randomized, but the actual relationship between treatment and outcome Y may still be confounded. For instance, the Oregon health insurance experiment (16), people were given access to health insurance (Z) by a lottery with aim
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to study the effect health insurance (A) on health outcome (Y) (16). Here, non-compliance information is observed because the lottery winners needed to sign up for health insurance.
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Objective: In this paper, we are interested in estimating the individual treatment effect (TTE)
(xx) (1) Y(0) = x].
(1)
If there is no unobserved confounding (U- 0), the ITE is identifiable from observational data under mild positivity assumptions (36). However, in practice, it is often unlikely that all confounders are observable. To account for this, leverage the instrument Z to identify the ITE. We state the following assumption for identifiability.
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Assumption (Identifiaiility of the ITE 45 ). At least one of the following two statements holds
true: (1) E[A Z = 1, U] - E
Z O, X. UJ = E[A| Z L, E[A Z 0, or
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(2) ([) - Y (0) X, X =
Y()
Example: Assumption 1 holds is when the function f(a, XU) (a) X,U is additive with respect to a and U, e.g. 7 (a. X,U) = g(a, X) + hU) for measurable functions h and
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Under Assumptions | and 2 the ITE is identifiable 45 It can be written as
(x) -
(x)
T(x) (x) - 16 (x) dΛ(x)*
where u. (x) = E|y = æ] and (z) EIA - = z]. Even if Assumption
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does not hold, the quantity on the right-hand side of Eq. D still allows for interpretation. If no unobserved Z-A confounders exist, it can be interpreted as conditional version of the local average treatment effect (LATE)
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[19 3 under monotonicity assumption. Furthermore. under a no-current- assumption, it can be interpreted as conditional treatment effect on the treated (ETT) 45). This has an important implication for our results: If Assumption 2 does not hold in practice our estimates still provide conditional LATE or ETT estimates under the respective assumptions because they are based on Eq. ( If Assumption 2 does hold, all three i.e., ITE, conditional LATE, and ETT coincide
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Related work
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ITE methods without unconfoundedness: Various machine learning methods for estimating ITES
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without unobserved confounding have been proposed in recent literature [ 15 25
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'The conditional LATE measures the ITE for individuals which are part of the complier subpopulation. i.c., the subpopulation for whom A(Z = > A(Z = 0). The conditional ETT measures the ITE for treated
individuals.
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To remove plug-in bias, the DR-learnrr performs a second stage regression on the uncentered
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ini uenee function of the average treatment effect [22] [4]. However, under unobserved confounding.
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all of these methods are biased (see Appendix). As result, this hampers their performance in our
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setting.
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ITE methods for unobserved confounding: There is a rich literature for causal effect estimation
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under unobserved confounding, Methods include deconfounding methods [4 18 proxy learning methods [3 causal sensitivity analysis 21 20 and IV methods. IV methods address the
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problem of unobserved confounding by exploiting the variance treatment and outcome induced by the instruments. Traditionally, two-stage least squares (2SLS) has been used for estimating causal effects [48] 4j. 2SLS was originally developed in economics, and follows a two-stage procedure: performs a histt stage regression of treatment A on the instrument Z, and then uses the fitted values for a second stage regression to predict the outcome Y. Several nonparametrie methods have been developed in econometric to generalize 2SLS in order to account for non-linearities within the data yet these are limited to low-dimensional settings.
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Only recently, machine learning has been integrated into IV methods. These are: 37 and (50] generalize 2SLS by learning complex feature maps using kernel methods and deep learning. re- spectively. (17) adopts a two-stage neural network architecture that performs the first stage via conditional density estimation. [6] and 440 leverage moment conditions for IV estimation. However, the aforementioned methods are not specifically designed for the binary IV setting but, rather, for multiple IVs or treatment scenarios. In particular, they impose stronger assumptions such as additive confounding in order to identify the ITT. Note that additive confounding is a special case of when our Assumption holds. Moreover, they are not multiply robust: Even though doubly robust IV methods have been proposed (e.g., Syrgkanis et al.
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[40] ) these methods are not consistent in the union of more than two model specifications (45). We provide more details below.
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Doubly robust estimators are commonly used in causal inference as they allow for consistent estimation under model misspecification
Doubly robust IV methods:
Table 1: Key methods for causal effect es- timation with IVs. This paper: Multiply robustness for ITEs.
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and fast convergence rates [22] Recently, they also have been adopted for IV settings:
Ruissttess
Exiiard
ATE
TTF
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23 proposes
Doubly nisus
Olii et
Satakanis et MRIV (ours)
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pseudo regression estimator for the local average treat-
Mulliply robust
Warz all
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ment effect using continuous instruments, which has been extended to individual effects by
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Furthermore, (38 uses a doubly robust approach to estimate average compiler parameters. Finally,
Ogburn et al. 29 and Syrgkanis et al. [40] propose doubly robust ITE estimators in the IV setting which both rely on doubly robust parametrizations of the uncentered efficient influence function (30] However, these estimators are not multiply robust in the sense that they are consistent in the union of
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more than two model specifications 45
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Multiply robust IV methods: Multiply robust estimators for IV settings have been proposed only
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for average treatment effects (ATEs) (45 and optimal treatment regimes [12] but not for ITEs, In
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particular, Wang et al. 45 derive multiply robust parametrization of the efficient influence function
for the ATE, However, there exists no similar approach for ITE estimation (see Table I
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We provide a detailed, technical comparison of existing methods and our framework in Appendix
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In the binary IV setting, current methods proceed by estimating nu (ar) and n (xr) separately, before plugging them in Eq, 3 5 As result, these suffer from plug-in bias and do not offer robustness properties.
Binary IVs:
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Research gap: To the best of our knowledge, there exists no method for ITE estimation under unobserved confounding that is multiply robust. To fill this gap. we propose MRIV: a multiply robust machine learning framework tailored to the binary IV setting. For this, we build upon the approach by Kennedy (22) to derive robust convergence rates, yet this approach has not been adapted to IV settings, which is our contribution.
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MRIV for estimating ITES using binary instruments
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In the following, we present our MRIV framework for estimating ITTEs under unobserved confounding
179 (Sec. J.T.. We then derive an asymptotic convergence rate for MRIV (Sec. 4.2p and finally use our framework to develop a tailored deep neural network called MRIV-Net (Sec.

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4.1 Framework
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Motivation: A naive approach to estimate the ITE is to leverage the identification result in Eq. (2)
Assuming that we have estimated the nuisance components ji and م for i E {0. 10.1, we can simply plug them into Eq. (2) to obtain the so-called (plug-in) Wald estimator fw(x)) [33]
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However, in practice, the true ITE curve 7(x) is often simpler (e.g., smoother, more sparse) than its complements n. (x)) 1 (xx 25). In this case, (w( is inefficient because it models all components separately, and, to address this, our proposed framework estimates т directly using a pseudo outcome regression.
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We now propose MRIV. MRIV is two-stage meta learner that takes any base method for ITE estimation input. For instance, the base ssssmethod could be the Wald estimator from Eq. (2) any other IV method such as 2SLS, or a deep neural network (as we propose in our MRIV-Net later
Overview:
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in Sec. 4.4). In Stage 1, MRIV produces nuisance estimators jo (x)) 16 (xr). (x), and # (xx) where
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At (x) an estimator of the propensity score T(x)) P(Z X x). Stage MRIV estimates
(a) directly using pseudo outcome YMR as a regression target.
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Given an arbitrary initial ITE estimator fini and nuisance estimates Po (x), A (x). 8x(x), and 5t (z), we define the pseudo outcome
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PMR = 2 Z (2-(1-2) (~-,(1)
(AO (X) + Pinit (A- (X))) Z+(X) + (1 Z) +(X))
+ Amit (X).
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The pseudo outcome MM in Eq. (3) is multiply robust parameterization of the (uncentered) efficient influence function for the average treatment effect Ex [T()) (see the derivation in [45] The initial estimator Pitt (X) is corrected by weighted difference of the observed outcome y and the term
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Py (X) + Amit (X) (A ig (X)). Individuals X with small (X) (large estimated compliance) or small/large (X) (i.e., low/high probability of receiving treatment Z) receive a larger correction.
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Once we have obtained the pseudo outcome YMR, we regress on X to obtain the Stage 2 MRIV estimator (z) for 7xx) The pseudocode for MRIV is given in Algorithm [ MRIV can be interpreted as way to remove plug-in bias from Aiit (x) via the efficient influence function 14]
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Algorithm 1: MRIV
Input: data (X, Z, A, AYYY iniitaal
initial ITE estimator finit(z))
Stage Rstinate muisance companents
#(x)* < E X = x/. (x) + BY X x, 0). pd(x) BJA x, 0
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(=) + 3/4 X = 2,
=1 2=
=xx -
Stage 2: Psaudo outcome regression
Yue 2-(1-2))
Y fiin (mmn
(X)
(A(X)
AMRIV (x) + B/S X =]
Using the fact that YMR is a multiply robust parametrization of the efficient influence function, we derive a multiply robustness property of (M))
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Theorem 1 (multiply robustness property). Let Pn (z), (a(x), 8A(2), R(x), and Amit (x) denote
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estimators of n/ (x), H (x), (z(z), TI)), and T TIx), respectively. Then, for all x E X, it holds that
211 B MR Xx x = (f) least one of the following conditions is satisfied: (1) juy = py 16 = pg
212 òA = 5A and Aiit = T; or (2) म = π and م = δ or (3) मे = π and Tiii = T.
Theorem D implies that )MRIV( is consistent for 7(x) if either condition (1), (2), or (3) holds. As result, our MRIV framework is multiply robust in the sense that our estimator, fMRIV (x), is
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consistent in the union of three different model specifications. Importantly, this is different from
doubly robust estimators which only consistent in the union of two model specifications 45
Example: We illustrate the robustness under model specification (2) in an example. Let A (z)
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A (x) = Aint (z) = be misspecified and let = π and 8 8 be correctly specified.
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follows E Mn X x] = 2x2xy E
2Y-(1-2)Y (=(=))(-2(1--<3)) | X = of (x)- = =(x). This
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6x(X)
justifies the pscudo-outcome regression in last step of MRIV.
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Our MRIV is directly applicable to RCTs with con-compliance: Then, the treatment assignment is randomized and the propensity score <(z) is known. Our MRIV framework can be thus adopted by plugging in the known (1)) into the pseudo outcome in Eq. 3 Moreover, FMRIV (z) is already consistent if either (x) (z) are.
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4.2
Theoretical analysis
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In the following, we derive the asymptotic convergence rate of MRIV under smoothness assumptions. For this, we define s-smooth functions as functions contained in the Hölder class H(s), associated with Stone's ' minimax rate (39 of n -2=+/(2++)), where is the dimension of X'
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Assumption 3 (Smoothness). We assume that (1) the nuisance components μu (-) are a-smooth, M (-) and (-) are B-smooth, and (T()) is 8-smooth; (2) all nuisance components are estimated with
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-2k their respective minimax rate of n 2htp, where E {a, 3, 6): and (3) the oracle ITE T-) is y-smooth and the initial ITE estimator finit converges with rate r- (n).
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Assumption 3 for smoothness provides with way to quantify the difficulty of the underlying nonparametric regression problems. Similar assumptions have been imposed for asymptotic analysis They can be replaced with other assumptions such as assumptions on the level of sparsity of the ITE components. We also provide an asymptotic analysis under sparsity assumptions (see Appendix B).
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of previous ITE estimators in 12 [5]
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We additionally impose the following boundedness assumptions on the the underlying data generating process and estimators.
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Assumption 4 (Boundedness). We assume that there exist constants C. e, K > 0 such that for all E X' it holds that: (1) (M) (x)] S C; (2) (0a(z)) (m((z) 1 (2)) 2 p and [xa(=)) 2 p.:
242 (3) E S (x) < 1 - E; and (4) Fiii (x) K.
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Assumptions 1. A 3. and 4 are standard and line with previous works on theoretical analyses of ITE estimators 15 22 Assumption 2 ensures that both the oracle ITE and the estimator are bounded. Violations of Assumption 2 may occur when working with so-called "weak" instruments, which are IVs that are only weakly correlated with the treatment. Using IV methods with weak instruments should generally be avoided (26). However, in many applications such as RCTs with non-compliance, weak instruments are unlikely to occur patients' decisions to follow the treatment are generally correlated with the initial treatment assignments.
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We state now our main theoretical result: an upper bound on the oracle risk of the MRIV estimator. To derive our bound, we leverage the sample splitting approach from [22] The approach in [22 has
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been initially used to analyze the DR-learner for ITE estimation under unconfoundedness and allows
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for the derivation of robust convergence rates. has later been adapted to several other meta learners
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yet not for IV methods.
Theorem 2 (Oracle upper bound under sample splitting). Let De for E E (,,2,3) be independent samples of size n. Let (x), Pn (x),, and jág (xx) be trained on Dr, and let 8a(x) and a (xx) be trained on Dz. We denote YAAR as the pseudo outcome from E.q. 3 and Yo as the corresponding
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orucle. Let fMRRI(()) (4) = [YMR X x] and "MRRV (x) E., Yo X = x] denote the (orucle)
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pseudo outcome regression on D: for some generic estimator E. X = x] of Xx=
We assume that the second-stage stage estimator En yields the minimax rate n 25t0 and satisfies the fol- lowing two assumptions from Kennedy [22 (1) E,, [
2"
+ X x]
Ê. W X x +
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for any random W and constant c. and (2) if E X = x] E
X
= x), then
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(
[[ |[ x P V
X = x]
E(V = =1)
2
Then, the
oracle risk is upper bounded by
-())) ((fMRIV(()) - [11))]] S n rn( (n
20
22
2( zamp 28
+ n 2( 10-pp 28-p)
+ n 2,
+ n
Proof. See Appendix A.

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Recall that the first summand of the lower bound in Eq. 2 is the minimax rate for the oracle ITE
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7(x) which cannot be improved upon. Hence, for fast convergence rate of MMRVV (z), it is sufficient if either: (1) rr (n) decreases fast and 5 is large; (2) r=(n) decreases fast and a and / are large; or (3) all a, 9, and 8 are large. This is in line with the multiply robustness property of MRIV and
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means that MRIV achieves fast rate of convergence even if the initial estimator or several nuisance estimators converge slowly.
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From the bound in Eq. (2). it follows that (2) improves on the convergence rate of the initial ITE estimator min (x) if its rate (n) is lower bounded by
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-2y (n) 2 n + +n +==22 (20tp - 24tp) +n - 2244 264))
(4) Hence, our MRIV estimator is more likely to improve on the initial estimator for large a, 3. and ie., if the nuisance components are smooth. Note that it is sufficient if either (1) only the propensity score T (2)) is relatively smooth (large d) or (2) that all other nuisance components are (large a and 3). In fact, this is widely fulfilled in practice. For example, the former is fulfilled for RCTs with non-compliance, where (2) is often some known, fixed number p E (0, 1). Hence, for RCT's with non-compliance, MRIV should (at least asymptotically) improve the performance of most estimators.
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4.3
MRIV vs. Wald estimator
In the following, we compare (MRIV (x) to the Wald estimator Aw (x)). First, we derive corresponding upper bound under smoothness.
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Theorem 3 (Wald oracle upper bound). Given estimators juţ (x) and iz Let Let))(x)= = -nf(()) 284 P (z) satisfy Assumption 4 Then, the oracle risk of the Wald estimator fw (z) is bounded by
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E (fw(z)->(x))) S
20 +n 2010.
(5)
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Proof. See Appendix A.
We now consider the MRIV estimator ÊMRIV (I:) with Aiit fw(x)) i.e., initialized with the Wald estimator (under sample splitting). Plugging the Wald rate from Eq. (5) into the Eq. (2) yields
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(+MRIV(()) T(x))²] S 2T-P +n2 +n 2( 2att 28 ) +n 2( 25ip 2otr +n 2( 204 2847
(6)
For = 3= ô, the rates of (x) and xw reduce to 288
22 E (fMRIV(()) ((x))] S noith nantz and E (fw(z))-(xt) S p2--p.
-4a
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Hence, fMRIV (I) outperforms fw (x) asymptotically for > a, i.e., when the ITE r(x) is smoother than its components, which usually the case in practice [25]. For = a, the rates of both estimators coincide. Hence, we should expect MRIV to improve on the Wald estimator in real-world settings with sufficiently large sample size.
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4.4 MRIV-Net
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Based on our MRIV framwork, we develop a tailored deep neural network called MRIV-Net for ITE estimation using IVs. Our MRIV-Net produces both an initial ITE estimator Amit (I) and nuisance estimators Ay (z), (z). (z).. and (z).
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297 For MRIV-Net, we choose deep neural networks for the nuisance components due to their predictive power and their ability to learn complex shared representations for several nuisance components. Sharing representations between nuisance components has been exploited previously for ITE estima- tion, yet only under unconfoundedness 15). Building shared representations is more efficient in finite sample regimes than estimating all nuisance components separately as they usually share some common structure.
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[5].
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In MeIV-Net, not all nuisance components should share a representation. Recall that, in Theorem 2 we assumed that (1) (z), (x), and (r);; and (2) (x(z) and #(x) trained two independent samples in order to derive the upper bound on the oracle
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306 risk. Hence, we propose to build two separate representations $ and Φ2, so that (i) $
307 is used to learn fun(()) po (z), and (x), and (ii) 42 is used to learn dz(x) and *(x).
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This ensures that the nuisance estimators (1) share minimal information with nuisance estimators (2) even though they are estimated on the same data. Intuitively, this should lead to faster decay of the oracle upper bound (cf. [T5).
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The architecture of MRIV-Net is shown in Fig. 2 MRIV-Net takes the observed covariates X as input to build the two representations & and
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2.. The first representation Φ is used to output estimates ji (2), Pg (x),
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i (z), and j (z) of the ITE components. The second representation
Figure 2: Architecture of MRIV-Net.
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22 is used to output estimates (z), Po (z), and (()). MRIV-Net is
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trained by minimizing an overall loss
[ = Σ (1) - u1)² BCE (2 + BCE
+ BCE (*(r;),),
(8)
318
where & denotes the neural network parameters and BCE is the binary cross entropy loss. After training MRIV-Net, we obtain the Amit (a)) Pi (a) P (a) and obtain the nuisance estimators io (z)) (x)
319
Ag (x). 8 (x) = (x) 76 and (xx). Then, we perform, we perform the pseudo regression (Stage 2) of MRIV to obtain (x).
320
322
Implementation: We use PyTorch Lightning for our implementation and train MRIV-Net with the Adam optimizer (24). Details on the network architecture and hyperparameter tuning are in Appendix G. We perform both the training of MRIV-Net and the pseudo outcome regression on the full training data. Needless to say, MRIV-Net can be easily adopted for sample splitting or cross-fitee fottss-itting procedures as in [TO), namely, by learning separate networks for each representation
323
324
325
326
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1 and Φ.. However, in our experiments, we do not use sample splitting or cross-fitting as this can
328
affect the performance in finite sample regimes. Of note, our choice consistent with previous work
15
Computational experiments
330
5.1
Simulated data
332
In causal inference literature, it is common practice to use simulated data for performance evaluations Simulated data offers the crucial benefit that it provides ground-trute -tuuh information on the counterfactual outcomes and thus allows for direct benchmarking against the oracle ITE.
13
334
Data generation: We generate simulated data by sampling the oracle ITE T (ar) and the nuisance components nu (x), p (æ)), and r(x) from Gaussian process priors. Using Gaussian processes has the following advantages: (1) It allows for fair method comparison, there is no need to explicitly specify the nuisance components, which could lead to unwanted inductive biases favoring a specific method; (2) the sampled nuisance components are non-linear and thus resemble real-world scenarios where machine learning methods would be applied; and, (3) by sampling from the prior induced by control the smoothness of the nuisance components, which allows us to confirm our theoretical results from Sec. 4.2 For a detailed description of our data generating process, refer Appendix C.
336
338
340
the Matérn kernel (32),
342
344
Baselines:
We compare our MRIV-Net with the following state-of-the-an baselines: (1) ITE methods
for unconfoundedness: TARNet (36] and TARNet combined with the DR-learner (22) ; (2) general
346
IV methods: 2SLS [48), kernel IV (KIV) (37, DFIV (50). DeepIV (17), DeepGMM 6). DMLIV
40 and DMLIV combined with DRIV (as described in [40];; (3) the (plug-in) Wald estimator using
348
linear models and Bayesian additive regression trees (BART) [ Of note, the DR-Iearner assumes unconfoundedness, which is why we only combine it TARNet in experiments. Implementation
350
details regarding baselines and nuisance parameter estimation are in Appendix E. Note that many of the baselines do not directly aim at ITE estimation but rather counterfactual outcome prediction.
352
We nevertheless use these methods as baselines and, for this, obtain the ITE by taking the difference between the predictions of the factual and counterfactual outcomes.
353
354
Performance evaluation: For all experiments, we use 80/20 split as training/test set. We calcalute the root mean squared errors (RMSE) between the ITE estimates and the oracle ITE on the test set.
8

---

356
We report the mean RMSE and the standard deviation over five data sets generated from random seeds.
357
358
Results: Table 2 shows the results for all baselines. Here, the DR-learner does
Table 2: Performance comparison: our MRIV-Net vs. ex- isting baselines.
359
380
not improve the performance of TAR-
Method
3000
5000
S000
361
Net, which is reasonable as both the
STANDADAR
TARN& TARNi
0.76 0.14
0.70 +
0.69 +
382
DR-learner and TARNet assume uncon-
22
0.78 = 0.10
0.66 : 0.09
0.70 :: 0.
363
foundedness and are thus biased in our
(2) GENERAL IV
384
setting. Our MRIV-Net outperforms all baselines. Our MRIV-Net also achieves
2SLS 47
KIV
1.22 + 0.23
1.54 0.53
0.79 + 0.37
1.18 1: 1.14
3.80 1 4.71
1.12 + 0.29
DRV
0.43
0.40 1: 0:21
0.45 1: 0.54
365
Despl
0.96 0.30
0.28 1: 0.09
0.23 0.04
386
a smaller standard deviation. For addi- tional results, we refer to Appendix H.
60
1.02
0.92 0.41
0.09
0.42 0.14
1.14 0.24
367
DMILI FDRIV
0.41 0.12
0.22 4 0.04
0.21 + 0.06
(33 WALD ESTIMATOR 44
368
We further compare the performance of
Lisea
1.06 H: 0.63
0.62 d: 0.22
0.81 d: 0.34
389
lwo different mett--lermer frameworks-
BART
0.96 0.30
0.63 + 0.33
0.88 + 0.28
370
DRIV [40] and our MRIV- across differ-
MRIV-Na (ours)
0.26 0.11
0.15 + 0.08
0.18 +: 0.08
RMSE methods
sanddr
better
371
ent base methods. The nuisance param-
372
eters are estimated using feed forward neural networks (DRIV) or TARNets with either binary or continuous outputs (MRIV). The results are in Table Our MRIV improves over the variant without
373
374
any meta- learner framework across all base methods (both in terms of RMSE and standard deviation).
Table 3: Base model with different meta- learners (i.e., none, DRIV, and our MRIV).
5000
Bisc melhads
Jeaeess
Noue
DRIV
(ouzs)
None
MRIV (ouns)
DRIV MRIV (ouEs)
STANDARD ITE
TARNa
0.76
0.51
0.05
0.34 0.13
0.12
0.06
0.17 0.06
0.60
0.16 0.04
GevauaL
2SLS 47
0:22
0.60 00 0.11
0.31 + 0.08
0.29
0.17 + 0.09
0.19:2:0.56
1.12 0.29
0.21 0.0.
0.16 0.02
KIV 3m
0.55
0.60 5: 0.10
0.39 + 0.11
L18 1.14
0.20 2:0:::000 0.08
0.17 + 0.08
3.80 4.71
0.31 0:15
0.28
0.19
DRIV
0.33
0.36 4: 0.06
0.27 2 0.07
0.20
0.18 2.022:007 0.00
0.16
0.04
0.46 0.54
2:21
0.18
0.05
Despit
177
0.96++00 0.30
0.27 + D05
0.26 2: 0.45
0.28
0.00
0.18 2: 0.08
0.18 0.05
0.25-0007 0.0
0.21 0.03
0.10 0.00
DeepChMar
0.85 0.336
0.10+ 0.15
0.36
2: 0.15
0.37
0.00
0.11
0.16.2:0.1 0.05
0.12-0031
0.21 +.05
0.17220.00
DMLIV 40
1.92+
0:41 + 0.12
0.57 + 0.11
0.52==0.41
0.22 6.05
0.16
1: 0.06
1.14=:021
0.21 + 0.05
0.18 2: 0.05
(3) WALD ESTIMATOR
45
innor
0.12 0:15
0.38 + 0.14
0.62 0.22
0.18 0.09
0:25 0.00
0:11+0031
0.19 0.00
+180
BART
0.85
+ 0.30
0.08 0.14
0.46 = 0.12
0.3
0.20
0:13
0.20 +: 0.07
0.88:0028
MRIV Neto networks only (ours)
0.30
0.35 0.12
0,26 + 0.11
0:04
0.19 0:13
0.15 0.00
0.26:0000
0.13 0.05
Repecia EMSE incan it standand devintion) Lawer beta thea ingrovemenn ove sone incertemmmer in bota))
375
Furthermore, MRIV is clearly superior
Table 4: Ablation study.
376
over DRIV. This demonstrates the effec- tiveness of our MRIV across different
377
Method
n 3000
0 5000
8000
378
base methods (note: MRIV with an ar-
MRIV-Nett network only
0.39 0.13
0.31 0.04
0.26 :: 0.06
379
bitrary base model is typically superior
Nell-Nellw singlo repe.
0.28 1: 0.12
0.21
0.04
0.32 4: 0.10
MRIV-Net (ours)
0,28 0.11
0.15 - 0.03
0.13 - 0.03
to DRIV with our custom network from above). MRIV-Net is overall best. We
Reported: RMSE (mean standard deviatio)) Lower better (best boll)
381
also performed additional experiments where we used cross-fitting approaches for both meta- neta-learrerr (see Appendix
383
384 A Alltion study: Table 4 compares different variants of our MRIV-Net. These are: (1) MRIV but network only; (2) Net with a single representation for all nuisance estimators; and (3) our MRIV-Net from above. We observe that MRIV-Net is best. This justifies our proposed network architecture for MRIV-Net. Hence, combing the result from above, our performance gain must be attributed to both our framework and the architecture of our deep neural network.
305
386
387
389
Robustness checks for unobserved confounding and smoothness: Here, we demonstrate the importance
3300
n- 5093
390
391
382
VAAGE CA
393
of handling unobserved
394
confounding (as we do in our
395
MRIV framework). For this,
Contucain
396
Fig. B plots the results for
397
Figure 3: Results over different levels of confounding ag. Shaded area shows standard deviation. for confounding (i.e., TARNet with and without the DR-learner) over over different levels of unobserved confounding. The RMSE of both TARNet variants increase almost linearly with
our MRIV-Net vs. standard
ITE without customization
399

---

401
older age. increasing confounding. In contrast, the RMSE of our MRIV-Net only marginally. Even for low confounding regimes, our MRIV-Net performs competitively.
402
Fig. varies the smoothness level. This is given by a of (.)()) (controlled by the Matérn kernel prior). Here, the performance decreases for the baselines, i.e., DeepIV and network without MRIV framework. In contrast, the peformance of our MRIV-Net remains robust and outperforms the baselines. This confirms our theoretical results from above Ir this indicates that our MRIV
403
404
405
408
407
framework works best when the oracle ITE r(x) is smoother th.
0.35
408
409
5.2
Case study with real-world data
0.25
Method
410 Setting: We demonstrate effectiveness of our framework using a case study with real-world, medical data. Here, we use medi- cal data from the so-called Oregon health insurance experiment (OHIE) [16]. It provides data for an RCT with non-compliance: In 2008, 30,000 low-income, uninsured adults in Oregon were offered participation in a health insurance program by lottery. Individuals whose names were drawn could decide to sign up for health insurance. After period of 12 months, in-person interviews took place to evaluate the health condition of the respective participant.
0.20-
only
411
0.15
412
0.30
413
414
Figure 4: Results over different lev- els of smoothness a of μu (-), sam- ple size smoother. Shaded areas show stan- dard deviation.
415
416
8000. Larger a
417
418
420
In our analysis, the lottery assignment is the instrument Z, the decision sign up for health insurance is treatment A, and an overall health score is the outcome Y. We also include five covariates X (age, gender, language, the number of emergency visits before the experiment, and the number of people the individual signed up with). It is important to include the latter in our analysis as it is the only variable influencing the propensity score. For details, we refer to Appendix D. We first estimate the
421
422
423
424
425
ITE function and then report the treatment effect heterogeneity w.r.t. age and gender, while fixing the other covariates (i.e., we consider the English-speaking subpopulation with one emergency visit that signed up alone). We repeat the same procedure for our neural network architecture without the
426
427
428
MRIV-Net framework and TARNet. The results are in Fig. 5
429
Results:
Our MRIV-Net estimates larger causal effects for an older age. In TARNet does not estimate positive ITES even for
430
contrast,
431
Even though we cannot evaluate the estimation quality on real-world data, our estimates seem reasonable in light of the medical literature: the benefit of health insurance should increase with volerr age. This showcases that TARNet may suffer from bias induced by unobserved con- founders. We also report the results for DRIV with DMLIV as base method, and observe that in contrast to MRIV-Net, the corresponding ITE does not vary much between ages. Interestingly, both MRIV-Net estimate somewhat smaller ITE for middle ages (around 30-50 yrs). One explanation might be that individual in this age group are more likely to have stable jobs and, thus, are also more likely to be able afford medical care, decreasing the direct effect of health insurance individuals health. In sum, the findings from our case study are of direct relevance for decision-makent in public health 19]. and highlight the practical value of our framework.
432
grnder male
creder ferale
433
434
Vettod
435
redownt
436
OMLY DRTV
437
20
40
60
20
60
438
Figure 5: Results on real-world medical data.
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6
Conclusion
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447
In this paper, we propose MRIV-Net: a novel ITE estimator based on deep neural network. Importantly, our estimator is consistent in the union of three models specifications and, therefore, multiply robust. This is crucial difference to existing works: previously, existing ITE estimators (such es DRIV from Syrgkanis et al. 40) were only doubly robust. We show both theoretically and
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empirically that MRIV-Net is state-of the for estimating ITEs using binary IVs. For future work, would be interesting to derive finite sample results for MRIV-Net, because our theoretical analysis is purely asymptotic. Furthermore, one could develop multiply robust estimators for other IV settings (e.g., multiple or continuous instruments and treatments).
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10

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Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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contributions and scope? [Yes] See here
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(b) Did you describe the limitations of your work? [Yes] See Sec. 5.2
(c) Did you discuss any potential negative societal impacts of your work? [No]
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] See See. and Appendix A..
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.
3. If you ran experiments...
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [Yes] (both).
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix E and G
(c) Did you report error bars (e.g., with respect the random seed after running ex- periments multiple times)? [Yes] One standard deviation using 5 random seeds, see Sec.
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix E.
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes] See Sec. 5.2
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(b) Did you mention the license of the assets? [No] No license is provided on the OHIE
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website.
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(c) Did you include any new assets either in the supplemental material or as URL? [Yes] See Appendix D
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(d) Did you discuss whether and how consent was obtained from people whose data you're re using/curating? [No] We only used simulated and publicly available data.
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [No] The OHIE data publicly available, personally identifiable information are censored.
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5. If you used crowdsourcing or conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, applicable? [No] Not applicable.
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] Not applicable.
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No] Not applicable.
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13

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Estimating individual treatment effects under unobserved confounding using binary instruments Appendix
Anonymous Author(s)
Affiliation
Address
email
Contents
Proofs
A.1 Proof of Theorem (multiple robustness property)
A.2 Proof of Theorem (Convergence rate MRIV)
A.3 Proof of Theorem 3 (Convergence rate of the Wald estimator)
Theoretical analysis under sparsity assumptions
Simulated data
Oregon health insurance experiment
E
Details for baseline methods
10
ITE methods for unconfoundedness
11
General IV methods
Wald estimator
11
13
Visualization of predicted ITES
12
Implementation details and hyperparameter tuning
13
15
Results for semi-synthetic data
15
Results for cross-fitting
16
Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

---

17 A
Proofs
We start by deriving auxiliary Lemma. That is, derive explicit expression for the Stage
19 oracle pseudo outcome regression E|Yo X æ] of MRIV.
Lemma 4.
B| X x]
x(x) (Mr (x) - p1 (z) Am)((())
(1- - T(()))
(x) (x) - -00 (x))
()))((())
8x(2))1 *(())
(I)
+ no (1) Fmiit (2) - P. (x)
1- T(x)
+ (1))
(x(x)
#(x)
- ftxx))
Proof.
X s²]
(2)
Y A (X) 146 (X) + iá (X) (X)
x= xz, 2= Y - A Amit (X) - -10 (X) + (x) mite 8x(x) (1 =*()))
(x( #(X)
+ (1 {<)))
X x, Z 0
+ Amit (I)
(3)
π(x)
(1( (xx) (ML (x) - (z))-1((x) (z) - A. (x) + Ao (x) Finit (z))
1 x(5)
(x(2) (1 - (x)) (MD (x) (x) (x) (x) (z) Ag (x) Amit(()) + (x)
(4)
20
Rearranging the terms yields the desired result.
21
A.1
Proof of Theorem (multiple robustness property)
We use Lemma
4 to show that under each of the three conditions it follows that EE(Y X = x] = T(x).
1.
[Yo X = x!
(5)
T(x)
(z) (()))===((((((+++()))))))+---)-- (z))
(x) (T))
(1 (1-7t) x())
+ 8x(2) (1 -1- a(x)) (z) - + +T(x) (6)
T(x)
(xy 6y (z)) + ((x) Txx).
(x) (1))
2.
(uf (x) (x)))) fimit(())
+ (10 (v)) Ami - n1 (x)) -x)x))
E x = x|
+ Aiit (x)
(x(z)
ÓΛ(x)
(8)
5y (x) Amit (x) (x())
+ (z) = T(z).
(9)
(x)
3.
B X = x] (pf (z) (x)
+ (16(2) r(z) - n² (z))
7(x)
(10)
(x(Z)
81(2)
8y(x)
T(x) (A(T)
+((z)= = T(x)
(11)
(x(
(x)

---

23
A.2 Proof of Theorem 2 (Convergence rate of MRIV)
24 To prove Theorem 2, we need an additional assumption on the second stage regression estimator E,,
25
We refer to Kennedy [ (Theorem 1) for detailed discussion on this assumption.
26
Assumption 5 (From Theorem 1 of Kennedy (8). The following two statements hold:
1. E., IW + c X = x] Em [W X = x] + e for any random W and constant e
27
2. If WW | X = xx| = V X = xi] then
28
27 27 x E (ENIV xx= = x] - E X = x!)
E BB [ X = xc| - E|W x = 211
(12)
29
Proof of Theorem?
Using Assumption we can apply Theorem 1 of Kennedy
8 and obtain
E (finit (x) (x)-==>))) ( S R(x) + E [+()]]
(13)
30 where R(x) = E (FM)(()) - -(x))² is the oracle risk of the second stage regression and r(x) =
31 EN% X x| T(x) We can apply Lemma 4 obtain
Txx) (MY (x) - mx( (x) + (1- T()))
f(x) (z)) ((x)
3 (1(x) (1 ffx))) (x) (x) fmni(()) nō (z))
-
+ 16 (x) Amit - A (1))
(())
1 1-
+ finit (z) - F(x)
(14)
(1(x)
A(x)
1 f(xx)
uu (x)) n (x)
x(x))
16 (z) 10--ih 120 (x)
(Tx))
1 1- 1-m(2))
+
+ (fiiit - Tx)))
3x(Z)
A(x)
(x(2)
((z)
1 1 --((x))
(46 111 (x:)) Finit
T(x) (x) 1b8(( (x:) Riuit (z)
[Tz))
- T(x)
8x(2)
(xx)
(x(2)
#(x)
4 (xx)
(15)
5y(x) T(x)
8A(z) + (w)
18 +xx))
+ ( (()
(z())((z) (1-#(z))
(x( T (x) (x)
(16(() - (x)) Amit (a) (T(x)-
(16)
(A(Z) #(x)
(2(2) (xx) (1 - =x)))
(T(x) - (x:)
(x) (xx) (1 - +(x)) (166 (x) - 180 (x)) + (16 1/ (x:)) (m)
+ (x) - [z)) + (x)) (F(x) (z))
(17)
*(z)&1(z)
(T(x)- - (z))
[H6 (x) - - (z)) + (n)(()) - Hz Am))(())]
(x(z) [(x) (1 - ((x))
+ (T(x) - Fuii (x)) (5a(1)- - )x( (0x) + (T(x) - Aim (x)) (T +(x)) (x)))
(18)
32 Applying the inequality (a+b)² < 2(aa + b²) together with Assumption 4 and the fact that T(x) < 1
33 yields
f(x)2 S cpp (T(x) - #(x))² (1u (x) - in (x)) + (R6(()) - 1 (x)) 2 k²)
+4 - xz)) (D1(z)- - )u(x)) +4(r(z) - Amit (2:)) (T(2) #(x))) (19)
3

---

By setting K- max K 1}, we obtain
34
4K 2
f(x)² - c²p² (अ(()) - a(x))² (1z (x) - Ao (x))² + (At(x) - mz(x)) 2 + (2) - =
+ (T(x) (z)) (6x(z)-8an) -8x()))
(20)
Applying expectations on both sides yields
E (finit (x) - [
(21)
2 R(x) + (fimit (2x) - (()))² (E (
-5a(x)) 21 + E (f(z)) - -(x))² (8 (A6 (x) - n² (x)) + +E - )
+ +B ((f())=<(z))))
(22)
36 because (f(z), (#()),,A()) TT (i) (z), A0 (z), (z)) due sample splitting. The claim follows now
37 by applying Assumption 3.
Li
38
A.3
Proof of Theorem 3 (Convergence rate of the Wald estimator)
39 Proof. We define Č = max{C, 1} and obtain the upper bound
(fw(x) - ) (if (1) - - (x))) + (1) (x) - A. (x)) - (5^(x) - 8x(x)) 8r (x)
(23)
2
(24)
6x(Z) (x))
4
[NN (x) (x)
- pz (x)) (1 (z) ne (x)) + (6Λ(2) - 81(=))²]
(25)
178
80
p T [nn (x) - (x)) + (16 (z) No (z)) + (1f(2)) - /x))²
+x) - 16 (1
(26)
where we used the inequality (a + b)2 < 2(a² - b2) several times. Taking expectations and applying
41 the smoothness assumptions yields the result.

---

50 In the following, we say function fx) is k-sparse, if it is linear in Rº and it only depends B Theoretical analysis under sparsity assumptions
42
43 In Sec. 4.2 we analyzed MRIV theoretically by imposing smoothness assumptions on the underlying
44 data generating process. In particular, we derived a multiple robust convergence rate and showed
45 that MRIV outperforms the Wald estimator the oracle ITE is smoother than its components. In
46 this section, we derive similar results by relying on different set of assumptions. Instead of using smoothness, we make assumptions on the level of sparsity of ITE components. This assumption is often imposed in high-dimensional settings (n < p) and is in line with previous literature on analyzing ITE estimators
47
49
51 on (n, p} predictors. (22) showed, that in this case the minimax rate of f(z) is given by kog(p)) The linearity assumption can be relaxed to an additive structural assumption, which we omit 53 here for simplicity. In the following. replace the smoothness conditions in Assumption with
52
54 sparsity conditions.
55 Assumption 6 (Sparsity). We assume that (1) the nuisance components μu (+) are a-sparse, (.) and (-) are 3-sparse, and rT-) ) is d-sparse; (2) all nuisance components estimated with their respective minimax rate (log(p) where k E {ax, 3, 6); and (3) the oracle ITE r()) is 7-sparse and the initial ITE estimator Aiit converges with rate (n(n).
57
n
59 We restate now our result from Theorem 3 for MRIV using the sparsity assumption.
60
Theorem 5 (MRIV upper bound under sparsity). We consider the same setting as in Theorem
under the sparsity assumption 6 If the second-stage estimator En yields the minimax rate 7 log(p)
61
and satisfies Assumption 5 the oracle risk upper bounded by
E (fMRIV (x) - (x))) s
y ylog(p)
(3 + 8) log(p) + +r((m)
+ (a + 3)8 (p)
n
n
n
Proof. Follows immediately from the proof of Theorem 2 i.e., from 21 by applying Ass- 6
64
Again, we obtain multiple robust convergence rate for MRIV in the sense that MRIV achieves a fast 85 rate even if the initial estimator or several nuisance estimators converge slowly. More precisely, for fast convergence rate of FMRIV (x), it is sufficient if either: (1) r, (n) decreases fast and ó is small; r, (n) decreases fast and a and are small; or (3) all a, 3, and à are small.
86
67
68
We derive now the corresponding rate for the Wald estimator.
Theorem 6 (Wald oracle upper bound). Given estimators juy (x) and in (x). Let (x)
An (x) satisfy Assumption Then, under Assumption 6 the oracle risk of the Wald estimator fw (x)
70
71
bounded by
(a + () log(p)
E [ff((x) [(=))²] S
(27)
72 Proof. Follows immediately from the proof of Theorem 3 i.e., from 2q. by applying Ass- 6
Ifa Ō, we obtain the rates
E (o) s llo(pp) a log)(p)
log(p)
and E (ffw(x) [x)))
n²
n
(28)
74 which means that (M) outperforms fw(x) for y < a, i.c., the oracle ITE is more sparse than its components.
5

---

78 C Simulated data
77 In the following, we describe how we simulate synthetic data for the experiments in Sec. [.1 from the
78 main paper. As mentioned therein, we simulate the ITE components from Gaussian processes using the prior induced by the Matern kernel [2]
79
V2v 21 -,,11
V2V ||,,
(z), ((v)2--1
Kr
Zz 2z112
(29)
80 where r(-) is the Gamma function and (-) is the modified Bessel function of second kind. Here, 8 81 is the length scale of the kernel and controls the smoothness of the sampled functions.
We & and sample functions dy GP(O, Ke,y), ply GP(O, fi gP(O, Ke,s). fo (P(O, Ke,s) and g GP(O, Ke Then, define pi by + MO μ 0.3 ao + 0.7, MD 0.3 a fo. òA Mf nb. Mo codA, and
Finally, we set the oracle ITE to
1 P4 dy
(30)
"1 14 ÖA
Note that we can create setup where the ITE is smoother than its components by using small a/8 ratio. An example is shown in Fig. !
86
YIZ=1
YI2=0
Figure 1: Gaussian process simulation for 1.5 and B = 50.
In the following, describe how generate data the (x, 2, using the ITE components
" (x), ua (x), and π (r). We begin by sampling n observed confounder X NN 1), unobserved (0, 0.23), and instruments ~ Bernoulli(m(X)) Then, we obtain treatments
88
89 confounders via
A Z (U + (x)) + (1 (1- + +
(x(X)}
(31)
91 with indicator function 1, noise Ед ~ N (0, 0.13). and ax(X) = - -- (1- V
92 where + denotes the quantile function of the standard normal distribution. Finally, we generate the outcomes via
93
Y = A (uf(X) - -X) (X) ( (X) +f 8x(X)
(32)
+ +(1 ((1-A) AA) (((X)) (X) n (X)xX(()
+a +
(33)
(X)
94 where cy ~ N (0. 0.3?) is noise and ag > is a parameter indicating the level of unobserved 35 confounding. This choice of and in Eq. (3T) and Eq. (32). respectively, implies that r(x) is indeed the ITE, i. e., it holds that 7(x) E (1) (0) X =]. z|.

---

97
Lemma 7. Ler Z, be sampled from the the previously described procedure. Then, it holds
96 that
(x) = E[A| 2 = =; a]
and 11 (x) = Z = x x²)
(34)
99
Proof. The first claim follows from
E[A| 2 = i, = - = (U + EA (c(x)) = 01- - (4))
(35)
100
because U + ~ (0, +
The second claim follows from
B|y 2 = i, x = 14x)) (uf((z) 1) (x) (o (x)
(36)
( (=) (1)) (5)) (z)--10
+ (1 (x))
(37)
6ạ(x)
M (1)
(x)
(38)
61(2)

---

D
Oregon health insurance experiment
102
The so-called Oregon health insurance experimen (OHIE) was an important RCT with non- compliance. It was intentionally conducted as large-scale effort among public health to assess the effect of health insurance several outcomes such as health economic status. 2008, lottery draw offered low-income, uninsured adults in Oregon participation in Medicaid program, providing health insurance. Individuals whose names were drawn could decide to sign up for the program.
103
104
105
106
107
108
In our analysis, the lottery assignment the instrument Z, the decision to sign up for the Medicaid program is the treatment A, and an overall health score is the outcome Y. The outcome after a period of 12 months during in-person interviews. We use the following covariates X: age, gender, language, the number of emergency visits before the experiment, and the number of people the individual signed up with. The latter is used to control for peer effects, and it is important to include this variable in our analysis as is the only variable influencing the propensity score (see below). We extract ~ 10,000 observations from the OHIE data and plot the histograms of all variables
109
obtained
110
112
113
114
115
in Fig. 2 We can clearly observe the presence of compolale within the data, because the
116
ratio of treated / untreated individuals is much lower than the corresponding ratio for the treatment
117
assignment.
20
Age
Frequancyy
10
15
Fwits
Otee
Engiihh
Nim kgrad
Num webx
Language
Figure 2: Histograms of each variable in sample from OHIE.
118
The data collection in the OHIE was done follows: After excluding individuals below the age of 19, above the age of 64, and individuals with residence outside of Oregon, 74,922 individuals were considered for the lottery. Among those, 29,834 were selected randomly and were offered participation in the program. However, the probability of selection depended on the number of household members on the waiting list: for instance, an individual who signed up with another person twice as likely to be selected. From the 74,922 individuals, 57,528 signed up alone. 17,236 signed up with another person, and 158 signed up with two more people on the waiting list. Thus, the probability of being selected conditional on the number of household members the waiting list follows the multivariate version Wallenius' noncentral hypergeometric distribution (2).
119
120
122
124
126
127
We computed the propensity score as follows. To account for the Wallenius' noncentral hypergeometric distribution, we use the R package BiasedU'n to calculate the propensity score π (xr) P(Z = X = z). We obtained
Propensity score:
128
0..35. if individual signed up alone,
(2x)
person, 0.719, if individual signed up with two more people.
0.571, if individual signed up with
(39)
During the training of both MRIV and DRIV, we use the calculated values from Eq. 39 for the propensity score.
131
Data available here: http:////wwwwweeooooommmeeepepee hnw. insurance- experiment

---

132
E
Details for baseline methods
In this section, we give brief overview on the baselines which used in our experiments. We
134
implemented: (1) ITE methods for unconfoundedness [8] 13; (2) general IV methods, i.e., IV
135
methods developed for IV settings with multiple or continuous instruments and treatments
14
136
15
20
21]; and (3) two instantiations of the Wald estimator for the binary IV setting [16].
137
F.1
ITE methods for unconfoundedness
Many ITE methods assume unconfoundeedress, i.e., that all confounders are observed in the data. Formally, the unconfoundedness assumption can be expressed in the potential outcomes framework
139
140
as
(1), (0) AA |
(40)
141
Under unconfoundedness, the ITE is identified as
T(z) = (1) - mo(z) with [1(x) = E|Y A- x].
(41)
142
Methods that assume unconfoundedness proceed by estimating pı(x)) = B[ AA 1, X
from
Eq.
41
However, if unobserved confounders U exist, it follows that
T(x) = E A = 1, 2, U] E|y A = 0, z, U] * p1(a) - Ho(z),
(42)
144
which means that estimators that assume unconfoundedness are generally biased. Nevertheless, we include baselines that assume unconfoundedness into experiments: TARNet [13] and the
145
146
DR-learner
8
TA RNet 13
TARNet 13 is neural network that estimates the ITE components ps(x)) from Eq. 41 by learning a shared representation (xx) and two potential outcome heads (x)). We train
148
TARN RNet by minimizing the loss
((0)- = Σ L (ha, (D(ri, 0s), On.) 9:),
(43)
2m
150
where 0 = , Oz) denotes the model parameters and L denotes squared loss if Y is continuous or binary cross entropy loss if Y is binary.
151
152
Note regarding balanced representations: In 13 the authors propose to add an additional regular- ization term inspired from domain adaptation literature, which forces TARNet to learn balanced representation (xx), i.e., that minimizes the distance the treatment and control group the feature
154
155
space. They showed that this approach leads minimization of generalization bound on the ITE estimation error if the representation is invertible.
156
In experiments, refrained from learning balanced representations because minimizing the regularized loss from [13] does not necessarily result in an invertible representation and thus may harm the estimation performance. For detailed discussion, refer to (4). Furthermore, by leaving out the regularization, we ensure comparability between the different baselines. If balanced representations desired, the balanced representation approach could also be extended to MRIV-Net, as we also build MRIV-Net on learning shared representations.
158
160
162
DR-leareer [8] The DR-leameer is meta learner that takes arbitrary estimators of the ITE componenets μi and the propensity score (xx) = IPAA = X = x) as input and performs a pseudo outcome regression by using the pseudo outcome
164
Y,
A
1- A
Y
A #(X)
(x(X)=
1- -11
10000(1111) (100-(11111 1-4
Po(X).
(44)
#(X)
+X))
In experiments, use TARNet as base method provide initial estimators (ax()) We further learn propensity score estimates it (X) by adding a seperate representation to TARNet as done in
187
168
E.2
General IV methods
170
2SLS [20]: 2SLS (20 is linear two-stage approach. First, the treatments A are regressed on the instruments Z and fitted values Å are obtained. In the second stage, the outcome Y regressed on Å. We implement 2SLS using the scikit-learn package.
171

---

173 KIV [14] Kernel IV
[4] generalizes 2SLS to nonlinear settings. KIV assumes that the data is
174
generated by
(A)(A)+U; +U; where U is an additive unobserved confounder and f is some unknown (potentially nonlinear) structural function. KIV then models the structural function via
(45)
176
f(a) = μ² (a) and Blp(A) Z = 2] Vo(=),
(46)
177
where & andø are feature maps. Here, kernel ridge regressions instead of linear regressions are used in both stages to estimate and V.
178
Following [4]
the exponential kernel [12 and set the length scale to the median inter-poin distance. KIV does not provide a direct way to incorporate the observed confounders X. Hence, we
180
augment both the instrument and the treatment with X, which is consistent with previous work [T]
182
21J. We also use two different samples for each stage as recommended in [4].
183
DFIV (1:: DFIV [1 is a similar approach KIV in generalizing 2SLS to nonlinear setting by assuming Eq. (45) and Eq. (46). However, instead of using kernel methods, DFIV models the features
184
maps 18, and ØØz as neural networks with parameters A and 0z. respectively. DFIV is trained by iteratively updating the parameters A and Oz. The authors also provide a training algorithm that incorporates observed confounders X, which we implemented for our experiments. During training, we used two different datasets for each of the two stages as described in in the paper.
186
188
189
DeepIV 7 DeepIV (7) also assumes additive unobserved confounding as in Eq. (5)). but leverages the identification result 10]
190
- hha, 2) dF x.
E|Y
x= = Z =] =
(47)
where liaa, x)) (a, x) EU X x= x] is the target counterfactual prediction function. DeeplV estimates F(a| x. 2), i.e., the conditional distribution function of the treatment A given observed covariates X and instruments Z, by using neural networks. Because we consider only binary treatments, simply implement (tunable) feed-forward neural network with sigmoid activation function. Then, DeepIV proceeds by learning a second stage neural network to solve the inverse problem defined by Eq. 47).
191
192
199
195
198
197
DMM M DM [ adopts neural networks for IV estimation inspired by the (optimally weighted) Generalized Method of Moments. The DeepGMM estimator is defined the solution of the following minimax game:
199
① arg min sup - Σ fz, T) g(a,, 0)) #Ee
Σ
0))
(48)
TET n i=1
200
where f2,, and g(,,,)) are parameterized by neural networks. As recommended in [T. we solve this optimization via adversarial training with the Optimistic Adam optimizer (5). where we set the parameter& 0 to the previous value of f.
201
203
DMLIV [15]: DMLIV [15] assumes that the data is generated via
Y= T==(X)A= s(X) + U,
(49)
where is the ITE f some function of the observed covariates. First, DMLIV estimates the functions 205 (X) = B[Y X]. MZ, X) = E[A| Z, X]. and p(X) = E[A X]. Then, the ITE learned by
minimizing the loss
(()) = Σ (y. - ((x:)) - f(x;, (h.,, x.)) ))- - p(x;))²
(50)
1=1
207
where i(X.-) -) is some model for T(X). In our experiments, we use (tunable) feed-forward neural networks for all estimators.
208
209
DRIV 15 DRIV [5] is meta learner, originally proposed in combination with DMLIV. requires
initial estimators for (XX), p(X), (X) = E|Z X = z²), and f(X) EjA- Z X x) well
211
as an initial ITE estimatior (X) (e.g., from DMLIV). The ITE is then estimated by a pseudo
212
regression on the following doubly robust pseudo outcome:
(Y - ((X) - Ama (A -(X))Z-A(XI)
YDR (X) +
(51)
f(X) - ((X)F(x
10

---

213
We implement all regressions using (tunable) ferd-forward neural networks.
Comparison between DRIV vs. MRIV: There two key differences between our paper and 15
215
(i) Our MRIV is multiply robust, while DRIV is only doubly robust. (ii) We derive a multiple robust convergence rate, while the rate in 15
216
is not robust with respect to the nuisance rates.
217 Ad (i): Both MRIV and DRIV perform a pseudo-outcome regression on the efficient influence function (EIF) of the ATE. The key difference: DRIV uses the doubly robust parametrization of the
EIF from whereas we use the multiply robust parametrization of the EIF from [17 Hence, MRIV frameworks extends DRIV in non-trivial way to achieve multiple robustness (rather than doubly robustness). Thus, our estimator is consistent in the union of three different model specifications rather than two
219
221
223
Ad (ii): Here, we compare the convergence rates from DRIV and our MRIV and, thereby, show the strengths of our MRIV. To this end, let us assume that the pseudo regression function is 7-smooth and
224
that we use the same second-stage estimator En, with minimax rate n for both DRIV and MRIV.
225
If the nuisance parameters ((X), ), p(X), f(X). and (X) are a-smooth and further are estimated
-In
227
with minimax rate n T Corollary from 15 states that DRIV converges with rate
(+) - -())) S nart + 22
228 In contrast, MRIV assumes estimation of the nuisance parameters MD (x) with rate n 20 N (1)
229 and dx(x) with rate n zo and T(()) with rate n 2..P. the initial estimator (I) converges with rate (n²) our Theorem yields the rate
E (FuRIv(#)) (())) S
(n)
-28 n 204
-2 22 + 20 20 + n 2 20 261p
If all nuisance parameters converge with the same minimax rate of n 20 the rates of DRIV and our MRIV coincide. However, different to DRIV, our rate is additionally multiple robust in spirit of Theorem 1. This presents a crucial strength of our MRIV over DRIV: For example, if is small (slow convergence of (x)), our MRIV still with fast rate long as a and / are large (i.e., if the other nuisance parameters are sufficiently smooth).
232
233
234
235
236
E.3
Wald estimator
Finally, we consider the Wald estimator [16] for the binary IV setting. More preciscly, we estimate 238
the ITE components n.! (x) and 11 (i:) seperately and plug them into
i (x) - P. (w)
T (x:
(52)
p (x) (x)
239
We consider two versions of the Wald estimator:
Linear: We use linear regressions to estimate the uu (x) and logistic regressions to estimate the UA (x).
242
BART: We use Bayesian additive regression trees trees to estimate the nu (I) and random forest
classifier to estimate the n (x:).
243
FFor a detailed discussion on multiple robustness and the importance of the EIF parametrization, we refer to 18].
Section 4.5.
On related note, a similar, important contribution of developing multiply robust method was recently made for the average treatment effect. Here, the estimator of []] was extended by the estimator of (17) to allow for multi robustness. Yet. this different from our work in that it focuses on the average treatment effect, while we study the individual treatment effect in our paper.

---

Visualization of predicted ITES
244
245
We plot the predicted ITEs for the different baselines and MRIV-Net Fig. (for n 3000). As expected, the linear methods (2SLS and linear Wald) are not flexible enough to provide accurate ITE estimates. We also observe that the curve of MRIV-Net without MRIV quite wiggly, i.e., the estimator has relatively large variance. This variance is reduced when the full MRIV-Net is applied. As result, curve much smoother. This is reasonable because MRIV does not estimate ITE components individually, but estimates the ITE directly via the Stage 2 pseudo outcome regression. Overall, this confirms the superiority of our proposed framework.
247
249
251
Nethod TARkiet
Method 2515
Nethod
Nethod DFV
Method
Method
Methed
Nethod (incar)
Mathod - Wold (BAAA)
Method Milli-bee (nntwork only)
Method MRUV-Net
-2
Figure 3: Predicted ITEss (blue) and oracle ITT (red) for different baselines.
12

---

G
Implementation details and hyperparameter tuning
252
253
Implementation details for deep learning models: To make the performance of the deep learning
254
models comparable, we implemented all feed-forward neural networks (including MRIV-Net) as follows: We use two hidden layers with RELU activation functions. We also incorporated a dropout
255
256
layer for each hidden layer. We trained all models with the Adam optimizer [9] using 100 epochs. Exceptions are only DFIV and DeepGMM, where we used 200 epochs for training, accounting for
257
258
slower convergence of the respective (adversarial) training algorithms. For DeepGMM, we further
used Optimistic Adam (5) as the original paper.
259
260
Training times:
We report the approximate times needed to train the deep learning models on our simulated data with 5000 in Table [ For training, we used an AMD Ryzen Pro 7 CPU.
261
262
Compared to DMLIV and DRIV, the training of MRIV-Net is faster because only a single neural
263
network is trained.
Table
Training times for deep learning models (in seconds).
TARNet
TARNet DR
DFIV
DeepIV
DeepGMM
DMLIV
DMLIV + DRIV MRIV-Net
10.62
288.7
1644.9
~30.21
17.31
74498
91.12
~32.00
We performed hyperparameter tuning for all deep learning models (including MRIV-Net), KIV, and the BART Wald estimator on all datasets. For all methods except KIV and DFIV, we split the data into training set (80%) and a validation set (20%). We then performed 40 random grid search iterations and chose the set of parameters that minimized the respective training loss on the validation set. In particular, the tuning procedure was the same for all baselines, which ensures that the performance gain of MRIV-Net is due to the method itself and not due larger flexibility. Exceptions are only KIV and DFIV, for which we implemented the customized hyperparameter tuning algorithms proposed in (14) and (2I) to ensure consistency with prior literature. For the meta learners (DR-learner, DRIV, and MRIV), we first performed hyperparameter tuning for the base methods and nuisance models, before tuning the pseudo outcome regression neural network by using the input from the tuned models. The tuning ranges for the hyperparameter are shown in Table 2 These include both the hyperparameter rangers shared across all neural networks and the model-specifie hyperparameters. For reproducibility purposes, we publish the selected hyperparameters in our GitHub project .yaml files
Hyperparameter tuning:
265
257
268
289
270
271
272
273
274
275
276
Table 2: Hyperparameter tuning ranges.
MODEL
HYpeRpARAMETE
TUNING RANGE
Feed-forward neural networks (Shared parameter ranges for all deep learning baselines)
Hidden layer size(es)
p. 5y, 10p, 20p, 30p (simulated data)
p. 3p. 5p. Sp, 10p (OHIE)
Learning rate
0.0001, 0.0005, 0.001, 0.005, 0.01
Butch size
64, 128, 256
Dropout probability
0, 0. 0.2, 0.3
KIV
X (Ridge penalty first stage)
5, 6.6. 7, 8, 9, 10, 12
{ (Ridge penalty second stage)
5, 6. 7, 8, 9, 10, 12
DFIV
A (Ridge penalty finst stage)
0.0001, 0.001, 0.01, 0.1 (simulated data)
0.01, 0.05. 0.1 (OHIE)
Ay (Ridge penalty second stage)
0.0001, 0.001, 0.01, (simulated data)
0.01, 0.05, 0.1 (OHIE)
DccpGMM Wald (BART)
xy (leamning rate multiplier) Number of trees (BART)
0.5, 1. 1.5, 2,5
20, 30, 40, 50
Number of trees (Random forest classitier)
20, 30, 40, 50
network input size
278 Hy Hyperparameter robustness checks: We also investigate the robustness of MRIV-Net with respect to hyperparameter choice. To to this, we fix the optimal hyperparameter constellation for our simulated 280 data for = 3000 and perturb the hidden layer sizes, learning rate, dropout probability, and batch size.
*Codes are in the supplementary https://anonymous.oo wwwp:oooouoooppscccceooee hwopeensciencc///MRiiiieeeee public GitHub repositoryy.
materials. (Upon acceptance, we replace the link and point
Codes are also available at
13

---

The results are shown in Fig. We observe that the RMSE only changes marginally when perturbing the different hyperparameters, indicating that our method
282
certain degree robust against hyperparameter misspecification. Furthermore, our results indicate that the performance improvement of MRIV-Net over the baselines observed in our experiments is not due to hyperparameter tuning,
284
but to our method itself.
Parameter
Hidden size
Parameter Learning rate
0.5
RMSE
0.2
0.1
20
25
30
35
0.0025 0.0050 0.0075 0.0100
Value
Value
Parameter Dropouu prob.
Parameter Batch
0.5
RMSE
0.1
0.0
0.1
0.2
50
100
150
200
250
Value
Value
Figure 4: Robustness checks for different hyperparameters of MRIV-Net.
14

---

H
Results for semi-synthetic data
286
287
In the main paper, we evaluated MRIV-Net both on synthetic and real-world data. Here, we provide additional results by constructing semi-synthetic dataset on the basis of OHIE, is common practice causal inference literature to use semi-synthetic data for evaluation, because it combines advantages of both synthetic and real-world data. On the one hand, the real-world data part ensures that the data distribution is realistic and matches those in practice. On the other hand, the counterfactual ground-truth is still available, which makes it possible to measure the performance of ITE methods.
289
290
291
We construct our semi-synthetic data as follows: First, we extract the covariates X IR and instru- ments Z E {0, 1} of our OHIE dataset from Sec. D Then, we construct the treatment components M (x) via
294
295
(x) 0.3 (X1)) + 0.7 and μ (X) 0.3 (X1),
(53)
where X1 is the (standardized) age and (-) is the sigmoid function. The outcome components constructed via
297
(XX)= = 0.5X,
Σ 1=2
x?
and pá (X) -0.5X
X
(54)
We then sample treatments A and outcomes Y as in Eq. 31 and Eq. 32 Lemma ensures that
n (x) Y Z i, X] and (x) = R[A 2
289
300
Given the above, the oracle ITE becomes
X7 0.7
(X)
(55)
Note that T(X) ) is sparse in the sense that it only depends on age, while the outcome components depend on all five covariates. Following our theoretical analysis in Sec. [ MRIV-Net should thus outperform methods that aim at estimating the components directly. This is confirmed in Table
302
303
304
where we show the results for all baselines and MRIV-Net on the semi-synthetic data. Indeed, we
305
observe that MRIV-Net outperforms all other baselines, confirming both the superiority of our method
306
as well as our theoretical results under sparsity assumptions from Sec.
Table 3: Results for semi-synthetic data.
Method
3000
5000
8000
(1) STANDARD ITE
TARNet 13
1.66 + 0.11
1.58 + 0.07 1.22 + 0.37
1.57 + 0.11
TARNet DR 13
8
1.31
0.28
1.12 0.15
(2) GENERAL IV
2SLS 19
0.06
1.31
1.32
KIV
1.97
1.92
1.05
DFIV
1.67
0.47
1.45
Deepl
1.24 +
0.22
0.84 184+0.10 0.19
DeepGMM
1.39 + 0.03
0.16
1.18 0.16
DMLIV 15
2.12 + 0.10
2.09 +
2.02 + 0.11
DMLIV DRIV 15
1.22 0.10
0.19
1.00 + 0.08
(3) WALD ESTIMATOR 166
Linear
1.42 + 0.24
1.28 + 0.07
1.32 + 0.07
BART
1.48 0.24
1.29 + 0.04
1.06 0.13
MRIV-Net (network only)
1.11 + 0.15
0.84 + 0.14
0.95 + 0.21 0.78 0.26
MRIV-Net (ours)
0.71 + 0.24
0.75 + 0.18
Reported: RMSE (mean + standard deviation).
Lower = better (best in bold)
15

---

I
Results for cross-fitting
307
308
Here, we repeat our experiments from the main paper but now make use of cross-fitting. Recall that, in Theorem 2 we assume that the nuisance parameter estimation and the peeudo-uuccome regression are performed on three independent samples. We now address this through cross-firting. To this end, our aim is to show that our proposed MRIV framework is again superior.
309
310
312
For MRIV, proceeded as follows: We split the sample DD into three equally sized samples D1, D2, and D3. We then trained (x), Ag (I)) and A (z) on Dr, 8 (z) and क (zz on D2, and performed the pseudo-outcome regression on D3. Then, we repeated the same training procedure two times, but
313
314
315
performed the eseudo-outcome regression on D2 and D1. Finally, we averaged the resulting three ITE estimators. For DRIV, we implemented the cross-fitting procedure described in [5] For the
316
317
De-learner, we followed g
318
The results are in Table H Importantly, the results confirm the effectiveness of our proposed MRIV. Overall, we find that our proposed MRIV outperforms DRIV for the vast majority of base methods when performing cross-fitting. Furthermore, MRIV-Net is highly competitive even when comparing with the cross-fitted estimators. This shows that our heuristic to learn separate representations instead of performing sample splits works in practice. In sum, the results confirm empirically that our
319
320
321
322
323
MRIV is superior.
Table 4: Results for base methods with different meta-learners (i.e., DRIV, and our MRIV) using cross-fitting and results for MRIV-Net without cross-fitting.
n 3000
5000
8000
Mota-leamers
DRIV
MRIV (ours)
DRIV
MRIV (ours)
DRIV
MRIV (ours)
Base methods
(1) STANDARD ITE
TARNet 13
0.30 + 0.02
0.36 0.16
0.18 0.06
0.16 - 0.03
0.21 = 0.08
0.13 + 0.04
TARNet DR-aaaerer
13
0.85
0.60
0.08
0.67
(2) GENERAL. IV
2SLS
19
0.33
+ 0.09
0.20 ::00::0 0.07
0.23 0.11
0.24
0.14 + 0.02
KIV
0.47
0.45
+ 0.15
0.20
0.19
0.22
0.15 0.03
DFIV
0.35 + 0.05
0.28 + 0.09
0.22
0.18 0.08
0.24
0.16 0.04
DecpIV
0.38 - 0.09
0.16
0.20
0.19 0.07
0.20
0.12 0.02
DeepGMM
0.42 - 0.09
0.42 0.16
0.19
0.19 0.00
0.22
0 0.06
0.13 0.02
DMLIV
0.44 + 0.09
0.46
0.166
0.21 0.004
0.19 - 0.07
0.21
0.05
0.14 0.02
(3) WALD ESTIMATOR 16
Linear
0.47 =: 0.23
0.36 + 0.12
0.01
0.20 + 0.08
0.22 002200 0.05
0.15 + 0.02
BART
0.43 + 0.12
0.39 + 0.12
0.14 0.05
0.13 - 0.05
0.23 0.08
0.15 + 0.02
MRIV-Netww netwerk only (ours)
0.35 4 0.12
0.26 + 0.11
0.19 0.13
0.15 - 0.03
0.18 0.08
0.13 0.03
Reported: RMSE (mean + standard deviation). Lower = hetter (hest bold)
16

---

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324
325
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66
Amy Finkelstein et al. "The oregon health insurance experiment: Evidence from the first year".
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17


Reviews:
[{"review_id": "x9w-2vkGEIQ", "reviewer": "Reviewer_biU1", "summary": "The authors propose an algorithm for estimating ITE for binary instrument and binary treatment. The procedure has three steps.\n-implement some initial estimator of ITE\n-construct pseudo-outcome as sum of the initial estimator and a term combining several nuisances\n-regress the pseudo-outcome on covariates\nFor this algorithm, the authors prove excess risk that scales as the sum of an oracle rate and several product rates.", "questions": "Please address the items in Originality, Quality, and Clarity and I will increase the score", "limitations": "N/A", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Originality\n\nI will raise the score if the authors improve the framing of the contribution in the context of previous work.\n\n1. The pseudo-outcome technique is relatively popular in causal inference; see e.g. Kennedy et al.’s work on dose response (2019) which Semenova and Chernozhukov (2021) extend to ITE. This paper extends those techniques, so I would like them to be cited.\n\n2. The main theoretical result extends an analogous result by Kennedy (2020), which somehow doesn’t get mentioned until Section 4.2. Moreover, citations for ITE without unobserved confounding could be more generous; it is a vast literature.\n\n3. To distinguish the originality of this paper relative to Syrgkanis et al (2019), I would like to see a discussion that compares the estimation procedures and their guarantees. If I understand correctly, the Syrgkanis et al (2019) estimator proposes a Neyman orthogonal loss while this work used the pseudo-outcome technique, yet both provide excess risk rates that involve products of nuisance rates. \n\n4. Estimation that uses the efficient influence function to study ITE in the IV setting has been proposed by Ogburn et al (2015). Estimation that combines the efficient influence function with sample splitting and machine learning has been analyzed in the semiparametric IV setting, e.g. Singh and Sun (2019). This paper goes further than these previous works in proposing an ML estimator for ITE. I would like these previous works to be cited as well.\n\nQuality\n\n1. I will raise the score if the experiments are made to align more closely with the theory.\n\n-line 284. Is there any sense in which the proposed nuisance estimators would satisfy Assumption 3?\n\n-line 294. I would like to see the experiments with sample splitting since that is the proposal in the theory section\n\n-line 313. The comparisons are hard to understand. The various estimators have different estimands. For example, DeepIV estimates a function that is not the ITE. So what is being reported here? These comparison estimators also involve sample splitting in some cases, to align with their theory, while the proposed estimator is not implemented with sample splitting (though its theory requires it)\n\n2. I would like more discussion in the case study. In line 386, at what values are these variables fixed? Simply fixing values means that the quantity visualized in Figure 1 is not the ITE. Nor is it the ITE conditional on some subset of covariates. Please interpret what is being displayed.\n\n3. The theoretical result, stated in Section 4.2 and explained in Section 4.3, is well written and interesting. I would like to see the additional assumptions of Theorem 2 stated in the main text, rather than pointing to another paper. Then the reader can assess whether they are “mild” or not.\n\nClarity\n\n1. There were several instances in which the authors made statements that were either unclear or untrue as written. I will increase the score when these are addressed.\n\n-line 25. The definition of unconfoundedness is incorrect. The given definition is an example of when unconfoundedness holds.\n-line 92. Exclusion is not unrestrictive.\n\n-line 107. The descriptions in Assumption 2 are misleading. The assumptions state that U can essentially be differenced out. It is an additive equi-confounding.  \n\n-line 117 (footnote). The definition of compliers is incorrect. It is the subpopulation for whom A(1)>A(0).\n\n-line 142. This statement is misleading. Arguably this paper requires a type of additive separability of confounding as well; see the comment above.\n\n-line 223. This is a misleading citation. Cross-fitting is a type of sample splitting that has the same theoretical analysis. It is not a practical shortcut. The cited paper does not advocate fitting all nuisances on the same data set; that approach is only valid when the function spaces have limited complexity.\n\n2. Please provide some intuition for the first term in equation 3. It helps to group \\hat{\\tau}_{init} and look at what residual is being extracted from Y before being reweighted.\n\nSignificance\n\nMy assessment of significance will hinge upon the authors’ improvements described above. There are currently too many issues for this paper to be a significant contribution, but I am optimistic that these issues can be addressed in the revision.\n", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "FfQknu4vi6I", "reviewer": "Reviewer_Fvf9", "summary": "This paper presents a two-stage regression method: In stage 1, they estimates nuisance components $\\{\\pi(x), \\mu^Y_0, \\mu^A_0, \\delta_A(x)\\}$; In stage 2, given an arbitrary initial ITE estimator ${\\tau}_{\\text {init }}(x)$ and  nuisance estimates $\\{\\pi(x), \\mu^Y_0, \\mu^A_0, \\delta_A(x)\\}$  (obtained from stage 1), they define a pseudo outcome for ITE estimation.  Theoretically, the framework proposed in this paper yields multiple robust convergence rates. \n\n----\nI thank the authors for addressing my comments. I increase my score to 6.", "questions": "In Figure 1, is it necessary to assume that X and U are independent? \n\nLines 79-82: Does this article focus on observational data or experiments data? \n\n$\\hat{Y}_0$ is usually used to denote $\\mathbb{E}[Y|T=0]$ instead of ITE $\\tau$, which is confusing. \n\nTypos: Line 157: “We then derive we derive …”\n\nMaybe the author can give a brief conclusion and future exploration. ", "limitations": "I do not foresee any major limitations and/or societal impacts. ", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper builds upon the theoretical results and propose a deep neural network architecture called MRIV-Net for ITE estimation. However, the proof of the theory and Appendices A-G are missing, which I cannot find in the supplemental material. This makes me doubt the solidity of the method and results. I will adjust my score according to the authors’ responses to my comments.\n\n**Strengths**\n\n- This paper considers a very important and interesting problem in Treatment Effect and proposes a novel estimator using binary instruments. \n- The paper is very well written and provides extensive evaluation and discussion/comparison of past work. \n- Theoretically, I believe that Theorem 1 is correct, and MRIV has Multiple robustness properties. But I think it's necessary to give a brief description or simply express the process with an example: \n\n(1) Assuming $\\hat{\\mu}_0^Y=\\mu_0^Y, \\hat{\\mu}_0^A=\\mu_0^A, \\hat{\\delta}_A=\\delta_A, \\hat{\\pi}=0.5, \\hat{\\tau}=0$: \n\n$\\hat{Y}_0 \\leftarrow\\left(\\frac{Z-(1-Z)}{\\delta_A(X)}\\right)\\left({Y-{\\mu}_0^Y(X)}\\right)$;\n\n(2) Assuming $\\hat{\\mu}_0^Y=0, \\hat{\\mu}_0^A=0, \\hat{\\delta}_A=\\delta_A, \\hat{\\pi}=\\pi, \\text{and } \\hat{\\tau}=0$: \n\n$\\hat{Y}_0 \\leftarrow \\left(\\frac{Z-(1-Z)}{\\delta_A(X)}\\right)\\left(\\frac{Y}{Z {\\pi}(X)+(1-Z)(1-{\\pi}(X))}\\right)$;\n\n(3) Assuming $\\hat{\\mu}_0^Y=0, \\hat{\\mu}_0^A=0, $ $\\hat{\\delta}_A=1, \\hat{\\pi}=\\pi, \\text{and } \\hat{\\tau}=\\tau$: \n\n$\\hat{Y}_0 \\leftarrow\\left({Z-(1-Z)}\\right)\\left(\\frac{Y-A \\tau}{Z {\\pi}(X)+(1-Z)(1-{\\pi}(X))}\\right)+{\\tau}$. \n\n$\\mathbb{E}[\\hat{Y}^0∣X=x]=τ$. \n\n**Weaknesses**\n\n- For Eq. (3), at least two models need to be correctly specified to guarantee robust learning, which is a stronger condition than robust learning. The latter only requires either the propensity model or the potential outcome estimation model to be specified accurately. \n- The proof of the theory and Appendices A-G are missing, which I cannot find in the supplemental material. This makes me doubt the solidity of the method and results. \n- This paper should introduce the concept of multiple robust in the Introduction Section, and roughly explain that the original problem is simplified to 5 nuisance estimators —— the solution is still robust even if some models are misspecified. ", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "3EagjYV3X2K", "reviewer": "Reviewer_2sYo", "summary": "This paper studies a multiply-robust estimate of the conditional average treatment effect (CATE), $ \\tau(x) = \\mathbf{E}[Y(1) - Y(0) \\mid X=x] $ using a binary instrumental variable (IV) for identification. The approach is to construct a transformed outcome, $\\hat{Y}_0$ such that $E[ \\hat{Y}_0 \\mid X=x] = \\tau(x)$, and then use a machine learning regression model to estimate the mean of $\\hat{Y}_0$ using $X$. The paper shows that such an approach can improve convergence rates over naive approaches such as plug-in estimates typically used for IVs.", "questions": "How is this work different from DRIV method of Syrgkanis et al., 2019?", "limitations": "N/A", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is reasonably well written. While it focuses a lot of attention on the convergence rates, which require following a lot of subtlety that makes it difficult for the reader to fully understand the main message, it is reasonably high quality and appropriate for publication on the topics of clarity and quality.\n\nUnfortunately, the authors seem to have missed a key paper on this topic. Their abstract says \"[t]o the best of our knowledge, our MRIV is the first multiple robust machine learning framework tailored to estimating ITEs in the binary IV setting.\" This is strange, as they cite  Syrgkanis et al., 2019 [29 in the paper], who provide almost an identical method for inference (up to some algebra and mild re-parameterization), show that it is multiply robust, and provide convergence rates under certain conditions, similarly emphasizing the dependence on the nuisance parameter convergence rates.\n\nEdit: it appears that \"multiple\" (meaning more than double) robustness is the key innovation here, and that there is a nontrivial practical difference between it and double robustness. See the conversation in the reviews for more details. While the transformed outcomes are very similar, and have the same conditional mean, suggesting some way of transforming from one to the next, the differences are nontrivial, and allow for construction of different nuisance parameter estimates that allow for improved performance. The authors have clarified this in their updated version, and therefore, I believe the paper is appropriate for publication.", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Reject