VCNet and Functional Targeted Regularization For Learning Causal Effects of Continuous Treatments
Authors: Lizhen Nie
Mao Ye
qiang liu
Dan Nicolae
Venue: ICLR 2021 Oral (2021)
Keywords: causal inference
continuous treatment effect
doubly robustness

Abstract: Motivated by the rising abundance of observational data with continuous treatments, we investigate the problem of estimating the average dose-response curve (ADRF). Available parametric methods are limited in their model space, and previous attempts in leveraging neural network to enhance model expressiveness relied on partitioning continuous treatment into blocks and using separate heads for each block; this however produces in practice discontinuous ADRFs. Therefore, the question of how to adapt the structure and training of neural network to estimate ADRFs remains open. This paper makes two important contributions. First, we propose a novel varying coefficient neural network (VCNet) that improves model expressiveness while preserving continuity of the estimated ADRF. Second, to improve finite sample performance, we generalize targeted regularization to obtain a doubly robust estimator of the whole ADRF curve.

Published as a conference paper at ICLR 2021
VCNET AND FUNCTIONAL TARGETED REGULARIZA- TION FOR LEARNING CAUSAL EFFECTS OF CONTINU- OUS TREATMENTS
Lizhen Nie" Mao Ye* Qiang Liu", Dan Nicolac*
1 DDpartment of Statistics, The University of Chicago
2 2Department of Computer Science. University of Texas at Austin
11 1i wch caaoc edu, my21@cs.. utexas. lqiangecs utexas edu, nicola@staaiistcas htw.:/aataatttttcs.............e
ABSTRACT
Motivated by the rising abundance of observational data with continuous treat- ments, we investigate the problem of estimating the average rese-ees curve (ADRF). Available parametric methods are limited their model space, and previ- ous attempts in leveraging neural network to enhance model expressiveness relied on partitioning continuous treatment into blocks and using separate heads for each block; this however produces in practice discontinuous ADRFs. Therefore, the question of how to adapt the structure and training of neural network to estimate ADRFs remains open. This paper makes two important contributions. First, we propose a novel varying coefficient neural network (VCNet) that improves model expressiveness while preserving continuity of the estimated ADRF. Second, improve finite sample performance, we generalize targeted regularization to obtain doubly robust estimator of the whole ADRF curve.
INTRODUCTION
Continuous treatments arise in many fields, including healthcare, public policy, and economics. With the widespread accumulation of observational data, estimating the average dose-response function (ADRF) while correcting for confounders has become important problem (Hirano & Imbens, 2004; Imai Van Dyk, 2004; Kennedy et al., 2017; Fong et al., 2018).
Recently, papers in causal inference (Johansson al., 2016; Alaa & van der Schaar, 2017; Shalit et al., 2017; Schwab et al., 2019; Farrell et al., 2018; Shi et al., 2019) have utilized feed forward neural network for modeling. The success of using neural network model lies in the fact that neural networks, unlike traditional parametric models, are very flexible modeling the complex causal relationship shown by the universal approximation theorem (Csáji al., 2001). Also, unlike traditional non-parametric models, neural network has been shown to be powerful when dealing with nigh-dimensional input (i.e., Masci et al. (2011); Johansson et al. (2016)), which implies its potential for dealing with high-dimensional confounders.
successful application of neural network to causal inference requires specially designed network structure that distinguishes the treatment variable from other covariates, since otherwise the treatment information might be lost in the high dimensional latent representation (Shalit et al., 2017). However, most of the existing network structures are designed for binary treatments and are difficult to generalize to treatments taking value in continuum. For example, Shalit et al. (2017); Louizos et al. (2017); Schwab et al. (2019); Shi al. (2019) used separate prediction heads for the two treatment options and this structure is not directly applicable for continuous treatments as there is an infinite number of treatment levels. To deal with continuous treatment, recent work (Schwab et al., 2019) proposed a modification called DRNet. DRNet partitions a continuous treatment into blocks and for each block, trains separate head, in which the treatment is concatenated into each hidden layer (see Figure 2). Despite the improvements made by the building block of DRNet, this structure does not
"Equal contribution and corresponding authors.

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Published as a conference paper at ICLR 2021
Respanse
Response
Response
Truth
Venet
Drnet
Treatment
Treatment
Treatment
Figure I: Estimated ADRF on testing set from a typical run of VCNet and DRNet. From left to right panels are results on simulation, IHDP and News dataset. Both VCNet and DRNet are well optimized. Blue points denote DRNet estimation and red points VCNet. The truth is shown in yellow solid line.
take the continuity of ADRF (Prichard & Gillam, 1971; Schneider et al., 1993; Threlfall & English, 1999) into account, and it produces discontinuous ADRF estimators in practice (see Figure 1).
We propose new network building block that able to strengthen the influence of treatment but also preserve the continuity of ADRF. Under binary treatment, previous neural network models for treatment effect estimation use separate prediction heads to model the mapping from covariates to (expected) outcome under different treatment levels. When it comes to the continuous treatment case, by the continuity of ADRF, this mapping should change continuously with respect to the treatment. To achieve this, motivated by the varying coefficient model (Hastic & Tibshirani (1993), Fan et al. (1999), Chiang et al. (2001)), one can allow the weights of the prediction head to be continuous functions of the treatment. This serves the first contribution here, called Varying Coefficient Network (VCNet). In VCNet, once the activation function is continuous, the mapping defined by the network automatically produces continuous ADRF estimators as shown in Figure but also prevents the treatment information from being lost in its high dimensional latent representation.
The second contribution of this paper generalize targeted regularization (Shi et al., 2019) obtain a doubly robust estimator of the whole ADRF curve, which improves finite sample performance. Targeted regularization was previously used for estimating scalar quantity (Shi et al., 2019) and associates an extra perturbation parameter to the scalar quantity of interest. While adapting it to a finite- curve is far less straightforward. Difficulties
dimensional vector is not difficult, generalization arise from the fact that ADRFs cannot be regularized at each treatment level independently because number possible levels infinite and, thus, the model complexity cannot be controlled with the introduction of infinite extra perturbations parameters. Utilizing the continuity (and smoothness) of ADRF (Schneider et al., 1993; Threlfall & English, 1999). we introduce smoothing to control model complexity. Its model size increases in specific manner to balance model complexity and regularization strength. Moreover, the original targeted regularization in Shi et al. (2019) is not guaranteed to obtain a doubly robust estimator. By allowing regularization strength depend on sample size, we obtain a consistent and doubly robust estimator under mild assumptions. Noticing the connection between targeted regularization and TMLE (Van Der Laan & Rubin, 2006), a by-product of this work is that we give the first (to the best of our knowledge) generalization of TMLE to estimating function.
We do experiments on both synthetic and semi-synthetic datasets, finding that VCNet and targeted regularization boost performance independently. Using them jointly consistently achieves state-of- the-art performance.
Notation We denote the Dirac delta function by 8(-). We use E to denote expectation, IP to denote population probability measure and we write P(J) = [== (=)dd((=) Similarly, we denote P,, as the empirical measure and we write P, (f) = f (z))dP.())) We denote [n]] as the least integer greater than or equal n, and we denote [n] the greatest integer less than equal to We Rademacher random variables. We denote Rademacher complexity of a function class F
to denote R Rad,, (F) E (sup)ez Σ (X:))). Given two functions f1, f2 X' H. define 1/2 For function class F. define ||Fl sup || We denote stochastic boundedness with O,
Ilf f2llo= supzex [f((z) f2(2)) and Ilfi fallz= (Jeex (fi(x) fz(x)) dz
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DRNet
VCNet
(r, X)
H (),
X
Z
E
(t.
( XX
Figure 2: Comparison of network structure between DRNet and VCNet.
and convergence in probability with Op. Given two random variable X, and X2, X1 X2 denotes X1 and X2 are independent. We use On b, to denote that both o,/ and bn/a,, are bounded.
PROBLEM STATEMENT AND BACKGROUND
Suppose observe an i.i.d sample ((1,, wi, (01,4 where 24 (5,,4444 is realization of random vector (Y, X, 7) with support (Y' x X' x T). Here X a vector of covariates, is continuous
treatment, and is the outcome. Without loss of generality, we assume [0, 1]. We want to estimate the Average Dose Response Function (ADRF)
(+) == E do(T== = t)),
which the potential expected outcome that would have been observed under treatment level t. Suppose the conditional density of ' given X is ((T X). Throughout this paper, make the following assumptions:
Assumption 1. (a) There exists some constant e > such that x x)) 2 c for all x X and t e T. (b) The measured covariate X blocks all backdoor paths of the treatment and outcome.
Remark 1. Assumption (a) implies that treatment is assigned in a way that every subject has some chance of receiving every treatment level regardless of covariates, which is standard assumption to establish doubly robust estimators. Assumption (b) implies that the casual effect is identifiable, iee. can estimated using observational data.
VCNET: VARYING COEFFICIENT NETWORK STRUCTURE
Under Assumption I, we have
(t) [Y x
Thus, a naive estimator for v is to obtain an estimator ğ of and use {(t)
Σn f(t,,,), Here ((t, 2)) == E{Y X = x, T t) and ja is its estimator. Following Shi al. (2019), we utilize the sufficiency of the generalized propensity score x(t X) for estimating z (Hirano & Imbens, 2004):
(t) = EY T(t
X), =t]
indicates that learning क X) helps the removal of noise and distillation useful information in X for estimating v.. Similar Shi al. (2019). we add separate head for estimating T(t X). and use the feature z extracted by it for downstream estimation of p(t, x) (see Figure 2). Our contribution here is propose the varying coefficient structure of the prediction head for p(1, æ). which addresses difficulties confronting continuous treatment as discussed in the following paragraph.
3.1 THE VARYING COFFFICIENT PREDICTION HEAD
A naive method is to train a neural network which takes (t, x) imput in the first layer and outputs put, x) in the last layer. However, the role of treatment is different from that of æ and the influence of might be lost the high-dimensional hidden features (Shalit al., 2017). Aware of this problem, previous work (Schwab et al., 2019)
Our aim is to predict ptt, æ) = E(Y -, x).
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divides the range of treatment into blocks, and then use separate prediction heads for each block (see Figure 2). To further strengthen the influence of treatment t, Schwab et al. (2019) appends t to each hidden layer. One problem of this structure is that it destroys the continuity of by using different prediction heads for each block of treatment levels. practice, DRNet indeed produces discontinuous curve (see Figure 1).
In order to simultaneously emphasize the influence of treatment while preserving the continuity of ADRF, we propose varying coefficient neural network (VCNet). In VCNet, the prediction head for is defined as
# NN (t. æ) = (z),
where input z is the feature extracted by the conditional density estimator, and fe(() is a (deep) neural network with parameter (()) instead of a fixed O. It means that the nonlinear function defined by the neural network depends on the varying treatment level and thus we call this structure the varying coefficient structure (Hastie & Tibshirani, 1993; Fan et al., 1999; Chiang et al., 2001). For example, if ff is an one-hidden-laye layer ReLU network, we have fa(e) (z) = =E a, (t (t) and (t) are weights of the neural network, and (tt [(a (t), b (t)), (tD,t), (t))) Here we use splines to model (tt.. Suppose that ((t) = [01 (t).... Od (t) E Rddoe,, where do(n) is the dimension of ((). We have
2), where
(t(t) = Σ a (t). 1=1
where {4t MEz1 are the spline basis and as,e 's are the coefficients. Thus, we have
01,1
a1,
(l) (I) where A
and +(1) (1) (1...... . (D)]
Od
dd
It is worth mentioning that by choosing spline basis of the form I(to S t < t1) with different to. f1. we recover the structure in Schwab et al. (2019), which has a separate prediction head for each block. indicates that DRNet can also be viewed as a special case of VCNet (under a suboptimal choice of basis functions).
In VCNet, the influence of treatment effect on the outcome directly enters through parameters O() of the neural network, which distinguishes treatment from the other covariates and avoids the treatment information from being lost. Under typical choices of spline basis such as B-spline, once the activation function is continuous, VCNet will automatically produce continuous ADRF estimators.
3.2 CONDITIONAL DENSITY ESTIMATOR
Recall that the input feature z to NN is extracted by the conditional density estimator for itt x). Here propose simple network to estimate which is direct generalization of the conditional probability estimating head in Shi et al. (2019). Notice that E [0, 1] and the conditional density (t æ:) is continuous with respect to treatment i. for any given effectively approximated by piecewise linear functions. Thus, we divide [0, 1] equally into B grids, estimate the conditional density 7-- x)) on the (B + 1) grid points, and the conditional density for other t's are calculated via linear interpolation. To be more specific, we define the network x NN
continuous function can be
"grd
NN "grid (x) softmax(wzz) E B+1 where z z= fw)
Here z E R& is the hidden feature extracted by the network, wi is the parameter for the nonlinear mapping fwy W2 E (B+1xxh, NN (x:) = grd
U.NN (m:) ..... ....,
B.NN (=N((=)) and T grd (xx) is the estimated ANNN
conditional density of T = i/B given X æ. This structure is analogous to classification network by removing the last layer which outputs the class with highest softmax score. Estimation of conditional density at other t's given X
obtained via linear interpolation:
I NN æ))
f1 giia
(x)+ B
12. (z) Tiid
='1 "gi
(a:) (t tz), where t = |Bt] to [Bt].
This estimator NN is continuous with respect to for any given and we finally rescale it to yield valid density, i.e., n NN (t) |x) 2 0, Vt. xa and 1 NN (t) a: 1, Va.
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There are other options to estimate the conditional density. Popular methods include the mixture density network (Bishop (1994)), the kernel mixture network (Ambrogioni et al. (2017)) and nor- malizing flows (Rezende & Mohamed (2015), Dinh et al. (2016). Trippe & Turner (2018)). Here treatment levels are bounded, and thus Gaussian mixtures are not applicable. In current estimator, the linear interpolation for estimating conditional density on non-grid points be replaced by kernel smoothing, which is more computationally intensive due to the calculation of normalizing constant. Other techniques including smoothness regularization and data normalization (Rothfuss al. (2019)) can be implemented to further enhance the performance. However, density estimation is not the main focus of this paper. Thus, for simplicity, in all experiments we use the aforementioned method without other techniques and it works quite well on datasets we tried.
3.3 TRAINING
Notice that our model requires AN to extract good latent features as the input for nM to predict. This can achieved by training NN to estimate the conditional density, which motivates us train n and a NN simultaneously by minimizing the following loss:
CIN π NN
(y)
1 NN (t,, xx1)
a
log( (ts x;))).
(1)
In loss (1). the first term measures the prediction loss from NN The second term measures the loss from x NN and is the negative log likelihood. And controls the relative weights of the two losses.
Denote A. as the optimal solution of the above empirical risk minimization problem (1). After getting fr, one can estimate (-)) by (i.) = Σ ji(, æz). The correctness this naive estimator relies on whether the truth y is in the function space defined by the neural network model. However, we can plug f and â into the non parametric estimating equation (to be introduced later) to obtain doubly robust estimator of (t). In this way, are able to produce (asymptotically) correct estimator any one of μ or æ is in the model space of neural network. The next section discusses challenges in obtaining such doubly robust estimator and provides our solution.
FUNCTIONAL TARGETED REGULARIZATION
In this section we improve upon the previous method by utilizing semiparametric theory on doubly robust estimators. Doubly robust estimators
built upon #(t x) and fi(t, x), and it yields consistent estimator for e' even if one of them is inconsistent. When both # x) and p(t,, x) are consistent, doubly robust estimator leads faster rates of convergence. Here our task is to estimate the whole ADRF curve, which comes with additional challenges. First, we need to find a doubly robust estimator of (t)) for any to E [0, 1].
4.1 DOUBLY ROBUST ESTIMATOR
Before we proceed, we define the following quantity:
Sto (Y, X, T, 2to (Y, x, 14, ,,, 11llll X) ((oo),
where Ito (Y, X, 14, 5(T tn) Y ((T,
(T |X)
The following theorem serves as the basis for our subsequent estimators:
Theorem 1. Under assumption and assume that it x)) 2 for all a E X and I E T. For any to E T, Sto is the efficient inftuence function for (to)). Moreover, Sta is doubly robust in the sense that
X, R, 3² =
if either मे or = p. Further, if u 11 Op(ri(n) (1)) and lll allo0 Op(r2(n)), we have
sup EET
|PG (Y, (V,X, T, म A, v²) (r)
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Theorem shows that for any E T, (qto (Y, X, T, fr fa(t. X)) is doubly robust estimator for (to). Under some mild assumptions, one way to obtain a doubly robust estimator of i is to utilize the two-stage procedure from Kennedy al. (2017) by regressing
Y - X)
J, #(T ())F ((T,X)
7 (T
on T using any nonparametric regression methods like kernel regression spline regression.
4.2 TARGETED REGULARIZATION FOR INFERRING A FINITE DIMENSIONAL QUANTITY
However, as discussed in Shi et (2019), when estimating the term F (Y, X,,, #)) the #(T X) in the denominator might make the finite sample estimator unstable, especially in cases where Assumption ((a) is nearly violated. Targeted regularization is proposed by Shi al. (2019) to solve this issue. The key intuition of targeted regularization is to learn jú and स such that P (%: (Y, X, T, p. +)) s and thus the estimation of this term is no more needed. In the binary treatment where T {00, 1}. if we want to estimate (1)), which is single quantity, targeted regularization simultaneously optimizes over j NN, π NN and an extra scalar perturbation parameter c using the following loss
LTR(MAN NN ,c] L(I NNN + BRRR Ju NN :cl,
(3)
1
t
where RI
NN ,T .c] .NNNNcc
(ts, - NN xx))
with CTN π N defined in (1). Assume the complexity of the function space of μ MN and n NN is finite and since the complexity of the function space of the introduced perturbation is also finite, we have
1 a
P Pla XY, +)] = P [g (Y, X, T. PTR. #)] + RIR [á, म. # le=ē
2 de
-P (qn X T, p2 68(T t X)) op(1),
where (tR(l, x) = i0(,, x)) [({( and (ú, म, è) is the minimizer of (3). Notice that the first equality holds because at the convergence of the optimization, a. RTR[A â, le=e= And the last equality by uniform concentration inequality. This implies that - fTTR( (1, x) is a doubly robust estimator for (1) and for this estimator, conditional density estimator presents at denominator and thus it has more stable finite sample performance.
4.3 FUNCTIONAL TARGETED REGULARIZATION FOR INFERRING THE WHOLE ADRF
Notice that in loss (3), the scalar e is associated with scalar quantity for inference. One can generalize targeted regularization to estimate d dimensional vector by using separate c's (see Theorem 3 in the Appendix). Generalizing to a curve, however, is more challenging. We need to optimize over a function 7 - T
where e(-) the perturbation associated with v--). Optimizing over the function space of all mappings from T to R is not feasible in practice, and its high complexity will lead to overfitting.
Our solution is to utilize the smoothness of and æ (Prichard & Gillam, 1971; Schneider al., 1993; Threlfall & English, 1999). which allows us to use splines {yx Kn with K,, basis functions to approximate e(.). Here the subscript n K,, denotes that the number of basis functions might change with the sample size Define (-) = Σk= axFx(.). We use the following loss with Functional Targeted Regularization (FTR).
NNN NN [n] CIPNN NN + RETRIEN NN En
(4)
(x()))
where NN NN En]
Vi (t.,
NNN(L: x3;)
Here RETR denotes the FTR term and Bn O when XX.

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Remark 2 (On 3n). The targeted regularization proposed by Shi et al. (2019) uses fixed B. However, using fixed 3 might lead to the estimator constructed by targeted regularization no more consistent when ed is mis mis-specified, which means the estimator is more doubly robust. To overcome this issue, we make a slight change on B by allowing 3 to depend on n. Specifically, we find that once 3n o(1), we able to make sure that targeted regularization gives doubly robust estimator. See discussion at Remark 4 and Appendix A.1 for more details.
Demonstrating the asymptotic correctness of FTR more challenging than analyzing traditional targeted regularization. One reason is that we no more have 8 additional efforts, we will establish convergence rate for our estimator using loss (4) in Theorem Before we proceed, let pause bit and introduce some definitions, which will be used in the main theorem. Denote fú, î and én as the minimizer of (4). We use т and j to denote fixed functions to which म and ji converge in the sense that || - ol oy(1) and / 0ll.0 Op)).. We define NN (æ, t). We denote G, Q the function space in which gt. Hº,, at NN lies. We denote Bx, as the closed linear span of basis sp K" {
0g RTR(A, î. le= 0. With some
9r X I,,
The key intuition of the asymptotic correctness of FTR is that: once NN and a are uniformly upper/lower bounded and some other weak regularization conditions hold, we can show that (1)x()) e* (-)l,, letting AFTR == B + én/ ft, we have
(1) where (-) := IE [(Y p)) / T /E 2 TT= And thus
P llao X.T, AFTRR, +)] ((Y (FTR) / T to) =(o)) 2 [E - i - / / T to)] ((6) = O.
Assumption 2. We consider the following assumptions:
(i) There exists constant e > 0) such that for any 1 E T, x E X, and NN E U, we have 1/05 S a AN
2) S c, 1/c S (t x) S c, ||
and ||14| S.
(ii) Y = p(X, T) + V where EV = V x, V T,, and V follows sub-Gaussian distribution.
(iii) n, ps, NN and 1ea have bounded second derivatives for any NN E Q and NN E U.
(iv) Either ग = π or jt And Radn(9), Rada (2), Rad, = 0 (n-1/2).
(v) BK, equals the closed linear span of B-spline with equally spaced knots, fixed degree, and dimension Kn x 1/6
Theorem 2. Under Assumption and 2, let (0(.) = 1. E (D(mz)))
(,,()) #(-(ae)
we have
18 - - = =Op (n0(n-11 n11133 / Vlgn + +1)()r2((1))
1/3
where || - 111~0 = = Op(ri (m)) and 1 1111-111=0 0r
Remark 3. In Theorem 2. assumption (i), (iii) and the first half of (v) are weak and standard condi- tions for establishing convergence rate of spline estimators (Huang et al., 2003; 2004). Assumption (ii) bounds the tail behavior V. The second half of (v) restricts the growth rate of Kn, which is a typical assumption ( (Huang et al., 2003; 2004) but with different rate in order to obtain uniform bound. The first half of assumption (iv) states that least one of p., should be consistent. The second half of assumption (iv) considers the complexity of model space, and is a common assumption for problems with nuisance functions (Kennedy et al., 2017).
Remark 4. We want to point out that adding targeted regularization does not affect the limit of j and क in large sample asymptotics. That is, the limit of fa and ā using loss (4) will be the same as using loss (1). We refer the reader Appendix A. for a more detailed discussion and proof.
Notice that our proof for Theorem 2 can also be adapted for analyzing modified one-step TMLE (Van Der Laan & Rubin, 2006). With very similar assumptions, we could obtain double robustness and the same consistency rate for TMLE estimator.
Theorem 2 guarantees that if appropriately control the model complexity, under some mild assumptions, the estimator & from targeted regularization is doubly robust, and when both è and ji are consistent, the rate of convergence of & to the truth is faster than the individual convergence rate of â or Thus, using targeted regularization theoretically helps obtain better estimator of vb.
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Dataset
Model
Naive
Doubly Robust
TMLE
TR
Dragonnet
0.045 + 0.00094
0.026 + 0.0012
0.037 + 0.00086
0.028 + 0.00088
Simulation
Drnet
0.042 + 0.00090
0.023 + 0.0011
0.035 + 0.00083
0.027 0.00086
Venet
0.018 + 0.00098
0.022 + 0.0013
0.016 + 0.00082
0.014 0.00091
Dragonnet
0.350 + 0.016
0.307 1 0.016
0.252 + 0.0087
0.208 1 0.0072
IHDP
DRnet
0.316 + 0.016
0.274 + 0.014
0.274 = 0.018
0.230 + 0,0086
Venet
0.189 0.013
0.190 0.013
0.148 0.010
0.117 1 0.0085
Dragonnet
0.180 = 0.0081
0.155 + 0.0057
0.179 + 0.0080
0.149 + 0.0051
News
DRnet
0.183 = 0.0084
0.141 + 0.0054
0.183 + 0.0083
0.114 + 0.0041
Venet
0.028 = 0.0011
0.023 + 0.0013
0.028 0.0010
0.024 + 0.0009
Table 1: Experiment result comparing neural network based methods. TR refers to targeted regular- ization. Numbers reported are AMSE of testing data based on 100 repeats for Simulation and IHDP and 20 repeats for News, and numbers after + are the estimated standard deviation of the average value.
5 RELATED WORK
The Varying Coefficient Structure. Varying coefficient (linear) model is first proposed an extension of linear model (Hastie & Tibshirani, 1993; Fan et al., 1999) and is usually used for modeling longitudinal data (Huang et al., 2004; Zhang & Wang, 2015; Li et al., 2017; Ye al., 2019). The key motivation of varying coefficient model is a dimension reduction technique that avoids the curse of dimensionality for statistical estimation. Different from existing models, our varying coefficient structure is applied on a complex neural network model with a different motivation of enhancing the expressiveness of the treatment effect. Besides, building a hierarchical structure on the network parameter is also explored by the HyperNetwork (Stanley et al., 2009; Ha et al., 2016). Hypernetworks provide an abstraction that mimics the biology structure: the relationship between genotype (the hypernetorkk), and a phenotype (the main network). The weight of the main network is also a function of a latent embedding variable, which is learned with end-to-end training. Hypernetwork trains much smaller network to generate the weights of larger main network in order to reduce search space, while our network directly trains the main network whose weights are linear combinations of spline functions of treatment. Moreover our network is proposed in order to appropriately incorporate treatment into modelling, which is not touched upon in HyperNetwork.
Neural Network Structure for Treatment Effect Estimation. We refer readers to the introduction for the connections and comparisons with previous developments using feed forward neural network for treatment effect estimation. In addition feed forward neural network, previous results also utilize other networks to learn treatment effects. For example, Yoon et al. (2018), 8); Bica et al. (2020) estimated the causal effect via learning generate the counterfactual. Louizos al. (2017) learned the causal effect by learning deep variable models using variational autoencoder. Compared with our method, their approaches are mainly heuristic and do not provide theoretical guarantees for the asymptotic correctness of the estimator.
Doubly Robustness, TMLE and Targeted Regularization. Cheroozuukvv al. (2017: 2018) developed theory for 'double machine learning" showing the convergence rate for doubly robust estimator. Despite its good asymptotic property, doubly robust estimators can be unstable due to the presence of conditional density estimator at denominator. Targeted Maximum Likelihood Estimation (TMLE) (Van der Laan & Rose, 2011) and targeted regularization (Shi et al., 2019) are then proposed to overcome this issue by introducing an extra perturbation parameter into the model. To the best of our knowledge, previous works on TMLE and targeted regularization focused on estimating a single quantity, such (1) (0) in binary treatment (Shi et al., 2019; Van der Laan & Rose, 201 1) or averaged treatment effect (t) for continuous treatment (Kennedy et al., 2017), while we give the first generalization of targeted regularization and TMLE for inferring the whole ADRF curve.
EXPERIMENTS
Dataset. Since the true ADRF are rarely available for real-world data, previous methods on treatment effect estimation often use synthetic/semi-syntheiii data for empirical evaluation. Following this convention, we consider one synthetic and two remi-synthetic datasets: IHDP (Hill, 2011) and News (Newman, 2008). The synthetic dataset contains 500 training points and 200 testing points, with
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Method
Simulation
IHDP
News
Causal Forest
0.043 0.0021
0.97 + 0.034
0.211 + 0.003
BART
0.040 + 0.0013
0.33 + 0.005
0.066 + 0.003
GPS
0.028 0.0016
0.67 = 0.025
0.022 + 0.001
VCNet+TR
0.014 0.0009
0.12 + 0.009
0.024 + 0.001
Table 2: Comparison of VCNet against non-neural-netee netwrrl based baselines. Reported AMSE are averaged over 100 experiments for simulation and IHDP, and 20 experiments for News. Numbers after + are estimated standard deviation of the average AMSE.
the detailed generating scheme included in the Appendix. IHDP contains binary treatment with 747 observations on 25 covariates, and News consists of 3000 randomly sampled news items from the NY Times corpus (Newman, 2008). Both IHDP and News are widely used benchmarking datasets for binary treatment effect estimation, but here we focus continuous treatment and thus we need to generate the continuous treatment as well as outcome by ourselves. The generating scheme the Appendix. For IHDP and news, we randomly split into training set (67%) and testing set (33%).
Baselines and Settings. For neural network baselines, we compare against Dragonnet (Shi et al., 2019) and DRNet (Schwab et al., 2019). We improve upon the original Dragonnet and DRNet by (a) using separate heads for 7 in different blocks for Dragonnet, and (b) adding a conditional density estimation head for DRNet, since it has been suggested by Shi et al. (2019) that adding conditional density estimation head improves the performance. For non-neural-netwonr causal forest (Wager & Athey, 2018), Bayesian Additive Regression Tree (BART) (Chipman et al., 2010), and GPS (Imbens, 2000).
baselines, we consider
For VCNet, we truncated polynomial basis with degree and two knots {1/3, 2/3) (thus altogether 5 basis). Dragonnet and DRNet use 5 blocks and thus the model complexity of neural- network models are the same. In practice we may vary the degree and number of knots in VCNet, here the choice is made simply for fair comparison against Dragonnet and DRNet, ensuring the number of parameters of the compared models is the same. The other apper-parameters of each method on each data are tuned on 20 separate tuning sets. Due to space limit, we refer readers to the Appendix A.4 for more details on experimental settings.
Estimator and Metrics. To evaluate the effectiveness of targeted regularization, for all neural- network methods we implement four versions: naive version (with conditional density estimator head, trained using loss (I)), doubly robust version (Kennedy et al., 2017) by regressing (2) on treatment with j. स trained using loss (1), TMLE (Van Der Laan & Rubin, 2006) version with initial estimator trained using loss (1), and TR version trained using loss (4). For non-neural-nee network based models, we use the usual estimator. For evaluation metric, following Schwab et al. (2019), we use the average mean squared error (AMSE) on test set, where AMSE = ΣS f (t) - (t))3 and &. (t) the estimated (t) in the s-th simulation.
Results. Table 1 compares neural network based methods. Comparing results in each column, we observe a performance boost from the varying coefficient structure. Comparing results in each row, we find that naive versions consistently perform the worst, targeted regularization often achieves the best performance, whereas performance of doubly robust estimator and TMLE varies across datasets. In Table 2, we compare our approach with traditional statistical models. We observe that in simulation and IHDP, VCNet + targeted regularization outperforms baselines by a large margin. In News, its performance is close to the best one. The implementation be found in open source repository.
CONCLUSION
This work proposes novel varying coefficient network and generalizes targeted regularization to continuous curve. We provide theorems showing its consistency and double robustness. Experiments show that VCNet structure and targeted regularization boost performance independently and when used together, it improves over existing methods by a large margin.
fi ent hent-netwwr with funct ona 1rtr

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A APPENDIX
A. ON THE CONSISTENCY OF J AND π
We show that adding targeted regularization does not affect the limit of f and ø in large sample asymptotics. That is, the limit of fi and it using loss (4) will be the same as using loss (1).
Denote
PÉ(U NN
IP [(V - 2ZNN² + a log log N x)]+
n
P. PRE NN a INNN
(w6- - 1 (ts, x;))² + an NN (t: x;)
Lemma 1. Suppose that (y', x²), is the minimizer of loss Pe( x and (it Rt, En) is the minimizer of LFTR, then we have
Pe(a, at) - Pe(pl =(()
Proof. We have
E() +)) Pf(p, 4)- Pely's x) S Pn 4) P,,e(p', T²) (P)
((P-- -P.) T')1
(0)
- Pn R) - 1-e--P x)) + (1
(Pallic, 4) + RITR jút, #, km (Pe()', x²) RETRITII x000) + Op(n-1/8
+ 8n (RFTR [p', ,T, 0] RETR [i,, म. ēnl)
(8)
S Bn (ReIR (p', n/, 0] RETR (it, n (nl) Op(n 1122
S Bn RFTR ju', , + On 111)
( (1)) + (n 1//3),
where (a) follows from uniform concentration inequality using the fact that f NN N is uniformly bounded, Rad,, (2), Rad., (24) O -1/2) and the Lipschitz constant of log(x) is bounded when x E [1/c, cl for some finite c. (b) follows from the fact that (β,, म, én) is the minimizer of the empirical risk (with FTR). (c) follows from the fact that
Rlu T 0]
Σ i=1
(y: 11')
E (y 1²)² + +0p(n-1/3)
E(V?) + BBH - 21112 + Op 1/2)
O(1).
Now we prove that
[/ / 1
|| - (1).
r''l
(5)
For simplicity, we ignore the unidentifiability of neural network parameterization and assume (a', .') the unique minimizer in the sense that for any e 0, there exists r7(c) > 0 such that
inf -oNN-'llaa-lil - - l'a
[IP([ NN pe(y', n²)] > n)()).
(6)
If Equation (5) is not true, then there exists > 0 such that for any N > 0, there exists N such that || - 1'll 52 + || - n z2 2 s. From Lemma 1, we know that P!(A, R) - Pl x') s (s)) when n is sufficiently large. Thus, there exists no such that
|| - م '2= + || T'l 2 s, Pe(p. #) Pe(p', x') ))==))))) S nss),
which contradicts (6).
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A.2 TECHNICAL PROOFS
In this section, we prove the two main theorems (Theorem I and Theorem 2) and give some additional results which are mentioned briefly in the main text.
A.2.1 NOTATIONS AND DEFINITIONS
We denote 5tg as the Dirac measure centered on ig and recall that 5(.) denotes the Dirac delta function.
We use On S bn to denote that on I Cb, for some C. > 0 for all sufficiently large n. We denote
1n 11,1,... 1)" E RR. For any function f and function spaces Fi, F2. we write Fi + F2 = {11 f2 fi E F1 E F2). FiJ2 = {fis2 fi E F1, E F2). JF = (sh h E F}.
fo {f ofoh: :hhhhEr FF and F {f f F). E.
We define
(- = P - Pnt/ I = 1//
where (Ma, An an) is the minimizer of loss (4). We denote (.).) = Σx= (uspp()) as the spline regression estimator of e(-). With some slight abuse of notation, we denote
8 *M (1) (421 (2(()) (K.. (1)) E ER""" x Kn
Bn = (t2 (t,), (tn))) " E Ruxk.
We define
II, = diag ( (t1 x1); At2 x2)) ... , (n xn)),
ñr diag
(IP (# (TT x) T = [n)] 1/2
[P
(T X) T = =b,)] 112)
We define
yi un (1,,,
Zn = (21,,2,"" ;,))
=n) E IR"
where
21
* (1: xes)
Y
Prn (T X)
Zn (31, "2, ",,)" E R"
where
= P
T ti X)
tn (T
Notice that
â = 11 Bn) -1 BTT Zn.
and we denote
ā = (BT
-1 B1 Bn) Ba) 2 2n.
A.2.2 USEFUL LEMMAS
This section gives lemmas which are used in our proofs of main theorems.
Recall that Theorem consists of two parts: the efficient influence function of (to) and its double robustness. Notice that vtoa) can be written as a special case of more general parameter r (r)(t:P) (see Remark 5). Here (t: PT) is a function of which depends on the probability measure Pr. For brevity we also write 7(t) := >(t: when the corresponding Pr is the true probability measure of treatment T.. The following Lemma gives the efficient influence function of T.
Lemma 2. The efficient inftuence function for I' is
(x + / plt, (T( ([(T;)) J P(T, x)dP()
-/ (()
EET (T((T) / u( ((( (T;(T))
where {(Y, X, T. n. I.
Y- Sx (T| x)dP(a) + (d) %(())
(T(
drlt::TTe)) 15-0, f. (t: 0)
Blo (t)
Ie-o and Pr,c is a parametric submodel with parameter
E R and (.) = Pr()).
de
de
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d'r(-)' we get I (to)) Setting 7(-) 1, we get get = _T (tt)dPr((t) which is the average outcome under randomized trial and a quantity of interest in its own right (Kennedy et al., 2017). Setting 7(-) aft .-
Remark 5. Setting 7())
d
we get = (01) - -0))), which is the average d d
treatment effect under binary treatment setting.
Recall that in Theorem 2, én is the spline regression estimator of e. In order to establish the convergence rate of our final estimator v. we need to establish the convergence rate of é,, first.
Lemma 3. Under assumptions in Theorem 2, we have
- Op (n-1/3 ||e, - -x1112 =
Remark 6. For fixed objective function, the convergence rate of the B-spline estimator to the truth standard result (Huang et al., 2004; 2003), which is (n 2/5) when choosing Kn. x n n=15 However, Lemma gives a uniform bound on a class of functions in order to deal with the fact that in and Pn are NOT fixed and dependent the observations. The bound is thus of larger order n 1/3 Vlogn n. when choosing the optimal Kn = n 1/6
Lemma 4. Under ussumption Theorem 2, there exist positive constants M1 and M2 such that except event whose probability tends to zero, all the eigenvalues of (Kn/n) n) B - B, fall between Mj and M2, and consequently, (Kn/n) B 11,11"2"00000 B,, invertible.
Lemma 5. Assume || < co and F2 < 00, we have
1
Rad, (FiF2) S (Rad Rad, Rad, [[5
A.2.3 PROOF OF THEOREM
Proof. Denote Pt;
do., (f) [P)(0] Then the efficient influence function St, of rb(ot is obtained by
plugging the definition of (t) in Lemma and some simplifications using
(t; Pr) (t;;;;
(1;0)
So here we only need to prove that the efficient influence function is doubly robust. We have
PG. (Y,
8(T - tv) Y -i((TXX)
P
+ Atto. X) - -vi(to)
(
/ (()=)T(( 2z)6(1 - 10) jlt,, x) fu(t, x)) dodd + (illo, ())())d - / u((o: x)) (x))da
#(t x))
(6)
ittol x)) -1) (p(to, x) - Pato, x:) d((()),
7 x))
(8)
where (a) follows from iterated expectation, (b) follows from << t) = x(t ())(x)/m(t). From the last line of Equation (8), it is obvious that the desired conclusions hold.
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A.2.4 PROOF OF THEOREM 2
Proof. First, from condition (i), we have
1
dP, (X) TP 6(T -,) x - ((TTx)
(()) x #(- X)
#[T X)
/x
1
dlP, (X) -()) (y=
An (x, T)
(())
IT
x a(:1 x)
ān (X,T) T)
1
< (8(+) {())
dP (X)
((+)
dP dXp(() x(-)PP
(3'(. X) T 1.2
x #(-1 X)
x #(-) X))
12
((() (/x .(.1 X)
1
T = =
$ 1 2 +
d (P,, P) (x)
+ +((-)
dP(x) TT-)
#(. X)
#2(. X)
x
12
S E1 E + |(()) (// #(-1 x))
(P,, - -P) (x)
12a
+ +|IP PP [xXX, T) pna(X (X, T)
1 1 #(.
X)
T(.
X) 1
dIP (X)
in (X,T)
#(.1
X)
#(: X
12
(0) Op n11// Vlogg n + (1)))2(()))
where (a) follows from Lemma 3, which says ||ē, - Enllı= = 1133 (o.nn1/3/logm
From generalization bound and condition (iv), we know
sup (0,10
Σ 1=1
Pm (0), (inn(X,(o) 40) = sup
[Pnnto - -Por (X)I = Opp(nn111
1oE(0.1]
Thus,
1
P (xx;)) - (x,)
-0ppn-1113)
(10)
i=1
Recall that Theorem 1 says that if su sup xEx |in(l X) - x((1] X) (p(r((n)).
suPrejo.1] supxex I (t, X) Q X) (p(
then
8(T to) Y ur (T, X)
+ jún (1o, x)
- vi(to) (p(m(())2(mm)). (11)
sup te
tn (T) X
Combining Equation (9), Equation (10) and Equation (11), using triangle inequality, we have
1 (() //x # xx) dx,(x)) (n (1))
1 n
=O 113 Vlog n + (1)))2(n))
n
L2
So if we set
((to)= = =(to) / Ato |x)
dP, + Σ
Am (2;, to) =
- i=1
n
(Ma;;)t) 1o) + E(4o)
#(to xx))
1=1
we have
25110 = Op On Vlogn+ +n1(n)r2(n))
A.2.5
PROOF OF LEMMA 2
Proof. The proof follows Kennedy al. (2017). Denote
((e) = J (6(() f yr(y a (e):)))(;;)) e) e) dyddddt
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where we write (+)+) := -= :Pr.e) for brevity. Then, by definition, the efficient influence function for I is the unique function č(Y, X, T) such that
(0) E (C(Y, X, m. lc-o. (y, a. 1) is parametric submodel with
(12)
where (v, xr (,,,,,100) parameter E R. and Py,x.T.o(y. 2. Prx.. x.t). and (0) dT) l==o- We have
de
(0) = J,71)) / J/S y ( (..))))((;))) + +(y a. 1)''' ( (a; 0)] x (()dydard
+ I (t(
VT a te, s) ()( O)dydædt
x 1
+ J (t) /
VT (ttc))((
= 11 + 12 + la.
From (v x. t; Tc 2x,1;0 1; and definition of (t), we have
h = | (1) / vl ( x,1:0) x. t))()))) + x(v æ t) (a; ())())) d()ddddxdt
7(1) [ExBrIX (W a,400) + Ex (n(a, (a; 0)) क (1)dl,
12 = (t))((t)((( (ttp:///0n0//oo0000000000o000000 7())()))(()))))))))))
13 = (((0)(07((0)d))
Thus, have
[. = /f Ey (v a,4;0) + Ex (u(a, (0 +000))00)))))))000000000+++00+)) (()dt.
+((00)))))))))
(13)
Mcanwhile, for the right hand size term Ex,ry ((XXXXX.))) in Equation (12), from (Y, X, T; = (Y +XXX,T;;) where (, 1/,TT,O and (x, 0) are defined similar to XY, know that (C(Y, X,
Ex,Ty (C(Y, (Y, X, O)) = Ex,T, ( (X,,
(14)
Now we bound each term in Equation (14) separately. Recall that
U(T, X)
/ ( x))P(x) + (T) / ((T, -
c(Y (2) = (T)
TT X)
/ m(l ()(dPPP(((()) + (T(T)
(T(T:)) / [(T, x))P((())
J () (dd - R, (T)
(T; / U(T a))))(())))
Thus, for the first term in Equation (14), we have
EX (C(Y, X, & (Y(,
Exx Ey X.T (C(Y, X. TY ( (X,T,
Y
T(( m))d(())) C((I,, T.00]
Ex EX.rExIX
(TT) (T X)
ByJ EY& (Y ac, [:
( F(t m))P(z)) p(x))((|| z)) dtdx (15)
(1))
TXX
T(t x))
frax Txx
IY (()TEY
(Y [x, 4;
= J1, EL)
[e
Y& (Y az, t: 0)] (t))dt
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where (a) follows from the fact that E (E,(Y]XTT))) = 0, (b) uses law of iterated expecta- tions.
For the second term in Equation (14), have
Ex,Ty
X,(Y,,, & (x,;;0) T;)))
(0) Ex,Ty
((T) x(T) X) / T(T ([T1x)dP(()) P. (Xx,T:0))
uTT,X x)
((T)) / (dP))) + 1((T)
+ Ex,r
((T;:) M(T, ((Tx))))(()) -((x)
+ Ex,r ( ((L1()))0())))) (((7));00) (x; ((((00)))+(((000
+ Ex.r I(- 1-- (- 100000000 Ey (T(T;;) 0) / M((,, ((T,)))dF(x)) -r) -T) (C((T.x:0)
((T
(b)
((()) / (TT: + +Er (T)
(2(T;:) / [(TT
ET
x
+ Ex Ur (I..X000000X00000
[Ex Bx
+ (000()0()))))))
(16)
where (a) follows from the fact that
= -X
(b) follows from the fact that
=0 and law of iterated expectations, (c) follows from the definition of () = fxx plt, x (x).
Ty X), E, x X (T| = (17 =xx
Comparing Equation (15), Equation (16) against Equation (13), we immediately get
(0) = Ex,ry
which implies that <(Y, X,Y,XTT T) is indeed efficient influence function of I.
A.2.6 PROOF OF LEMMA
Proof. This proof follows from Huang et al. (2004). Let us start with the following decomposition:
|| én 11. S é, - Enll a + l| Enlln,
where č, (m (t)) ā. The first term ||ž, - En le a is the bias and the second term !|2, Enllze the variance.
Bound on bias term Let á E RK be such that (a)" kn - Ealloo SEBK, II - En loc.
Then we have
É Enll = |i (à) th + (ār) kk - EE
< | (a)² Kn = + (0x) *k - En 122.
By definition of & and properties of B-spline space, we have a bound on the first term
|| (a)) *kKnllz Op (Pm).
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where Pn. bounded:
inf EESpan ( suP1E[0,1] [e((t) f(t)). Notice that the second term also be
(á)) T &kth - Enll1a
l 0 dl12/VKn,
-11 B IT BB,))
TT, (B, -122.n 12/ V Kn
(6) Kn
B ITm
(Br - n 2m) 212/ VKn
(c) Kn
12 BB B 1
(17)
Pr
n
Kn
Kn
px((t)
Pu
मं (ti xz:
k=1
K
KaPus
pp((tt
where (a) follows from properties of B-spline basis functions, (b) follows from Lemma 4, (c) follows from properties of B-spline space such that B, I Zn = -p(pn) because
(4)) & (1,)) I Z., (d) is from the upper and lower boundedness of u.. Follow- ing proof of Lemma A.6 of Huang et al. (2004), for any >> [Tep (7) Kn, we have
K
Prob
(k(tt)
>a a
Kn
1 a
Prob
(t+)
>
V Kn
Prob
n Σ 8Px (4:) (T)
[Bxpx(T)
Kn
Kn
Prob
Σ i=1
4px (t:) - (T)
V K, |Bpx((T)
2K exp
22 (Va/K, - [EPx
where (a) uses union bound,
(b) follows from Hoeffding's Inequality for bounded random
variables. Since (T)(T) x 1/Kn, we can pick
([)4)x(T)]=X) Kn 1/K, and thus,
Σkn EH =0p k..) Plugging into Equation (17), get
(a) *k En.l1.22 = Op (pm),
Thus, we can bound the bias term
112 = (pn). - -
Bound on variance term
From properties of B-spline space, we have
||εn - ēnll S ||á - 0/l2/ Knn
Notice that
| a012 = | (B - Bn)
-1 BTT Zn 11, 27? 2... 11
1B, -1 BTd Zn
2, + 2, -I, 2 2n. ))1
/12
11.2 -1 Bdd Zn 2n)
+
BB 11 B.))
By i... - 11, 31
(18)
19

---

Published as a conference paper at ICLR 2021
Control of the first term in Equation (18): denote ô = (61, "6n"" = Zn - -Zn. then we have
B 2
Bn) -1
Zn
Zn
Zn
Zn
Bn (B " -2 *By Zn Zn
2
(a) K2 n²
K² n2²
K2
Kn
8 B B, 5
Σ 6
Ku
(t;)os
Σ
Pa (ti)
n²
i=1
km 1
1=1
K
1 n
2
< K
sup R.p
Ph
(1:)8;
1=1
(19)
where (a) is from Lemma 4. By definition we know
6, yi Mu (ax(,))4)) P
in (xT)
|T= =14) 111)
At; xi)
#(T X)
(1 14) iin (n((2441))
- (u( X, T) (
1T TT= t.) +
*(t; xii)
(TT X)
#(z xx))
u, + üis
where p( (. P [XXX,T) x
T T-tt)) t.). E (u; (t)) = 0 and 0; =(1))z)))
(1,1)22))
E (o: ti x2;) = O. Thus, from union bound, we have
K=
Prob
Σ
sup
(t:6
> a
km at
1=1
Kn
2
a
Prob
sup f.
ph (4:)0:
>
Kn
1=1
1=1
Kn
Prob
sup 7,P
(6
V Kn
k=1
K
Prob
sup a.
9h (u: + +ū,)
n
K
Kn
Kr
Prob
sup f.A
7
(t,)))
2\
Kn
a
Prob
sup 5.
Σ
pp(tt)
>
2V (20)
K
a
1=1
From Lemma 5, we know that
1
Rad,, ([0 + 1)
<
Ol + 1|2r- (Rad,, (Q) + Rad,, (1-
2
(a) 1
c²
2
1
+ 2c
s 2
111 + (1xo) Rad (Q) + max
Rad,, (U
2'
1/c]2
2c
n
= On 1/2),
where (a) follows from plugging h + --1/ 2c in Theorem 12(4) in Bartlett & Mendelson (2002). Similarly, write A = (2 + u)ll from Lemma 5, we have
(11pm + (Rad,,) + Rad, (A)) = (nn Rad, (P) < 2
Thus, we bound the first term of (20) using
a
(a) E (supa, 1 E (t)))a)
(b)
Kn
Prob
sup R
Pa (tt)
2\ Kn
a K.
V
an
(21)
20

---

Published as a conference paper at ICLR 2021
where (a) follows Markov Inequality, and (b) follows from the definition of Rademacher complexity.
We bound the second term of Equation (20) using union bound: for any Mr > 0,
Prob
sup
Ph (1)
2 Kn
R,p
<Prob
supp
PM (t,)6,,II(14) > Mn)
1
+
Prob
sup f,f
Σ
px
(t)
S M,)
1
4
Kn
n2
4
K
(22)
We have from Markov Inequality that
Prob
sup 5.A
Σ
(x(tt) Ma)
4V Kn
(23)
1 Esupa.a in Ei px(tt)/A((t 2.)oed((ee)) wil
S Mn)
(3)
Kn Mn:
Va/Kn
where (a) follows from Lemma 5. Also have
Prob
sup a,
Σ i=1
(x(()),(((b))) Mn)
> V K
Esupa,4
| Di= Pp (t, (t,))o[((es)) > M,)
Va Kn
Esup -i=1 At (o:))
(e:(t:)
2y [[[us]] M,)
Va/Kn Kn
E |T0|I((1 > Mn)]
Va/Ka
(0) S0 (1 (w)) dw- 1 Fw (w)dw
(24)
Va/Kn
S0 PP[||] 2 max(M, w))dw
Va/K, Kn 00 -olmaxx (Maw))? dw
Va/Kn S000 to] du
Va/Kn
2Λ -
VKn
1
Mn
Va
where (a) uses the formula EW = (1- F(w)dw and we set [w|[[[|] > M). (b) utilizes the fact that e follows sub- Gaussian distribution, (c) uses Mills ratio.
Plugging Equation (21), (23), (24) into (20), and taking M Vlg a Ka logn
we get
Kn.
sup A
1
Σ
(tt))
= Op
Kn log n
(25)
A=1
which, when plugging back into Equation (19), gives
Em)11s =Op (V
( (B. 2
Byh² Zn,
En
K3 log
(26)
n
21

---

Published as a conference paper at ICLR 2021
Control of the second term in Equation (18): Notice that each coordinate of Z, bounded, thus using similar arguments as that Equation (21), we know that
11, 2 Z, is
(V
Zn - -, D (,)
K log
(27)
Combining Equation (26) and (27) into (18), we know that
| cb|l2 =0 Op (v
K log
and thus,
K2 logn
1|,, 8 Op
(28)
n
Combining the rate on bias and variance term, we get
| - En = Op 07(
(pn Pn +
K2 logn
(0) Op k==² +
Kn Vlognnn
n
where (a) follows from assumption (iii,, giving
||ēn Enlle2 Op (n-1/3 ViiV/oog Vlogn)
when taking K,, x n 1/6
A.2.7 PROOF OF LEMMA
Proof. Suppose the SVD decomposition of B, = UAVT where U E Raxn A E P"*K,, V E XKn,K, From Lemma A.3 of Huang et al. (2004), we know that all diagonal ele- A fall between some positive constants. Notice that the eigenvalues of
ments of (Kn/v) II, B,, are the diagonal elements of (K,/n) 111,, From the upper and lower boundedness of tn. get the desired conclusion.
A.2.8 PROOF OF LEMMA 5
Proof. Write F3= = -2 F2. Notice that FiF2 [ h
(1( + - f 12(4) of Bartlett & Mendelson (2002) know that
F F Let : x², from Theorem
Rad, (FFFa) Rad, (h Fz) S F3 Rad,, (Fz).
Thus,
Rad,, (FIF2)
-Rad,,
52
F2)
1
1
<Rad,,
(T) + Rad, ( - r2)
$
Rad, (F?)
Rad, (F2)
4
4
1
1
[a Rad, (F3) + 2 [F Rad, (Fa)
2
1
$
(1 + [FFalloc) (Fa) + Rad,, (Fa)).
22

---

Published as a conference paper at ICLR 2021
A.3 ADDITIONAL RESULTS
This section formally states and proves the efficient influence function of a multidimensional vector, which is briefly mentioned in the main text. Suppose I =
E R² where
r r f (t;PT) Jxe J, uply (r.t)((x)((()ddddaa))
Then have the following theorem:
Theorem 3. The efficient influence finction for the d-dimensional vector r is
C(Y, X, T, n, H, D)
X, x, ( C2 X, T,
pr, T2); Co(Y, X, T,,, n. (,,) B
where GY, ,4, is the efficient influence f for J, 1,2, ... .d
Proof. Define r(e) (Ti(e), (2(e), (Ta())))),,,(())) ERd where for i 1,2, ,d
(e())) (,²0) (t: J
700)
J, sply a. ( e) e dyddadt.
Define 8(Y, TT;;) = log Py; where XY;, is a parametric submodel with parameter E E Rd and Py. = PY. Then, the efficient influence function Ç is defined as the unique
function such that
de k==
)
dT de
E
c=
i.e.,
df
dr,
E
Ci
dep
c=
de,
c=
Vi,j ...
Notice that the efficient influence function (,Y, for I does not depend on Thus for each i.j = 1, 1, ... , d, the above equation can be proved using similar arguments as that in Section A.2.5.
A.4 EXPERIMENTAL DETAILS
A.4.1 NETWORK STRUCTURE
For all methods, we implement the conditional density estimator as a neural network with two hidden fully connected layers, each consisting 50 hidden units using ReLU activation. Hidden feature is defined as the latent representation extracted after the second ReLU activation. We set the number of grids B 10. The estimation of x (t x)) is computed as introduced in Section Following Schwab et al. (2019), we use 5 blocks for Dragonnet and DRNet. Structure of prediction head for each block is the same as the prediction head μ for VCNet, except that Dragonnet and DRNet do not use treatment-depender two hidden fully connected layers stacking over the hidden feature z. Each hidden layer consists of 50 hidden units with ReLU activation. We use B-spline with degree two and two knots placed at (1/3, 2/3} (altogether 5 basis). In this way, all methods have the same complexity, i.e., the number of parameters. We also tried different structures and found the relative performance of different methods to be similar. Thus, all reported results below are based on this structure. All networks are trained for 800 epochs.
weights. In VCNet, the prediction head for n(l, a)) is neural network with
A.4.2 PARAMETER SETTING
For each dataset we tune parameters based on 20 runs. In each run simulate data, randomly split into training and testing, and use AMSE on testing data for evaluation. We tune the following parameters. For all methods: network learning rate Ir E {0.05, 0.005, 0.001, 0.0005, 0.0001} and E {1, 0.5). For TR: learning rate for e(t): Ir. E {0.001, 0.0001). 8 E {20, 10, 5) x n -1/2 We found that performance not sensitive to a. In estimator of c(t), we use B-spline with degree and tune the number of knots across (5, 10, 10,20) (all equally spaced at [0, 1)). For TMLE and doubly robust estimator: we tune parameters of B B-spline in the same way as in TR version. During tuning, all networks are trained for 800 epochs.
23

---

Published as a conference paper at ICLR 2021
A.4.3 STATISTICAL BASELINES
We implement Causal forest (Wager & Athey, 2018) using package "grf" (Tibshirani et al., 2018), BART using R package 'bartMachine' (Kapeiner el al., 2016), and GPS using R package 'causaldrf" (Galagate et al., 2015). We tune the paramters of each method on each dataset using 20 separate tuning sets, including the number of trees for BART, the number of trees and minimum node size for causal forest, and the number of knots for GPS. The other eeper-parameters are set to the default value of the R packages.
A.4.4 DATASET
Synthetic Dataset We generate data as follows: æj iid Unif(O, 1J, where æj is the j-th dimension of x E R. and
22, +
+ sin + exp(24 0.5xr))
1 (x1 + 25)²
23 + 2 (z))
20z 6.5 (00,0.25.
+ 4 max sin((z)) + (0(00,
xr, 1. cos
0.5))
2xg
where t= (1 + exp(-t)) Notice that a(t x) only depends on I1, 22, 33, 21, 2y while Q(t, x) only depends 31, 33, F4, As discussed in Shi al. (2019), this allows to observe the improvement using VCNet when noise covariates exist. Results are reported in Table 1.
The original symi-syn IHDP dataset from Hill (2011) contains binary treatments with 747 observations on 25 covariates. To allow comparison on continuous treatments, we randomly generate treatment and response using:
IHDP
2x1
2 max ra) + tanh
Eieson (a; 02)
7 7|
4 0.25).
(1+ x2)
0.2 + min( F5, x8)
S
sin(3nt)
Eiesa,, (1=-
ex.( -x)))
y xx /
Lanh 5
O.0,0..)5)
1.2
Sais,
0.5 + 5 min(12, 33, m85)
where t (1 +11+ cxx(-t)) 1)) Seon = (1,223,5,6) is the index set of continuous features, Sai,11 = {4. 7,8,9, 10,10,1,110
11, 12, 13. 14, 15), Sais (16, 17,188111111 18, 19, 20, 21, 22, 23, 24, 25) and Sus,1U/aiss3
[25] - -Sco- Here 61 = Eieso.
Notice that all continuous features are [ C2 - [Sa.2]
useful for (( æ) and Qtt, x)) but only Sais, is useful for Q and only Sas is useful for Following Hill (2011), covariates are standardized with mean and standard deviation 1 and the generated treatments normalized to lie between (0, 1). Results are summarized in Table 1.
The News dataset consists of 3000 randomly sampled news items from the NY Times corpus (Newman, 2008), which was originally introduced as a benchmark in the binary treatment setting (Johansson et al., 2016). We generate the treatment and outcome in a similar way as Bica et al. (2020). We first generate v/, vi and v/g from (/0,, 1) and then set v; = v ||0/ for = {1, 2,33. Given
News
ar, we generate from Beta (2,
And we generate the outcome by
20, m
1 0.3
y x, t exp
03 x
x, t 2 (max( 10022xmxxxx
-2, m2nn(2, v')) + 200 x)) (A 0.5) sin (2) + +N(000.5).
24


Reviews:
[{"review_id": "uu55CNxwRwk", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Solid theoretical results are provided to confirm the doubly robustness of the treatment estimator and outcome estimator. This paper is well-written, however, I still have some concerns about the contributions. \n1.\tThe author claimed that confounder is one challenge for treatment effect estimation. Some existing deep learning methods resort to balanced learning or reweighting. For the continuous treatment, how the proposed method addresses the confounder factors?\n2.\tAnother big concern is the varing coefficient neural network. Why the VCnet designed to be dependent of treatment information? The dependence is not theoretically discussed in this paper.\n3.\tThe experimental results are not convincing for me. Only two baselines are compared with the proposed method. Also the sufficient theoretical discussion is given which however is mainly about the doubly robust estimators. But the design of VCnet needs the numerical results to confirm its effectiveness for ADRF.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "ajLRo_Qtacg", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 9, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This problem is well-motivated --- estimating dose-response is a challenging and practically important problem.\nThe paper is extremely well written.  It explained complex (poor written) ideas in the semiparametric literature clearly. \nThe comparison against existing works is clear.\nThe theory, as far as I can tell, is sound. It improves the existing results in targeted regularization and can be adapted to analyze one step TMLE.\nThe experiment shows that the model outperforms existing benchmarks on the task it set out to do. \n\nMy main suggestion to improve the paper is to use some datasets that actually have continuous treatment in evaluating the method.  In particular, I would be interested in seeing an application of the methods on real-world datasets. That being said, while it will improve the paper, it's probably asking too much for an 8-page conference submission. I believe the paper at its current state is sufficient for acceptance.\n\nI want to thank the authors for writing such an elegant paper. I really enjoyed reading it. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "YG8UY8wP5D8", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper is to develop a varying coefficient neural network to estimate average dose-response curve (ADRF).  Although this paper has several interesting results, the paper is full of many typos and small errors.  The current paper needs substantial improvement. Here are some detailed comments. \n\n1. The introduction section is not well written since the logic does not flow very smooth. \n2. The motivation for （1）can be improved. You miss period in (1).  \n3. In Theorem 2, please not use (1)-(6) to itemize the assumptions, since you use them to denote the equations. \n4. In the proof of all theorems, there are some obvious mistakes inside. ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper designs a new way (in some sense a new perspective) on how neural networks can be used to model intervention variables when the goal is to estimate ADRF.  Basically, the idea is to emphasize the importance of the intervention variable by ensuring that it appears not just in every layer but also in every neural of a neural network.  

Reviewers mostly agree that this is a good paper with varying degrees, although there are some criticisms on e.g., assuming away the confounders.  However, I believe the authors address the criticisms of R4 satisfactorily. 

Overall I find the idea new and interesting and the experimental results strong, hence I happily recommend accepting the paper.

I do have a few quips myself and some comments that may help the authors to further improve the paper.

1. Re: the design that models each parameter as a spline. 
This is equivalent to introducing additional parameters (coefficients for spline basis) and adding a fixed linear layer (spline basis themselves) to every layer of the neural networks. t is taken as an input in all layers thus it makes sure that the model prioritizes on learning the impact of t. 

2. If you use a B-spline basis (that comes with kernels of bounded support), then the proposed method is very similar to stratifying the data according to different bins of t, and then fitting a separate model for each t. The only difference is that the different bins are now smooth kernels and they overlap somewhat. As a side note,  the authors should clearly write out how they are choosing the knots to specify the basis functions. Otherwise the paper will not be reproducible. 

3. I am not sure how this method would compare to naive (non-deep) baselines. Maybe this was considered in a prior work? If not, then I tend to side with Reviewer 4 that the evaluations are mostly ablation studies and they are not really comparing to representative work in this domain. Given that there is a large body of work on this before deep learning takes over, it is important to somehow compare with the right baselines.


Decision: Accept (Oral)