V ARYING C OEFFICIENT N EURAL N ETWORK WITH F UNCTIONAL T ARGETED R EGULARIZATION FOR E STIMATING C ONTINUOUS T REATMENT E FFECTS
Anonymous authors
Paper under double-blind review
A BSTRACT
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 ADRF 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.
I NTRODUCTION
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 an important problem (Hirano & Imbens, 2004;
Imai & Van Dyk, 2004; Kennedy et al., 2017; Fong et al., 2018).
Recently, papers in causal inference (Johansson et 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 in modeling the complex causal relationship as shown by the universal approximation theorem (Csáji et al., 2001).
Also, unlike traditional non-parametric models, it has been shown to be powerful when dealing with high-dimensional input (i.e., Masci et al.
; Johansson et al.
(2016)), which implies its potential for dealing with high-dimensional confounders.
A successful application of neural network model requires a 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.
; Louizos et al.
Schwab et al.
; Shi et al.
(2019) use 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 a 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 a 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 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 a new network building block that is able to attenuate the inﬂuence of treatment but also preserve the continuity of ADRF.
Previous neural network models for treatment effect estimation use separate prediction heads to model the mapping from covariates to (expected) outcome under different
1.0 Treatment 2.
R e s [Figure] s e
Truth Vcnet
Drnet 0.
1.0 Treatment 1.
R e s pon s e 0.
1.0 Treatment 0.
R e s pon s e 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 (Hastie & Tibshirani (1993), Fan et al.
(1999), Chiang et al.
, one can allow the weights of the prediction head to be continuous functions of the treatment.
This serves as the ﬁrst contribution of this work 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 1 but also avoids the treatment being lost in high dimensional latent representation.
The second contribution of this paper is to generalize targeted regularization to obtain a doubly robust estimator of the whole ADRF curve, which signiﬁcantly improves ﬁnite sample performance.
Targeted regularization was previously used for estimating a scalar quantity (Shi et al., 2019) and it associates an extra perturbation parameter to the scalar quantity of interest.
While adapting it to a ﬁnitedimensional vector is not difﬁcult, generalization to a curve is far less straightforward.
Difﬁculties arise from the fact that we cannot regularize ADRF at each treatment level independently because the number of possible levels is 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 a 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 to 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 a function.
We do experiments on both synthetic and real 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 δ ( · ) .
We use E to denote expectation.
Denote P as the probability measure and we write P ( f ) =
(cid:82) f ( z ) d P ( z ) .
Similarly we denote P n as the empirical measure and we write P n ( f )
=
(cid:82) f ( z ) d P n ( z ) .
We denote (cid:100) n (cid:101) as the least integer greater than or equal to n , and we denote (cid:98) n (cid:99) as the greatest integer less than or equal to n .
We use τ to denote
Rademacher random variables.
We denote Rademacher complexity of a function class F : X → R as Rad n ( F ) =
E (cid:0) sup f ∈F (cid:12) (cid:12) 1 n (cid:80) n i =
τ i f
( X i ) (cid:12)(cid:12)(cid:1) .
Given two functions f 1 , f 2 : X → R , we deﬁne (cid:107) f 1 − f 2 (cid:107) ∞ = sup x ∈X | f 1 ( x ) − f 2 ( x ) | and (cid:107) f 1 − f 2 (cid:107) L 2 = (cid:16)(cid:82) x ∈X ( f 1 ( x )
− f 2 ( x )) 2 dx (cid:17) 1 / 2 .
For a function class F , we deﬁne (cid:107)F(cid:107) ∞ =
sup f ∈F (cid:107) f (cid:107) ∞ .
We denote stochastic boundedness with O p and convergence in probability with o p .
Given two random variable X 1 and X 2 , X 1 ⊥ X 2 denotes
X 1 and X 2 are independent.
We use a n ∼ b n to denote that both a n /b n and b n /a n are bounded.
[Figure]
P ROBLEM S TATEMENT AND B ACKGROUND
Suppose we observe an i.i.d sample { ( y i , x i , t i ) } ni =1 where ( y i , x i , t i ) is a realization of random vector ( Y, X , T ) with support ( Y × X × T ) .
Here X is a vector of covariates, T is a continuous treatment, and Y ∈ Y is the outcome.
Without loss of generality, we assume T =
We want to estimate the Average Dose Response Function (ADRF)
[Equation]
, which is the potential expected outcome that would have been observed under treatment level t .
Suppose the conditional density of T given X is π ( T | X ) .
Throughout this paper, we make the following assumptions:
Assumption 1.
(a) There exists some constant c >
such that π ( t | x ) ≥ c for all x ∈ X and t ∈ T .
(b) The measured covariate X blocks all backdoor path 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 its covariates, which is an standard assumption to establish doubly robust estimator.
Assumption (b) implies that the casual effect is identiﬁable, i.e., can be estimated using observational data.
VCN ET : V ARYING C OEFFICIENT N ETWORK S TRUCTURE
Under Assumption 1, we have
[Equation]
Y | X , T = t )] .
Thus, a naive estimator is to obtain an estimator ˆ µ of µ , and use ˆ ψ ( t )
= 1 n (cid:80) ni
=1 ˆ µ ( t, x i ) .
Here µ ( t, x ) :=
E ( Y | X = x , T = t ) and ˆ µ is its estimator.
Following Shi et al.
(2019), we utilize the sufﬁciency of the generalized propensity score π ( t | X ) for estimating ψ (Hirano & Imbens, 2004):
[Equation]
It indicates that learning π ( t | X ) helps the removal of noise and distillation of useful information in
X for estimating ψ .
Similar to Shi et al.
(2019), we add a separate head for estimating π ( t | X ) , and use the feature z extracted by it for downstream estimation of µ ( t, x ) (see Figure 2).
Our contribution here is to propose the varying coefﬁcient structure of the prediction head for µ ( t, x ) , which addresses difﬁculties confronting continuous treatment as discussed in the following paragraph.
3. T HE V ARYING C OEFFICIENT P REDICTION H EAD
Our aim is to predict µ ( t, x ) =
E ( Y | T = t, X = x ) .
A naive method is to train a neural network which takes ( t, x ) as input in the ﬁrst layer and output µ ( t, x ) in the last layer.
However, the role of treatment t is different from that of x and the inﬂuence of t might be lost in the high-dimensional hidden features (Shalit et al., 2017).
Aware of this problem, previous work (Schwab et al., 2019) divides the range of treatment into blocks, and then use separate prediction heads for each block (see Figure 2).
To further attenuate 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 µ (Meretoja &
Wirtavuori, 1989; Desquilbet & Mariotti, 2010) by using different prediction heads for each block of it.
In practice, DRNet indeed produces discontinuous curve (see Figure 1).
We propose a varying coefﬁcient neural network (VCNet), which simultaneously emphasizes the inﬂuence of treatment while utilizing and preserving the continuity of ADRF.
In VCNet, the prediction head for µ is deﬁned as
[Equation]
where input z is the feature extracted by the conditional density estimator, and f θ ( t ) is a (deep) neural network with parameter θ ( t ) , which is a nonlinear function depending on the treatment level t .
For example, if f is an one-hidden-layer ReLU network, we have f θ ( t ) =
(cid:80) i a i ( t ) ReLU (
b i ( t ) (cid:62) z ) .
We use splines to model θ ( t ) .
Suppose that θ ( t ) =
[ θ 1 ( t ) , ..., θ d θ ( t )]
∈ R d θ , where d θ is the dimension of θ .
We have
[Equation]
where { ϕ NN l } L(cid:96)
=1 are the spline basis and a i,(cid:96) ’s are the coefﬁcients.
Thus, we have θ ( t ) =
A Φ ( t ) where A =   a 1 ,· · · a 1 ,L .
a d θ ,· · · a d θ ,L   and Φ ( t ) = (cid:2) ϕ NN 1 ( t ) , · · · , ϕ NN L ( t ) (cid:3) T .
It is worth mentioning that by choosing spline basis of the form ( t − t 0 ) + with different offset t 0 , we recover the structure in Schwab et al.
(2019), which has a separate prediction head for each block.
It indicates that DRNet can also be viewed as a special case of VCNet (under a suboptimal choice of basis functions).
3. C ONDITIONAL D ENSITY E STIMATOR
Recall that the input feature z to µ NN is extracted by the conditional density estimator for π ( t | x ) .
Here we propose a simple network to estimate π , which is a direct generalization of the conditional probability estimating head in Shi et al.
Notice that t ∈ [0 , 1] and the conditional density π ( t | x ) is continuous with respect to treatment t for any given x .
A continuous function can be effectively approximated by piecewise linear functions.
Thus, we divide [0 , 1] equally into B grids, estimate the conditional density π ( · | 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 π NNgrid as
[Equation]
where z = f ω 1 ( x ) .
Here z ∈ R h is the hidden feature extracted by the network, ω 1 is the parameter for the nonlinear mapping f ω 1 , ω
2 ∈ R ( B +1)
× h , π NNgrid ( x ) =
[ π 0 , NN grid ( x ) , ...., π B, NN grid ( x )] and π i, NN grid ( x ) is the estimated conditional density of T
=
i/B given X = x .
This structure is analogous to a 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 = x are obtained via linear interpolation:
[Equation]
where t 1 = (cid:98) Bt (cid:99) , t 2 = (cid:100)
Bt (cid:101) .
Notice that this linear interpolation automatically yields a valid continuous density.
i.e., π NN ( t | x ) ≥ 0 , ∀ t, x and (cid:82) 1 t =
π NN ( t | x ) dt = 1 , ∀ x .
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.
and normalizing ﬂows (Rezende & Mohamed (2015), Dinh et al.
(2016), Trippe & Turner (2018)).
Here treatments are bounded, and thus Gaussian mixtures are not applicable.
And linear interpolation can 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 et 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. T RAINING
Notice that our model requires π NN to extract good latent features z as the input for µ NN to predict.
This can be achieved by training π NN to estimate the conditional density, which motivates us to train π NN and µ NN simultaneously by minimizing the following loss:
[Equation]
In loss (1), the ﬁrst term measures the prediction loss from µ NN .
The second term measures the loss from π NN and is the negative log likelihood.
And α controls the relative weights of the two losses.
Denote ˆ µ , ˆ π as the optimal solution of the above empirical risk minimization problem (1).
After getting ˆ µ , one can estimate ψ ( · ) by ˆ ψ ( · ) = 1 n (cid:80) ni
=1 ˆ µ ( · , x i ) .
The correctness of this naive estimator relies on whether the truth µ is in the function space deﬁned by the neural network model.
However, we can plug ˆ µ and ˆ π into the non-parametric estimating equation to obtain a doubly robust estimator of ψ ( t ) .
In this way, we are able to produce an (asymptotically) correct estimator if any one of µ and π is in the model space.
The next section discusses challenges in obtaining such a doubly robust estimator and provides our solution.
F UNCTIONAL T ARGETED R EGULARIZATION
In this section we improve upon the previous method by utilizing semiparametric theory on doubly robust estimators.
Doubly robust estimators are built upon ˆ π ( t | x ) and ˆ µ ( t, x ) , and it yields a consistent estimator for ψ even if one of them is inconsistent.
When both ˆ π ( t | x ) and ˆ µ ( t, x ) are consistent, a doubly robust estimator leads to 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 ψ ( t 0 ) for any t 0 ∈ [0 , 1] .
4. D OUBLY R OBUST E STIMATOR
Before we proceed, we deﬁne the following quantity:
[Equation]
The following theorem serves as the basis for our subsequent estimators:
Theorem 1.
Under assumption 1 and assume that ˆ π ( t | x ) ≥ c > 0 for all x ∈ X and t ∈ T .
For any t 0 ∈ T , ζ t 0 is the efﬁcient inﬂuence function for ψ ( t 0 ) .
Moreover ζ t 0 is doubly robust in the sense that
[Equation]
if either ˆ π = π or ˆ µ = µ .
Further, if (cid:107) ˆ π − π (cid:107) ∞ =
O p ( r 1 ( n )) and (cid:107) ˆ µ − µ (cid:107) ∞ =
O p ( r 2 ( n )) , we have
[Equation]
Theorem 1 shows that for any t 0 ∈ T , P ( q t 0 ( Y, X , T, ˆ µ, ˆ π ) + ˆ µ ( t 0 , X ))
− ψ ( t 0 ) is a doubly robust estimator for ψ ( t 0 ) .
Under some mild assumptions, one way to obtain a doubly robust estimator of ψ is to utilize the two-stage procedure from Kennedy et al.
by regressing
[Equation]
on T using any nonparametric regression methods like kernel regression or spline regression.
4. T ARGETED R EGULARIZATION FOR I NFERRING A F INITE D IMENSIONAL Q UANTITY
However, as discussed in Shi et al.
(2019), when estimating the term P ( q t 0 ( Y, X , T, ˆ µ, ˆ π )) , the ˆ π ( T | X ) in the denominator might make the ﬁnite sample estimator unstable, especially in cases where Assumption 1(a) is nearly violated.
Targeted regularization is proposed by Shi et al.
(2019) to solve this issue.
The key intuition of targeted regularization is to learn ˆ µ and ˆ π such that P ( q t 0 ( Y, X , T, ˆ µ, ˆ π ))
≈ 0 and thus the estimation of this term is no more needed.
In the binary treatment case where T = { 0 , 1 } , if we want to estimate ψ (1) , which is a single quantity, targeted regularization simultaneously optimizes over µ NN , π NN and an extra scalar perturbation parameter (cid:15) using the following loss
[Equation]
) where R TR [ µ NN , π NN , (cid:15) ] = 1 n n (cid:88) i =
[Equation]
Here L [ µ NN , π NN ] is deﬁned in (1).
Assume the complexity of the function space of µ NN and π NN is ﬁnite and since the complexity of the function space of the introduced perturbation (cid:15) is also ﬁnite, we have
[Equation]
TR [ˆ µ, ˆ π, ˆ (cid:15) ]
= o p (1) , where ˆ µ TR ( t, x ) :
= ˆ µ ( t, x )+ˆ (cid:15) t ˆ π ( t | x ) and (ˆ µ, ˆ π, ˆ (cid:15) ) is the minimizer of (3).
Notice that the ﬁrst equality holds because at the convergence of the optimization, ∂∂(cid:15) R TR [ˆ µ, ˆ π, ˆ (cid:15) ] = 0 .
And the second equality is by uniform concentration inequality.
This implies that 1 n (cid:80) ni
=
ˆ µ TR (1 , x ) is a doubly robust estimator for ψ (1) and for this estimator, no conditional density estimator presents at denominator and thus it has more stable ﬁnite sample performance.
4. F UNCTIONAL T ARGETED R EGULARIZATION FOR I NFERRING THE W HOLE ADRF
Notice that in loss (3), the scalar (cid:15) is associated with a scalar quantity for inference.
One can generalize targeted regularization to estimating a d dimensional vector by using d separate (cid:15) ’s (see
Theorem 3 in the Appendix).
Generalizing to a curve, however, is more challenging.
We need to optimize over a function (cid:15) :
T → R where (cid:15) ( · ) is the perturbation associated with ψ ( · ) .
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 µ and π (Prichard & Gillam, 1971; Schneider et al., 1993; Threlfall & English, 1999), which allows us to use splines { ϕ k } K n k =1 with K n basis functions to approximate (cid:15) ( · ) .
Here the subscript n in K n denotes that the number of basis functions might change with the sample size n .
Deﬁne (cid:15) n ( · ) =
(cid:80) K n k
=
α k ϕ k ( · ) .
We give the following loss with
Functional Targeted Regularization (FTR).
[Equation]
where R FTR [ µ NN , π NN , (cid:15)
[Equation]
=
[Equation]
n
[Equation]
Here R FTR denotes the FTR term and β n → 0 when n → ∞ .
Remark 2 ( On β n ) .
The targeted regularization proposed by Shi et al.
(2019) uses a ﬁxed β .
However, using ﬁxed β might make the estimator constructed by targeted regularization no more consistent when µ NN is mis-speciﬁed, which makes the estimator no more doubly robust.
To overcome this issue, we make a slight change on β by allowing β to depend on n .
Speciﬁcally, we ﬁnd that once β n = o (1) , we are 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 is more challenging than analyzing traditional targeted regularization.
One reason is that we no more have the property that ∂∂(cid:15) R TR [ˆ µ, ˆ π, ˆ (cid:15) ]
= 0 in targeted regularization.
We establish convergence rate for our estimator using loss (4) in Theorem 2.
Before we proceed, let us pause a bit and introduce some deﬁnitions, which will be used in the main theorem.
Denote ˆ µ , ˆ π and ˆ (cid:15) n as the minimizer of (4).
We use ¯ π and ¯ µ to denote ﬁxed functions to which ˆ π and ˆ µ converge in the sense that (cid:107) ˆ π − ¯ π (cid:107) ∞
=
o p (1) and (cid:107) ˆ µ − ¯ µ (cid:107) ∞ =
o p (1) .
We deﬁne g t : X → R , x (cid:55)→ µ NN ( x , t ) .
We denote G , Q , U as the function space in which g t , µ NN , π NN lies.
We denote B K n as the closed linear span of basis ϕ K n = { ϕ k } K n k =1 .
The key intuition of the asymptotic correctness of FTR is that: once π NN and π are uniformly upper/lower bounded and some other weak regularization conditions hold, we can show that (cid:107) ˆ (cid:15) n ( · ) − (cid:15) ∗ ( · ) (cid:107) L 2 = o p (1) where (cid:15) ∗ ( · ) :
=
E [( Y − ¯ µ ) / ¯ π | T = · ] / E (cid:2) ¯ π − 2 | T = · (cid:3) .
And thus letting ˆ µ FTR :
= ˆ µ + ˆ (cid:15) n / ˆ π , we have
[Equation]
Theorem 2.
In addition to Assumption 1, suppose the following assumptions hold:
(i) There exists constant c > 0 such that for any t ∈ T , x ∈ X , and π NN ∈ U , we have 1 /c ≤ π NN ( t | x ) ≤ c , 1 /c ≤ π ( t | x ) ≤ c , (cid:107)Q(cid:107) ∞ ≤ c and (cid:107) µ (cid:107) ∞ ≤ c .
(ii) Y = µ ( X , T )
+ V where E V = 0 , V ⊥ X , V ⊥ T , and V follows sub-Gaussian distribution.
(iii) B K n equals the closed linear span of B-spline with equally spaced knots, ﬁxed degree, and dimension K n ∼ n 1 / 6 .
(iv) Either ¯ π = π or ¯ µ = µ .
(v) Rad n ( G ) =
O (cid:0) n − 1 / 2 (cid:1) , Rad n ( Q ) =
O (cid:0) n − 1 / 2 (cid:1) and Rad n ( U ) =
O (cid:0) n − 1 / 2 (cid:1) .
(vi) π , µ , π NN and µ NN have bounded second derivatives for any π NN ∈ Q
and
µ NN ∈
U .
Let (cid:98) ψ ( · ) :
= 1 n (cid:80) ni
=1 (cid:16) ˆ µ ( x i , · ) + ˆ (cid:15) n ( · ) ˆ π ( ·| x i ) (cid:17) , we have
[Equation]
where (cid:107) ˆ π − π (cid:107) ∞ =
O p ( r 1 ( n )) and (cid:107) ˆ µ − µ (cid:107) ∞ =
O p ( r 2 ( n )) .
Remark 3 ( On the assumptions ) .
In Theorem 2, assumption (i), the ﬁrst half of (iii), and (vi) are weak and standard conditions for establishing convergence rate of spline estimators (Huang et al., 2003; 2004).
Assumption (ii) bounds the tail behavior of V .
The second half of (iii) restricts the growth rate of K n , which is a typical assumption (Huang et al., 2003; 2004)
but with different rate in order to obtain uniform bound.
Assumption (iv) states that at least one of ˆ µ, ˆ π should be consistent, which implies the double robustness of ˆ ψ .
Assumption (v) considers the complexity of model space, and is a common assumption for problems with nuisance functions (Kennedy et al., 2017).
Remark 4 ( On the consistency of ¯ µ and ¯ π ) .
We want to point out that adding targeted regularization does not affect the limit of ˆ µ and ˆ π in large sample asymptotics.
That is, the limit of ˆ µ and ˆ π using loss (4) will be the same as using loss (1).
We refer the reader to Appendix A.1 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.
R ELATED W ORK
The Varying Coefﬁcient Structure Varying coefﬁcient (linear) model is ﬁrst proposed as an extension of linear model (Hastie & Tibshirani, 1993; Fan et al., 1999), which is usually used for modeling longitudinal data (Huang et al., 2004; Zhang & Wang, 2015;
Li et al., 2017; Ye et 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 those models, our varying coefﬁcient structured 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
[Table]
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 a genotype (the hypernetwork), 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 a much smaller network to generate the weights of a 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.
And our network is proposed in order to appropriately incorporate treatment into modelling, which is not touched on in HyperNetwork.
Neural Network Structure for Treatment Effect Estimation We refer readers to introduction for the connections and comparisons with previous works using feed forward neural network for treatment effect estimation.
In addition to feed forward neural network, previous works also utilize other networks to learn treatment effects.
For example, Yoon et al.
; Bica et al.
(2020) estimate the causal effect via learning to generate the counterfactual.
And Louizos et al.
(2017) learns the causal effect by learning deep variable models using variational autoencoder.
Compared with our method, their approaches are mainly heuristic and lack theoretical guarantee for the asymptotic correctness of the estimator.
Doubly Robustness, TMLE and Targeted Regularization Chernozhukov et al.
(2017; 2018) develop 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 focus on estimating a single quantity, such as ψ (1) − ψ (0) in binary treatment (Shi et al., 2019; Van der Laan & Rose, 2011) or averaged treatment effect E ψ ( t ) for continuous treatment (Kennedy et al., 2017), while we give the ﬁrst targeted regularization and TMLE for inferring the whole ADRF curve.
E XPERIMENTS
Baseline This paper is focused on adapting neural nets to estimate ADRF.
Thus, we compare against two neural network based methods: Dragonnet (Shi et al., 2019) and DRNet (Schwab et al., 2019).
Dragonnet is SOTA among neural network methods for estimating ATE in binary setting.
Following (Schwab et al., 2019), we concatenate T as an extra input and further improve by using separate heads for T in different intervals.
DRNet is SOTA in estimating individual dose-response curve.
The estimation of ADRF is provably and empirically improved by adding a conditional density estimator head (Shi et al., 2019).
Thus, for fairness, we add a conditional density estimating head for DRNet.
To demonstrate the effectiveness of targeted regularization (TR), for all methods we implement four versions: naive version (with conditional density estimator head, trained using loss (1)), doubly robust version (Kennedy et al., 2017) by regressing (2) on treatment with ˆ µ, ˆ π 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).
Experiment Setting Following Schwab et al.
(2019), we use the average mean squared error (AMSE) on test set for evaluation: (cid:92) AMSE = 1 S (cid:80) Ss =1 (cid:82) T (cid:16) (cid:91) ψ s ( t ) − ψ ( t ) (cid:17) 2 π ( t ) dt where (cid:91) ψ s ( t ) is the estimated ψ ( t ) in the s -th simulation.
For synthetic dataset, training size is 500 and test size is 200.
For
IHDP and News, we randomly split the data into training (67%) and testing (33%).
Due to space limit, we refer reader to Appendix A.4 for other details of the experiment.
Results In Table 1, by comparing results in each column, we observe a signiﬁcant 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.
C ONCLUSION
This work proposes a novel varying coefﬁcient network and generalizes targeted regularization to a continuous curve.
We give theorem showing its consistency and double robustness.
Experiments show that VCNet structure and targeted regularization boost performance independently and when used together, state-of-the-art performance is achieved.
A A PPENDIX A. O N THE CONSISTENCY OF ¯ µ AND ¯ π
We show that adding targeted regularization does not affect the limit of ˆ µ and ˆ π in large sample asymptotics.
That is, the limit of ˆ µ and ˆ π using loss (4) will be the same as using loss (1).
Denote
[Equation]
Suppose that ( µ (cid:48) , π (cid:48) ) is the minimizer of loss P (cid:96) ( µ NN , π NN ) and (ˆ µ, ˆ π, ˆ (cid:15) n ) is the minimizer of L FTR .
For simplicity
[Equation]
we ignore the unidentiﬁability of neural network parameterization and assume ( µ (cid:48) , π (cid:48) ) is the unique minimizer in the sense that for any (cid:15) > 0 , there exist η ( (cid:15) ) > 0 such that
[Equation]
We have
[Equation]
Here ( a ) uses the fact that µ NN , π NN is uniformly bounded, Rad n ( Q ) , Rad n ( U )
=
O ( n − 1 / 2 ) and the Lipschitz constant of log( s ) is bounded when s ∈ [1 /c, c ] for some ﬁnite c .
( b ) comes from the fact that (ˆ µ, ˆ π, ˆ (cid:15) n ) is the minimizer of the empirical risk (with FTR).
( c ) is because of
R [ µ (cid:48) , π (cid:48) , 0]
= 1 n n (cid:88) i = (cid:0)
y i − µ (cid:48)
(cid:1)= E (cid:0) y − µ (cid:48) (cid:1) 2 + O p ( n − 1 / 2 ) =
E ( V 2 ) + E ( µ − µ (cid:48) )
2 + O p ( n − 1 / 2 ) =
O (1) .
Now we prove that (cid:107) ˆ µ − µ (cid:48) (cid:107)
L 2 + (cid:107) ˆ π − π (cid:48)
(cid:107) L 2 = o p (1) .
If it is not true, then there exists s > 0 , when n is sufﬁciently large, (cid:107) ˆ µ − µ (cid:48)
(cid:107)
L 2 + (cid:107) ˆ π − π (cid:48)
(cid:107) L 2 ≥ s and P (cid:96) (ˆ µ, ˆ π ) − P (cid:96) ( µ (cid:48) , π (cid:48) )
≤ η ( s ) .
It makes contradiction of the assumption that ( µ (cid:48) , π (cid:48) ) is the unique minimizer of P (cid:96) ( µ NN , π NN ) .
A. T ECHNICAL P ROOFS
In this section, we prove the two main theorems (Theorem 1 and Theorem 2) and give some additional results which are mentioned brieﬂy in the main text.
A.2. E XTRA N OTATIONS
Here we introduce some additional notations useful for our subsequent proofs.
We use p and π to denote density.
We denote δ t 0 as the Dirac measure centered on t 0 .
We use a ∼ b to denote that a has the same order of magnitude as b , and a (cid:46) b to denote that the order of magnitude of a is no larger than that of b .
We write ˇ (cid:15) n ( · ) :
= P [( Y − ˆ µ n ) /
ˆ π n | T = · ] and ˜Π n = diag (cid:16)(cid:2) P (cid:0) ˆ π − 2 n ( T | X ) | T = t 1 (cid:1)(cid:3) − 1 / 2 , · · · , (cid:2) P (cid:0) ˆ π − 2 n ( T | X ) | T = t n (cid:1)(cid:3) − 1 / 2 (cid:17) .
We denote
Z n = ( z 1 , z 2 , · · · , z n ) T ∈ R n where z i = y
i − ˆ µ n ( t i , x i )
ˆ π n ( t i | x i ) , and ˜ Z
n =
(˜ z 1 , ˜ z 2 , · · · , ˜ z n ) T ∈ R n where ˜ z i = P (cid:16)
Y − ˆ µ n ( T, X ) ˆ π n ( T | X ) | T
=
t i (cid:17) .
We write α =
( α 1 , α 2 , · · · , α K ) T ∈ R K , ˆ α = (cid:0) B Tn Π − 2 n B n (cid:1) − 1 B Tn Z n and ˜ α
= (cid:0) B Tn Π − 2 n B n (cid:1) − 1 B Tn Π − 2 n ˜Π 2 n ˜ Z n .
We denote 1 n = (1 , 1 , · · · , 1) T ∈ R n and ρ n = K − 2 n .
For any function f and function spaces F 1 , F 2 , we write F 1 + F 2 = { f 1 + f 2 : f 1 ∈ F 1 , f 2 ∈ F 2 } , F 1 F 2 = { f 1 f 2 : f 1 ∈ F 1 , f 2 ∈ F 2 } , f F = { fh : h ∈ F} , f ◦ F = { f ◦ h : h ∈ F} , and F a = { f a : f ∈ F} , ∀ a ∈ R .
A.2. U SEFUL L EMMAS
This section gives lemmas which are used in our proofs of main theorems.
Recall that Theorem 1 consists of two parts: the efﬁcient inﬂuence function of ψ ( t 0 ) and its double robustness.
Notice that ψ ( t 0 ) can be written as a special case of a more general parameter Γ = (cid:82) T γ ( t ; P T ) ψ ( t ) d P T ( t ) (see Remark 5).
Here γ ( t ; P T ) is a function of t which depends on the probability measure P T .
For brevity we also write γ ( t ) :
=
γ ( t ; P T ) when the corresponding P T is clear.
The following Lemma gives the efﬁcient inﬂuence function of Γ .
Lemma 1.
The efﬁcient inﬂuence function for Γ is
[Equation]
where
[Equation]
| ε =0 and P T,ε is a parametric submodel with parameter ε ∈ R and P T
[Equation]
0 ( · ) = P T ( · ) .
Remark 5.
Setting γ ( · )
= dδ t 0 ( · ) d P T ( · ) , we get the efﬁcient inﬂuence function of ψ ( t 0 ) .
Setting γ ( · ) = 1 , we get Γ = (cid:82) T ψ ( t ) d P T ( t ) , which is the average outcome under a randomized trial and is a quantity of interest in its own right (Kennedy et al., 2017).
Setting γ ( · )
= dδ 1 ( · ) d P T ( · ) − dδ 0 ( · ) d P T ( · ) , we get Γ = ψ (1) − ψ (0) , which is the average treatment effect under binary treatment setting.
Recall that in Theorem 2, we use B-spline regression to estimate ˇ (cid:15) n .
In order to establish the convergence rate of our ﬁnal estimator ˆ ψ , we need to establish the convergence rate of the spline estimator ˆ (cid:15) n ﬁrst.
Lemma 2.
Under assumptions in Theorem 2, we have
[Equation]
Remark 6.
For ﬁxed objective function, the convergence rate of the B-spline estimator to the truth is a standard result (Huang et al., 2004; 2003), which is O p ( n − 2 / 5 ) when choosing K n (cid:16) n − 1 / 5 .
However, Lemma 2 gives a uniform bound on a class of functions in order to deal with the fact that ˆ π n and ˆ µ n are NOT ﬁxed and dependent on the observations.
The bound is thus of a larger order n − 1 / 3 √ log n when choosing the optimal K n (cid:16) n − 1 / 6 .
Lemma 3.
Under assumption in Theorem 2, there are positive constants M 1 and M 2 such that except on an event whose probability tends to zero, all the eigenvalues of ( K n /n )
B Tn Π − 2 n B n fall between M 1 and M 2 , and consequently, ( K n /n )
B Tn Π − 2 n B n is invertible.
Lemma 4.
Assume (cid:107)F 1 (cid:107) ∞ < ∞ and (cid:107)F 2 (cid:107) ∞ < ∞ , we have
[Equation]
A.2. P ROOF OF T HEOREM
Proof.
First, the efﬁcient inﬂuence function of ψ ( t 0 ) is a direct consequence of plugging in γ ( · ) =
dδ t 0 ( · ) d P T ( · ) in
Lemma 1.
So here we only need to prove that the efﬁcient inﬂuence function is doubly robust.
Notice when γ ( · ; ˆ P T ) =
dδ t 0 ( · ) d ˆ P T ( · ) , we have
[Equation]
By repeatedly using iterated expectation and deﬁnition of l
[Table]
since ˆ π ( T ) =
[Table]
ˆ π
[Table]
T
[Table]
x )
[Table]
P
[Table] [Table]
[Equation]
where (a) follows from iterated expectation and Equation (5), (b) follows from the fact that π ( x | t ) =
π ( t | x ) π ( x ) /π
( t ) , (c) uses the deﬁnition γ ( · ; P T )
= dδ
[Equation]
0 ( · ) /d P T ( · ) and γ ( t ; ˆ P T ) =
dδ t
[Equation]
( · ) /d ˆ P T ( · ) .
From the last line of Equation (6), it is obvious that the desired conclusions hold.
A.2. P ROOF OF T HEOREM
Proof.
From Lemma 2, we have (cid:107) ˆ (cid:15) n − ˇ (cid:15) n (cid:107) L 2 = O p ( n − 1 / 3 (cid:112) log n ) and thus, (cid:13)(cid:13)(cid:13)(cid:13) ˆ (cid:15) ( · ) (cid:90)
X
[Equation]
From generalization bound and condition (v), we know
[Equation]
t
[Equation] [Equation]
1] (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)
[Table]
Thus,
[Equation]
Recall that Theorem 1 says that if sup
[Equation]
∈
[Equation]
[Equation] [Equation]
= O
[Equation]
sup
[Equation]
∈
[Equation] [Equation]
, then
[Equation]
Combining Equation (7), Equation (8) and Equation (9), using triangle inequality, we have
[Equation]
So if we let
[Equation]
we have
[Equation]
A.2. P ROOF OF L EMMA
Proof.
The proof follows Kennedy et al.
Deﬁne
[Equation]
where we write γ ε ( · ) :
= γ ( · ; P T,ε ) for brevity.
Then, by deﬁnition, the efﬁcient inﬂuence function for Γ is the unique function ζ ( Y, X , T ) such that
[Equation]
where (cid:96) (cid:48)
ε ( y, x , t
[Equation] [Equation]
=
[Equation] [Equation]
[Equation] [Equation]
| ε =0 , P Y, X ,T,ε ( y, x , t ) is a parametric submodel with parameter ε ∈ R , and P Y, X ,T, 0 ( y, x , t )
=
P Y, X ,T ( y, x , t ) .
By deﬁnition Γ (cid:48) (cid:15) (0) =
d Γ( ε ) /dε
| ε =0 , we have
[Equation]
where
[Equation]
Thus, we have
[Equation]
Meanwhile,
[Table]
right
[Table] [Table]
[Table] [Table]
) in Equation
[Table]
10), from
[Table]
where (cid:96) (cid:48)
ε ( Y | X , T ; 0) and (cid:96)
[Table]
ε ( X , T
[Table]
0) are deﬁned similar to (cid:96)
[Table]
ε ( Y, X , T
[Table]
0) , we know that
[Table]
Recall
[Table]
[Equation]
and thus, for the ﬁrst term in Equation (12)
[Equation]
we have
[Equation]
where ( a ) follows from the fact that E Y | X ,T ( (cid:96) (cid:48)
(cid:15)
( Y | X , T ; 0))
= 0 , ( b ) uses iterated expectation, ( c ) and ( d ) follows immediately by utilizing the deﬁnition of expectations.
For the second term in Equation (12), we have
[Equation]
where ( a ) follows from the fact that
[Equation]
(b) follows from the fact that
[Equation]
(cid:48) ε ( X | T ; 0) =
E X ,T (cid:96) (cid:48)
ε ( X , T ; 0) =and repeated application of iterated expectation, (c) follows from the deﬁnition of ψ ( t )
= (cid:82) X µ ( t, x ) d P ( x ) .
Comparing Equation (13), Equation (14) against Equation (11), we immediately get Γ (cid:48) ε (0) =
E X ,T,Y (cid:0) ζ ( Y, X , T ) (cid:96) (cid:48)
ε ( Y, X , T ; 0) (cid:1) , which implies that ζ ( Y, X , T ) is indeed an efﬁcient inﬂuence function of Γ .
A.2. P ROOF OF L EMMA
Proof.
This proof follows from Huang et al.
Let us start with the following decomposition:
[Equation]
where ˜ (cid:15) n = (cid:0) ϕ K n ( t ) (cid:1) T ˜ α where ˜ α = (cid:0) B Tn Π − 2 n B n (cid:1) − 1 B Tn Π − 2 n ˜Π 2 n ˜ Z n .
The ﬁrst term (cid:107) ˇ (cid:15) n − ˜ (cid:15) n (cid:107) L 2 is the bias and the second term (cid:107) ˆ (cid:15) n − ˜ (cid:15) n (cid:107) L 2 is the variance.
Bound on bias term Let ˇ α ∈ R K n be such that (cid:107) ( ˇ α ) T ϕ K n − ˇ (cid:15) n (cid:107) ∞ =
inf f ∈B Kn (cid:107)
f − ˇ (cid:15) n (cid:107) ∞ .
Then we have
[Equation]
L
[Equation]
By deﬁnition of ˇ α and properties of B-spline space, we have a bound on the ﬁrst term (cid:107) ˇ (cid:15)
[Equation]
Notice that the second term can also be bounded:
[Equation]
where (a) is from properties of B-spline basis functions, (b) is from Lemma 3, (c) is from properties of B-spline space such that
[Equation]
∞ =
O p ( ρ n ) because (ˇ (cid:15) n ( t 1 ) , ˇ (cid:15) n ( t 1 ) , · · · , ˇ (cid:15) n ( t 1 )) T = ˜Π 2 n ˜ Z n , (d) is from the upper and lower boundedness of ˆ π n .
Following proof of Lemma A.6 of Huang et al.
(2004), we have
Prob
[Equation]
[Table]
[Equation] [Table] [Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
where (a) uses union bound, (b) follows from Hoeffding’s Inequality for bounded random variables.
Since
E ϕ k ( T ) ∼ 1 /K n , we can pick a ∼ 1 /K n and thus, (cid:80) K n k =1 (cid:0) 1 n (cid:80) ni
=1 ϕ k
( t i ) (cid:1)
= O p (cid:16) 1 K n (cid:17) .
Plugging into
Equation (15), we get
[Equation]
Thus, we can bound the bias term
[Equation]
Bound on variance term From properties of B-spline space, we have
[Equation]
Notice that
[Equation]
[Table]
[Equation]
where δ = ( δ 1 , · · · , δ n ) T :
= Z n − ˜ Z n
[Equation]
(a) is from Lemma 3.
By deﬁnition we know
[Equation]
=
u i + ˜ v i where u i = µ ( x i ,t i ) − ˆ µ n ( x i ,t i )
ˆ π ( t i | x i ) − P (cid:16) µ ( X ,T ) − ˆ µ n ( X ,T )
ˆ π ( T | X ) | T =
t i (cid:17) , E ( u i | t i )
=
and ˜ v i
=
v i
ˆ π ( t i | x i ) , E (˜ v i | t
i , x i ) = 0 .
Thus, from union bound, we have
Prob (cid:32) K n (cid:88) k
= sup ˆ π, ˆ µ (cid:32)n n (cid:88) i =
ϕ k
( t i )
δ i (cid:33)> a (cid:33) ≤
K n (cid:88) k
=
Prob (cid:32) sup ˆ π, ˆ µ (cid:32)n n (cid:88)
i =
ϕ k
( t i )
δ i (cid:33)> a K n (cid:33) =
K n (cid:88) k
=
Prob (cid:32) sup ˆ π, ˆ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n (cid:88) i =
ϕ k
( t i )
δ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > (cid:114) a K n (cid:33) =
K n (cid:88) k
=
Prob (cid:32) sup ˆ π, ˆ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n (cid:88) i =
ϕ k
( t i )( u i + ˜ v i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
[Equation]
We could bound the ﬁrst term of equation 18 from Markov Inequality:
[Equation] [Equation]
an (
[Equation]
where (a) follows from Lemma 4:
Rad n (( Q + µ ) U − 1 ) ≤
[Equation]
and (b) follows from plugging h :
x (cid:55)→
[Table]
+
[Table]
in Theorem 12(4) in Bartlett & Mendelson (2002).
Similarly, write A =
( Q + µ ) U − 1 , from Lemma 4, we have
[Equation]
Thus (a) follows immediately from the deﬁnition of Rademacher complexity.
We can bound the second term of Equation (18) using union bound:
[Table]
where
[Table]
from Lemma 4.
Also we have
[Table]
[Equation]
where (a) uses the formula E W = (cid:82) ∞ 0 (1 − F ( w ))
dw − (cid:82)
−∞ F ( w ) dw setting W = | v | I ( | v | > M ) , (b) utilizes the fact that v follows from sub-Gaussian distribution, (c) uses Mills ratio.
Plugging Equation (19), (21), (22) into (18), and taking M ∼ √ log n , a ∼ K n log n n , we get
K n (cid:88)
[Equation]
(cid:19) (23) which, when plugging back into Equation (17), gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:16)
[Equation]
Tn
[Equation]
n B n (cid:17) −
[Equation]
Tn (cid:16) Z
[Equation]
n (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)
[Equation]
O
[Equation]
(cid:32)(cid:114)
[Equation]
Control of the second term in Equation (17): Notice that each coordinate of ˜ Z n − Π − 2 n ˜Π 2 n ˜ Z n is bounded, thus using similar arguments as that of Equation (19), we know that
[Equation]
Combining Equation (24) and (25) into (16), we know that
[Equation]
and thus,
[Equation]
Combining the rate on bias and variance term, we get
[Equation]
where (a) follows from assumption (vi), giving
[Equation]
when taking K n = n 1 / 6 .
A.2. P ROOF OF L EMMA
Proof.
Suppose SVD decomposition of B n = U Λ V T where U ∈ R n × n , Λ ∈ R n × K n , V ∈ R K n × K n .
From Lemma A.3 of Huang et al.
(2004), we know that all diagonal elements of ( K n /n )
Λ T Λ fall between some positive constants.
Notice that the eigenvalues of ( K n /n )
B Tn Π − 2 n B n are the diagonal elements of ( K n /n )
Λ T Π − 2 n Λ .
From the upper and lower boundedness of ˆ π n , we can get the desired conclusion.
A.2. P ROOF OF L EMMA
Proof.
Write F 3 = F 1 + F 2 , F 4 = F 1 − F 2 .
Notice that F 1 F 2 = { f 1 f 2 : f 1 ∈ F 1 , f 2 ∈ F 2 } = { 14 ( f 1 + f 2 ) 2 − 14 ( f 1 − f 2 ) 2 : f 1 ∈ F 1 , f 2 ∈ F 2 } = 14 F 23 − 14 F 24 .
Let h : x
(cid:55)→ x 2 , from Theorem 12(4) of Bartlett & Mendelson (2002) we know that
[Equation]
Thus,
[Equation]
A. A DDITIONAL R ESULTS
This section formally states and proves the efﬁcient inﬂuence function of a multidimensional vector, which is brieﬂy mentioned in the main text.
Suppose Γ =
(Γ 1 , Γ 2 , · · · , Γ d ) T ∈ R d where
[Equation]
Then we have the following theorem:
Theorem 3.
The efﬁcient inﬂuence function for the d -dimensional vector Γ is ζ ( Y, X , T, π, µ, Γ ) =
( ζ 1 ( Y, X , T, π, µ, Γ 1 ) , ζ 2 ( Y, X , T, π, µ, Γ 2 ) , · · · , ζ d ( Y, X , T, π, µ, Γ d ))
T ∈ R d where ζ j ( Y, X , T, π, µ, Γ j ) is the efﬁcient inﬂuence function for Γ j , j = 1 , 2 , · · · , d .
Proof.
Deﬁne Γ ( ε ) :
= (Γ 1 ( ε ) , Γ 2 ( ε ) , · · · , Γ d ( ε ))
T ∈ R d where for i = 1 , 2 , · · · , d ,
[Equation]
Deﬁne (cid:96) ( Y, X , T ; ε ) = log P Y, X ,T ; ε where P Y, X ,T ; ε is a parametric submodel with parameter ε ∈ R d and P
Y, X ,T ; 0 = P Y, X ,T .
Then, the efﬁcient inﬂuence function ζ is deﬁned as the unique function such that
[Equation]
i.e.,
[Equation]
Notice that the efﬁcient inﬂuence function ζ i ( Y, X , T, π, µ, Γ i ) for Γ i does not depend on ε .
Thus for each i, j = 1 , 2 , · · · , d , the above equation can be proved identically to that in Section A.2.5.
A. E XPERIMENTAL D ETAILS A.4. N ETWORK S TRUCTURE
For all methods, we implement the conditional density estimator as a neural network with two hidden fully connected layers, each consisting of 50 hidden units using ReLU activation.
Hidden feature z is deﬁned as the latent representation extracted after the second ReLU activation.
We set number of grids B = 100 .
The estimation of π ( t | x ) is computed as introduced in Section 3.
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 Q for VCNet, except that Dragonnet and DRNet do not use treatment-dependent weights.
In VCNet, the prediction head for µ ( t, x ) is a neural network with 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.
A.4. P ARAMETER S ETTING
For each dataset we tune parameters based on 20 runs.
In each run we 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 lr ∈ { 0 . 05 , 0 . 005 , 0 .
and α ∈ { 1 , 0 . 5 } .
For TR: learning rate for (cid:15) ( t ) : lr (cid:15) ∈ { 0 . 001 , 0 . 0001 } , β ∈ { 20 , 10 , 5 } × n − 1 / 2 .
We found that performance is not sensitive to α .
In estimator of (cid:15) ( t ) , we use B-spline with degree 2 and tune the number of knots across { 5 , 10 , 20 } (all equally spaced at [0 , 1] ).
For TMLE and doubly robust estimator: we tune parameters of B-spline in the same way as in TR version.
During tuning, all networks are trained for 800 epochs.
A.4. D ATASET
Synthetic Dataset We generate data as follows: x j i.i.d.
∼ Unif [0 , 1] , where x j is the j -th dimension of x , and
[Equation]
where t = (1 + exp( − ˜ t )) − 1 .
Notice that π ( t | x ) only depends on x 1 , x 2 , x 3 , x 4 , x 5 while Q ( t, x ) only depends on x 1 , x 3 , x 4 , x 6 .
As discussed in Shi et al.
, this allows us to observe the improvement using VCNet when noise covariates exist.
Results are reported in Table 1.
IHDP The original semi-synthetic 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:
[Equation]
where t = (1 + exp( − ˜ t )) − 1 , S con =
{ 1 , 2 , 3 , 5 , 6 } is the index set of continuous features, S dis , 1 = { 4 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 } , S dis , 2 = { 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 } and S dis , 1 ∪ S dis , 2 =
[25] −
S con .
Here c 1 = E (cid:80)
[Equation]
dis
[Equation] [Equation]
i
[Equation]
dis
[Equation] [Equation]
Notice that all continuous features are useful for π ( t | x ) and Q ( t, x ) but only S dis , 1 is useful for Q and only S dis , 2 is useful for π .
Following Hill (2011), covariates are standardized with mean 0 and standard deviation 1 and the generated treatments are normalized to lie between [0 , 1] .
Results are summarized in Table 1.
News
The News dataset consists of 5000 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 (cid:48) 1 , v (cid:48) 2 and v (cid:48)from N ( 0 , 1 ) and then set v i = v (cid:48)
i / (cid:107) v (cid:48)
i (cid:107) 2 for i = { 1 , 2 , 3 } .
Given x , we generate t from Beta (cid:16) 2 , (cid:12)(cid:12)(cid:12)
[Table]
And we generate the outcome by
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]