Partial Identification of Treatment Effects with Implicit Generative Models
Authors: Vahid Balazadeh Meresht
Vasilis Syrgkanis
Rahul G Krishnan
Venue: NeurIPS 2022 Accept (2022)
Keywords: partial identification
average treatment effect
generative adversarial networks

Abstract: We consider the problem of partial identification, the estimation of bounds on the treatment effects from observational data. Although studied using discrete treatment variables or in specific causal graphs (e.g., instrumental variables), partial identification has been recently explored using tools from deep generative modeling. We propose a new method for partial identification of average treatment effects (ATEs) in general causal graphs using implicit generative models comprising continuous and discrete random variables. Since ATE with continuous treatment is generally non-regular, we leverage the partial derivatives of response functions to define a regular approximation of ATE, a quantity we call uniform average treatment derivative (UATD). We prove that our algorithm converges to tight bounds on ATE in linear structural causal models (SCMs). For nonlinear SCMs, we empirically show that using UATD leads to tighter and more stable bounds than methods that directly optimize the ATE.

Partial Identification of Treatment Effects with Implicit Generative Models
Vahid Balazadeh' University of Toronto, Vector Institute
Vasilis Syrgkanis'
Rahul G. Krishnan Stanford University
{vahid, rahulgk)0cs toronto... edu veyrgk@stanford. edu
Abstract
We consider the problem of partial identification, the estimation of bounds on the treatment effects from observational data. Although studied using discrete treatment variables or in specific causal graphs (e.g.. instrumental variables), partial identification has been recently explored using tools from deep generative modeling. method for partial identification of average treatment effects (ATEs) in general causal graphs using implicit generative models comprising continuous and discrete random variables. Since ATE with continuous treatment is generally non-regular, we leverage the partial derivatives response functions define regular approximation of ATE, quantity we call uniform average treatment derivative (UATD). We prove that our algorithm converges to tight bounds on ATE in linear structural causal models (SCMs). For nonlinear SCMs, we empirically show that using UATD leads to tighter and more stable bounds than methods that directly optimize the ATE,
We propose
1 Introduction
Estimating average treatment effects (ATEs) a common task that arises in fields involving decision- making, such as healthcare and economics. In the presence of the gold-standard randomized controlled trial (RCT) data, one can compare the outcome variable between treated and control groups to make decisions. But RCTs can be costly to set up and run and are, in many circumstances, infeasible. Consequently, communitics are using observational data to assist in decision decision-making.
Identification of treatment effects from observational data is tied to the structure of the causal graph. For example, the treatment and outcome Y in Figure 1b are confounded by an unobserved random variable, making it impossible to find the causal effect of T on Y only from observational data. On the other hand, Figure 1e is identifiable, and one can adjust for confounders using the Back-door formula [Pearl, 2009]. Even in identifiable settings, non-parametric estimations such Back-door adjustment formula can point-identify
ATE only with additional assumptions such as positivity, i.e., P(T 4XX)= >> 0 for all values of covariate X. Observational data is finite, high-dimensional, and consequently can suffer from severe violations of such assumptions [D' Amour et al., 2021].
In lieu of the challenges of point-identification, there has been recognition that decisions can be justified using reliable bounds on the ATE rather than its exact value. For an oncologist treating cancer patient, knowing that drug has a significant, positive reduction in the patient's 's risk of progression may suffice as a rationale to prescribe that drug. This problem is known as partial identification [Manski, 2003]. Most existing methods for bounding the ATES are only applicable discrete/binary treatment variables [Makar et al., 2020, Zhang et al., 2021, Duarte et al., 2021, Guo et al., 2022]. There has been recent interest in continuous treatment settings. However, such
Our code is accessible https://github. httpp//////tttt.oooooa/aaaaaaaaa interiiiaalddettiiiatattoooo
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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X
T
X
Y
T
T
X
(a) Leaky mediation
(b) IV
(c) Back-door
(d) Front-door
bound
upper bound
response
Target, Y
-0
-2
0
2
-5
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Treatmeut.
(e) Partial identification of ATE in finite linear Back-door dataset.
Figure 1: The causal graphs for non identifiable (a) Leaky mediation and (b) Instrumental Variable (IV), and identifiable (c) Back-door and (d) front-door settings. T, X. and Y represent treatment, covariates, and target variables. The dashed double-arrows represent latent factors. (e) The response functions corresponding to partial identification T2[ - =
Back-door linear SCM after training a generative model match the distrihution. (Teftt) shows the results for directly optimizing the ATE, which leads to a non-informative bound due to the irregularity of ATE with continuous treatments. (Right) is our solution by optimizing the UATD, which results in tighter bound. Each point y) represents the expected outcome after intervention T ( in the learned generative model.
methods are applicable for special causal graphs such as the instrumental variables (IV) setting [Gunsilius, 2020, Kilbertus al., 2020] or make parametric assumptions on the family of treatment- response functions [Padh et al., 2022]. An exception is the work by Hu et al. [2021] which provides a non-parametric approach for partial identification using generative adversarial networks (GANs). However, they only provide convergence guarantees for the special case of IV causal graphs.
Using the framework of structural causal models (SCMs) and causal graphs, one see partial identification as a constrained optimization problem, where the objective, i.c., maximizing/minimizinn the ATE, can be written as a post-intervention function of exogenous noise (a.k.a response function) and the constraint is to match the generated samples with the observational distribution. This naturally leads using generative neural networks such as neural causal models (NCMs) [Xia al., 2021]. We find that directly solving the ATE optimization using flexible generative models such as GANs can lead to con-informative and degenerate solutions. The flexibility afforded by generative models such as GANs allows them to deviate significantly from the true response curve in the neighborhood of intervention points to maxinize/minimize the ATE while continuing to generate samples akin to the data distribution. Figure le (left) showcases typical solution to the ATE optimization.
Our insight that the ATE between any two points can be approximated as an integral over the derivatives of the response function w.r.t. the treatment variable. Rather than directly optimizing the ATE, we optimize the partial derivatives of the response function, a quantity that we refer as the uniform average treatment derivative (UATD). By optimizing the UATD, the model is required maximize/minimize the partial derivatives for all points within the treatment support, avoiding extreme local solutions as shown in Figure le (right). Our contributions are as follows:
We formally define the partial identification of average treatment effects as a distributionally- constrained optimization problem, where we choose Wasserstein distance as our constraint metric.
For the class linear SCMs, we prove that the solution to our optimization problem converges to optimal bounds the true value of ATE in infinite data for general causal graphs.
We use the solution partial identification of UATDs to find informative bounds on the value of ATE. We introduce a practical algorithm to solve the problem using the Lagrange multiplier formulation with alternating optimization. We empirically
optimization
Average treatment derivative is also known average partial effect in the literature [Powell et al., 1989, Wooldridge, 2005, Rothenhäusler and Yu, 2019].

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show that our algorithm results in tighter and more stable bounds than methods that directly optimize the ATE.
Problem Setup & Background
We introduce the definitions and assumptions will use throughout the paper. Consider the observed data as (possibly continuous) random variables V [X,, .mm T, Y E RR, where T, Y. and (X1," X denote the treatment variable, target variable, and covariates, respectively.
Data generating model. Our approach will be based on the framework of Structural Causal Models (SCMs). An SCM is tuple M = (V, F, Pv), where each observed variable V; E V is deterministic function of a subset of variables pa(V)) S V and latent variables Uv, S U, i.c.,
V2 = (pa(V). Uv;) where fv. E F,, V1 & pa(V.)
The only source of randomness are latent variables U with probability space (S, E, Po). This induces a probability law over the observed variables PM. We may omit the subscript M and denote the observational probability distribution by P throughout the text. Given M, one can construct graph with nodes V YUU U and directed edges from nodes in pa (V,) U Uv; to V. We call this graph the causal graph corresponding and known. Moreover, We will assume each node in the graph is 1-dimensional. For random variable V in the SCM M, let (u be its deterministic value after fixing realization u of latent variables U. The causal effect of treatment T on target Y is:
SCM M and denote it by 9M simply G. We assume 9 is acyelic
Definition (Causal Effect). Let (M(T=)) (u) be the value of Y by fixing = u and changing the function fr a constant function fr = in M. Then, we call the random variable MMT==t) the causal effect of treatment T = on target Y. We may simplify the notation and write it as Y, if the SCM M and treatment variable T are known from context. Note that Yríu) Y(u).
When T is continuous, then we can view {Y: : t E (T)p(T)) } as a stochastic function defined on (2, Σ, Pu). This is referred to the response function, partial dependence plot, and dose-response curve in the literature [Zhao and Hastie, 2021, Ritz et al., 2015, Chemozhukov et al., 2018].
Average treatment effect, average treatment derivative, and partial identification. Our goal is to estimate bounds on the effectiveness of treatment regime on population from the observational distribution P and the causal graph G. In the continuous treatment case, where there is no "on"/"off" notion of treatment, we can compare the average causal effect of an arbitrary treatment (dosage) to the average causal effect at a fixed point T = to. For example, to indicate the effect relative to not prescribing any treatment, we can choose to = O. This quantity is known as the average treatment effect, average level effect, or average dose effect in the literature on continuous treatment setting [Hirano and Imbens, 2004, Kennedy et al., 2017, Callaway al., 2021].
Definition (Average Treatment Effect). For SCM M, the average treatment effect (ATE) at T d w.E.t. fixed point T = to is defined as
(d) = Eu-Po (VM[T=a) - (M)
(2)
Note that estimating the ATE and finding bounds it only depends on the value of the average response function in T' = d and T = to. As pointed in Gunsilius [2020], this quantity can take arbitrary values if we do not make any assumptions on the set of response functions. Here, we assume the partial derivative of the response function w.r.t. the treatment, i.c., 0Y,/ōr exists and is a bounded continuous function. We then define the average treatment derivative as the following:
Definition 3 (Average Treatment Derivative). For the trealment regime fr in SCM M, define the average treatment derivative (ATD) as
(() Ōt
ATD M = EurPu
(3)
[=T(())
Estimating the ATD can seen as a proxy for the effectiveness of the prescribed treatment, where we consider the population-level average effect of an infinitesimal increase in the treat- ment/dosage [Rothenhäusler and Yu, 2019]. In this work, however, we leverage the regularity of this quantity to achieve smoother solutions to the ATE estimation. We will expand on this in section 4.
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Note that we cannot readily use eq. 2 (or eq. 3) to estimate the ATE (or ATD), as we only have access to the observational distribution P and the causal graph G and not the latent distribution Pu. In fact, ATES are generally non-identifiable, i.e., there exist multiple SCMs with the same causal graph 9 and generated distribution P that result in different values of ATE. For some graphs, however, one can use parametrati identification algorithms like do-calculus to identify the causal effect from the observational distribution [Pearl, 2009]. In practice, even for identifiable causal graphs, we cannot pinpoint the true ATE due to the uncertainty caused by sampling variation and finite sample errors. Instead, we are interested in finding a tight set of possible solutions that will contain the true value of ATE (or ATD) with high probability. This is known the partial identification problem in the literature [Manski, 2003].² More formally, the partial identification of ATDs/ATEs is defined as: Definition 4 (Partial Identification of ATD/ATE). Partial identification of ATD is the solution to the following optimization problem:
min ATD max Mr) ) s.t. PMe = P G M
(4)
M'
where Mi is the set of all SCMs on random variables V. We denote the solution to the above problem as (ATD, ) Similarly, we can define the partial identification of ATEs by replacing ATD with Mr (d) in eq. 4. We refer to the solution to the latter problem as (TTEd))ATEE(
Implicit generative models. To solve the partial identification problem, we use the expressive power of generative models to satisfy the distribution constraint in eq. 4. Choosing distance measures such as -hasen-Shannon divergence or Wasserstein metric results in models such as GANs or Wasserstein GANs (WGANs) [Goodfellow et al., 2014, Arjovsky al., 2017]. The typical way to implement these models is to solve minimax game between the generator and discriminator. However, adding the ATD training. Instead, in our practical algorithm, we will use Sinkhorn Generative Networks (SGNs) that use Sinkhorn divergence S., differentiable e-approximation of Wasserstein metric, as the distance measure between generated and true samples [Cuturi, 2013, Genevay et al., 2018, Feydy et al., 2019]. Due to the differentiability of Sinkhorn divergence, we will only need train a generator network enabling us to sidestep much of the unstable minimax training in (W)GANs.
term to the minimax loss function will result in unstable
3 Related Work
This work builds upon partial identification and generative causal models.
Partial identification. Finding informative bounds on treatment effects has been well-studied in the existing literature ([Robins, 1989, Manski, 1990, Evans, 2012, Ramsahai, 2012, Richardson et al., 2014, Miles et al., 2015, Finkelstein et al., 2021, Zhang and Bareinboim, 2021a.b]). Balke and Pearl [1997] find the tightest possible bound for the discrete instrumental variable setting by converting it to a linear programming problem. For the backdoor setting and binary treatments, Makar et al. [2020] provide probabilistic upper/lower bounds potential outcomes in the finite sample regime. Recently, Zhang et al. [2021] and Duarte et al. [2021] independently describe a polynomial programming approach to solve the partial identification for general causal graphs. They both use the notion of canonical SCMs to map the latent variables to the space of functions from treatment T' to outcome Y. Though they show their polynomial programming formulation finds the optimal bound, their approach is only applicable to discrete random variables with small support. In fact, the time complexity of their algorithm grows exponentially with the size of the support set of variables, making their algorithm intractable for continuous settings.
Gunsilius [2019] extends the commonly-used linear programming approach partial identification of IV graphs with continuous treatments. They use a stochastic process representation of the variables and solve the linear programming via sampling. However, their method suffers from stability issues, as discussed in Kilbertus et al. [2020] and is only applicable for the IV setting. Kilbertus et al. [2020]. Padh et al. [2022] parameterize the space of response functions by assuming them as linear combinations of a set of fixed basis functions. Then, they match the first two moments of observed the ATE. However, they do not provide any theoretical guarantees on the tightness of their derived bounds.
distribution while weapiiizing/maxmmmziiii
LIn the literature, partial identification is not concemed with sampling uncertainty and is detined population- wise for non-ie oonidentiffialle causal effects. However, in this paper, we abuse the terminology and use partial identification for both non- non-identifiable quantities and identifiable effects with finite samples.

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Most similar to our work is Hu et al. [2021] who use generative adversarial networks (GANs) to match the observed distribution and search for response functions with maximun/minimum ATEs. They provide convergence guarantees for the instrumental variable causal graph with linear models. Their approach is also based on the minimax game between generator and discriminator, which can result in unstable training. Our work differs in a few important ways. We focus on partial identification of average derivatives and use that to find bounds over the ATE. Using this approach, we show that our derived bounds converge to the optimal bounds for linear SCM with general causal graphs, including both identifiable and non-identifiable settings. We use Sinkhorn divergence, a differentiable approximation of Wasserstein distance, to train our implicit generative models. Empirically, we find that this avoids the unstable training of GANs. Guo et al. [2022] studied the partial identification of ATE with noisy covariates. Their work is similar to our approach in that we both use similar robust optimization formulation. However, they focus on identifiable causal graphs, where one can use adjustment formulas such as the Back-door formula and make parametric assumptions on the joint distribution of observed variables.
Generative causal models. [Goudet et al., 2017, Yoon et al., 2018, Kocaoglu et al., 2018, Sauer and Geiger, 2021] use generative models to capture a causal perspective on evaluating the effect of interventions on nigh-dimensional data such as images. They do not consider the problem of bounding treatment effects. Xia et al. [2021] introduced Neural Causal Models (NCMs) that leverages the universal approximability of neural networks to learn the SCM. Although it is not generally possible to learn the true SCM by training on the observational data, they prove that NCMs can be used to test the identifiability of causal effects and propose an algorithm to estimate identifiable causal effects. Their work's theory and empirical instantiation are in the context of discrete random variable datasets. Our work builds upon NCMs for partial identification with both continuous and discrete random variables.
4 Partial Identification using Implicit Generative Models
We explain our method to solve the partial identification problem in Def. 4 using implicit generative models. In subsection 4.1, we describe partial identification of ATDs as a constrained optimization problem using G-constraint generative models [Xia et al., 2021]. Then, in subsection 4.2, we show that the solution to this constrained optimization problem converges to the optimal bounds on the ATD in infinite data samples. We prove our results for linear SCMs with general causal graphs, ie.. both identifiable and -on-identifiable settings. Next, we propose our approach to extend the partial identification of ATDs to ATEs. Finally, we describe a practical algorithm solve our isstrbutioonally-contrrrai optimization problem in subsection 4.3.
4.1 G-constraint generative models
To solve the partial identification problem, we need search over the set of all possible SCMs 21. This is generally not feasible, as there is no constraint on the distribution of the latent variables Pt. as well the function family F. Instead, we parameterize the space of all SCMs that are consistent with causal graph G using neural networks. More specifically, we use G-constraint generative models:
Definition 5 (G-constraint Generative Models (Def. 7 in Xia et al. [2021])). For a given causal graph G. a G-constraint generative model is a tuple Ma = (V, U, (o)) where each V, E V is
generated from
V12 f (pa (V.), Úc) for fy. F
(5)
where pa( V3) is the observed parents of node V; in 9 and Uc E Us the latent noise corresponding maximal C2 -Component C C V containing node Vú, i.e. each pair of variables in C have common latent parent nodes. In addition, Pú Uni for each U E Ü.
G-constraint generative models make the search over the set Di feasible by limiting it to generative models with uniformly distributed latent variables that consistent with causal graph G. In fact, in their Theorem 3, Xia et al. [2021] show that for any discrete SCM M with causal graph G. there exists G-constrained generative model MÓ that generates the same causal effect, i.e.. YM-(F=)) = Yug (T=() a.s. Their proof technique, however, only works for SCMs with discrete variables. Here, we do not prove the expressiveness of g Gooonstradde generative models for continuous
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SCMs. Instead, for simplicity and completeness of our theoretical results in subsection 4.2, we assume that the true SCM is a G-constrained generative model itself. In our experiments, we empirically show that our results hold even for SCMs with different latent distributions, such as Gaussian noise.
Assumption 1. The true SCM M is a G-constrained generative model. In other words, there exist such that M MMg
Under Assumption 1, we reformulate the problem in eq. 4 using generative models, i.e.,
(min ATD mg. max ATD
ATD,Mg) mgg s.t. PMg = P
(6)
In practice, we never have access to true distribution as we only observe a finite number of samples corresponding the empirical distribution lM 1 Σس= 8y00 for given dataset {v()).... v(()), J.
Also, the observed variables may be biased due to noisy measurements. Therefore, we reformulate the problem in eq. our distance measure, which naturally results in generative models such as WGANs. We will state our theory in subsection 4.2 based on this metric. However, in subsection 4.3, we will propose a practical algorithm that uses Sinkhorn divergence, a differentiable approximation of 1- 1-Wasserstein distance, for more stable results. Our constrained optimization problem is as follows:
constrained optimization problem. We choose the I-Wasserstein metric as
mjn ATD Mg. max ATD
s.t. Wi (PMg, Pm S On
where Os, is a solution to eq. 7 as LATD,ATD) In the case of noisy measurements, of how noisy the data is to determine the value of ax.- Otherwise, we can use the sampl convergence rate of empirical Wasserstein distance to choose an appropriate value of a,, [Weed and Bach, 2019]. As our theoretical results are concerned with the infinite-sample case, we will assume that there exist values of Ot, such that the true distribution lies within the Wasserstein ball. Assumption 2. For each n E N, there exist an > 0 such that W1 P*) S ag.
that specifies the level of tightness of the bounds. We denote the
need domain knowledge
4.2 Theoretical guarantees and extension to ATES
Assumptions and 2 ensure that the bound derived by eq. 7 contains the true value of ATD. However, we do not know how informative/tight the derived bounds are. In fact, one can always return (-o0, +00) one solution to partial identification. This part gives theoretical guarantees that our algorithm can result in tight bounds over ATD. In particular, we focus on linear SCMs and show that, under the infinite number of samples, our algorithm converges the optimal bound (ATD, ATD) for both identifiable and non-identifiable causal graphs. See Appendix A for the proof.
Definition 6 (Linear SCMs). SCM M (V, U. FP) is linear, if
pa(i) + bJ, Uv, for vectors av, bv, F
(8)
Theorem (Tight Bounds). Assume the dataset (v()).... ..... vnn) is generated from linear SCM. Then, under assumptions 1, and 2, the solution to the constrained optimization problem in eq. 7 converges to the optimal bound over the ATD in infinite samples, ie., AÎD + ATD and ATD + ATD.
Up until now, we have only focused on partial identification of ATDs. Here, we discuss how to extend our results to find bounds on ATEs. A naive solution is to replace ATD with ATE in eq. 7 and directly optimize However, as demonstrated in the experiments, this approach can result in non-informative bounds. In fact, ATE with continuous treatments is non-regular quantity [Kennedy et al., 2017].
Instead, we claim that we can use the same generative model trained for partial identification of ATD to bound the value of ATE. In particular, we define a new objective function by uniform intervention on the treatment, which we call uniform average treatment derivative (UATD), and show that the solution to partial identification of UATD matches the solution to partial identification of ATES.
Definition (UATD), For an SCM M, we define the uniform average treatment derivative (UATD) at interval [to, d as
0Y
UATD,M[to d] == Eu-Po Ba-
(9)
Øt

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response
upper bound
treatment density fi
0.5
Target
0
-5
5
-5
5
5
Treatment. T
Figure 2: Maximizing the UATD between
and T with approximation densities μ. standard deviation of μı increases from left to right, resulting in smoother response curves. All three plots are trained on a linear data generating process Y = T + +0, 1) using 5,000 samples
Now, we state our result on using average derivatives to solve partial identification of ATES:
Corollary 1. Let 8t be the solution to the partial identification of UATD at interval (ta, d]. Then, 6" also solution to the partial identification of ATE(d). If the true SCM M is linear, then the bound is light, i.e., ATE(d) ATE(d) d) and ATE(d) + ATE (d) as
The proof is given in Appendix B.
Remark. ATD and ATE are generally different quantities and finding bounds on one does not necessarily results in bounds on the other. In fact, ATE not pathwise differentiable for continuous treatments [Diaz and van der Laan, 2013, Kennedy et al., 2017, Chernozhukov et al., 2018] (See Ap- pendix C). We define UATD as quantity relate ATE and ATD. The formal definition of UATD in Def. 7 is the same as ATE up to a scale factor, and one would expect to see similar issues with ATE here as well. Therefore, we approximate UATD with version that, instead of uniform distribution of treatment within interval [to, d] with zero density outside, the treatment distribution has continuous differentiable non-zero density defined over the whole support of T. More concretely,
0Y Mg (T
ATE Mg (d) x UATD (4u d = EurPo
dull)
eeppp(T)
Ō1.
0Y Mg = dīa(t)
~ Eu-Po Po
(10)
supp(TT)
where y is the uniform measure within interval [fo, d] and ji is its approximation with a non-zero density over the full support of Choosing trades off between regularity of the response function and the approximation error. The more jı is close to the uniform measure the more irregularity we allow the response curve and the less informative the bound will be, while the objective function is closer to ATE. Adding more density outside of interval [to. d imposes regularity on the response curve (Figure 2). It important treatment distribution) in Corollary since, in linear SCMs, it is not possible for the generator model attain arbitrarily large values in points to and as the derivative of linear functions remains fixed outside of interval [to, d]. This is why we state our theoretical results based on the uniform intervention.
that we use the formal definition of UATD (with uniform
4.3 Our algorithm
We describe algorithm to solve the optimization problem in eq. 7. We will focus finding ATD, similar approach can be taken for ATD. A general strategy is to convert the constrained problem to its unconstrained version using the method of Lagrange multiplier:
min max ATD ATDMg mg + X(W (P (PMg. P") - -a.)
(11)
A20
cannot directly gradient descent solve eq. 11. A common approach is to use the dual formulation of Wasserstein distance W, (PMg..P") = max 51 Ep- (v)) Ep,,, [v)( and solve eq. 11 using WGANs, similar solution used
As the Wasserstein distance is not differentiable,
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Extimated Bound
Thee ATD
11.0
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500
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(a)
(b) Number of samples,
(c)
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Figure 3: Our derived bounds ATD for linear Back-door, (b) Front-door, (c) linear IV and (d) leaky mediation settings. As number samples increases, our algorithm pin-points the ATD identifiable settings and leads to tight bounds on it in non-identifiable cases.
in Hu et al. [2021]. However, this min-max-mas formulation can result in unstable bounds as show in our experiments. Instead, we use Sinkhorn divergence, a differentiable approximation to Wasserstein distance, as the measure of distance between distributions and solve the following:
min max ATD MY + ((S.(PMg: P*) an))
(12)
A20
To solve eq. 12, we need to evaluate ATD .y and calculate its gradient w.r.t 0. As using g-constrained generative models, we can calculate the value of y (T= (u) by hard intervention
T i.e., fixing the output of function 19 i and computing Y through topological order of calculations. Then, we estimate ATD as follows:
1 Σ (uger= (M (T= (1) (n())
ATD
(13)
where [== are samples from the treatment variable, and {u() Ji=1 are the latent variables generated from uniform distribution. To choose appropriate value of on we first train our generator without the ATD term until the Sinkhorn loss converges to some value and use that as our choice of a., We then continue training by adding the ATD term.
We note that using algorithms such as projected gradient descent solve the constrained optimization problem requires us to project the weights of our network into the Wasserstein (Sinkhorn) ball in each step. This can be computationally infeasible, and current methods are mainly focused on special loss functions [Mohajerin Esfahani and Kuhn, 2018, Li et al., 2019, Wong et al., 2019]. Instead, we consider an alternating optimization procedure, in which we alternate hetween updating the gradients for the ATD and the Sinkhorn loss. The full details of our algorithm, its extension to ATEs, and the alternating optimization are described in Appendix D.
Experiments
We run our partial identification algorithms on a variety of simulated settings. We mainly focus on the synthetic data generating processes as the ground truth must be known to evaluate our derived bounds properly. Our primary goal is to show that using uniform average treatment derivatives instead of directly optimizing the average treatment effect will result in tighter and more stable bounds. We first run our algorithm to estimate bounds on the value of ATDs for both identifiable and non non-identifiable causal graphs. We show that, as the number of samples increases, our algorithm converges to tight bounds over the true value of ATD (Figure 3). We then focus on partial identification of ATES and demonstrate that using partial derivatives of the response function leads to more informative bounds, while being valid (Figures 4a, 4b).
Additionally, as sanity check, consider two breaty-t datasets where the optimal bounds are known and show that our approach can reach the optimal solution. We also test our method on an ACIC dataset, a case study with real-world covariates, to illustrate the performance of our algorithm on higher-dimensional datasets. We include these additional experiments in Appendix E. Our implementation details, well as cupee-parametert can be found Appendix
5.1 Datasets and Baseline
Datasets. We consider various data generating processes for different causal graphs, including a linear SCM with
covariates and a quadratic SCM with nonlinear interaction between
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Our Bounds
GAN Bounds
Real samples
37
0.5
3.5
Target,
25
2
2
170
34
-99
65
2008
15
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Treatment. T
(a) GAN baseline. (top left) Nonlinear Back-door, (top right) linear IV, (bottom left) nonlincar IV, and (bottom right) leaky mediation settings. to is chosen as the maximum treatment value in each data. Our derived bounds are tighter and more stable than the GAN baseline, which directly optimizes the ATE.
E[Yoo]
Our Bounds
Padh al.
Samples
50
100
Target, Y
-50
Treatment. T
(b) Padh et al. [2022]. (left) Nonlinear IV, (right) linear with strong confounding. to is chosen as the minimum treatment value in each data. While resulting tighter bounds in the linear setting, the baseline leads to invalid bounds in the nonlinear dataset.
Figure 4: Comparing our results for partial identification of E[Y] B[Y.]] for 10 different values of treatment.
the covariates and treatment for the Back-door causal graph (Figure Ic), as well as a nonlinear SCM for the Front-door setting (Figure 1d). For the IV graph (Figure lb), use four different SCMs, two linear and two nonlinear datasets, based on the strength of the instrument variable and the confounding. Finally, we generate two-dimensional linear dataset for the leaky mediation causal graph (Figure la). The full details of our lata-generating processes can be found in Appendix G.
Baselines. Our first baseline is the algorithm in Hu al. [2021] that directly optimizes the value of ATE using GANs. We use their default hyper-parameters with a tolerance of 0.0001. Similar to their experimental setup, we consider 50 intermediate solutions where the distance is within the tolerance and compute the bounds using the mean and one-sided confidence intervals.
We also compare algorithm with the moment-matchinn method in Padh et al. [2022]. They parameterize the first two moments of the generated distribution and match them with the observed samples. They choose response functions as linear combinations of fixed number of basis functions. We consider the neural basis functions for our experiments with their default hyper- setee--aramterrs. In particular, train 3-hidden layer MLP with 64 neurons in each layer to learn the target variable given the treatment. We then use the activation of the kth neuron in the last hidden layer as the kth basis function. Instead of maximizing/minimiznn our goal is to bound the ATE between two points. We leave other implementation details untouched.
E consider [ E( as
5.2 Results
Bounding average treatment derivatives. We generate data with sample sizes (500, 1000, 2000, 5000) from the nonlinear Front-door and linear Back-door SCMs (identifiable), as well as linear IV with strong confounding and leaky mediation settings (non-identifiable). We run our algorithm with ten different random seeds for each setting/sample size. Then, choose

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the five runs with the lowest tolerance parameter on and choose the upper (lower) bound as the maximum (minimum) value of the ATD within these five runs. Figure 3 shows our derived bounds. As expected, the algorithm is able to point-identify the value of ATD for identifiable scenarios as the number of samples increases (Figures 3 (a) and (b)). In non-identifiable cases, our algorithm leads to tight bounds containing the true value of ATD by increasing the number of samples as depicted in Figures (c) and (d). This is in line with our results in Theorem 1.
Bounding average treatment effects. Here, we aim to demonstrate the effectiveness of using partial derivatives for bounding the ATE compared to the direct optimization approach. We consider six different settings and run our algorithm for 10 different values of treatment [t in each setting, We compute the value of ATE w.r.t. a fixed point to. For each value of T' we generate N = 5,000 samples and run each experiment five times. Then, we select the maximum (minimum) value of ATE within the five runs as the upper (lower) bound. We follow the same procedure for the baselines.
To find the bounds on ATE using our approach, we sample from a distribution with uniform density within [to, t4] and Gaussian tails outside of [to, ti], and maximize/minimize the partial derivatives. Figure 4a shows the effectiveness of this approach in comparison to the GAN baseline, Our algorithm produces stable and tight bounds containing the true value of ATE, while the GAN baseline, which relies on the direct optimization of ATES, results in unstable loose bounds that may not include the true value of the treatment effect.
Figure 4b compares our algorithm with the method in Padh et al. [2022]. First, we find that their method do not contain the actual response curve in the nonlinear IV setting (left). This is not surprising result; since they consider a fixed set of hasis functions, the set of possible response curves is more restricted than ours. If the basis functions of the response curve, the resulting bounds can be invalid, thus not containing the actual value of ATE even if there are regimes where the approach provides tighter fit. On the other hand, our approach considers a larger family of response functions, enabling us to both capture shape and variation in the response curve while bounding the effects.
not carefully designed capture the true form
6 Conclusion, Limitations and Future Work
Our work introduces novel method to estimate bounds on average treatment effects from obser- vational data. Specifically,, we propose optimizing the average treatment derivative, which in turn can be used to estimate the average treatment effect in treatment response curves. Empirically we find that the use of our method recovers known bounds on treatment effects in the discrete case and outperforms other methods based implicit models for partial identification in the continuous case.
There remain several limitations of this work. Our work builds on the constrained optimization problem defined by Xia et al. [2021] instantiated in the context of the ATD. Developing new methods for function maximizzaion/mmiimizaaiio
approaches under distributional constraints remains an important direction for future work. Our work primarily uses carefully designed synthetic datasets to evaluate our method under different constraints on the data distribution. A larger-scale evaluation of our approach on real-world benchmarks will better help us assess the method's practicality. Moreover, the theory of our work restricted to the linear SCM scenario. We have also made regularity assumptions throughout the paper, including Assumption 1, that the true SCM can be modeled using implicit generative models with uniform confounding distribution, as well as the approximation of UATD with regular treatment distributions. Consequently, practitioners must exercise caution when deploying this method when there are nonlinear or irregular structures among the random variables. Finally, we have stated our results (Theorem I and Corollary 1) without quantifying the finite-sample estimation uncertainty. One can use the existing theory of Enite-sample convergence of empirical Wasserstein distance in Weed and Bach [2019] to extend our results with high-rrobbiiltty bounds.
Acknowledgments
We thank Tom Ginsberg and Michael Cooper for many helpful discussions. We also thank David Alvarez Melis for his suggestion on using Sinkhorn divergence instead of Wasserstein distance. This research was supported by NSERC Discovery Award RGPIN SEFIN-2022-00 2-2206446 and a CIFAR AI Chair. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute.
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Paper checklist
1. (a) Do the main claims made in the abstract and introduction accurately reflect the paper's s contributions and scope? [Yes] We have listed the key contributions of our work the introduction. In the experimental section we verify our theoretical contributions and perform comparison the closest baseline. Our theoretical results are in idealized linear settings though we do experiment beyond that in our work. We highlight the limitations of our theoretical results and explain the rationale for the approximations that we make in practice.
(b)
Did you describe the limitations of your work? [Yes] We explicitly highlight the limitations of our work in the context of related work, in our exposition of the theoretical results and the computation necessary for method. Our theoretical results make explicit the assumptions that we make and full proofs are presented in the supplementary material.
(c)
Did you discuss any potential negative societal impacts of your work? [Yes] Our paper explicitly tackles the problem of bounding treatment effects from observational data. We have highlighted that our theory is focused on the case of linear models and provided empirical results supporting the method's 's utility in the general, non-linear case. If the method is deployed in high-risk settings such as healthcare, there may be times where the method fails to estimate bounds correctly from real-data in non-linear settings. We have encouraged practitioners to excercise caution in this regard.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] We include the complete proofs of our results in the supplementary material.
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [Yes]
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes] Our supplemental material makes clear the training procedure for our method, the main sets of hyperparameters and any heuristics for optimizing the models. We will release code to reproduce our results. Our work made no use of participants.
(c) Did you report error bars (e.g., with respect to the random seed after running exper- iments multiple times)? [Yes] Our method finds maximum/ninimum values on the quantity of interest. Instead of reporting error bars, we consider maximum/minimum values among multiple runs with different random seeds.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Experiments were run on two single machine with 2 Nvidia RTX A5000 GPUs, 2 RTX A6000 cards GPUs, 96 CPUs and 188GB of RAM and on a larger internal cluster comprising hundreds of GPUs.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [Yes] Our evaluation is primarily on synthetic data for which we have cited the original authors who created them.
(c) Did you include any new assets either in the supplemental material or as URL? [No]
(d) Did you discuss whether and how consent was obtained from people whose data you's 're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We use publicly available datasets that do not contain unanonymized personal data.
14


Reviews:
[{"review_id": "yHR79TjZioI", "reviewer": "Reviewer_LsyV", "summary": "The paper proposed to perform partial identification of average treatment effect using average treatment derivatives and implicit generative models. The paper argues that quantities like ATE can be hard to optimize due to certain regularities and resorting to average treatment derivatives and their expectations make things much more amenable to computation. Similarly, implicit generative models are also more amenable than GANs in modeling observed data distribution.", "questions": "Can you please clarify the points in the strengths and weaknesses section?", "limitations": "The paper discusses the limitations.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n+ The problem of partial identification with continuous variables is important.\n\n+ The approach of using constrained optimization with implicit generative models is sound (under assumptions).\n\nWeaknesses:\n\n+ It is unclear why the idea of average treatment derivatives resolves the computational issues in partial identification. In particular, it seems like average treatment derivatives work in the proposed setting because it implicitly imposes regularity or smoothness assumptions like continuously differentiable log-density. Thus it is not the average treatment derivatives that are helping with the computational issues, but rather these additional regularity and smoothness assumptions that are helping with the computational issues. In principle, one should also be able to impose these assumptions to achieve computational gains without invoking average treatment derivatives.\n\n+ Across the paper, there are (regularity) assumptions made, e.g.implicit generative models, in general, can only approximate a subset of all distributions; the paper also considers smooth densities for average treatment derivatives. While these assumptions make sense, they usually have non-trivial implications on the partial identification bounds. For example, it is well known that low-complexity unobserved confounders or functions (e.g. smooth functions) restrict how much the unobserved variables can induce confounding. Thus these regularity conditions are non-trivial, and their implications on the partial identification bounds need to be carefully and transparently studied. The validity of the resulting partial identification bounds must be qualified.\n\n+ While the paper focuses on using implicit generative models for partial identification, it has limited applicability to usual sensitivity analysis assumptions like marginal sensitivity conditions. The algorithm can only work with the constraints that the implicit generative models induce a population distribution that is close to the observed data distribution. However, It appears very challenging to work with classical sensitivity assumptions: how could one impose the marginal sensitivity assumption (Tan, 2006) on the implicit generative model and obtain a partial identification bound? Without applicability to these standard sensitivity assumptions, the applicability of the proposed algorithm can be limited, since the ATE bound tends to be wide in practice.\n\n+ Despite the asymptotic guarantees, the proposed algorithm cannot account for sampling uncertainty. It does not quantify the estimation uncertainty of the min and max ATE/ATD. (Also unlike L116, sampling variation does not motivate nor necessitate partial identification.)\n\n\nTan, Z. (2006). A distributional approach for causal inference using propensity scores. Journal of the American Statistical Association, 101(476), 1619-1637.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "rUFsQcExnUp", "reviewer": "Reviewer_jbMK", "summary": "This work proposed a new method for partial identification. The new method uses GAN for modeling, with the novelty of replacing the usual objective function by a new one that is based on average treatment derivatives, which is shown in this paper to have superior performance than the vanilla average treatment effects. The core is by putting assumption of G-constraint generative models with uniform distribution of latent variables, which leads to Equation (9) connecting ATE with ATD. ", "questions": "1. In experiments the author seems to compare only with one baseline -- are there any other baselines that can be used? For this paper, experimental evaluations are very important, but the current experiment section seems a little bit insufficient.\n\n2. It is not clear to me why using ATD is better than ATE? Some intuition should be added in the paper.", "limitations": "Yes", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Originality:\nI am not very familiar with related literature. Based on section 3, I think the paper has sound originality compared with existing works.\n\nQuality:\nThe quality of this work is acceptable.\n\nClarify:\nThe presentation of this paper is good. \n\nSignificance:\nThe new objective function (ATD) proposed in this work is not dramatically different from the existing ones, but seems to result in great improvement in experiments. One concern on the experiment section is raised in the next section. \n\nI believe the main strength of this paper is the author's clever use of the existing ideas in literature, especially the concept of G-constraint generative models in Xia et al [2021]. This observation itself leads to a simple but seems effective method. This contribution is significant, beyond that other technical ones are less so. \n\nSome minor comments:\n1. In Line 241, the convergence of a interval is not defined\n2. Assumption2 seems a little bit weird — is it a trivial assumption?\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "cXckxguYwF", "reviewer": "Reviewer_sNpo", "summary": "The manuscript proposes a novel method for bounding average treatment effects (ATEs) with continuous treatments by optimizing the average treatment derivative (ATD) using Sinkhorn generative networks. The authors empirically show this to result in stabler, more informative bounds than attempting to directly optimize the ATE itself. ", "questions": "N/A", "limitations": "The limitations are well covered in §6.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Partial effect identification with continuous treatments is an important and challenging problem, of great interest to methodologists and practitioners alike. The results here are quite impressive, including a theoretical guarantee in the case of linear SCMs and strong empirical performance under a range of different data generating processes. The manuscript is well-researched, building on a number of recent papers in this area. The writing is generally strong and easy to follow.\n\nMy only real critique is one readily acknowledged in the paper’s conclusion, namely that the empirical examples are drawn from a range of small-dimensional simulations rather than more challenging semi-synthetic or real-world datasets with more covariates. Generative modeling gets exponentially difficult in high dimensions, so methods based on GANs or other generative approaches may struggle in this setting. This is one motivation for Padh et al.’s (2022) partial identification approach, which relies instead on moment matching techniques. I am unsure whether their algorithm is publicly available for comparison, but if possible I think this would make a valuable baseline, especially on more complex data.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "1XkH1wxd4dV", "reviewer": "Reviewer_3JcQ", "summary": "This paper proposes a new method the partial identification of causal effect for when the variables might be continuous. There has been a lot of progress in the identification of causal effect in the discrete and finite settings, but bounding for continuous treatments is relatively new. \n\nSimilar to most recent work on bounding treatment effects for continuous variables, the paper defines a generative model and bounds effect by minimizing and maximizing the quantity of interest (in this case Average Treatment derivatives (ATD)) with the constraint that the the generated data should match the observed data. The authors then show how bounding ATD also helps to bound ATE without doing a separate optimization, under certain assumptions such as the distribution of the confounding being uniform for example. The authors then compare their work to to the work of Hu. et al and show that they get tighter bounds.", "questions": "* Is it a new idea to use ATD to bound ATE or has this been done before?\n* l544 (appendix). 'Our proof can be extended to multi-dimensional random variables by assuming each dimension as a separate variable.'. Could the authors elaborate on this a bit?\n* Figure 2: Have the authors also done this experiment with the Hu et al. comparison?\n* Is there a particular reason why Padh et al. was not used as one of the baselines as well?\n", "limitations": "The authors have generally addressed the limitations well. Though I would suggest to add Assumption 1 as a limitation as I have detailed in the main section of the review.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\nI find the paper to be generally well written and easy to follow. The assumptions are also clearly stated. And while the use of a distributionally constrained optimization formulation for partial identification of continuous treatment effects is not new (Hu et al., Padh et al., Kilbertus et al.), I find the idea to use ATD to bound ATE quite nice and certainly relevant and interesting to the community. The authors also provide a very nice overview of the related work. It is also worthwhile and interesting that the authors provide a proof of validity and tightness of the bounds for continuous variables, even if in limited settings and under strong assumptions.\n\n**Weaknesses**\n* Assuming a uniform distribution for the confounders, and Assumption 1: In justifying assumption 1, the authors mention that a similar result holds for discrete variable as shown by Xia et al. and the authors believe  'a similar approach can be used to show their results for general SCMs'. However, this is far from trivial. Proofs in causal inference which assume discrete variables generally revolve around some kind of counting arguments, and are not easily extendible to the continuous setting, and might not necessarily hold. It is not obvious that $\\hat{P}_{\\hat{U}} \\sim Unif(0, 1)$ is reasonable to assume when the variables are not just continuous but also high dimensional. However, granting that a lot of causal inference is a tradeoff between assumptions and the strength of statements we can make, perhaps this can be considered a relatively small weakness. However, it would be useful to mention this clearly in the limitations section at the very least.\n\nSome comments\n* l36: 'Padh et al. make parametric assumptions on the distribution of treatment-response functions'. From my understanding, this might be a slight mischaracterization since they don't seem to make assumptions on the distribution of the response functions but rather parametrize the mean and covariance of the response functions.\n* l173: Section 'Generative causal models' and comparison to NCM: The authors mention that their work 'can be seen as an extension of NCM'. However Xia et al. provide a sound and complete algorithm for effect identifiability in NCMs, which is why the NCMs are interesting. This paper provides a very limited setting in which their bounds are valid, and while making assumptions (such as Assumption 1) which would be quite non-trivial to justify (see weakness section). Therefore, in my opinion it might be better not to call this method an extension of NCMs.\n\nOverall, I find that this work contributes some new new ideas to the topic of partial identification which could be interesting to the community. 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Author rebuttal:

Decision: Accept