1 Introduction
Consider an experiment to verify a drug’s efficacy for an aggressive cancer type. It would be extremely unethical to split the patients into two groups and give only a placebo as drug therapy for one of them, considering that the lack of proper medication could lead to the patients to death. The more suitable approach for such experiments is adopting causal inference methods for observational studies. In observational studies, the counterfactual data is missing due to several factors, such as cost or ethical concerns. The core idea of the causal inference methods for observational data is very similar: overcome the lack of counterfactual data by estimating the conditional treatment effects. However, different methods are based on different assumptions. Some of the common assumptions among these methods are regarding the nature of the treatment (binary versus continuous), the number of treatments (single versus multiple), the occurrence of a single treatment at any given time, and causal sufficiency.
Our work focuses on developing a causal inference method to estimate treatment effects in observational studies with realistic assumptions. A typical example is an application with millions of covariates, and yet, missing confounders, and several continuous (or sometimes binary) treatments applied simultaneously, such as studies about adverse side effects in patients under cancer treatment. These patients often receive a drug cocktail, and these are considered a set of treatments in the causal inference context. Sometimes, it is possible to observe an additive effect: a single drug would not result in adverse side effects; however, the adverse side effect is observed when considering all the drugs together. Such an effect is observed because many of the drugs adopted in cancer therapy contain heavy metals, and their accumulation on the patients’ bodies might be one of the causes of the adverse side effects. Inspired by such applications, we propose a method that uses proxies to decrease confounding biases, and most importantly, can estimate the effect of multiple treatments.
There are only a few methods that can estimate the effect of multiple treatments. HiCI (Sharma et al. (2020)) considers and models multiple treatments but assumes that only one is assigned to a unit at any given time. The Deconfounder Algorithm (DA) (Wang and Blei (2019)), a probabilistic graphical model, is our main baseline. DA works for multiple continuous and binary treatments. Singletreatment effect methods, such as Dragonnet (Shi et al. (2019)), and CEVAE (Louizos et al. (2017)), can be adapted to the multipletreatments context by fitting an independent model for each treatment. Nevertheless, similarly to the singletask versus multitask learning approach (Ruder (2017); Vandenhende et al. (2020)), singletreatment methods may fail to capture the synergy of multiple treatments, e.g. an additive effect.
Contributions: In this work, we focus on applications with multiple treatments and potential confounding biases. The main contributions of this paper are as follows:

We propose M3E2, a method to estimate multiple treatment effects with a multitask learning neural network architecture.

We validate M3E2 in three synthetic datasets where the true causal effects are known. We also compare our method with three existing methods.
2 Related Work
This work combines estimation of treatment effects and multitask learning (MTL).
Estimating Treatment Effects BART (Chipman et al. (2010); Hill (2011)), CEVAE (Louizos et al. (2017)), and Dragonnet (Shi et al. (2019)
), have explored the estimation of a single treatment effect, using Bayesian Random Forests, VAEs, and neural networks (NN) respectively.
Hernán and Robins (2006) present an inverse propensity weighting based method, also focused on single treatments. Other methods such as the Deconfounder Algorithm (Wang and Blei (2019)), HiCI (Sharma et al. (2020)), and approaches based on the propensity score (Lechner (2001); Lopez and Gutman (2017); McCaffrey et al. (2013)) aim to estimate multiple treatment effects. However, many of these methods assume that only one treatment is applied at any given time. The causal sufficiency assumption also poses a challenge. Causal sufficiency assumes that all confounders, variables that affect both treatment assignment and outcomes, are observed. This assumption often fails in realworld settings. The Deconfounder Algorithm, HiCI, and CEVAE are robust to unobserved confounders under certain conditions. Using an approach similar to HiCI and CEVAE, M3E2 adopts latent variables to obtain treatment effect estimates robust to hidden confounders. Finally, M3E2 estimates the multiple treatment effects through an outcome model in a multitask learning neural network architecture. The outcome model approach is similar to the one adopted by the Deconfounder Algorithm.Multitask learning MTL neural network architectures aim to simultaneously optimize a single model for two or more tasks. Hardparameter sharing NN, first proposed by Caruana (1993), is one of the MTL pillars. Such architecture is composed of a set of layers shared among all tasks and a set of taskspecific layers on the top. From the MTL perspective, the Dragonnet (Shi et al. (2019)) has a hardparameter sharing architecture. Building upon the hardparameter sharing architectures, Ma et al. (2018)
proposed a Multigate MixtureofExperts (MMoE) architecture, where each expert can be seen as a hardparameter sharing NN, and all the experts are combined through a gate function, which is also trainable. The core idea of such an approach is to improve the model’s generalization; plus, it allows experts to specialize in one of the tasks. To put into perspective, an MMoE is to hardparameter sharing NN what a Random Forest Model is to a Decision Tree. Our proposed method M3E2 is built upon a MMoE (
Ma et al. (2018)) architecture with an autoencoder to address hidden confounders. For more details on MTL architectures and optimization methods,
Ruder (2017) and Vandenhende et al. (2020) present an overview of the most recent works. Our work expands the MMoE architecture to satisfy causal inference assumptions and estimate multiple treatment effects simultaneously.3 Multigate Mixtureofexperts for Multitreatment Effect Estimation (M3E2)
This section describes our proposed method, the Multigate Mixtureofexperts for Multitreatment Effect Estimation (M3E2). Its multitask learning architecture allows simultaneous prediction of the multiple treatments assignment and the outcome .
Figure 1 illustrates the proposed neural network architecture, with a MMoE (Ma et al. (2018)), and an autoencoder as subcomponents. The autoencoder is responsible for learning latent variables, used as proxies of hidden confounders. In the MTL context, each propensity score prediction is a task, and the prediction of the outcome of interest is also a task. Therefore, if a given application has treatments, M3E2 would have tasks. Note that, in the testing phase, the propensity score predictions are not used (Section 1  Sup. Material).
One of the strengths of M3E2 is its capacity to estimate the combined effect of a large number of treatments: the M3E2 network only grows linearly with the number of treatments, handling all potential combinations, something that other multitreatment methods typically struggle to accomplish. Furthermore, the autoencoder can handle different data types by dividing the input covariates into two groups, and . While the autoencoder handles the covariates in , the covariates are fed directly to the experts. For example, consider an application whose input data mixes clinical information (few columns) and gene expression (thousands of columns). In this case, splitting the data into clinical covariates and genomic information can improve the results and handle the different data types better. In applications with only one data type, both and , and and are acceptable. In summary, the proposed architecture of M3E2 extends the MMoE architecture by adding meaning to the tasks, incorporating causal sufficiency assumptions through suitable regularizers, and adding the autoencoder component.
The following sections explain each component of the architecture in a bottomup approach.
3.1 Autoencoder
In observational studies, it is always important to describe how one is addressing confounders. Some works assume no unobserved confounders (Shi et al. (2019)), others try to reduce the bias through latent variables (Louizos et al. (2017); Wang and Blei (2019); Sharma et al. (2020)); while others question if the latent variables are solving the problem at all (Zheng et al. (2021)). We believe that the choice of the best approach for handling confounders is still an open question. In this paper, our focus is to propose a methodology to estimate the effects of several simultaneous treatments which is robust to large datasets. A user of M3E2 might use the autoencoder component to estimate latent variables used to reduce confounding biases (Louizos et al. (2017); Wang and Blei (2019); Sharma et al. (2020)). Alternatively, a user can choose not to adopt the autoencoder (by setting ), which is the same as assuming no unobserved confounders or not using latent variables.
Following Louizos et al. (2017), Wang and Blei (2019), Sharma et al. (2020), and others (D’Amour (2019); Li et al. (2020); Ranganath and Perotte (2018); Sharma et al. (2018)), M3E2 also builts its robustness to hidden confounders through the adoption of latent variables. The overall idea of CEVAE, DA, HiCI, and M3E2 is very similar: learn latent variables to replace hidden confounders. The latent features and the treatment assignment are used to predict the outcome , as shown in Equation 1, and to estimate the treatment effects (Section 3.6).
(1) 
While these methods core idea is very alike, their assumptions and path to Equation 1 differ. The DA, a multitreatment method, uses a probabilistic factor model to estimate the latent variables. Their main assumption is that their robustness to unobserved confounders raises from them affecting multiple causes. CEVAE uses Pearl’s backdoor adjustment. In this case, , where represent latent variables estimated. Through a variational autoencoder, CEVAE obtains the required distributions to estimate an approximate distribution for , and later use to estimate the outcomes . CEVAE’s main limitation is to consider only one binary treatment. HiCI, another multitreatment method, uses an autoencoder to estimate
with a loss function that enforces decorrelation. M3E2 is somewhere in between CEVAE and HiCI. Like CEVAE, M3E2 aims to predict
with the difference that M3E2 does not use the propensity score as an auxiliary distribution. Instead, similarly to HiCI, M3E2 uses to predict the propensity score.The use of latent variables to replace unobserved confounders still has its limitations. Similar to CEVAE and HiCI, M3E2 assumes robustness only to observed confounders in the input data or unobserved confounders correlated with observed variates in . Otherwise, it would be impossible for the factor models to estimate proxies for the unobserved confounders. To the best of our knowledge, DA is the only method that claims to be robust to unobserved confounders even if they are uncorrelated with observed variates in , under the condition that these unobserved confounders affect multiple causes. However, there has been intense debate on the validity of its assumptions (Ogburn et al. (2019)).
M3E2 adopts an autoencoder with two linear encoder layers and two linear decoder layers. Consider an application with samples, columns in , as the latent variables size, and the input data as a matrix . The function returns , a representation of in a lower dimension. Finally, returns the reconstructed data , back on space. The autoencoder’s loss is the mean squared error between the input and the reconstructed input .
3.2 Experts and Gates
In multitask learning, a hardparameter sharing network is a combination of shared layers and taskspecific layers. In a multigate mixtureofexpert (MMoE) architecture, a hardparameter sharing network can be interpreted as a single expert model. Therefore, a model with several experts is a model with several independent shared layers components.
Ma et al. (2018) shows that a set of experts can generalize better than hardparameter sharing architectures. The MMoE architecture is more flexible than a traditional hardparameter sharing architecture, which allows some of the experts to specialize in a single task. The user defines the number of experts and the architecture, with the possibility of adopting experts with different architectures. In the context of multiple treatment effect estimation, the tasks are the treatment assignment and the outcome prediction. The experts’s input data is . The ideal number of experts depends a lot on the tasks’ nature. Homogeneous tasks might not benefit from many experts and might overfit if the number of experts is too large; heterogeneous tasks, on the other hand, tend to benefit from a larger number of experts.
The gates control the contribution of each expert to each task. There is a gate per treatment defined as:
where is a trainable matrix of weights, is the number of experts defined by the user, and the number of columns in .
3.3 Taskspecific layers and
The taskspecific layers are responsible for predicting the propensity score and the outcome of interest . Each treatment taskspecific layer receives as input a weighted average of the experts, where the weights come from the gates associated with that given task. This relationship is formally defined as:
In the training phase (Figure 1), the treatment assignment is predicted with the propensity score , estimated as (for discrete treatments) or (for continuous treatments using the conditional density (Hirano and Imbens (2004); Nie et al. (2021))). To estimate the treatment assignment of we only use
. For binary treatments, a softmax activation function will outputs, for each sample, the probability of
and . These predictions are used to calculate the loss of the neural network, as described in the Section 3.4. The propensity score losses are used to drive to be sufficient (Assumption 1  Section 3.5). Note that can be a combination of one or more layers.Finally, a layer with trainable weights is used to predicts the final outcome. Consider the input data of this layer as , where are the observed treatment assignments, , and is the number of columns. The trainable weights layer estimates the final outcome as . In the testing set, if the true treatment assignment is unknown, it is also possible to adopt the propensity score predictions as a replacement of the observed treatment assignments. The works as an outcome model and the treatment effects are estimated as in Wang and Blei (2019). In our context of treatment effect estimation, is the treatment effect of the treatment .
Note that our approach targets additive effects, which are fairly common in realworld applications. For instance, consider adversarial side effects in cancer therapy. Cancer therapy usually involves several treatments (drugs) simultaneously or through a certain period. Many of these drugs contain heavy metals, and their cumulation can result or contribute to observed adversarial side effects. The inclusion of nonlinear effects on the outcome model would require a different approach to estimate the treatment effects. Plus, it could result in a combinatorial explosion if all potential combinations of treatments are considered, something avoided on M3E2. Therefore, the consideration of nonlinear effects is left as future work.
3.4 Loss function
M3E2’s loss function is composed of:

Root mean square error loss for continuous outcomes and binary crossentropy for binary outcomes.

Similar to the outcome loss functions, we adopt or/and as the propensity score losses.

is the the autoencoder loss function.

as the regularization.
The modification of both and to other loss functions are quite straightforward.
As a reminder, while our architecture minimizes the propensity score and the outcome losses, our main target is to obtain estimates of the treatment effects robust to hidden confounders. The treatment effects are a coproduct of this model, being the weights associated with the treatments in the trainable layer . This last layer is responsible for estimating ; therefore, poor estimates of would directly reflect on
. Furthermore, M3E2 robustness to confounders comes from getting unbiased estimates of the outcome by conditioning on the treatment assignments T and H. Therefore, it is fundamental that the propensity score components ensure that the experts learn a meaningful representation of the confounders. The model also learns weights in
associated with the confounders; however, these are not considered treatment effects. The final loss is shown in Equation 2.(2) 
The , and from Equation 2
are weights. There are two possible ways to define these weights: adopting them as a hyperparameter or through the adoption of an MTL task balancing approach. For example,
Liu et al. (2019) dynamically sets these weights based on the losses values to minimize negative transfer between tasks. For more heterogeneous treatments, such as a mix between continuous and discrete treatments, adopting a weight per propensity score can also help with taskbalancing.3.5 Assumptions
We aim to develop a method that makes more realistic assumptions, such as considering multiple treatments and robust to their data type (binary or continuous). However, like any other causal inference method, there is no free lunch.
Assumption 1 Sufficiency of Propensity Score (Rosenbaum and Rubin (1983); Shi et al. (2019); Nie et al. (2021)): If the average treatment effect is identifiable from observational data by adjusting for X, i.e., , then adjusting for the propensity score also suffices:
In other words, this assumption means that it suffices to adjust only the information in that is relevant for predicting the treatment , and this information is the output of . For multiple treatments, the generalization goes as follows(Imbens (2000); Imai and Van Dyk (2004)):
Assumption 2 Stable Unit Treatment Value Assumption (SUTVA): the response of a particular unit depends only on the treatment(s) assigned, not the treatments of other units.
Assumption 3 Common Confounders and conditional independence: Treatments share confounders. Given the shared confounders, the treatments are independent. (Ranganath and Perotte (2018))
Assumption 4 (Weak) Ignorability Assumption: There are no uncorrelated unobserved confounders.
Assumption 2 guarantee that the elements are independent and do not interfere in each other. Assumption 3 is specific for multiple treatment settings, where the dependence between the treatments might result in biased treatment effects estimates. Finally, the original ignorability assumption considers that all confounders are observed. However, due to the adoption of proxies to account for correlated unobserved confounders, we make a weak ignorability assumption. The weak is to account for unobserved confounders correlated with observed covariates. These unobserved confounders would be made accountable through the autoencoder and experts. Therefore, our method is not robust to unobserved confounders uncorrelated to observed covariates, or whose strong effect is not sufficiently captured by the latent variables learned.
3.6 Estimation of the treatment effect
There are several metrics to measure the treatment effect. The perfect metric choice depends on the nature of the treatment, the method’s architecture, and the outcome’s datatype. Among our baselines, CEVAE and Dragonnet use the average treatment effect (ATE), defined as . While widely adopted, ATE is suitable for applications where only one binary treatment. As previously mentioned, the intend of M3E2 is to be a method with realistic assumptions and a wide range of possible applications. Therefore, similarly to the Deconfounder Algorithm, we adopt an outcome model (Wang and Blei (2019)), with is a more flexible alternative to estimate the treatment effects. The outcome model fits a predictive model on the augmented observed data to estimate the quantity . In our architecture, this is done by our last layer , as previously described in Section 3.3.
4 Experiments
In causal inference, the lack of ground truth for realworld applications poses a challenge to its evaluation. Therefore, we adopt three synthetic benchmark datasets^{1}^{1}1The code to generate the datasets is available at github.com/raquelaoki/M3E2. that have known treatment effects:

IHDP (Hill (2011); Louizos et al. (2017); Shi et al. (2019)): this is a traditional benchmark dataset for single binary treatments. It has 24 covariates and a continuous outcome. We adopt this dataset to compare with some of our singletreatment estimation baselines that have been evaluated on the IHDP dataset in their publications.
For more details about the synthetic benchmark datasets, check Section 2  Sup. Material.
We adopted the mean absolute error (MAE) to compare the real treatment effects, , with the estimated values :
(3) 
The goal is to minimize the difference between estimated and true treatment effects; therefore, low MAE values are desirable.
We adopt an experimental setting similar to the multitask learning settings (Ma et al. (2018)), where the proposed multitask learning method is compared with other multitask learning methods and singletask learning models, specialized and designed to optimize a single task. Among our baselines, the DA(Wang and Blei (2019)) is the only method that can estimate the effect of multiple treatments. Similarly to M3E2, there is one model for each dataset simulated that estimates all treatment effects. The CEVAE(Louizos et al. (2017)) and Dragonnet (Shi et al. (2019)), on the other hand, are singletreatment methods. In this case, we ran one independent model for each treatment. We utilized the author’s implementation of the baselines when available. We also performed experiments with BART. However, since CEVAE and Dragonnet achieved better performance results in the recent publications (Louizos et al. (2017); Shi et al. (2019)), and since BART performed poorly on the GWAS and Copula datasets, we decided not to discuss BART in the experimental section. We also performed experiments with HiCI, a multitreatment method. We reimplemented HiCl, since, unfortunately, no implementation is publicly available. In our experiments, HiCI produced very poor results and frequently did not converge. Consequently, we did not include HiCl in our experimental comparison.
We generate four datasets using different seeds for each configuration explored in the GWAS and Copula datasets. For the IHDP, we adopted the ten replications previously generated by Louizos et al. (2017). We fitted the models eight times in each dataset generated, also using different seeds. These repetitions were important to remove potential biases from random seeds and to construct error bars. All methods were run on the same virtual environment setting, with 1 GPU and 16 GB. Table 1 shows the settings explored in the experiments.
Setting  Data  Sample Size  #T  #C 

a  GWAS  2500  10000  5  995 
b  GWAS  10000  5  100  1000 
c  GWAS  10000  3, 6, 9  1k 
d  Copula  2500  10000  4  10 
e  Copula  10000  4  5  1000 
f  IHDP  747  1  24 
4.1 Overall performance
Figure 2 shows, for each dataset, the average MAE across all settings. Our proposed method, M3E2, clearly outperforms all baselines on the multitreatment datasets GWAS and COPULA. DA obtains a smaller error bar, indicating more consistent treatment effect estimates. On IHDP, a singletreatment dataset, M3E2 was outperformed by Dragonnet, but was comparable or better than the other two baselines. We note that our results for Dragonnet and CEVAE on IHDP match the results previously reported by Shi et al. (2019)
. The larger variance on the IHDP dataset can be explained by the larger scale of the true treatment effect.
To better illustrate the methods performance, Figure 3 shows a deeper analysis of the results on one of the Copula datasets. Figure 3.a shows M3E2 has the lowest MAE values in comparison with the other baselines. Figure 3.b shows the total run time of each method in seconds. As a reminder, both DA and M3E2 fit one model for all treatments; Dragonnet and CEVAE, on the other hand, fit one model for each treatment. DA, a probabilistic model, has the fastest running time; among the NN methods, M3E2 has the lowest running time. A comparison between the true (line in red) and the estimated treatment effects (dots) is shown in Figures 3.cf. As previously mentioned, M3E2 has more assertive treatment effect estimates; on the other hand, it also has more variability when compared with the baselines. For , M3E2 is the only method capable of recovery the true effect; for the remaining treatment effects, CEVAE and M3E2 have both strong results, being at a similar distance from the true effect on , M3E2 having better results for , and CEVAE having the best results in .
To investigate the fit of the baselines and M3E2, we checked their F1 score or the root mean square error (RMSE) when predicting the outcome of interest on the testing set. We observed fair values for all methods, indicating the methods were well fitted.
4.2 Impact of dataset parameters
The following experiments explore the impact of the dataset parameters in the estimation of the treatment effects. As Table 1 shows, we explored three parameters: the sample size, number of treatments, and number of covariates.
Figure 4 shows, in details, the average MAE and the 95% confidence interval (colored area) for the settings shown in Table 1. Settings and , for instance, explore the sample size, and the results are shown in Figure 4 and , respectively. Our proposed method, M3E2, is the method that benefits the most from increasing the sample size. Increase the number of treatments and confounders on the GWAS simulated dataset (Figure 4 ) did not seem to affect the baselines’ MAE and M3E2’s MAE. On the other hand, in the Copula synthetic dataset, when the number of covariates grows (Figure 4 ), there was an increase in CEVAE’s variability and Dragonnet MAE. In this same setting, both DA and M3E2 are robust to the number of covariates. Note that we followed the experimental design of Zheng et al. (2021) for the COPULA dataset, which did not vary the number of treatments.
5 Discussion and Conclusion
In this work, we proposed M3E2, a multitask learning neural network model to estimate multiple treatment effects with an autoencoder component to improve its robustness to hidden confounders. The proposed neural network extends the MMoE model by adding meaning to the tasks and by incorporating causal inference assumptions in the regularizers. Compared to existing methods, M3E2 makes only a weaker ignorability assumption, which is more realistic in many applications. We experimentally compared our method against three baselines on three synthetic benchmark datasets and demonstrated its superior performance. The online repository github.com/raquelaoki/M3E2 contains the code to replicate all the experiments. We put extra effort into making the M3E2 implementation agnostic to the application; therefore, its adoption in other applications should be straightforward.
A potential negative societal impact of estimating treatment effects from observational data is the lack of a standard approach to evaluate such results in realworld applications. Most publications evaluate their methods using the method’s predictive power or expert knowledge. However, both are susceptible to mistakes because they are an indirect and nonobjective measure of the treatment effect, which opens room for potential misuse. Furthermore, the success of observational causal inference methods might lead researchers to rely on such methods even in settings where they are not needed. Randomized controlled trials are still the gold standard for treatment effect estimation, and if available, should always be the first choice.
M3E2 demonstrated promising experimental results and strong evidence that the joint learning contributed to better estimate the treatment effects; however, it still has a few limitations. As discussed in Section 3.5
, our method is robust only to observed confounders and unobserved confounders correlated with observed covariates. M3E2 inherits the limitations of other multitask learning models, in particular, the susceptibility to imbalanced tasks and overfitting. Plus, like many other machine learning methods, its generalization depends on the quality of its data. All strengths and limitations considered, we believe that M3E2 has a very good use case with manageable limitations. We want to explore applications with very heterogeneous treatments, such as temporal versus nontemporal treatments, and nonlinear effects in future research. We also want to apply our proposed method to a realworld dataset that explores adverse side effects in therapies for treating cancer in infants.
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