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Efficient and Scalable Estimation of Distributional Treatment Effects with Multi-Task Neural Networks
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Randomized experiments, also known as A/B tests, have served as a cornerstone for understanding intervention effects and shaping policy decisions since Fisher's seminal work Fisher1937. The framework of causal inference through randomized experiments has become foundational across diverse scientific disciplines Rubin1980, Imbens2015, Rubin1974 and has been widely adopted by technology companies as a standard practice in their decision-making processes tang2010overlapping, xie2016improving, Kohavi2020.
While the Average Treatment Effect (ATE) is the conventional measure in randomized experiments, it is subject to two significant limitations: imprecise estimates with high variance relative to their magnitudes lewis2015unfavorable, and incomplete characterization of intervention impacts across the outcome distribution. Although numerous studies have proposed methods for capturing distributional impacts of treatments Abadie2002, Firpo2007, ROTHE201056, Athey2006, FORTIN20111, Wang03072018, Chernozhukov2005, addressing both estimation precision and distributional characterization remains a challenge.
In this paper, we propose a novel approach to estimating Distributional Treatment Effects (DTE) using multi-task neural networks (NN). The DTE captures comprehensive treatment effects by measuring changes across the outcome distribution induced by treatments. While DTE estimation can be performed by directly comparing empirical outcome distributions between treatment and control groups, this approach often overlooks valuable pre-treatment auxiliary information commonly available in randomized experiments. Our method extends existing regression-adjustment techniques that leverage pre-treatment covariates for ATE estimation, an approach widely adopted in industry to enhance experimental sensitivity Deng2013, Xie2016. By incorporating regression-adjustment methods within a neural network architecture for distributional regression, we develop a more powerful framework for DTE estimation.
Our proposed method enhances estimation precision, particularly in the distribution tails, through a multi-task neural network architecture with shape restrictions that jointly learns from multiple parts of the outcome distribution. By estimating distributional features simultaneously rather than solving separate regression problems, our method achieves significant computational efficiency. These methodological advances prove particularly valuable for analyzing large-scale datasets commonly encountered in industrial applications, as demonstrated in a US water conservation experiment and a large-scale A/B test on a Japanese streaming platform.
Our primary contributions are summarized as follows:
The rest of this paper is structured as follows. Section (ref) describes related research. We explain the problem setup in Section (ref) and the proposed methodology in Section (ref). Section (ref) presents simulation studies and two real-world applications. We conclude the paper in Section (ref). The Appendix in the paper includes additional details and results.
Distributional Treatment Effect: Although the ATE remains a standard metric in treatment evaluation, the distributional characterization of treatment effects provides more comprehensive insights into treatment heterogeneity. This has motivated extensive research on both Quantile Treatment Effects (QTE) and Distributional Treatment Effects (DTE) Abadie2002, Chernozhukov2005, Athey2006, Callaway2019, Chernozhukov2013, Chernozhukov2020, kallus2024localized, 11fde4a6-7813-3966-82c1-2aeb85f068b8, callaway2018quantile, xu2018. Unlike QTE which requires continuous outcome distributions, DTE can be applied to both discrete and continuous outcomes.
Regression Adjustment: When estimating the treatment effect of RCT, the sensitivity of the estimation is critical. For example, failure to detect an effect due to a large variance leads to potential revenue loss in industry. Regression adjustment is a method that utilizes pre-treatment covariates to estimate the treatment effect more precisely. Various researches have explored the usage of regression adjustment to gain precise estimation for the ATE including Freedman2006, Berk2013, Lin2013, Rosenblum2010, Tsiatis2008, Imani2018, Rosenbaum2002, Yang2001. Despite the challenges in learning a correct relationship between pre-treatment covariates and outcome variables, it has been proved that the regression-adjusted estimation is unbiased, consistent, and practically useful. Recent studies have extended regression adjustment beyond ATE to distributional treatment effects, establishing theoretical foundations for its application jiang2023regression, oka2024, Byambadalai2024, byambadalai2025efficient. Building on this line of work, we enhance regression adjustment for distributional treatment effects by improving the underlying algorithm and integrating recent advancements in deep learning.
Multi-Task Learning: Multi-task learning enables a single model to learn multiple related tasks simultaneously, leveraging shared representations to enhance performance caruana1997multitask,zhang2018overview, ruder2017overviewmultitasklearning,Michael2020. This joint learning strategy improves generalization and mitigates overfitting zhang2021survey. The framework proves particularly valuable in multi-label classification, where outputs can belong to multiple categories simultaneously tarekegn2024deeplearningmultilabellearning, zhang2014reviewmlc. By treating each label prediction as a distinct yet related task, multi-task learning effectively captures label dependencies Min2006, dembczynski2012label, su2022zlprnovellossmultilabel and leverages shared patterns within a unified framework argyriou2007multitaskfeature. The deep neural network approach has been studied to estimate distributional regression based on binary cross-entropy li2019deepdistributionregression and quantile regression tagasovska2019singlemodeluncertainties,white1992nonparametric. Our approach reformulates distributional regression as a multi-label classification task, utilizing multi-task learning to share representations across the distribution.
Shape Restriction: Shape restriction on statistical models has a long history in statistics and econometrics, and there is a broad range of research about various shape restrictions such as concavity or monotonicity for parametric or semiparametric statistical models Barlow1972,Hastie1990,Matzkin1994, Horowitz2017,Chetverikov2018,Dümbgen2024. These shape restrictions are used for stabilizing the semiparametric models or increasing the efficiency of the estimation. They work particularly well if simple estimation is difficult due to a small sample size. Recent studies explored the shape restriction on machine learning (ML) models with higher dimensions of covariates gupta2016monotonic,You2017, gupta2018diminishing, cotter2019shape, ensuring the monotonic relationship between input variables and outputs of neural networks. Wu et al. dnet incorporated a cumsum function to ensure the monotonicity of individualized treatment effects (ITEs) estimation across different quantiles. In this study, we introduce a monotonic constraint on the output of neural networks to enhance the stability and precision of unconditional treatment effects.
We consider a randomized experiment with \( K \) treatment arms, where each unit is assigned to treatment \( w \) with probability \( \pi_w : = P(W = w) \) such that \( \sum_{w \in \mathcal{W}} \pi_w = 1 \). The observed data consist of \( n \) independent samples \( \{(X_i, W_i, Y_i)\}_{i=1}^n \) from \((X, W, Y) \), where \( X \in \mathcal{X} \subseteq \mathbb{R}^{d_x} \) is a vector of pre-treatment covariates, \( W \in \mathcal{W} := \{1, \dots, K\} \) is the treatment assignment, and \( Y \in \mathcal{Y} \subseteq \mathbb{R} \) is the observed outcome. Under the potential outcomes framework Rubin1974, Rubin1980, Imbens2015, each unit has potential outcomes \( Y(1), \dots, Y(K) \), but only the outcome corresponding to the assigned treatment is observed, denoted as \( Y = Y(W) \). Each treatment group contains \( n_w \) observations, satisfying \( \sum_{w \in \mathcal{W}} n_w = n \).
Throughout the paper, we maintain the following assumptions:
Assumption (ref) ensures treatment assignment is independent of potential outcomes and pre-treatment covariates, while Assumption (ref) guarantees non-zero assignment probabilities. These assumptions are valid under randomized experiments.
We define the cumulative distribution functions (CDFs) of potential outcomes as: \[ F_{Y(w)}(y) := \Pr \big ( Y(w) \le y \big), \] for $y \in \mathcal{Y} \subseteq \mathbb{R}$ and $w \in \mathcal{W}$. While potential outcomes are unobservable, the potential outcome distribution $F_{Y(w)}(y)$ becomes identifiable when Assumptions (ref) and (ref) are satisfied, as it equals the observed outcome distribution $F_{Y|W}(y|w) := \Pr(Y \leq y \mid W = w)$ for each treatment $w$. In practice, we usually have a set of scalar-valued locations $\tilde{\mathcal{Y}} \subseteq \mathcal{Y}$ for which cumulative distribution functions are estimated and the following set of parameters are computed together: \[ \mathcal{F}_{Y(w)} := \big \{F_{Y(w)}(y) \mid y \in \tilde{\mathcal{Y}} \big\}. \] This collection of CDFs provides a distributional characterization of the treatment effect across the outcome distribution.
We formally define the Distributional Treatment Effect (DTE) between treatments $w, w' \in \mathcal{W}$ as
for $y\in \mathcal{Y}$. The DTE measures the difference between CDFs of the potential outcomes under the two treatments.
While our primary focus is the DTE, our framework extends to any parameters obtained through transformations of CDFs. An important example is the Probability Treatment Effect (PTE) between treatments $w, w'\in \mathcal{W}$, defined as
where
for $y_j\in \mathcal{Y}$. Each quantity $f_{Y(w)}(y_j)$, which we refer to as the interval probability, represents the difference between consecutive CDF values at $y_j$ and $y_{j-1}$. The PTE measures the differences in these interval probabilities between treatments.
In this section, we describe our proposed method and its advantages. As a prerequisite, the existing algorithm is explained below.
We describe the existing regression-adjusted estimation procedure for distribution functions, which is outlined in Algorithm (ref). The procedure begins with estimating the conditional distribution function of $Y(w)$ given $X$, defined as
and let $\widehat\gamma_y^{(w)}(X)$ denote its estimator for each $y \in \mathcal{Y}$ and $w \in \mathcal{W}$. The estimation can be reformulated as a prediction problem by noting that $\gamma_y^{(w)}(X)= E[\mathbf{1}_{\{Y(w)\leq y\}}|X]$, wherein we predict binary outcomes $\mathbf{1}_{\{Y(w)\leq y\}}$ using pre-treatment covariates $X$ through a supervised learning algorithm $\mathcal S$ (e.g., linear regression, gradient boosting, random forests, single-task neural network). The estimation procedure relies on a sample-splitting method called cross-fitting, which, combined with the Neyman orthogonality, enables us to derive the asymptotic distribution of the regression-adjusted estimator under mild conditions chernozhukov2018debiased.
Then, the regression-adjusted estimator of $F_{Y(w)}(y)$ is computed as follows:
This estimator modifies the empirical distribution function by subtracting the treatment group average of $\widehat{\gamma}_y^{(w)}(\cdot)$ and adding its overall average. The resulting estimator is unbiased since the expected values of these adjustment terms cancel out.
Moreover, the following theorem holds for the adjusted estimator in equation (ref) highlighting the efficiency gain through the regression adjustment compared with the empirical estimation defined by $\widehat{\mathbf{F}}^{empirical}_{Y(w)}(y) := \sum_{i: W_i=w} \mathbf{1}_{\{Y_i \leq y\}}/n_w.$ Consider the scenario where the true conditional distribution function, $\gamma_{y}^{(w)}(\cdot)$, is employed in (ref), leading to the idealized form of the regression-adjusted estimator, denoted by $ \widetilde{\theta}_{y}^{(w)}:= \widetilde{F}_{Y(w)}(y)$. Let the vector version be denoted by $\widetilde{\theta}_{y}:= (\widetilde{\theta}_{y}^{(1)}, \dots,\widetilde{\theta}_{y}^{(K)})^\top$.
Theorem (ref)(a) states the variance reduction accomplished by applying regression adjustment to distribution functions. Additionally, Theorem (ref)(b) elaborates on this variance reduction as a vector of regression-adjusted estimators, indicating the efficiency gain of the adjusted estimator for the DTE as a special case.
Despite effectively leveraging semiparametric models in conjunction with cross-fitting chernozhukov2018debiased, chernozhukov2022locally, the algorithm has several limitations. First, the algorithm has a high computational cost due to the need for iterations of single-target regression problems. The computational complexity of the algorithm is $O(M \cdot D \cdot L \cdot P)$, where $M$ indicates the number of locations, $D$ indicates the number of treatment levels, $L$ indicates the number of folds, and $P := Cost(\mathcal S(1))$ indicates the computational cost of one iteration of the supervised statistical model training $\mathcal S$ for one target variable. When the number of locations is large, this algorithm takes a long time to compute the adjusted distributional parameters due to many iterations of ML model training. Moreover, estimating the conditional distribution function is particularly challenging in the distribution tails due to label imbalance. For instance, when estimating the cumulative probability at the 90th percentile, 90% of the training data satisfies $\mathbf{1}_{\{Y \leq y\}} = 1$, leading to difficulties in handling the imbalance. A similar issue arises at lower percentiles, where the majority of labels take the opposite value.
In this study, we reformulate the distribution regression into a multi-label classification task and propose a novel algorithm to estimate the regression-adjusted distribution function by leveraging a multi-task statistical model for estimating conditional distribution function $\widehat{\gamma}_y^{(w)}(X)$ for all $y \in \tilde{\mathcal{Y}}$ simultaneously. The procedure estimates a set of conditional distribution functions, defined as
and let $\mathbf{\widehat{\Gamma}}^{(w)}$ be its estimator for each $w \in \mathcal{W}$ instead of estimating $\widehat{\gamma}_{y}^{(w)}(X)$ for each $y\in\tilde{\mathcal{Y}}$ individually.
The proposed method aims to overcome the high computational cost of the existing algorithm based on the following Assumption (ref), where $\mathcal S(M)$ indicates the multi-task supervised learning algorithm with the number of target variables equal to $M$. This assumption posits that the computational cost of training the machine learning model increases at a sublinear rate for the size of the target variables.
If a single multi-task ML model is trained for all locations together and the computation cost for the single model is $\tilde{P}$, the total computation cost of the entire estimation process is $O(D \cdot L \cdot \tilde{P})$, which is lower than the existing algorithm $O(M \cdot D \cdot L \cdot P)$ when Assumption (ref) is met. Note that Assumption (ref) is not strong as many statistical models that can estimate multiple target variables together satisfy the assumption, including Linear Regression, Lasso, and Neural Networks. Appendix (ref) presents the theoretical analysis of the sub-linear assumption for linear regression and the empirical results of computational complexity across multiple neural network models. However, some algorithms, such as gradient-boosting trees or random forests, do not fulfill this condition since these algorithms need one model per target and usually the One vs Rest strategy is adopted for multi-output learning. The proposed algorithm for estimating the regression-adjusted distribution function with a multi-task ML model is available in Algorithm (ref) where $\mathcal{\widehat{F}}_{Y(w)} := \{\widehat{F}_{Y(w)}(y) \mid y \in \tilde{\mathcal{Y}}\}$.
Beyond computational efficiency, the multi-task model can learn richer representations compared to a single model trained for a single location. By simultaneously estimating multiple locations, the multi-task approach helps mitigate the label imbalance issue discussed in the previous section. Since the underlying relationship between covariates and the outcome remains consistent across locations, sharing representations through multi-task learning may improve estimation precision, as suggested in previous research Michael2020.
Moreover, using a single multi-task ML model for estimating cumulative probability for each $y \in \tilde{\mathcal{Y}}$ enables us to enforce a monotonicity constraint across estimated probabilities in multiple locations. Specifically, for all $w\in\mathcal W$, the true conditional distribution function satisfies the following:
Since the true function satisfies the monotonicity property above, enforcing this constraint not only enhances interpretability gupta2018diminishing,You2017, but also has the potential to improve training efficiency, as suggested by previous research Chetverikov2018. Specifically, for all $w \in \mathcal{W}$ and $X_i\in\mathcal{X}$, the estimated probabilities must satisfy
This monotonicity constraint, which may not be ensured when training separate single-task ML models for each target variable, can be enforced through a multi-task ML model. In this study, the DTE regression adjustment model is implemented based on a fully connected neural network. The architecture is illustrated in Figure (ref). In this figure, $\sigma:\mathcal{X} \rightarrow\mathbb{R}$ is an activation function (e.g., ReLU or Sigmoid) used within the hidden layers, $g:\mathbb{R}\rightarrow[0, \infty)$ is a monotonic non-negative function for the last hidden layer, and $f:[0,\infty) \rightarrow [0,1]$ represents a monotonic map to compute the output probabilities.
In our proposed model, the monotonic constraint of the outputs is accomplished by the combination of $g$ and $f$. Let $H$ denote the number of hidden layers, $h_{j}^{k}$ be the $j$th element of the hidden layer $k \in \{1, 2, \dots, H\}$, and $o_{j}:=\widehat\gamma_{y_j}^{(w)}(X)$ be the $j$th element of the final output. Also, suppose $\tilde{\mathcal{Y}}$ is sorted before training the model. The layers up to the final hidden layer are standard fully-connected neural network layers. In the final hidden layer $H$, a vector of non-negative values is obtained by applying $g$ to its output. Then, a vector of non-negative values $\tilde{h}^{H} = \{ \tilde{h}_{1}^{H},\tilde{h}_{2}^{H}, \dots, \tilde{h}_{M}^{H} \}$ is computed where each component $\tilde{h}_{j}^{H}$ is obtained by accumulating the non-negative components up to that point from the previous layer as
Then, by construction, its elements satisfy the monotonicity property:
Finally, the final output is obtained by applying $f$ function to each component in the last hidden layer as $o_{j} = f(\tilde{h}_{j}^{H})$ for $j\in\{1, \dots, M\}$, and satisfies the monotonicity property given in (ref) by applying the monotonic function $f$ to the monotonically increasing vector.
In this section, we evaluate our proposed multi-task regression-adjusted estimators against other methods in three distinct settings: simulation studies, a water conservation experiment, and a content promotion A/B test at a Japanese video streaming platform.
We conduct a Monte Carlo simulation study to assess the finite-sample performance of our adjusted estimators. Specifically, we evaluate the effectiveness of the proposed multi-task neural networks model adjustment in variance reduction, comparing it with the empirical distribution function, linear regression adjustment and the single-task NN model adjustment. The details of the data generation process (DGP) are provided in Appendix (ref).
For each simulation iteration, we generate a sample of size \( n = 1000 \) following the specified DGP and compute the distributional treatment effects (DTE) at quantiles \( \{0.05, 0.10, \dots, 0.95\} \). This process is repeated \( S = 500 \) times. To establish a ground truth for evaluating estimation efficiency, we separately generate \( 10^5 \) samples from the same DGP and compute the corresponding DTE, which is used to calculate bias and mean squared error (MSE). The details of the multi-task NN model are described in Appendix (ref).
Figure (ref) illustrates the reduction rate of pointwise MSE for each method compared to the empirical method across different locations, where higher values indicate greater precision. The results indicate that the multi-task NN adjustment method, both with and without the monotonic constraint, reduces pointwise MSE by 50--65%. This reduction is greater than that of the single-task NN adjustment and linear regression adjustment, which achieve reductions of 12--50%. Notably, at the tails of the distribution, such as \( q = 0.05 \) and \( q = 0.95 \), the reduction in MSE is particularly pronounced.
Since all estimators used in the experiment are unbiased, there is no substantial difference in bias values. Therefore, the observed differences in MSE are primarily attributable to variance reduction. These results indicate that multi-task learning significantly contributes to variance reduction. Moreover, since the monotonic multi-task NN adjustment method demonstrates the best performance, the monotonic restriction also contributes the variance reduction.
To evaluate the efficiency gains achieved through multi-task learning, we measured the average execution time of the estimation over 500 repetitions for each method. As presented in Table (ref), the average execution time of the multi-task NN adjustment is significantly lower than that of the single-task NN adjustment method.
By leveraging the multi-task learning, the computation time for adjusted DTE estimation is reduced by around 80% over the single-task NN adjustment under this simulation setting, highlighting the efficiency of the proposed approach.
We also validate whether the improvement in the precision of distribution function estimation can be attributed to the enhancement of model prediction performance. To investigate this, we compare the prediction performance of linear regression, the single-task NN model, the multi-task NN model, and the monotonic multi-task NN model on a binary classification problem. We use the same data and locations of the simulation study and aggregate the prediction performance across all locations.
The prediction performance of each model obtained from 2-fold cross-validation is presented in Table (ref). As shown in the numerical results, the multi-task and the monotonic multi-task NN models outperform other models. This result suggests that multi-task learning contributes to prediction performance at each location. Additionally, although the contribution is smaller compared to multi-task learning, the monotonic restriction also improves prediction performance. This result suggests that the improvement in precision for distribution function estimation is primarily driven by the superior predictive capability of the proposed monotonic multi-task NN model.
In this section, we evaluate the performance of the proposed method using publicly available real randomized experiment data.
Specifically, we reanalyze data from a randomized experiment conducted by Ferraro and Price Ferraro2013 in 2007, which examined the impact of norm-based messages, or “nudges,” on water consumption in Cobb County, Atlanta, Georgia. The experiment involved three distinct interventions designed to reduce water usage, each compared to a control group that received no nudge.
List et al. List2024 re-estimated the ATE for the intervention labeled “strong social norm,” which incorporated both a prosocial appeal and social comparison, making it the most intensive intervention applied. Building on their analysis, we estimate the regression-adjusted DTE for this intervention relative to the control group, utilizing the same set of pre-treatment covariates, \(X\), representing monthly water consumption during the year preceding the experiment. Consequently, the covariate space \(X\) has a dimensionality of \(d_x = 12\). The outcome variable \(Y\) represents water consumption levels from June to September 2007, measured in thousands of gallons and treated as discretely distributed.
Figure (ref) presents the DTE estimation results using the empirical cumulative distribution function (CDF), linear regression adjustment, single-task NN adjustment, and the proposed multi-task NN adjustments. We calculate and compare standard errors (SE) using a multiplier bootstrap with \(B=5,000\) repetitions for each method as explained in Appendix (ref). Regardless of the method used, the results show an increase in the number of users with a consumption level of around 10-50 in the treatment group, while the number of users with higher consumption levels has decreased.
In real-world data, the true effect cannot be observed, so we cannot calculate MSE as in the previous section. Instead, we examine whether SE has been reduced. To examine the reduction in SE, we calculated the pointwise SE reduction relative to the empirical DTE for each location and presented it in Figure (ref). The proposed method achieves a standard error reduction of 0--30%, outperforming both the linear regression adjustment and the single-task NN adjustment. The methods using multi-task learning show a smaller difference in SE than single-task learning for the location between 0 and 100. On the other hand, for locations greater than 100, SE is more significantly reduced with multi-task learning. Predictions in the range of 100 to 200 are more challenging, as more than 95% of records fall within the range of 0 to 100, as illustrated in Figure (ref) of Appendix (ref).
The enhanced precision in this region suggests that learning labels jointly across all locations enables the model to better capture the underlying distribution and address the imbalanced data issue effectively. Though not significant, SE is reduced further around the tail of the distribution by the monotonic constraint.
Lastly, the proposed regression adjustment methods are applied to evaluate the result of an A/B test at ABEMA, a leading video streaming platform in Japan with over 30 million weekly active users. The A/B test was conducted to investigate the impact of top-of-screen content promotions in the ABEMA mobile application on viewer engagement over a four-week period. In the experiment, users were randomly assigned to one of two treatment groups: those exposed to content promotions and those exposed to advertisements. Figure (ref) illustrates these two treatment conditions with screen captures.
The outcome variable in this experiment is the total viewing time (in minutes) for a selected series over the four-week experiment period. The pre-treatment covariates include age, gender, and various historical user activities recorded during the three weeks preceding the experiment. See Appendix (ref) for experiment settings. For this analysis, we focus on a 46-minute comedy program. The experiment included a total of 4,311,905 users, which is a subset of total platform users.
The estimated ATE of the content promotion is 0.0145 (SE = 0.006, p = 0.01) minutes, corresponding to a 5.9% increase in the average viewing time. This result indicates that the promotion successfully increased users' average viewing duration. However, an increase in average viewing time can arise from multiple scenarios, such as a rise in new users watching the content, a significant increase in users who watch for a short duration, or a growth in users who watch for a relatively long duration. The estimated average treatment effect does not provide particularly useful information in distinguishing between these scenarios.
To examine the heterogeneous effects, we estimate the DTE. Figure (ref) presents the DTE estimation results using the empirical CDF, linear regression adjustment, and the proposed multi-task NN adjustments. Regardless of the method used, it is shown that there are fewer users in the treatment group at \( y = 0 \). This indicates that content promotion primarily encouraged users to start watching the series and reduced the number of users who did not engage with the content at all. Additionally, for \( y > 0 \), the number of users in the treatment group has increased, with a particularly larger increase among users who watched for a short duration. This suggests that while some users exposed to content promotion complete viewing, many others engage only briefly.
As illustrated in the SE reduction comparisons in Figure (ref), all methods improve estimation precision relative to the empirical estimation. The SE reduction is relatively small at \( y = 0 \) but increases for \( y > 0 \), where the number of observations is lower. The proposed multi-task NN adjustment and monotonic multi-task NN adjustment achieve the highest SE reduction across most locations, reducing SE by 0--16%.
Finally, we compare computational times across methods using two real-world datasets in Table (ref). Our multi-task neural network implementation demonstrates substantial computational advantages, yielding execution time reductions of 93% and 71% in the first and second experiments, respectively, compared to the single-task neural network implementation.
In this study, we introduce a novel multi-task neural network approach with monotonic shape constraints for efficiently estimating distributional treatment effects in randomized experiments. Our method reduces computational costs drastically by leveraging multi-task learning, while enhancing precision through the monotonic shape constraint and multi-label learning. Theoretical analysis establishes the efficiency gains of our approach, and empirical results confirm significant reductions in execution time alongside improved estimation precision.
The method's computational advantages and shared representations are most pronounced when estimating cumulative distribution functions at multiple locations, though these benefits may be more modest for scenarios with fewer estimation points. Additionally, the computational efficiency gains depend on Assumption (ref), and may vary under different model architectures such as boosting trees. Nevertheless, our approach represents a significant advancement in the estimation of distributional treatment effects, offering a scalable and practical solution for modern large-scale randomized experiments. The framework we establish opens new possibilities for efficient treatment effect analysis across diverse experimental settings.