Estimating heterogeneous single and interaction treatment effects from observational data under multiple simultaneous treatments is crucial for decision-making. To mitigate estimation variance, previous studies balance representation distributions between every pair of treatment patterns. However, such pairwise balancing scales quadratically with the number of treatment patterns and fails to preserve consistent local proximity structures across patterns, which degrades counterfactual estimation. To address these challenges, we propose the Causal Inference for Heterogeneous Single and Interaction Treatment Effects Network (CIHSI-Net), a deep learning framework built on a novel Barycentric Fused Gromov-Wasserstein Balancing (BFG-WB) objective. BFG-WB aligns the representation distribution of each treatment pattern with a shared Wasserstein barycenter, achieving global alignment while reducing the computational complexity from quadratic to linear, and its Fused Gromov-Wasserstein discrepancy preserves the local proximity structures essential for reliable heterogeneous effect estimation. Simulation studies show that CIHSI-Net consistently outperforms state-of-the-art baselines, and an application to real-world marketing data demonstrates its practical utility in complex multi-treatment scenarios.
Figures & tables
Figure 1: Illustration of neighborhood switching under Wasserstein-based representation balancing. The gray arrows show OT correspondences. Wasserstein minimization aligns the two distributions but can map neighboring points to distant locations, distorting within-distribution neighborhood relations; a structure-preserving match such as j↔j′ would be desirable.
Figure 2: Conceptual illustration of BFG-WB. The gray arrows show a subset of OT correspondences. By evaluating the FGW discrepancy, the alignment is encouraged to preserve within-distribution neighborhood relations.
Figure 3: The architecture of CIHSI-Net. The model consists of three components: the representation learning network ϕ , the task embedding network tw , and the outcome prediction network h . BFG-WB is a regularization term applied to the latent representation ϕ(x) .
ϵCASE
ϵCAIE
Sim.
Method
k=1
k=2
k=3
S={1,2}
S={2,3}
S={1,3}
S={1,2,3}
1
TECE-VAE
1.82 ± 0.09
1.91 ± 0.10
1.80 ± 0.14
3.58 ± 0.11
3.27 ± 0.10
3.30 ± 0.10
6.80 ± 0.18
NCoRE
0.28 ± 0.06
0.30 ± 0.06
0.25 ± 0.05
0.52 ± 0.10
0.33 ± 0.10
0.37 ± 0.10
0.56 ± 0.14
CISI-Net
0.22 ± 0.04
0.24 ± 0.03
0.19 ± 0.06
0.31 ± 0.07
0.11 ± 0.04
0.15 ± 0.08
0.31 ± 0.08
CIHSI-Net
0.19 ± 0.04
0.21 ± 0.04
0.18 ± 0.10
0.28 ± 0.05
0.08 ± 0.05
0.12 ± 0.04
0.24 ± 0.04
2
TECE-VAE
1.60 ± 0.05
1.65 ± 0.09
1.58 ± 0.16
3.01 ± 0.05
2.65 ± 0.08
2.69 ± 0.09
5.50 ± 0.15
Table 1: Comparison of mean estimation errors and standard deviations for ϵCASE and ϵCAIE in two simulation datasets. Here, k∈{1,2,3} indexes individual treatments, and S⊆{1,2,3} (with ∣S∣≥2 ) denotes treatment combinations.
No.
WB
Penalty type
Average ϵCASE
Average ϵCAIE
1
✗
None
0.27 ± 0.06
0.24 ± 0.02
2
✗
W
0.22 ± 0.03
0.22 ± 0.03
3
✗
GW
0.25 ± 0.06
0.24 ± 0.03
4
✗
FGW
0.22 ± 0.02
0.23 ± 0.02
5
✓
W
0.22 ± 0.04
0.21 ± 0.02
6
✓
GW
0.22 ± 0.04
0.22 ± 0.03
Table 2: Ablation study results on simulation dataset 1. Average ϵCASE and Average ϵCAIE denote the means of the reported ϵCASE and ϵCAIE values, respectively. WB indicates whether the Wasserstein barycenter is used. Penalties: Wasserstein (W), Gromov-Wasserstein (GW), Fused GW (FGW), and None.
Figure 4: Estimated CASE (top) and CAIE (bottom) across user groups stratified by pre-promotion service usage.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Hyperparameter
Candidate values
Selected value
BFG-WB coefficient α
{0.1,0.5,1.0,1.5,2.0,5.0}
1.0
FGW trade-off η
{10−5,0.1,0.2,…,0.9,1.0}
0.6
Number of units per FC hidden layer
{20,50,100}
100
Appendix
Table A1: Hyperparameter search ranges and selected values.
Method
K=2
K=3
K=4
K=5
K=6
K=8
CISI-Net
5.98
15.08
31.09
138.54
125.62
1529.83
CIHSI-Net
8.21
9.02
13.92
22.71
23.01
48.32
Appendix
Table A2: Wall-clock training time per epoch (s/epoch) for CISI-Net (pairwise balancing) and CIHSI-Net (BFG-WB), evaluated with K∈{2,3,4,5,6,8} treatments under the same hardware and training configuration. For each K , both methods are trained with the same batch size, while the batch size is adjusted across different values of K . Times are averaged across all observed epochs in 100 trials with early stopping (max 300 epochs, patience = 15).
Figure A1: Average ϵCASE (left) and ϵCAIE (right) under varying numbers of treatments K∈{2,4,6,8} .
ϵCASE
ϵCAIE
No.
WB
Penalty type
k=1
k=2
k=3
S={1,2}
S={2,3}
S={1,3}
S={1,2,3}
1
✗
None
0.28 ± 0.08
0.29 ± 0.04
0.25 ± 0.09
0.36 ± 0.03
0.12 ± 0.04
0.16 ± 0.03
0.31 ± 0.04
2
✗
W
0.22 ± 0.04
0.24 ± 0.03
0.19 ± 0.06
0.31 ± 0.07
0.11 ± 0.04
0.15 ± 0.08
0.31 ± 0.08
3
✗
GW
0.25 ± 0.03
0.28 ± 0.04
0.23 ± 0.10
0.35 ± 0.03
0.12 ± 0.06
0.16 ± 0.04
0.32 ± 0.05
4
✗
FGW
0.23 ± 0.04
0.25 ± 0.03
0.19 ± 0.05
0.33 ± 0.04
0.12 ± 0.06
0.16 ± 0.04
0.31 ± 0.04
5
✓
W
0.22 ± 0.03
0.24 ± 0.03
0.19 ± 0.08
0.30 ± 0.04
0.10 ± 0.05
0.15 ± 0.09
0.28 ± 0.07
Appendix
Table A3: Ablation study result on simulation dataset 1. WB indicates whether the Wasserstein barycenter is used. Penalties: Wasserstein (W), Gromov-Wasserstein (GW), Fused GW (FGW), and None.
Figure A2: Distribution of pairwise Wasserstein (left) and Gromov-Wasserstein distances (right) between representation distributions for all unique treatment pairs. We compare CISI-Net (without balancing), CISI-Net (with pairwise balancing), and CIHSI-Net (ours). Lower values indicate better alignment across treatment patterns.
Figure A3: Sensitivity analysis of hyperparameters α and η on simulation dataset 1. The top row shows ϵCASE(1) - ϵCASE(3) , the middle row shows ϵCAIE({1,2}) - ϵCAIE({1,3}) , and the bottom panel shows ϵCAIE({1,2,3}) .
ϵCASE
ϵCAIE
Method
k=1
k=2
k=3
S={1,2}
S={2,3}
S={1,3}
S={1,2,3}
TECE-VAE
1.23 ± 0.09
1.25 ± 0.06
1.28 ± 0.07
3.22 ± 0.46
1.82 ± 0.08
1.62 ± 0.17
4.00 ± 0.34
NCoRE
0.98 ± 0.16
0.98 ± 0.07
0.78 ± 0.05
2.94 ± 0.48
1.29 ± 0.28
1.37 ± 0.14
5.73 ± 1.20
CISI-Net
0.79 ± 0.01
1.13 ± 0.04
0.96 ± 0.12
1.49 ± 0.14
1.21 ± 0.21
0.96 ± 0.02
1.49 ± 0.11
CIHSI-Net
0.81 ± 0.02
1.02 ± 0.07
0.78 ± 0.07
1.45 ± 0.07
1.09 ± 0.08
0.96 ± 0.04
1.45 ± 0.06
Appendix
Table A4: Semi-synthetic experiment results using Dataset A’s real covariates and observed treatment assignments. Bold values indicate the best performance, and underlined values indicate the second-best performance for each metric.
Figure A4: Sensitivity and uncertainty analysis of the estimated causal effects in the real-world application on Dataset A. The rows correspond to the BFG-WB regularization strength α∈{0.5,1.0,1.5} , and the columns correspond to the FGW trade-off parameter η∈{0.5,0.6,0.7} . In each panel, the top subpanel reports the estimated CASE and the bottom subpanel reports the estimated CAIE across user groups. The shaded regions represent bootstrap confidence intervals.
Figure A5: In dataset B, estimated CASEs for CP 4 and CP 5 (left) and the estimated CAIE for {CP\textsubscript4,CP\textsubscript5} (right) across user groups stratified by pre-promotion service usage. The outcomes are standardized.
Estimating Individual Treatment Effects (ITE) in multi-treatment scenarios faces two critical challenges: the Hyperparameter Selection Dilemma for balancing weights and the Curse of Dimensionality in computational scalability. This paper derives a novel multi-treatment generalization bound and proposes a theoretical estimator for the optimal balancing weight α, eliminating expensive heuristic tuning. We investigate three balancing strategies: Pairwise, One-vs-All (OVA), and Treatment Aggregation. While OVA achieves superior precision in low-dimensional settings, our proposed Treatment Aggregation ensures both accuracy and O(1) scalability as the treatment space expands. Furthermore, we extend our framework to a generative architecture, Multi-Treatment CausalEGM, which preserves the Wasserstein geodesic structure of the treatment manifold. Experiments on semi-synthetic and image datasets demonstrate that our approach significantly outperforms traditional models in estimation accuracy and efficiency, particularly in large-scale intervention scenarios.
Wanting Liang, Haoang Chi, Zhiheng Zhang
Shanghai University of Finance and Economics, Shanghai, China.
Estimating causal effects with continuous treatments in observational studies is challenging due to confounding, model misspecification, and high-dimensional covariates. We propose the Weighted Spline-Expanded Network (WSENet), an end-to-end neural framework that addresses these challenges by combining covariate balancing, structured treatment embedding, and bias-corrected outcome estimation. WSENet first applies Distance Covariate Optimal Weights to induce distributional independence between covariates and treatment without relying on parametric models. It then learns the conditional outcome via a structured network that fuses outcome-relevant representations of covariates with a spline-expanded treatment input, enabling smooth and flexible modeling of the dose-response relationship. To mitigate residual bias, we introduce Weighted Targeted Regularization, a correction technique based on efficient influence functions that yields a doubly robust estimator. Extensive evaluations on semi-synthetic and real-world datasets, including high-dimensional genomic and environmental health data, demonstrate that WSENet consistently outperforms existing baselines in both accuracy and stability.
Shucheng Liu, Chan Park, Guanhua Chen
University of North Carolina at Chapel Hill · University of Illinois Urbana-Champaign · University of Wisconsin–Madison
Causal inference has become a central issue across various fields, including computer science, statistics, economics, education, healthcare, and medicine. The broad applicability of this discipline has garnered increased research funding and attention. In recent years, the estimation of causal effects from observational data has gained traction due to the vast amounts of collected data and the lower costs compared to randomized controlled trials. Advances in causal effect estimation methods have enhanced service personalization tools. For instance, these tools can help identify the most effective type of treatment (considering both cost and success rate) for each patient among different medical service options. This paper proposes an innovative method for estimating the heterogeneity of treatment effects. The structure of the proposed model is based on a deep neural network and a pseudo-single learner. The proposed method has been compared with other state-of-the-art methods on the IHDP benchmark. Acceptable results have been obtained by using one estimator to estimate the potential outcomes of two treatment groups. Accordingly, this paves the way for further development and improvement of the proposed method.
Ali Haghpanah Jahromi, Mohammad Taheri, Zohreh Azimifar
Department of Electrical and Computer Engineering, University of Shiraz, Shiraz, Iran