Estimating Long-term Heterogeneous Dose-response Curve: Generalization Bound Leveraging Optimal Transport Weights

Zeqin Yang; Weilin Chen; Ruichu Cai; Yuguang Yan; Zhifeng Hao; Zhipeng Yu; Zhichao Zou; Jixing Xu

cs.LGJun 27, 2024

Estimating Long-term Heterogeneous Dose-response Curve: Generalization Bound Leveraging Optimal Transport Weights

Authors: Zeqin Yang, Weilin Chen, Ruichu Cai, Yuguang Yan, Zhifeng Hao, Zhipeng Yu, Zhichao Zou, Jixing Xu,

Organizations: School of Computer Science, Guangdong University of Technology · Didi Chuxing · Pazhou Laboratory (Huangpu) · College of Science, Shantou University

Abstract

Long-term treatment effect estimation is a significant but challenging problem in many applications. Existing methods rely on ideal assumptions, such as no unobserved confounders or binary treatment, to estimate long-term average treatment effects. However, in numerous real-world applications, these assumptions could be violated, and average treatment effects are insufficient for personalized decision-making. In this paper, we address a more general problem of estimating long-term Heterogeneous Dose-Response Curve (HDRC) while accounting for unobserved confounders and continuous treatment. Specifically, to remove the unobserved confounders in the long-term observational data, we introduce an optimal transport weighting framework to align the long-term observational data to an auxiliary short-term experimental data. Furthermore, to accurately predict the heterogeneous effects of continuous treatment, we establish a generalization bound on counterfactual prediction error by leveraging the reweighted distribution induced by optimal transport. Finally, we develop a long-term HDRC estimator building upon the above theoretical foundations. Extensive experiments on synthetic and semi-synthetic datasets demonstrate the effectiveness of our approach.

Figures & tables

Figure 1. Causal graphs of two types of data.

Figure 2. Model architecture of the proposed LEARN.

Dataset	Synthetic					News	TCGA
Dataset	$\beta_{U}=1$	$\beta_{U}=1.5$	$\beta_{U}=2.0$	$\beta_{U}=2.5$	$\beta_{U}=3$	News	TCGA
DRNet	$26.4\pm 7.3$	$29.0\pm 7.5$	$33.8\pm 8.8$	$39.4\pm 9.4$	$47.9\pm 10.5$	$1.97\pm 0.27$	$1.08\pm 0.51$
SCIGAN	$26.5\pm 6.9$	$30.5\pm 7.5$	$34.9\pm 8.4$	$40.5\pm 8.4$	$48.1\pm 9.7$	$2.45\pm 0.32$	$1.42\pm 1.21$
VCNet	$24.2\pm 7.2$	$27.9\pm 8.0$	$31.9\pm 9.0$	$36.4\pm 9.9$	$43.1\pm 11.2$	$1.52\pm 0.22$	$0.51\pm 0.61$
ADMIT	$23.0\pm 7.0$	$27.0\pm 6.8$	$30.8\pm 7.9$	$34.1\pm 8.6$	$41.4\pm 10.2$	$1.41\pm 0.17$	$0.49\pm 0.58$
ACRF	$22.9\pm 7.1$	$26.8\pm 6.6$	$30.7\pm 7.1$	$34.3\pm 7.2$	$41.0\pm 11.0$	$1.54\pm 0.27$	$0.53\pm 0.57$

Table 1. The MISE of long-term HDRC estimation is reported as

\mbox{Mean}\pm\mbox{Std}

. The best-performing method is bolded.

Batch Size	512	1024	2048	4096
Relative Error	$14.51\%$	$7.68\%$	$3.45\%$	$1.03\%$

Table 2. The relative approximation error of different batch size about

m\mbox{-}OT

problem in Theorem 5 .

Number	100	250	500	1000	2000
MISE	$22.52\pm 7.40$	$22.18\pm 7.19$	$21.98\pm 6.81$	$21.63\pm 6.83$	$21.43\pm 6.74$

Table 3. The impact of experimental data’s sample size.

Panel of Figure 3 — Figure 3. Impact of the hyperparameters: (a) IPM $\lambda_{b}$ , (b) short-term loss $\lambda_{o}$ , and (c) negative entropy regularization $\lambda_{e}$ .

Dose-Response Estimation

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Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects
Apr 1, 2026Haorui Ma, Dennis Frauen, Valentyn Melnychuk +1Causal Effect Estimation Conditional Average Treatment Effect Estimation
AI in Management, LMU Munich; Munich Center for Machine Learning; Munich, Germany
HiSTOR: Structured Treatment-outcome Representation Learning for High-Dimensional Longitudinal Data
Oct 13, 2026Chuanqi Lao, Junzhou Huang, Qiao LiuCausal Effect Estimation Individualized Treatment Effect Estimation
Department of Computer Science and Engineering, The University of Texas at Arlington · Department of Biostatistics, Yale School of Public Health, Yale University
Estimating Continuous Treatment Effects with Two-Stage Kernel Ridge Regression
Apr 15, 2026Seok-Jin Kim, Kaizheng WangCausal Effect Estimation Kernel Ridge Regression
Department of IEOR, Columbia University · Department of IEOR and Data Science Institute, Columbia University

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Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects

Haorui Ma, Dennis Frauen, Valentyn Melnychuk, Stefan Feuerriegel

AI in Management, LMU Munich; Munich Center for Machine Learning; Munich, Germany

Causal Effect Estimation Conditional Average Treatment Effect Estimation Robust Estimation

Open paper AskPDF arXiv

Estimation of heterogeneous long-term treatment effects (HLTEs) is relevant for personalized decision-making in marketing, economics, and medicine, where short-term observational datasets are often combined with long-term observational datasets. However, HLTE estimation is challenging due to limited overlap in treatment assignments or in long-term outcomes for certain subpopulations, which can lead to unstable HLTE estimates with large finite-sample variance. To address this challenge, we introduce the LT-O-learners (Long-Term Orthogonal Learners), a set of novel orthogonal learners for HLTE estimation in the canonical HLTE setting with surrogacy. The key idea of our LT-O-learners is to retarget the loss via custom overlap weights that downweight low-overlap samples. We show that the retargeted loss recovers the true HLTE pointwise and satisfies Neyman-orthogonality. We further prove two key theoretical results: (i) The nuisance error enters the error bound only through higher-order terms, which means our learners are robust to nuisance estimation error. (ii) Under a linear function class, the retargeting effectively controls the asymptotic variance of the HLTE estimator via the overlap weights in low-overlap regimes. We conduct experiments on synthetic and real-world datasets to confirm the theoretical properties of our LT-O-learners, particularly robustness in low-overlap regimes. To our knowledge, ours are the first orthogonal learners for HLTE estimation robust to low overlap in long-term settings.

Abstract

Dose-Response Estimation

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