Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects

Haorui Ma; Dennis Frauen; Valentyn Melnychuk; Stefan Feuerriegel

cs.LGApr 1, 2026

Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects

Authors: Haorui Ma, Dennis Frauen, Valentyn Melnychuk, Stefan Feuerriegel

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

Abstract

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.

Figures & tables

Figure 1: Our HLTE setting: The yellow and green box illustrate our two-sample setting.

\mathcal{D}_{1}

and

\mathcal{D}_{2}

share covariates

X

and surrogates

S

. The surrogate index (i.e.

\mathbb{E}_{\mathcal{D}_{2}}[Y|S,X]

), learned in the long-term dataset, is used as a proxy for long-term outcomes in the short-term dataset. The left plot shows the two overlap issues.

Meta-learner	Analogy in standard HTE	Neyman orthogonal	Robust to low treatment overlap	Robust to low outcome overlap
LT-T-learner	T-learner [ 38 ]	✗	✗	✗
LT-RA-learner	RA-learner [ 15 ]	✗	✗	✗
LT-IPW-learner	IPW-learner [ 15 ]	✗	✗	✗
LT-O-DR-learner ( ours , $\omega\equiv 1$ )	DR-learner [ 37 ]	✓	✗	✗
LT-O-TO-learner ( ours , $\omega=\pi^{2}(1-\pi)^{2}$ )	R-learner [ 42 ]	✓	✓	✗
LT-O-LO-learner ( ours , $\omega=\rho$ )	–	✓	✗	✓

Table 1: Overview of meta-learners for HLTE estimation. Our LT-O-learners are the only approaches that are both Neyman-orthogonal and robust to different types of low-overlap scenarios.

Table 3

Figure 2: Variance of estimator across different overlap scenarios (

\gamma

): Low overlap leads to a high finite-sample variance of the DR-learner, while our LT-O-learners maintain a low variance.

Figure 3: Heterogeneity on GAIN. Subgroup HLTEs on

Q_{13:36}

earnings of the LT-O-DO-learner against the pseudo ground-truth. Our method captures the known heterogeneity across age, education, prior labour status, and ethnicity.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Causal diagram for the canonical surrogacy setting. The short-term experimental dataset $\mathcal{D}_{1}$ observes $(X,A,S)$ while $Y$ is unobserved (dashed node); the long-term observational dataset $\mathcal{D}_{2}$ observes $(X,S,Y)$ with treatment $A$ absent. The dotted arrow $A\to Y$ marked with $\times$ indicates that the surrogacy assumption rules out a direct effect of $A$ on $Y$ given $(S,X)$ .

Setting

Paper

Short-term

\mathcal{D}_{1}

Long-term

\mathcal{D}_{2}

Identification through

Target

Benefits

X

A

S

X

A

S

Y

Neyman orthogonality

Robustness to low overlap

Latent unconfoundedness

Chen and Ritzwoller [9] (latent unconf.)

✓

Latent unconfoundedness

ALTE

✓

✗

Yang et al. [61]

✓

Latent unconfoundedness

HLTE (dose-response)

✗

Equi- confounding

Ghassami et al. [24]

✓

Additive equi-conf. †

ALTE, ATT

✓

✗

Chen et al. [10]

✓

CAECB

HLTE

✓

✗

Proximal

Imbens et al. [32]

✓

Outcome + selection bridge functions

ALTE

✓

✗

Appendix

Table 4: Comparison of settings for long-term causal effect estimation. Methods are grouped by their high-level identification setting. ✓/ ✓ indicates that a variable is observed; for the treatment columns, ✓ denotes binary/categorical treatment and ✓ denotes continuous treatment. ✗ indicates that the variable is not observed. ✓ / ✗ in the Benefits columns indicate whether each method satisfies the corresponding property.

Figure 5: Benefit of Neyman-orthogonality: ( Left ) PEHE vs. sample size. The orthogonal LT-O-DO-learner outperforms the weighted (but non-orthogonal) DR/RA-learner. ( Right ) Mean squared error (MSE) of the nuisance estimation. Taken together, high errors in the nuisance functions at small sample sizes drive the instability of non-orthogonal baselines.

Figure 6: PEHE stratified by overlap on

\mathcal{D}_{t+o}

: The boxplot shows the 0%, 25%, 50%, 75%, and 100% quantiles, as well as the mean (marker) of the square root pseudo-PEHE within each overlap stratum. The plots show that our LT-O-learners (LT-O-DO-learner, LT-O-TO-learner, LT-O-LO-learner) remain accurate even in low-overlap strata, despite the sample retargeting.

Figure 7: PEHE stratified by overlap on

\mathcal{D}_{t}

: The boxplot shows the 0%, 25%, 50%, 75%, and 100% quantiles, as well as the mean (marker) of the square root pseudo-PEHE within each overlap stratum. The plots show that our LT-O-learners (LT-O-DO-learner, LT-O-TO-learner) remain accurate even in low-overlap strata, despite the sample retargeting.

Figure 8: PEHE stratified by overlap on

\mathcal{D}_{o}

: The boxplot shows the 0%, 25%, 50%, 75%, and 100% quantiles, as well as the mean (marker) of the square root pseudo-PEHE within each overlap stratum. The plots show that our LT-O-LO-learner remains accurate even in low-overlap strata, despite the sample retargeting.

Figure 9: PEHE stratified by overlap on

\mathcal{D}_{*}

Figure 10: Surrogacy violation sensitivity analysis on GAIN: The left (right) figure shows the mean (variance) of pseudo-PEHE of the LT-T-learner, LT-O-DR-learner, and LT-O-DO-learner across different surrogate dimensions

T

. The surrogacy violation is more severe at lower

T

, which leads to higher pseudo-PEHE for all methods. However, our LT-O-DO-learner remains more robust than the baselines across all

T

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Estimating heterogeneous survival treatment effects (HSTEs) from time-to-event data under time-varying treatments is important in fields such as medicine (e.g., to decide which sequence of treatments is better in terms of survival probability). However, this task is challenging because treatment decisions, patient characteristics, and censoring probability often evolve together, requiring sequential adjustment for both treatments and censoring. Existing survival learners mostly consider a single baseline treatment assignment, while longitudinal treatment-effect learners typically assume uncensored outcomes. In this paper, we introduce a novel doubly robust learner for heterogeneous survival treatment effects under time-varying treatments and history-dependent censoring. Our framework separates the intervention window from the survival horizon, with treatment prescribed for a finite period, after which care follows its natural course. We further establish the favorable theoretical properties of Neyman-orthogonality, rate double robustness and quasi-oracle efficiency. The framework accommodates flexible machine learning models for nuisance estimation and treatment-effect regression. We evaluate our learners through several numerical experiments, demonstrating their effectiveness.

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