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Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects

Haorui Ma, Dennis Frauen, Valentyn Melnychuk, Stefan Feuerriegel

Abstract

Estimation of heterogeneous long-term treatment effects (HLTEs) is widely used for personalized decision-making in marketing, economics, and medicine, where short-term randomized experiments are often combined with long-term observational data. However, HLTE estimation is challenging due to limited overlap in treatment or in observing 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. The learners are designed for the canonical HLTE setting that combines a short-term randomized dataset $\mathcal{D}_1$ with a long-term historical dataset $\mathcal{D}_2$. The key idea of our LT-O-Learners is to retarget the learning objective by introducing custom overlap weights that downweight samples with low overlap in treatment or in long-term observation. We show that the retargeted loss is equivalent to the weighted oracle loss and satisfies Neyman-orthogonality, which means our learners are robust to errors in the nuisance estimation. We further provide a general error bound for the LT-O-Learners and give the conditions under which quasi-oracle rate can be achieved. Finally, our LT-O-learners are model-agnostic and can thus be instantiated with arbitrary machine learning models. We conduct empirical evaluations on synthetic and semi-synthetic benchmarks to confirm the theoretical properties of our LT-O-Learners, especially the robustness in low-overlap settings. To the best of our knowledge, ours are the first orthogonal learners for HLTE estimation that are robust to low overlap that is common in long-term outcomes.

Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects

Abstract

Estimation of heterogeneous long-term treatment effects (HLTEs) is widely used for personalized decision-making in marketing, economics, and medicine, where short-term randomized experiments are often combined with long-term observational data. However, HLTE estimation is challenging due to limited overlap in treatment or in observing 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. The learners are designed for the canonical HLTE setting that combines a short-term randomized dataset with a long-term historical dataset . The key idea of our LT-O-Learners is to retarget the learning objective by introducing custom overlap weights that downweight samples with low overlap in treatment or in long-term observation. We show that the retargeted loss is equivalent to the weighted oracle loss and satisfies Neyman-orthogonality, which means our learners are robust to errors in the nuisance estimation. We further provide a general error bound for the LT-O-Learners and give the conditions under which quasi-oracle rate can be achieved. Finally, our LT-O-learners are model-agnostic and can thus be instantiated with arbitrary machine learning models. We conduct empirical evaluations on synthetic and semi-synthetic benchmarks to confirm the theoretical properties of our LT-O-Learners, especially the robustness in low-overlap settings. To the best of our knowledge, ours are the first orthogonal learners for HLTE estimation that are robust to low overlap that is common in long-term outcomes.

Paper Structure

This paper contains 33 sections, 7 theorems, 146 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem formulation
  4. T-, RA-, and IPW-learners for HLTE estimation
  5. Our LT-O-learners
  6. Orthogonalizing the weighted oracle loss
  7. Algorithm
  8. Instantiations
  9. Convergence analysis
  10. Experiments
  11. Theory
  12. Identification
  13. Efficient influence function
  14. What is the efficient influence function (EIF)?
  15. Properties of EIF
  16. ...and 18 more sections

Key Result

Theorem 5.1

The proposed loss $\mathcal{L}_\omega(g, \eta)$ is orthogonal w.r.t. the nuisance $\eta$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • 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 experiment. The left plot shows the two overlap issues.
  • 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 are robust and maintain a low variance.
  • Figure 3: 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.

Theorems & Definitions (11)

  • Theorem 5.1: Neyman-orthogonality
  • proof
  • Theorem 5.2: Oracle equivalence
  • proof
  • Theorem 5.3: Error bounds
  • proof
  • Corollary 5.4: Quasi-oracle rate condition
  • Lemma A.1: Identification of HLTE
  • proof
  • Lemma A.2: Neyman-orthogonality
  • ...and 1 more