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OrthoGen: A Generative Orthogonal Learner for Time-Varying Treatments

Tomàs Garriga, Valentyn Melnychuk, Konstantin Hess, Eduard Serrahima de Cambra, Axel Brando, Gerard Sanz, Stefan Feuerriegel

arXiv:2610.10210v1cs.LG

TL;DR

Patient-specific outcome distributions under time-varying treatment sequences are difficult to estimate because changing patient histories create confounding and available adjustment strategies are limited. The paper introduces generative recursive g-computation and ORTHOGEN, an orthogonal, doubly robust generative learner. It derives rate double robustness and quasi-oracle efficiency under suitable conditions.

  • Problem

    Estimating conditional distributions of potential outcomes under time-varying treatment sequences matters for patient-specific risks, but time-varying confounding and limited adjustment strategies make the task challenging.

    Source: §1 · pp. 1–2, §3 · p. 4

  • Method

    The paper introduces generative recursive g-computation to propagate conditional outcome distributions and combines it with propensity-based adjustment in ORTHOGEN.

    Source: §1 · p. 2

  • Results

    ORTHOGEN is Neyman-orthogonal and doubly robust, with rate double robustness and quasi-oracle efficiency derived under suitable conditions.

    Source: §1 · p. 2, §6.1 · pp. 7–8

  • Takeaways & Limitations

    The approach directly targets CDPOs over time and can use generative backbones such as normalizing flows and diffusion models.

    Source: §1 · p. 2

    The plug-in and regression-adjusted learners and IPTW rely on a single nuisance component, making target risk sensitive to nuisance-estimation error at first order.

    Source: §5 · p. 7

Abstract

from arXiv · show

Estimating conditional distributional potential outcomes (CDPOs) over time is important in medicine (e.g., to estimate patient-specific risks under different treatment sequences). However, this task is challenging because of time-varying confounding, yet existing adjustment strategies for this task are limited. In this paper, we aim to learn CDPOs under time-varying treatments using flexible generative models. Our contributions are two-fold. (1) We introduce a tailored adjustment strategy for our setting, namely, generative recursive g-computation. Our adjustment strategy recursively propagates full conditional outcome distributions rather than conditional means, modeling the variables of interest directly rather than full trajectories. Building on our adjustment strategy, we formulate simple generative learners for CDPO estimation. However, these learners can be sensitive to nuisance estimation errors, which motivates an orthogonal learner. (2) We thus introduce OrthoGen, a Neyman-orthogonal and doubly robust generative learner. Importantly, we show that OrthoGen further achieves rate double robustness and quasi-oracle efficiency under suitable conditions. Our learners are flexible and can be instantiated with different generative backbones (e.g., normalizing flows and diffusion models). Across experiments with synthetic, semi-synthetic and real-world datasets, we find that OrthoGen is highly effective. To the best of our knowledge, we are the first to propose a generative orthogonal learner for estimating CDPOs under time-varying treatments.

1 INTRODUCTION

The paper targets patient-specific outcome distributions under time-varying treatment sequences, where changing patient histories create confounding that existing adjustment strategies address only in limited ways. It proposes generative recursive g-computation and ORTHOGEN, an orthogonal, doubly robust learner for this task.

Motivation and gap

  • The target is the conditional distribution of future outcomes under a treatment sequence, given the patient's observed history.Examples include patient-level risks for tumor size, hypoglycemia, or toxicity.Source: §1 · p. 1
  • Time-varying confounding means adjustment must account for future treatment sequences, while IPTW can have unstable weights under limited overlap, especially at large horizons.Patient characteristics can respond to earlier treatment and affect later treatment decisions and outcomes.Source: §1 · pp. 1–2

Contributions

  • Generative recursive g-computation propagates full conditional outcome distributions through generative models rather than recursively propagating conditional means.It recovers the target distribution at the prediction origin without simulating complete future covariate trajectories at inference time.Source: §1 · p. 2
  • ORTHOGEN combines generative recursive g-computation with propensity-based adjustment to provide Neyman orthogonality and double robustness for CDPO estimation over time.The paper also derives rate double robustness and quasi-oracle efficiency under suitable conditions, and allows different generative backbones.Source: §1 · p. 2

2 RELATED WORK2

Prior work leaves gaps in directly learning full conditional outcome distributions under time-varying treatments. The paper positions ORTHOGEN as the first learner for this task that is both Neyman-orthogonal and doubly robust.

Static versus time-varying treatments

  • Generative CDPO methods for static treatments do not address treatment sequences, which introduce time-varying confounding and require tailored adjustment [Frauen et al., 2025].The related-work discussion names GDR-learners as a general class for static settings (Melnychuk & Feuerriegel, 2026).Source: §2 · p. 3

Means versus distributions

Modeling target

  • Full longitudinal data-generating-process models simulate complete trajectories, while this work directly targets the CDPO of the outcome [Xiong et al., 2024; Wu et al., 2026b].The table is identified as a comparison of related approaches for direct learning of distributional potential outcomes over time.Source: §2 · p. 3, Table 1 · p. 3

Adjustment strategies

  • The closest prior approach combines IPTW with an expert-guided diffusion model, but IPTW can have high variance and be sensitive to nuisance misspecification (Mu et al., 2025; Frauen et al., 2025).The review identifies alternative adjustment strategies, such as g-computation, as missing for this task.Source: §2 · p. 3

Research gap

  • To the authors' knowledge, ORTHOGEN is the first learner for CDPOs under time-varying treatments that is both Neyman-orthogonal and doubly robust.The paper contrasts this with a literature where alternative adjustment strategies and learners with these properties are missing.Source: §2 · p. 3

3 PROBLEM SETTING

The problem is to learn a future outcome distribution under a specified treatment sequence, conditional on history at the prediction origin. Because later histories can confound subsequent treatment and outcomes, identification and estimation require sequential adjustment.

Estimand

  • The estimand is the conditional distribution of a future potential outcome under a specified treatment sequence, given history at the prediction origin.The setup fixes a prediction origin and horizon and uses the potential-outcomes framework for time-varying settings.Source: §3 · p. 4

Identifiability

  • Identification assumes recursive consistency, positivity at each treatment step, and sequential exchangeability conditional on the history available at that step.These are the stated assumptions for the time-varying treatment setting.Source: §3 · p. 4

Time-varying confounding

  • Later confounders can be affected by earlier treatments while influencing subsequent treatment and terminal outcomes, so adjustment must use updated histories at each treatment step.Adjusting only for history at the prediction origin leaves later treatment decisions confounded.Source: §3 · p. 4

Naive adjustment

  • Conditioning on prediction-origin history and selecting patients whose observed future treatment sequence matches the target sequence generally does not recover the interventional distribution.The passage identifies this naive history adjustment as biased.Source: §3 · p. 4

Generative target

  • The generative-model target is the best approximation to the true CDPO within a predefined model class, but its oracle risk depends on an unobserved counterfactual outcome.Consequently, the oracle risk cannot be evaluated directly from observational data.Source: §3 · pp. 4–5

4 GENERATIVE RECURSIVE G-COMPUTATION

Generative recursive g-computation identifies the target outcome distribution by propagating conditional outcome distributions backward across treatment times. Generative models fit this recursion using backward pseudo-sample fitting.

4 GENERATIVE RECURSIVE G-COMPUTATION

  • The backward recursion propagates the terminal outcome distribution across treatment times to identify the target CDPO at the prediction origin.Each continuation density identifies the terminal potential-outcome distribution under the remaining treatment sequence.Source: §4 · p. 5
  • At each recursive step, the successor distribution already incorporates future treatments, so the update conditions on the current treatment and averages over the next history.This extends recursive g-computation from conditional means to full conditional outcome distributions.Source: §4 · p. 5
  • At the final treatment time, the terminal density is fitted from observed outcomes; earlier densities are fitted using pseudo-outcomes sampled from the fitted successor density.Earlier outcomes under the remaining treatment sequence are not observed as direct training targets.Source: Figure 2 · p. 6, §4 · p. 6

5 NA¨IVE GENERATIVE LEARNERS BASED ON RECURSIVE G-COMPUTATION

The plug-in and regression-adjusted learners use estimated continuation densities from generative recursive g-computation in different ways. Both, like IPTW, rely on a single nuisance component and can be sensitive to nuisance-estimation error at first order.

5 NA¨IVE GENERATIVE LEARNERS BASED ON RECURSIVE G-COMPUTATION

  • The plug-in learner directly uses the fitted continuation density at the prediction origin as its CDPO estimate.The population continuation density at that time equals the target CDPO.Source: §5 · p. 6
  • The regression-adjusted learner uses continuation-density estimates to define a second-stage generative risk and fit a target model in G.This two-stage approach can incorporate additional regularization, architectural constraints, or interpretability requirements.Source: §5 · pp. 6–7
  • PI, RA, and IPTW rely on a single nuisance component, leaving their target risks sensitive to nuisance-estimation error at first order.This motivates ORTHOGEN’s orthogonal adjustment.Source: §5 · p. 7

6 ORTHOGEN

ORTHOGEN combines generative recursive g-computation with propensity-based adjustment for CDPO estimation under time-varying treatments. Under stated conditions, it is Neyman-orthogonal, doubly robust, and quasi-oracle efficient, with nuisance errors entering through products of propensity and continuation-density errors.

6 ORTHOGEN

  • ORTHOGEN combines continuation-density estimates and propensity-based adjustment in a generative learner that can use normalizing flows or diffusion models.Its estimation fits nuisance functions first, then fits the target generative model using the resulting objective.Source: §6 · p. 7

6.1 THEORETICAL PROPERTIES

  • Under the causal assumptions, ORTHOGEN’s objective is Neyman-orthogonal, or first-order insensitive to nuisance-function errors.The stated derivative condition applies across target models and admissible nuisance perturbations.Source: §6.1 · p. 7
  • The target-model error bound implies double robustness and quasi-oracle efficiency under the stated regularity conditions.The bound separates target-model learning error from a higher-order nuisance error contribution.Source: §6.1 · p. 8
  • Nuisance error enters through products of stagewise propensity and continuation-density errors, allowing faster convergence of one to offset slower convergence of the other.If the product is o(ρ_n), nuisance estimation is asymptotically negligible and ORTHOGEN attains the oracle learner’s first-order rate; two n^-1/4 nuisance rates yield an n^-1/2 product term.Source: §6.1 · p. 8

7 EVALUATION

Across synthetic, semi-synthetic, and real-world benchmarks, ORTHOGEN performs effectively across generative backbones, with its advantages becoming more pronounced at longer treatment horizons.

7 EVALUATION

  • For τ>1, ORTHOGEN outperforms PI on synthetic data; it achieves the lowest costs across semi-synthetic horizons and the lowest real-world CRPS with NFs.The semi-synthetic comparison holds for both NFs and DMs.Source: Table 2 · p. 8, Table 3 · p. 9, Table 4 · p. 9

A EXTENDED RELATED WORK

  • Prior generative CDPO methods mainly address static treatments, while many time-varying methods target conditional mean outcomes rather than full distributions.The related work also describes methods for time-varying conditional average potential outcomes using weighting, g-computation, and history-adjustment approaches.Source: A EXTENDED RELATED WORK · p. 15

B GENERATIVE MODEL INSTANTIATIONS

  • The framework uses conditional generative-model scores for continuation models and the target CDPO generator, with RA, IPTW, and ORTHOGEN risks providing observational representations.Continuation models condition on the evolving history; the target generator conditions on the history at the prediction origin.Source: B GENERATIVE MODEL INSTANTIATIONS · p. 15, §B.2 · p. 17

B.1 CONDITIONAL NORMALIZING FLOWS

  • Conditional flows transform a base variable through an invertible differentiable map, and maximum likelihood yields a forward-KL projection of the CDPO onto the flow class.The flow permits sampling and density evaluation without integrating over the base variable.Source: §B.1 · pp. 15–16

B.2 CONDITIONAL DIFFUSION MODELS

  • The diffusion ELBO includes a posterior gap that vanishes when the fixed forward law agrees with the model posterior; the uniform-weight denoising objective is a simplified surrogate.The exactly weighted objective, including endpoint terms, is the negative ELBO.Source: §B.2 · p. 17

C.1 IPTW LEARNER

  • The IPTW learner avoids continuation nuisances but requires consistent estimation of the treatment mechanism..Source: §C.1 · p. 17

C.2 ORACLE-RISK REPRESENTATION OF THE RA LEARNER

  • Under the stated causal assumptions and integrability, the population risk underlying the RA learner equals the oracle target generative risk..Source: §C.2 · p. 17

C.3 ORACLE-RISK REPRESENTATION OF THE IPTW LEARNER

The IPTW learner’s population risk equals the oracle target generative risk under the stated conditions. The appendix proves this equivalence and presents pseudocode for recursive g-computation.

C.3 ORACLE-RISK REPRESENTATION OF THE IPTW LEARNER

  • Under the stated causal assumptions and integrability, the population IPTW risk equals the oracle target generative risk..Source: §C.3 · p. 18
  • The proof peels cumulative weights backward using the propensity-weighting identity, tower property, recursive consistency, and sequential exchangeability..Source: §C.3 · pp. 18–19

D.1 PSEUDOCODE FOR GENERATIVE RECURSIVE G-COMPUTATION

  • The recursive g-computation pseudocode fits a final-stage generator, then works backward, training earlier generators on pseudo-outcomes sampled from frozen successor models..Source: §D.1 · p. 20

D.2 PROOF OF PROPOSITION 1

The proof derives the target CDPO by recursively propagating conditional distributions under positivity and causal assumptions. It also establishes Neyman-orthogonality for the objective under those assumptions.

D.2 PROOF OF PROPOSITION 1

  • Starting at the terminal stage, recursion, recursive consistency, iterated expectations, and sequential exchangeability establish the target CDPO at the prediction origin.Positivity ensures the relevant conditional distributions are identified on the support under consideration.Source: §D.2 · pp. 20–21

E.1 NEYMAN-ORTHOGONALITY

  • The derivative argument assumes locally admissible paths and sufficient differentiability and integrability to justify the required exchanges and differentiations..Source: §E.1 · p. 22
  • Under the causal assumptions, the objective is Neyman-orthogonal at the target model and true nuisances for every admissible nuisance perturbation..Source: §E.1 · p. 22

E.2 DOUBLE ROBUSTNESS AND QUASI-ORACLE EFFICIENCY

Under the stated causal and regularity conditions, OrthoGen has exact stagewise double robustness and an error bound separating target-model learning from higher-order nuisance error. When nuisance-error products are sufficiently small, it attains the oracle learner’s rate.

E.2 DOUBLE ROBUSTNESS AND QUASI-ORACLE EFFICIENCY

  • The population objective has exact stagewise double robustness: at each treatment step, either the propensity or continuation-density nuisance can be correct, with the choice varying by stage.Under this condition, the objective equals the one using true nuisances for any fixed target generator.Source: §E.2 · pp. 24–25
  • The theorem’s target-model regularity conditions are described as compatible with broad classes of probabilistic models, including normalizing flows and diffusion models.The interpretation notes that smoothness is needed along the selected path and curvature only in the realized direction at an intermediate model.Source: §E.2 · pp. 25–26
  • OrthoGen’s target-model error is bounded by a target-learning term plus a higher-order nuisance-error term, yielding double robustness and quasi-oracle efficiency.The bound is stated under mild target-model regularity conditions.Source: §E.2 · p. 26
  • Nuisance estimation enters through products of propensity and continuation-density errors, so faster convergence of one nuisance can offset slower convergence of the other.The exact remainder result assumes the causal assumptions, positive estimated propensities, and integrability.Source: §E.2 · pp. 24, 28
  • When the product of stagewise nuisance errors is o(ρn), nuisance estimation is asymptotically negligible and OrthoGen matches the rate of an oracle learner with known nuisances.More generally, a product error of O(ρn) does not worsen the target-model learning rate.Source: §E.2 · p. 28

F.1 FULLY SYNTHETIC DATASET

The fully synthetic benchmark tests joint distributional learning with multivariate outcomes and treatment-responsive covariates. Its treatment and outcome processes encode treatment–confounder feedback, and fitted distributions are compared with oracle distributions.

F.1 FULLY SYNTHETIC DATASET

  • The benchmark generates trajectories with a time-varying state, binary treatment, and three-dimensional continuous outcome, with treatment affecting both subsequent state and outcome.The data-generating process includes explicit treatment–confounder feedback because state and outcomes influence treatment assignment.Source: §F.1 · pp. 28–29
  • Each outcome coordinate follows a history- and treatment-dependent Gaussian/asymmetric-Laplace mixture, while a Gaussian-copula construction induces contemporaneous dependence across coordinates.The three outcome coordinates are explicitly not conditionally independent.Source: §F.1 · p. 29
  • Across horizons, the oracle fixes factual history, applies an alternating treatment sequence, and simulates future states and outcomes for comparison with fitted distributions.Evaluation compares empirical distributions using squared 2-Wasserstein transport cost on complete outcome vectors.Source: §F.1 · p. 29

F.2 SEMI-SYNTHETIC MIMIC-III DATASET

The experiments use semi-synthetic and real-world MIMIC-III data, alongside Transformer history encoding and normalizing-flow and diffusion generators. The real-world benchmark evaluates factual prediction because counterfactual ground truth is unavailable.

F.2 SEMI-SYNTHETIC MIMIC-III DATASET

  • The semi-synthetic benchmark combines MIMIC-III covariate trajectories with simulated treatments and outcomes, using patient-level splits and an all-treated regime for oracle evaluation.Previous outcomes affect later treatment assignment, while treatment, current covariates, and the previous outcome jointly determine the next outcome.Source: §F.2 · pp. 29–30

F.3 REAL-WORLD MIMIC-III DATASET

  • The real-world benchmark retains observed mechanical-ventilation treatments and diastolic-blood-pressure outcomes, evaluating factual prediction with CRPS because counterfactual ground truth is unavailable.Evaluation uses held-out patients whose observed treatment sequence matches the target sequence.Source: §F.3 · pp. 30–31

G.1 TRANSFORMER HISTORY ENCODER

  • A causal Transformer history encoder supplies context representations to propensity or generative heads, with a shared specification but separately fitted encoder-head models.The encoder uses observed pre-interventional history and is trained jointly with its head, without separate representation pretraining.Source: §G.1 · p. 31

G.2 GENERATIVE MODELS

  • Generative models receive a 20-dimensional history context; continuation models condition on the stage-specific history, while target generators condition on the prediction-origin history.The intervention is fixed for each fitted model and is not included again in the conditioning input.Source: §G.2 · p. 31

G.2.1 NORMALIZING-FLOW ARCHITECTURE

  • The normalizing-flow implementation uses a conditional spline transform, with a joint autoregressive spline for the three-dimensional synthetic outcome.For scalar MIMIC-III outcomes, a one-hidden-layer ELU network maps the history context to spline parameters.Source: §G.2.1 · p. 32

G.2.2 DIFFUSION-MODEL ARCHITECTURE

  • The diffusion model adapts its denoising map to patient history and generates samples through ancestral DDPM sampling across all reverse transitions.Training adds Gaussian noise at a uniformly sampled diffusion step and minimizes a denoising objective.Source: §G.2.2 · p. 32

G.2.3 TRAINING WITHIN THE LONGITUDINAL LEARNERS

The longitudinal learners fit continuation models backward and then train target generators, with hyperparameters selected using terminal-nuisance validation. Runtime comparisons include the complete experiment and identify diffusion instantiations as more demanding than normalizing flows.

G.2.3 TRAINING WITHIN THE LONGITUDINAL LEARNERS

  • Training fits continuation models backward to pseudo-outcomes from frozen successor models, then trains IPTW, RA, and OrthoGen target generators with their respective objectives.Pseudo-outcomes are redrawn each epoch; cumulative inverse-propensity ratios for IPTW and OrthoGen are clipped for numerical stability.Source: §G.2.3 · p. 32

G.2.4 HYPERPARAMETER SELECTION

  • Generative hyperparameters are selected separately by dataset using terminal-nuisance validation loss, without using test outcomes or counterfactual oracle samples.Normalizing flows are ranked by conditional NLL, while diffusion models are ranked by denoising loss.Source: §G.2.4 · p. 33

H COMPUTATIONAL RUNTIME

  • End-to-end runtime covers preprocessing, nuisance estimation, fitting all four learners, and evaluation; diffusion-model instantiations are more computationally demanding than normalizing flows.Reported runtimes average five replicates at horizon τ = 3.Source: H COMPUTATIONAL RUNTIME · p. 34, Table 8 · p. 34
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