Overlap-weighted orthogonal meta-learner for treatment effect estimation over time

## Abstract

Abstract Estimating heterogeneous treatment effects (HTEs) in time-varying settings is particularly challenging, as the probability of observing certain treatment sequences decreases exponentially with longer prediction horizons. Thus, the observed data contain little support for many plausible treatment sequences, which creates severe overlap problems. Existing meta-learners for the time-varying setting typically assume adequate treatment overlap, and thus suffer from exploding estimation variance when the overlap is low. To address this problem, we introduce a novel overlap-weighted orthogonal (WO) meta-learner for estimating HTEs that targets regions in the observed data with high probability of receiving the interventional treatment sequences. This offers a fully data-driven approach through which our WO-learner can counteract instabilities as in existing meta-learners and thus obtain more reliable HTE estimates. Methodologically, we develop a novel Neyman-orthogonal population risk function that minimizes the overlap-weighted oracle risk. We show that our WO-learner has the favorable property of Neyman-orthogonality, meaning that it is robust against misspecification in the nuisance functions. Further, our WO-learner is fully model-agnostic and can be applied to any machine learning model. Through extensive experiments with both transformer and LSTM backbones, we demonstrate the benefits of our novel WO-learner.

## 1 Introduction

Estimating heterogeneous treatment effects (HTEs) *over time* from patient trajectories is central for advancing personalized medicine (Allam et al., 2021; Battalio et al., 2021; Bica et al., 2021; Feuerriegel et al., 2024). Such estimates can, for example, guide treatment adaptation in chronic disease management or inform personalized intervention strategies in digital health.

Unlike standard predictive tasks, HTE estimation is inherently causal and, in time-series settings, requires adjustments for time-varying confounders (Bica et al., 2020). Without such adjustments, time-varying confounding can induce infinite-sample bias and lead to incorrect estimates. To address this, model-agnostic *meta-learners* (Frauen et al., 2025) have been proposed to provide principled strategies for causal adjustments in time-varying settings.(^1^11The term *meta-learner* refers to general estimation strategies (“recipes”) to learn causal quantities (Künzel et al., 2019), which can be instantiated with different machine learning backbones (e.g., neural networks).) Prominent examples include adaptations of the inverse propensity weighting (IPW) learner and the doubly robust (DR) learner to time-varying settings (Frauen et al., 2025).

However, existing meta-learners (Frauen et al., 2025) for estimating HTEs over time suffer from *instabilities in settings with low overlap*, which renders them inapplicable to medical scenarios. Here, *overlap* refers to the probability of receiving each of the two different treatment sequences of interest; if overlap is low, standard meta-learners have severe estimation variance (Jesson et al., 2020). This issue is especially serious in time-varying settings: with increasing prediction horizons, the probability of a treatment sequence consists of products of propensity scores (Hess & Feuerriegel, 2025), which makes the treatment overlap *decrease exponentially* (see Fig. [1](https://arxiv.org/html/2510.19643v1#S0.F1)). As a result, in low-overlap regimes, adjustment strategies based on inverse propensity weighting (IPW) will lead to extreme weights due to the product structure and, hence, division by values that are close to zero.

In this work, we propose a novel *overlap-weighted orthogonal meta-learner* (WO-learner) for estimating HTEs over time:

- 1.
Our novel WO-learner addresses low-overlap regimes with a carefully designed population risk function that *minimizes a novel, overlap-weighted oracle risk*. Hence, by focusing on high-overlap regions in the data, our WO-learner avoids extreme inverse propensity weights and unreliable response function estimates. As a result, our meta-learner provides stable HTE estimates, *even* in low-overlap regimes.
- 2.
We further ensure that our weighted population risk function is *Neyman-orthogonal* with respect to all nuisance functions. Hence, our WO-learner is robust to misspecification in the nuisance parameters, meaning that estimation errors in the nuisance functions do not propagate as first-order biases into the final HTE estimate, which is a crucial advantage over simple plug-in estimators (Hines et al., 2022; Kennedy, 2022).
- 3.
Our WO-learner is *fully model-agnostic* and can be used in combination with any machine learning backbone, such as transformers or LSTMs. Further, we derive our WO-learner for estimating *conditional average treatment effects* (CATEs). On top of that, we also extend our theory to conditional average potential outcome estimation (CAPOs); see Section [4](https://arxiv.org/html/2510.19643v1#S4).

We make three key contributions:(^2^22Code is available at [https://github.com/konstantinhess/wo_learner_timeseries](https://github.com/konstantinhess/wo_learner_timeseries).) (1) We introduce a novel overlap-weighted meta-learner for HTE estimation over time that minimizes the *overlap-weighted oracle risk*. (2) We further derive a *Neyman-orthogonal population risk* that eliminates first-order bias from the nuisance functions in the HTE estimates. (3) Through extensive experiments, we demonstrate that our WO-learner outperforms existing meta-learners, especially in settings with low overlap. In addition, we demonstrate the benefits of our meta-learner in settings where Neyman-orthogonality is crucial, such as limited sample size and complex nuisance functions.

## 2 Related work

We now review prior literature on treatment effect estimation *over time*, namely: 1 average treatment effect (ATE) estimation, 2 model-based HTE estimation, and 3 meta-learners for HTE estimation. *Our WO-learner belongs to the latter category, and this is where our primary contributions are.*

ATE vs. HTE estimation *over time*: The literature on estimating ATEs over time dates back to works in epidemiology and classical statistics (Robins, 1986; 1999; Robins et al., 2000). Examples are G-computation (Bang & Robins, 2005; Robins, 1999; Robins & Hernán, 2009), marginal structural models (Robins & Hernán, 2009; Robins et al., 2000) and structural nested models (Robins, 1994; Robins & Hernán, 2009), which belong to the broader class of so-called G-methods. More recently, targeted maximum likelihood has been adapted for the time-varying setting (van der Laan & Gruber, 2012; van der Laan & Rose, 2018). There is also some literature on model-based methods for estimating ATEs over time (Frauen et al., 2023; Shirakawa et al., 2024). Importantly, all of these works focus on *average* potential outcomes estimation and, therefore, *ignore patient heterogeneity*, because of which these works are not suitable for personalized medicine.

Limitations of model-based HTE estimation *over time*:
There has been much research on *model-based* estimation of HTEs over time (Bica et al., 2020; Hess et al., 2024a; b; Hess & Feuerriegel, 2025; Li et al., 2021; Lim et al., 2018; Melnychuk et al., 2022; Seedat et al., 2022; Wang et al., 2025). Importantly, the focus in this literature stream is primarily on how to adapt the underlying neural backbone, but not how to find the best adjustment strategy (i.e., the learning strategy to address time-varying confounding). Further, the above model-based methods are known to be *instantiations* of different meta-learners (see Frauen et al. (2025) for a discussion). Importantly, none of these model-based methods relies on either overlap-weighted or Neyman-orthogonal meta-learners. In contrast, we design a *overlap-weighted* meta-learner that is *Neyman-orthogonal* with respect to all its nuisance functions and that can be applied to *any* neural backbone.

Meta-learners for HTE estimation *over time*: Research on meta-learners for HTE estimators over time is still very limited, and we are aware of only a few works. Lewis & Syrgkanis (2021) developed a method that, however, relies on *parametric assumptions* on the data-generating process and is, therefore, *not* fully model agnostic.

Table: Table 1: Meta-learners for HTE estimation over time. Our WO-learner is the only method that adjusts for time-varying confounding, is Neyman-orthogonal with respect to all its nuisance functions, and avoids extreme weights in low-overlap regimes.

Recently, Frauen et al. (2025) formalized a suite of meta-learners for the time-varying setting, namely: (a) history adjustment (HA), (b) regression adjustment (RA), (c) inverse propensity weighting (IPW), (d) doubly-robust (DR), and an (e) inverse-variance-weighted (IVW) learner. However, they all have important *shortcomings* (Table [1](https://arxiv.org/html/2510.19643v1#S2.T1)): First, the HA is biased and does not target the correct estimand. Second, the RA, IPW, and IVW learners have plug-in bias and are not Neyman-orthogonal with respect to their estimated nuisance functions. Third, the IPW and DR learners rely on inverse propensity weighting, which can lead to extreme weights for long treatment sequences, especially in settings with low overlap. In contrast, we develop a novel WO-learner that is Neyman-orthogonal, and designed to deal with low overlap.

*Why low overlap is a non-trivial challenge for existing meta-learners:* When the interventional treatment sequences have low overlap, inverse propensity scores may lead to extreme weights. Further, small errors in estimated propensity scores lead to large errors in the constructed pseudo-outcomes and, therefore, to extreme variance for the IPW and the DR learner. Importantly, this issue is even more pronounced in the time-varying setting, where inverse propensity weighting relies on *products* of propensity scores and, thereby, the treatment propensity *decreases exponentially* with longer prediction horizons (Frauen et al., 2025; Hess et al., 2024a; Lim et al., 2018). Similar issues also arise for the response functions learned in the RA and in the DR learner, where low overlap leads to poorly learned, biased response surfaces, especially in high-dimensional covariate spaces. Finally, as the IVW learner is not Neyman-orthogonal w.r.t. its weight functions, estimation errors propagate as first-order bias through all time steps, which makes it highly unstable in low-overlap regimes.

Research gap: To the best of our knowledge, there is no meta-learner designed to counteract low-overlap regimes while being Neyman-orthogonal. As a remedy, we propose a novel *overlap-weighted*, *orthogonal* meta-learner (WO-learner) for HTE estimation over time.

## 3 Problem formulation

Setup:
Let $t\in\mathbb{N}_{0}$ be the time index. Further, let $Y_{t}\in\mathbb{R}^{d_{y}}$ the outcome variable of interest (e.g., a variable indicating the health status of a patient), $X_{t}\in\mathbb{R}^{d_{x}}$ the covariates that contain relevant patient information (including static features), and $A_{t}\in\{0,1\}^{d_{a}}$ the treatment variable. For any stochastic process $V_{t}\in\{Y_{t},X_{t},A_{t}\}$, we write $\bar{V}_{t}=V_{0:t}=(V_{0},\ldots,V_{t})$ for the history of $V_{t}$ up to time $t$. Then, let $\bar{H}_{t}=(\bar{Y}_{t-1},\bar{X}_{t},\bar{A}_{t-1})$ be the collective history observed at time step $t$, and $\bar{Z}_{t}=(\bar{Y}_{t},\bar{X}_{t},\bar{A}_{t})$ the history including the final treatment and outcome. Finally, $\tau$ is the prediction horizon. We further build upon the potential outcomes framework (Neyman, 1923; Rubin, 1978) for the time-varying setting (Robins, 1999; Robins & Hernán, 2009). Formally, let $Y_{t+\tau}[a_{t:t+\tau}]$ be the potential outcome that would have been observed under the *interventional* treatment sequence.

Estimation task:
Given a history $\bar{H}_{t}=\bar{h}_{t}$ and two interventional sequences of treatments $a_{t:t+\tau}=(a_{t},\ldots,a_{t+\tau})$ and $b_{t:t+\tau}=(b_{t},\ldots,b_{t+\tau})$, our main objective is to estimate the CATE

$$ $\displaystyle\mu_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})=\mathbb{E}\Big[Y_{t+\tau}[a_{t:t+\tau}]-Y_{t+\tau}[b_{t:t+\tau}]\mid\bar{H}_{t}=\bar{h}_{t}\Big].$ (1) $$

Later, we extend our theory to the conditional average potential outcome (CAPO), which is defined as

$$ $\displaystyle\mu_{t}^{\bar{a}}(\bar{h}_{t})=\mathbb{E}\Big[Y_{t+\tau}[a_{t:t+\tau}]\mid\bar{H}_{t}=\bar{h}_{t}\Big].$ (2) $$

Figure: Figure 2: Causal graph. Shown are the observed history, the interventional treatment sequence, and the potential outcome, along with their causal connections. When intervening on future treatments, we do not observe the future covariates (in purple), which leads to *time-varying confounding*.
Refer to caption: x2.png

Identifiability: In order to ensure identifiability from observational data, we need to make the following assumptions that are standard in the literature (Bica et al., 2020; Frauen et al., 2025; Hess et al., 2024b; Li et al., 2021; Melnychuk et al., 2022): (i) *Consistency:* Whenever the observed treatment $A_{t}$ equals the interventional treatment $a_{t}$, the observed outcome $Y_{t}$ corresponds to the potential outcome $Y_{t}[{a}_{t}]$. (ii) *Positivity:* Given a history $\bar{H}_{t}=\bar{h}_{t}$ with $\mathbb{P}(\bar{H}_{t}=\bar{h}_{t})>0$, there is non-zero probability $\mathbb{P}(A_{t}=a_{t}\mid\bar{H}_{t}=\bar{h}_{t})>0$ of receiving any treatment $A_{t}=a_{t}$. (iii) *Sequential ignorability:* Given a history $\bar{H}_{t}=\bar{h}_{t}$, the treatment assignment $A_{t}$ is independent of the potential outcome $Y_{t+\tau}[a_{t:t+\tau}]$.

Time-varying confounding: The key difficulty in estimating HTEs over time lies in *time-varying confounding* whenever $\tau>0$, i.e., when we are interested in a *sequence* of future treatments $a_{t:t+\tau}$ (see Fig. [2](https://arxiv.org/html/2510.19643v1#S3.F2)). Then, at time $t$, we intervene both on the current treatment $A_{t}$ and treatments $A_{t+\delta}$ for $0<\delta\leq\tau$ that lie several time steps in the future. However, future covariates $X_{t+\delta}$ and outcomes $Y_{t+\delta-1}$, that are *unobserved during inference time*, will confound the treatment assignment of $A_{t+\delta}$. This induces a feedback loop that needs to be accounted for (see Fig. [2](https://arxiv.org/html/2510.19643v1#S3.F2)).(^3^33In the static setting, a similar issue is known as runtime confounding, where not all confounders are observed during inference time (Coston et al., 2020).) Importantly, simply conditioning on the observed history $\bar{H}_{t}$ (i.e., a backdoor-type history-adjustment) is *biased* in this scenario. Specifically, for $\tau>0$,

$$ $\displaystyle\underbrace{{\color[rgb]{0,0.296875,0.6015625}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.296875,0.6015625}\mathbb{E}\Big[Y_{t+\tau}[a_{t:t+\tau}]\mid\bar{H}_{t}=\bar{h}_{t}\Big]}}_{\text{proper adjustment (=our method)}}{\neq}\underbrace{{\color[rgb]{0.698,0.132,0.203}\definecolor[named]{pgfstrokecolor}{rgb}{0.698,0.132,0.203}\mathbb{E}\Big[Y_{t+\tau}\mid\bar{H}_{t}=\bar{h}_{t},A_{t:t+\tau}=a_{t:t+\tau}\Big]}}_{\text{na\"{i}ve history adjustment}},$ (3) $$

which means that methods for the right-hand side (i.e., as in the HA-learner) target an incorrect estimand that is different from our causal quantity of interest. Instead, proper adjustments for time-varying confounding are required, such as in the RA, IPW, or DR learner. However, these adjustment strategies lead to poor performance when the interventional treatment overlap is low.
The inverse variance weighted adjustment in the IVW learner tries to circumvent this issue but suffers from first-order plug-in bias from its estimated weights that propagate through all timesteps. As a remedy, we develop our novel *weighted orthogonal (WO) meta-learner*.

Standard nuisance functions: To perform proper adjustments for time-varying confounding, all meta-learners rely on so-called nuisance functions; that is, functions that are not of direct interest but must be estimated accurately to enable valid estimation of the target parameter. Both existing meta-learners and, later, also our WO-learner rely on estimating response functions and/or propensity scores, which we define below.

###### Definition 3.1 (Response functions and propensity scores) .

*For interventional treatment sequences $\bar{a}=a_{t:t+\tau}$ and $\bar{b}=b_{t:t+\tau}$, let the response functions for CATE be defined as*

$$ $\displaystyle\mu_{j}^{\bar{a},\bar{b}}\left(\bar{h}_{j}\right)=\mu_{j}^{\bar{a}}\left(\bar{h}_{j}\right)-\mu_{j}^{\bar{b}}\left(\bar{h}_{j}\right),$ (4) $$

*where $\mu_{j}^{\bar{a}}(\bar{h}_{j}),\mu_{j}^{\bar{b}}(\bar{h}_{j})$ are the response functions for the CAPOs with*

$$ $\mu_{t+\tau}^{\bar{a}}\left(\bar{h}_{t+\tau}\right)=\mathbb{E}\left[Y_{t+\tau}\mid\bar{H}_{t+\tau}=\bar{h}_{t+\tau},A_{t+\tau}=a_{t+\tau}\right]$ (5) $$

*and, recursively, for $j\in\{t,\ldots,t+\tau-1\}$, *

$$ $\mu_{j}^{\bar{a}}\left(\bar{h}_{j}\right)=\mathbb{E}\left[\mu_{j+1}^{\bar{a}}(\bar{H}_{j+1})\mid\bar{H}_{j}=\bar{h}_{j},A_{j}=a_{j}\right].$ (6) $$

*Further, let the propensity scores for $j\in\{t,\ldots,t+\tau\}$ be*

$$ $\pi_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{P}(A_{j}=a_{j}\mid\bar{H}_{j}=\bar{h}_{j}).$ (7) $$

Finally, we introduce the pseudo-outcomes of the DR learner, which are a subcomponent of our weighted population risk. Here, pseudo-outcomes are variables that are estimated from nuisance functions, for which the conditional expectation equals the target causal estimand (in our case: CATE / CAPO) and, hence, enable consistent estimation.

###### Definition 3.2 (DR pseudo-outcomes (Frauen et al., 2025 ) ) .

*For interventional treatment sequences $\bar{a}=a_{t:t+\tau}$ and $\bar{b}=b_{t:t+\tau}$, let the DR pseudo-outcomes for CATE be*

$$ $\displaystyle\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\gamma_{t}^{\bar{b}}(\bar{Z}_{t+\tau}),$ (8) $$

*where the corresponding DR pseudo-outcomes for CAPO are*

$$ $\displaystyle\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}Y_{t+\tau}+\sum_{j=t}^{t+\tau}\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})}.$ (9) $$

Different from existing meta-learners, the population risk function we minimize in our WO-learner minimizes a *weighted, Neyman-orthogonal risk* to address low-overlap regimes.

###### Definition 3.1 (Response functions and propensity scores) .

###### Definition 3.2 (DR pseudo-outcomes (Frauen et al., 2025 ) ) .

## 4 Weighted orthogonal meta-learner

Figure: Figure 3: Weight functions. Our weight functions are designed to up-weight samples with large treatment overlap (CATE), or treatment propensity (CAPO). Thereby, we effectively overcome estimation variance issues in low-overlap and low-propensity regimes in a fully data-driven way.
Refer to caption: x3.png

The key idea of our WO-learner is to up-weight samples in the training data that have a higher probability of receiving the interventional treatment sequences. By up-weighting samples with *larger overlap* (in the case of CATE estimation) / *larger propensity* (in the case of CAPO estimation), we ensure that we target samples that are more relevant for estimating the HTE of interest (Figure [3](https://arxiv.org/html/2510.19643v1#S4.F3)). This requires non-trivial derivations to guarantee that our WO-learner (i) *correctly adjusts for time-varying confounding*, (ii) *minimizes the weighted oracle risk*, and (iii) is *Neyman-orthogonal* with respect to all its nuisance functions.

Below, we first introduce our novel weight functions in Section [4.1](https://arxiv.org/html/2510.19643v1#S4.SS1).
Then, we present our pseudo-outcomes for both CATE and CAPO estimation, and develop our weighted population risk; by minimizing our population risk, we adjusts for time-varying confounding while minimizing the weighted oracle risk. Finally, we ensure Neyman-orthogonality of our population risk with respect to all nuisance functions in Section [4.2](https://arxiv.org/html/2510.19643v1#S4.SS2).

### 4.1 Weighted population risk

We now introduce our weighted population risk for both CATE and CAPO estimation. We start by defining the weight functions, followed by our weighted population risk function. We then show that it is guaranteed to minimize the oracle risk ($\rightarrow$ Proposition [4.3](https://arxiv.org/html/2510.19643v1#S4.Thmtheorem3)) and, additionally, that it properly adjusts for time-varying confounding ($\rightarrow$ Corollary [4.4](https://arxiv.org/html/2510.19643v1#S4.Thmtheorem4)).

###### Definition 4.1 (Weight functions) .

Intuitively, our weight functions work as follows. For CATE, they act as *overlap weights*, i.e., they up-weight samples with a higher probability of receiving both interventional treatment sequences. Likewise, for CAPO, the weight functions correspond to *propensity weights*. That is, they seek to up-weight samples in the data that have a higher probability of receiving a single interventional treatment sequence of interest (see Fig. [3](https://arxiv.org/html/2510.19643v1#S4.F3)). Thereby, we effectively circumvent key issues of existing meta-learners that lead to highly unstable inverse propensity weights and response function estimates.

Next, we develop our weighted population risk that minimizes the oracle risk; specifically, our weighted risk function minimizes a weighted error term that up-weights samples that have a higher probability of receiving the interventional treatment sequences. Thereby, we counteract the issue in existing meta-learners (IPW, RA, DR) that suffer from poor data support when the overlap is low. For this, we first define the pseudo-outcomes for our WO-learner, which we will later need to satisfy Neyman-orthogonality in Section [4.2](https://arxiv.org/html/2510.19643v1#S4.SS2).

###### Definition 4.2 ( WO pseudo-outcomes) .

We now state our first theorem, which guarantees that our *weighted population risk minimizes the weighted oracle risk*. That is, our weighted population risk assigns the appropriate weights from Definition [4.1](https://arxiv.org/html/2510.19643v1#S4.Thmtheorem1) while correctly adjusting for time-varying confounding.

###### Theorem 4.3 (Weighted population risk) .

Let $\circ\in\{(\bar{a},\bar{b}),\bar{a}\}$ for CATE and CAPO, respectively. Then, the population risk function

$$ $\displaystyle\mathcal{L}(g;\eta^{\circ})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\circ}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\rho_{t}^{\circ}(\bar{Z}_{t+\tau})\Big({\color[rgb]{0,0.296875,0.6015625}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.296875,0.6015625}\xi_{t}^{\circ}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (16) $$

minimizes the oracle risk

$$ $\displaystyle\mathcal{L}^{*}(g;\eta^{\circ})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\circ}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\omega_{t}^{\circ}(\bar{H}_{t})\Big(\mu_{t}^{\circ}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg].$ (17) $$

###### Proof.

The proof for CAPO can be found in Supplement [A.1](https://arxiv.org/html/2510.19643v1#A1.SS1) and for CATE in Supplement [A.2](https://arxiv.org/html/2510.19643v1#A1.SS2). Therein, we derive several helping lemmas, including the result that $\mathbb{E}[\rho_{t}^{\circ}(\bar{Z}_{t+\tau})|\bar{H}_{t}]=\omega_{t}^{\circ}(\bar{H}_{t})$, which we leverage to prove our main theorem.
∎

Finally, we show that minimizing our weighted population risk guarantees that we target the correct estimand and, therefore, that our WO-learner *adjusts for time-varying confounding*.

###### Corollary 4.4 (Time-varying adjustment) .

The minimizer of the weighted population risk $\mathcal{L}(g;\eta^{\circ})$ adjusts for time-varying confounding.

###### Proof.

We leverage Theorem [4.3](https://arxiv.org/html/2510.19643v1#S4.Thmtheorem3) and notice that, since $\omega_{t}^{\circ}(\bar{H}_{t})>0$ by positivity, $\mathcal{L}^{*}(g;\eta^{\circ})$ (and, hence, $\mathcal{L}(g;\eta^{\circ})$) is minimized if and only if $g=\mu_{t}^{\circ}$, which is exactly the target estimand.
∎

Weighted orthogonal learning: We summarize the training of our WO-learner with the *empirical* weighted risk in Algorithm [1](https://arxiv.org/html/2510.19643v1#alg1). For this, we first learn the response functions $\hat{\mu}_{j}^{\bar{a}}$ and the propensity scores $\hat{\pi}_{j}^{\bar{a}}$, $j\in\{t,\ldots,t+\tau\}$, and then, using the pull-out property of expectations, the weights via

$$ $\displaystyle\hat{\omega}_{j}^{\bar{a}}(\bar{h}_{j})=\hat{\mathbb{E}}\bigg[\prod_{k=j}^{t+\tau}\hat{\pi}_{k}^{\bar{a}}(\bar{H}_{k})\;\;\Big|\;\;\bar{H}_{j}=\bar{h}_{j}\bigg]=\hat{\mathbb{E}}\bigg[\prod_{k=j+1}^{t+\tau}\hat{\pi}_{k}^{\bar{a}}(\bar{H}_{k})\;\;\Big|\;\;\bar{H}_{j}=\bar{h}_{j}\bigg]\hat{\pi}_{j}^{\bar{a}}(\bar{h}_{j}).$ (18) $$

###### Definition 4.1 (Weight functions) .

###### Definition 4.2 ( WO pseudo-outcomes) .

###### Theorem 4.3 (Weighted population risk) .

###### Proof.

###### Corollary 4.4 (Time-varying adjustment) .

###### Proof.

### 4.2 Neyman-orthogonality

Figure: Algorithm 1 WO learning

In the following, we show that our weighted population risk from Theorem [4.3](https://arxiv.org/html/2510.19643v1#S4.Thmtheorem3) is Neyman-orthogonal with respect to all its nuisance functions. This is *different* from IPW, RA, and IVW, all of which suffer from severe plug-in bias, which means that estimation errors in their estimated nuisance functions propagate as first-order biases into their final estimated pseudo-outcomes (Kennedy, 2022). Bias propagation is *even more severe in the time-varying setting*, where multiple nuisance functions are learned on top of each other, and incorrectly estimated nuisance functions in earlier stages lead to even worse bias in nuisance functions at later stages (and, hence, the final target estimate). In contrast, our WO-learner is *robust against estimation errors in its nuisance functions*; $\Rightarrow$ bias from nuisance function estimates only propagates as lower-order errors to the final HTE estimate.

If we directly minimize the oracle risk $\mathcal{L}^{*}(g;\eta^{\circ})$ as in Theorem [4.3](https://arxiv.org/html/2510.19643v1#S4.Thmtheorem3), we do not achieve Neyman-orthogonality with respect to the nuisance functions $\eta^{\circ}$. Instead, we need our tailored population risk $\mathcal{L}(g;\eta^{\circ})$.

###### Theorem 4.5 (Neyman-orthogonality) .

Let $\circ\in\{(\bar{a},\bar{b}),\bar{a}\}$ for CATE and CAPO, respectively. The weighted population risk

$$ $\displaystyle\mathcal{L}(g;\eta^{\circ})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\circ}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\rho_{t}^{\circ}(\bar{Z}_{t+\tau})\Big({\color[rgb]{0,0.296875,0.6015625}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.296875,0.6015625}\xi_{t}^{\circ}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (19) $$

is Neyman-orthogonal with respect to all nuisance functions $\eta^{\circ}$.

###### Proof.

The proof for CAPO is in Supplement [A.1](https://arxiv.org/html/2510.19643v1#A1.SS1) and for CATE in Supplement [A.2](https://arxiv.org/html/2510.19643v1#A1.SS2). Therein, we first calculate ${D_{g}\mathcal{L}(g;\eta^{\circ})[\hat{g}-g]}$, i.e., the path-wise derivative of $\mathcal{L}(g;\eta^{\circ})$ w.r.t. the target parameter $g$. Then, to establish Neyman orthogonality, we check that the cross-derivative with respect to any nuisance function vanishes, i.e., that the second-order derivative $D_{h_{j}}D_{g}\mathcal{L}(g;\eta^{\circ})[\hat{g}-g,\hat{h}_{j}^{\circ}-h_{j}^{\circ}]=0$ for all $h_{j}\in\eta^{\circ}$. Intuitively, this ensures that the influence of nuisance function estimation errors enters only at second order, which makes the score function locally robust to small perturbations (i.e., estimation errors) of $\eta^{\circ}$.
∎

Remark: For a single-step-ahead prediction $\tau=0$ (i.e., when there is no time-varying confounding as in the static setting), the R-learner (Nie & Wager, 2021) has the same overlap weights as our WO-learner for CATE. In other words, our WO-learner for CATE is a non-trivial generalization of the R-learner to the time-series setting. We show this property formally in Supplement [A.3](https://arxiv.org/html/2510.19643v1#A1.SS3).

Implementation: All meta-learners, including our WO-learner, can be implemented with any state-of-the-art neural network. Analogous to Frauen et al. (2025), our main results in Section [5](https://arxiv.org/html/2510.19643v1#S5) are with transformers (Vaswani et al., 2017) as neural backbones for both the nuisance function estimators and the second-stage estimators. To ensure a fair comparison, all nuisance estimators and second-stage regressions share the same transformer-based architecture (implementation details are in Supplement [C](https://arxiv.org/html/2510.19643v1#A3)).
Following Frauen et al. (2025), we highlight that these implementations serve as an example; the optimal model architecture depends on many different factors such as sample size or data dimensionality (Curth & van der Schaar, 2021).

###### Theorem 4.5 (Neyman-orthogonality) .

###### Proof.

## 5 Numerical experiments

In this section, we empirically evaluate our WO-learner against existing meta-learners. The purpose of our experiments is two-fold: (i) to *verify the theoretical insights* developed in Section [4](https://arxiv.org/html/2510.19643v1#S4); and (ii) to show that *our WO-learner consistently improves upon standard meta-learners* across a diverse set of settings with low overlap or where Neyman-orthogonality is crucial such as limited sample size and complex nuisance functions. Our main results are with transformer instantiations (see Supplement [C](https://arxiv.org/html/2510.19643v1#A3) for *implementation details*, *hyperparameters*, and *runtime*). We provide *additional ablations with LSTM instantiations* in Supplement [D](https://arxiv.org/html/2510.19643v1#A4). All experiments are repeated over five different seeds.

Datasets: We provide experimental results across several datasets, including synthetic, semi-synthetic, and real-world data. $\bullet$ We simulate four synthetic datasets, where we isolate different complexities for CATE estimation in the time-varying setting. $\bullet$ We then show that our WO-learner can deal with real-world covariates, and provide experiments on semi-synthetic data based on the MIMIC-III dataset (Johnson et al., 2016). Different from observational data, the advantages of both synthetic and semi-synthetic data are that we have access to the *ground-truth CATEs* and, thereby, can *correctly validate all meta-learners*. $\bullet$ We provide results on a real-world observational dataset in Supplement [E](https://arxiv.org/html/2510.19643v1#A5).

$\bullet$ Synthetic data:
In order to *isolate the different complexities in the time-varying setting*, we run experiments on different synthetic datasets $\mathcal{D}^{*}\in\{\mathcal{D}^{\gamma},\mathcal{D}^{\pi},\mathcal{D}^{\mu},\mathcal{D}^{N}\}$. Therein, we show that our WO-learner (1) benefits from its overlap weights in low-overlap regimes ($\mathcal{D}^{\gamma}$); (2) has crucial advantages over propensity based-methods when the propensity score function is complex ($\mathcal{D}^{\pi}$); (3) outperforms regression adjustments when the response function is complex ($\mathcal{D}^{\mu}$); and (4) remains robust even in low-sample settings ($\mathcal{D}^{N}$). On a high level, (1) and (2) highlight the importance of our overlap weights, while (2)–(4) show the benefits of Neyman-orthogonality. All experiments are conducted for multi-step-ahead predictions, and all datasets have time-varying confounding. Details about the data-generating processes are in Supplement [B.1](https://arxiv.org/html/2510.19643v1#A2.SS1).

**Table 2: Low-overlap regime $\mathcal{D}^{\gamma}$: Reported are the average RMSEs $\pm$ standard deviation for CATE estimation, and the relative improvement of our WO-learner over the best performing baselines across different levels of overlap (larger values of $\gamma$ correspond to lower overlap). Due to our weighted population loss, our WO-learner is highly stable even when the overlap is very low.**
| Overlap $\gamma$ | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| (a) HA | $0.17\pm 0.03$ | $0.19\pm 0.05$ | $0.20\pm 0.07$ | $0.21\pm 0.07$ | $0.23\pm 0.05$ | $0.22\pm 0.07$ | $0.24\pm 0.06$ | $0.36\pm 0.06$ | $0.22\pm 0.04$ | $0.22\pm 0.04$ | $0.25\pm 0.03$ | $0.25\pm 0.03$ | $0.25\pm 0.03$ |
| (b) RA | $0.10\pm 0.04$ | $0.11\pm 0.04$ | $0.09\pm 0.02$ | $0.11\pm 0.03$ | $0.10\pm 0.02$ | $0.10\pm 0.02$ | $0.11\pm 0.03$ | $0.12\pm 0.04$ | $0.09\pm 0.03$ | $0.11\pm 0.03$ | $0.09\pm 0.02$ | $0.10\pm 0.04$ | $0.09\pm 0.03$ |
| (c) IPW | $0.09\pm 0.02$ | $0.10\pm 0.04$ | $0.13\pm 0.04$ | $0.10\pm 0.04$ | $0.12\pm 0.05$ | $0.19\pm 0.08$ | $0.30\pm 0.20$ | $0.70\pm 0.76$ | $0.28\pm 0.15$ | $0.33\pm 0.12$ | $0.45\pm 0.29$ | $0.67\pm 0.32$ | $0.47\pm 0.54$ |
| (d) DR | $0.06\pm 0.01$ | $0.06\pm 0.01$ | $0.08\pm 0.04$ | $0.08\pm 0.01$ | $0.10\pm 0.04$ | $0.13\pm 0.05$ | $0.13\pm 0.03$ | $0.26\pm 0.22$ | $0.17\pm 0.07$ | $0.17\pm 0.10$ | $0.20\pm 0.10$ | $0.32\pm 0.15$ | $0.20\pm 0.10$ |
| (e) IVW | $0.06\pm 0.01$ | $0.05\pm 0.02$ | $0.07\pm 0.04$ | $0.07\pm 0.03$ | $0.08\pm 0.05$ | $0.12\pm 0.06$ | $0.11\pm 0.03$ | $0.62\pm 0.72$ | $0.13\pm 0.05$ | $0.17\pm 0.07$ | $0.08\pm 0.02$ | $0.15\pm 0.05$ | $0.16\pm 0.05$ |
| $(*)$ WO (*ours*) | $\mathbf{0.03\pm 0.01}$ | $\mathbf{0.02\pm 0.01}$ | $\mathbf{0.04\pm 0.02}$ | $\mathbf{0.05\pm 0.02}$ | $\mathbf{0.07\pm 0.04}$ | $\mathbf{0.07\pm 0.03}$ | $\mathbf{0.08\pm 0.03}$ | $\mathbf{0.10\pm 0.07}$ | $\mathbf{0.07\pm 0.03}$ | $\mathbf{0.05\pm 0.02}$ | $\mathbf{0.06\pm 0.02}$ | $\mathbf{0.08\pm 0.02}$ | $\mathbf{0.08\pm 0.02}$ |
| Rel. improv. (%) | $54.4\%$ | $58.4\%$ | $39.7\%$ | $21.0\%$ | $6.5\%$ | $25.9\%$ | $27.6\%$ | $13.6\%$ | $23.1\%$ | $50.2\%$ | $26.8\%$ | $16.2\%$ | $13.5\%$ |

Figure: Figure 4: Complex propensity score $\mathcal{D}^{\pi}$: Reported are the average RMSEs for CATE estimation. Our WO-learner significantly improves upon existing meta-learners, even for large prediction horizons.
Refer to caption: x4.png

(1) Low-overlap regime $\mathcal{D}^{\gamma}$: We control the overlap in the data generating process with an overlap parameter $\gamma$. By increasing $\gamma$, we decrease the overlap in the observed data, i.e., the probability of receiving both interventional treatment sequences ${a}_{t:t+\tau}$ and ${b}_{t:t+\tau}$.

Results: Table [2](https://arxiv.org/html/2510.19643v1#S5.T2) shows that our WO-learner *outperforms all existing meta-learners over all levels of overlap*. This confirms the *effectiveness of our overlap weights*. We further find that the RA-learner performs fairly stable, as it avoids inverse propensity weights that become increasingly more unstable in low-overlap regimes. Nonetheless, our WO-learner consistently outperforms the best baseline with a relative improvement of up to $58.4\%$.

(2) Complex propensity score $\mathcal{D}^{\pi}$: We simulate data with a complex propensity score. We further increase the prediction horizon $\tau$, and study the performance of meta-learners that are based on the propensity score. Thereby, we gain insight into how errors propagate over time, and how the meta-learners behave under exponentially decreasing overlap for increasing prediction horizons.

Results: In Figure [4](https://arxiv.org/html/2510.19643v1#S5.F4), we can clearly see the benefits of our WO-learner over propensity-based baselines due to its *overlap weights*, and its *Neyman-orthogonal population risk function*. Our WO-learner is robust against estimation errors in the estimated *propensity scores and weight functions*, and errors do not propagate as for the IPW and the IVW learner. Further, the performance deteriorates for the IPW and DR learners is due to the exponentially decreasing overlap for increasing prediction horizons $\tau$. In contrast, our WO-learner remains highly stable.

Figure: Figure 5: Complex response function $\mathcal{D}^{\mu}$: Reported are the average RMSEs for CATE estimation for increasing dimensionality of the covariate space. Our WO-learner clearly outperforms the non-orthogonal RA learner.
Refer to caption: x5.png

(3) Complex response function $\mathcal{D}^{\mu}$: In this dataset, we simulate a complex response function, and further vary the complexity by increasing the dimension of the covariate space. The purpose of this experimental setup is to empirically validate that our WO-learner is Neyman-orthogonal with respect to its estimated response functions. Hence, we compare it with the RA learner, which solely relies on estimating the response functions and is not Neyman-orthogonal.

Results: Figure [5](https://arxiv.org/html/2510.19643v1#S5.F5) shows the results for the complex response function. As our WO-learner is also *Neyman-orthogonal with respect to its estimated response functions*, we can clearly see that we outperform a simple RA-learner. As the response function becomes more complex with increasing dimension of the covariate space, our WO-learner stays largely unaffected.

**Table 3: Low-sample setting $\mathcal{D}^{N}$: Reported are the average RMSEs $\pm$ standard deviation for CATE estimation, and the relative improvement of our WO-learner with decreasing sample size. Due to Neyman-orthogonality, our WO-learner remains stable across all sample sizes.**
| $n_{\text{train}}$ | $8,000$ | $7,000$ | $6,000$ | $5,000$ | $4,000$ | $3,000$ | $2,000$ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| (a) HA | $0.33\pm 0.05$ | $0.38\pm 0.05$ | $0.39\pm 0.06$ | $0.42\pm 0.03$ | $0.50\pm 0.04$ | $0.49\pm 0.04$ | $0.58\pm 0.10$ |
| (b) RA | $0.15\pm 0.02$ | $0.16\pm 0.03$ | $0.14\pm 0.02$ | $0.15\pm 0.04$ | $0.23\pm 0.05$ | $0.20\pm 0.02$ | $0.30\pm 0.10$ |
| (c) IPW | $0.13\pm 0.04$ | $0.14\pm 0.07$ | $0.16\pm 0.04$ | $0.17\pm 0.06$ | $0.24\pm 0.09$ | $0.58\pm 0.35$ | $0.41\pm 0.26$ |
| (d) DR | $0.11\pm 0.04$ | $0.18\pm 0.08$ | $0.13\pm 0.03$ | $0.17\pm 0.03$ | $0.20\pm 0.04$ | $0.52\pm 0.42$ | $0.26\pm 0.10$ |
| (e) IVW | $0.10\pm 0.03$ | $0.13\pm 0.05$ | $0.13\pm 0.05$ | $0.13\pm 0.05$ | $0.13\pm 0.05$ | $0.19\pm 0.04$ | $0.30\pm 0.11$ |
| $(*)$ WO (*ours*) | $\mathbf{0.06\pm 0.02}$ | $\mathbf{0.04\pm 0.01}$ | $\mathbf{0.06\pm 0.03}$ | $\mathbf{0.04\pm 0.01}$ | $\mathbf{0.09\pm 0.04}$ | $\mathbf{0.13\pm 0.06}$ | $\mathbf{0.18\pm 0.07}$ |
| Rel. improv. (%) | $37.4\%$ | $65.8\%$ | $49.4\%$ | $66.9\%$ | $27.7\%$ | $30.6\%$ | $28.7\%$ |

(4) Low-sample setting $\mathcal{D}^{N}$: We study the performance in settings with low sample size. By decreasing the number of samples for training, we demonstrate the importance of *Neyman-orthogonality with respect to all estimated nuisance functions*. That is, we show empirically that increasing errors in the nuisance function estimates due to decreasing sample size only propagate as lower-order errors to the CATE estimates of our WO-learner.

Results: Table [3](https://arxiv.org/html/2510.19643v1#S5.T3) shows that, for decreasing sample size, our WO-learner maintains a stable performance, even as errors in its estimated nuisance functions increase. This is because our learner is Neyman-orthogonal with respect to all nuisance functions, and, thereby, errors do not propagate as first-order biases through all time steps up to the final CATE estimate.

**Table 4: Semi-synthetic data: Reported are the average RMSEs for CATE estimation. With all complexities in the time-varying setting combined, our WO-learner is the only meta-learner with consistent performance over all prediction horizons.**
| Prediction horizon | $2$ | $3$ | $4$ |
| --- | --- | --- | --- |
| (a) HA | $0.11\pm 0.00$ | ${0.19\pm 0.02}$ | ${0.24\pm 0.03}$ |
| (b) RA | $0.12\pm 0.01$ | $0.27\pm 0.04$ | $0.40\pm 0.00$ |
| (c) IPW | $0.66\pm 0.50$ | $0.94\pm 0.62$ | $5.23\pm 6.55$ |
| (d) DR | ${0.04\pm 0.02}$ | $0.20\pm 0.11$ | $1.57\pm 1.79$ |
| (e) IVW | $0.04\pm 0.02$ | $31.27\pm 44.02$ | $879.80\pm 1243.54$ |
| $(*)$ WO (*ours*) | $\mathbf{0.03\pm 0.01}$ | $\mathbf{0.15\pm 0.01}$ | $\mathbf{0.17\pm 0.07}$ |
| Rel. improv. (%) | $23.1\%$ | $22.4\%$ | $28.5\%$ |

$\bullet$ Semi-synthetic data: We analyze the performance of all meta-learners for increasing prediction horizons on semi-synthetic data. For this, we simulate treatments and outcomes based on real-world patient covariates of the MIMIC-III dataset (Johnson et al., 2016). Therein, we (i) show that our WO-learner can easily handle the complexities of observational covariates, while we (ii) ensure that we have access to ground-truth values for CATE in order to properly validate our results. We provide details on the data setup in Supplement [B.2](https://arxiv.org/html/2510.19643v1#A2.SS2). In short, the data combines all the difficulties from our synthetic experiments: the data has *low overlap*, a *complex propensity score*, a *complex response function*, *low sample size*, and *time-varying confounding*.

Results: Table [4](https://arxiv.org/html/2510.19643v1#S5.T4) shows the results for semi-synthetic data. Our WO-learner is the only method that remains stable for all prediction horizons. This confirms that our learner can deal with all the aforementioned difficulties in the time-varying setting, and clearly outperforms existing meta-learners.

$\bullet$ Real-world data: We report results for real-world outcome estimation in Supplement [E](https://arxiv.org/html/2510.19643v1#A5).

Conclusion: In this work, we present a novel weighted orthogonal meta-leaner for HTE estimation over time.
Our experiments confirm that our WO-learner presents an important step towards reliable decision-making for domains such as personalized medicine.

## References

- Allam et al. (2021)
Ahmed Allam, Stefan Feuerriegel, Michael Rebhan, and Michael Krauthammer.
Analyzing patient trajectories with artificial intelligence.
*Journal of Medical Internet Research*, 23(12):e29812, 2021.
- Bang & Robins (2005)
Heejung Bang and James M. Robins.
Doubly robust estimation in missing data and causal inference models.
*Biometrics*, 61(4):962–973, 2005.
- Battalio et al. (2021)
Samuel L. Battalio, David E. Conroy, Walter Dempsey, Peng Liao, Marianne Menictas, Susan Murphy, Inbal Nahum-Shani, Tianchen Qian, Santosh Kumar, and Bonnie Spring.
Sense2Stop: A micro-randomized trial using wearable sensors to optimize a just-in-time-adaptive stress management intervention for smoking relapse prevention.
*Contemporary Clinical Trials*, 109:106534, 2021.
- Bica et al. (2020)
Ioana Bica, Ahmed M. Alaa, James Jordon, and Mihaela van der Schaar.
Estimating counterfactual treatment outcomes over time through adversarially balanced representations.
In *ICLR*, 2020.
- Bica et al. (2021)
Ioana Bica, Ahmed M. Alaa, Craig Lambert, and Mihaela van der Schaar.
From real-world patient data to individualized treatment effects using machine learning: Current and future methods to address underlying challenges.
*Clinical Pharmacology and Therapeutics*, 109(1):87–100, 2021.
- Chernozhukov et al. (2024)
Victor Chernozhukov, Christian Hansen, Nathan Kallus, Martin Spindler, and Vasilis Syrgkanis.
*Applied Causal Inference Powered by ML and AI*.
CausalML-book.org, 2024.
- Coston et al. (2020)
Amanda Coston, Edward H. Kennedy, and Alexandra Chouldechova.
Counterfactual predictions under runtime confounding.
In *NeurIPS*, 2020.
- Curth & van der Schaar (2021)
Alicia Curth and Mihaela van der Schaar.
Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms.
In *AISTATS*, 2021.
- Feuerriegel et al. (2024)
Stefan Feuerriegel, Dennis Frauen, Valentyn Melnychuk, Jonas Schweisthal, Konstantin Hess, Alicia Curth, Stefan Bauer, Niki Kilbertus, Isaac S. Kohane, and Mihaela van der Schaar.
Causal machine learning for predicting treatment outcomes.
*Nature Medicine*, 2024.
- Fisher (2024)
Aaron Fisher.
The connection between r-learning and inverse-variance weighting for estimation of heterogeneous treatment effects.
In *ICML*, 2024.
- Frauen et al. (2023)
Dennis Frauen, Tobias Hatt, Valentyn Melnychuk, and Stefan Feuerriegel.
Estimating average causal effects from patient trajectories.
In *AAAI*, 2023.
- Frauen et al. (2025)
Dennis Frauen, Konstantin Hess, and Stefan Feuerriegel.
Model-agnostic meta-learners for estimating heterogeneous treatment effects over time.
In *ICLR*, 2025.
- Hess & Feuerriegel (2025)
Konstantin Hess and Stefan Feuerriegel.
Stabilized neural prediction of potential outcomes in continuous time.
In *ICLR*, 2025.
- Hess et al. (2024a)
Konstantin Hess, Dennis Frauen, Valentyn Melnychuk, and Stefan Feuerriegel.
G-transformer for conditional average potential outcome estimation over time.
*arXiv preprint*, arXiv:2405.21012, 2024a.
- Hess et al. (2024b)
Konstantin Hess, Valentyn Melnychuk, Dennis Frauen, and Stefan Feuerriegel.
Bayesian neural controlled differential equations for treatment effect estimation.
In *ICLR*, 2024b.
- Hines et al. (2022)
Oliver Hines, Oliver Dukes, Karla Diaz-Ordaz, and Stijn Vansteelandt.
Demystifying statistical learning based on efficient influence functions.
*The American Statistician*, 76(3):292–304, 2022.
- Hochreiter & Schmidhuber (1997)
Sepp Hochreiter and Jürgen Schmidhuber.
Long short-term memory.
*Neural Computation*, 9(8):1735–1780, 1997.
- Jesson et al. (2020)
Andrew Jesson, Sören Mindermann, Uri Shalit, and Yarin Gal.
Identifying causal effect inference failure with uncertainty-aware models.
In *NeurIPS*, 2020.
- Johnson et al. (2016)
Alistair E. W. Johnson, Tom J. Pollard, Lu Shen, Li-wei H. Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G. Mark.
MIMIC-III, a freely accessible critical care database.
*Scientific Data*, 3(1):160035, 2016.
- Kennedy (2022)
Edward H. Kennedy.
Semiparametric doubly robust targeted double machine learning: A review.
*arXiv preprint*, 2203.06469, 2022.
- Künzel et al. (2019)
Sören R. Künzel, Jasjeet S. Sekhon, Peter J. Bickel, and Bin Yu.
Metalearners for estimating heterogeneous treatment effects using machine learning.
*Proceedings of the National Academy of Sciences (PNAS)*, 116(10):4156–4165, 2019.
- Lewis & Syrgkanis (2021)
Greg Lewis and Vasilis Syrgkanis.
Double/debiased machine learning for dynamic treatment effects via g-estimation.
In *NeurIPS*, 2021.
- Li et al. (2021)
Rui Li, Stephanie Hu, Mingyu Lu, Yuria Utsumi, Prithwish Chakraborty, Daby M. Sow, Piyush Madan, Jun Li, Mohamed Ghalwash, Zach Shahn, and Li-wei Lehman.
G-net: A recurrent network approach to g-computation for counterfactual prediction under a dynamic treatment regime.
In *ML4H*, 2021.
- Lim et al. (2018)
Bryan Lim, Ahmed M. Alaa, and Mihaela van der Schaar.
Forecasting treatment responses over time using recurrent marginal structural networks.
In *NeurIPS*, 2018.
- Luedtke (2024)
Alex Luedtke.
Simplifying debiased inference via automatic differentiation and probabilistic programming.
*arXiv preprint*, 2405.08675, 2024.
- Melnychuk et al. (2022)
Valentyn Melnychuk, Dennis Frauen, and Stefan Feuerriegel.
Causal transformer for estimating counterfactual outcomes.
In *ICML*, 2022.
- Morzywolek et al. (2023)
Pawel Morzywolek, Johan Decruyenaere, and Stijn Vansteelandt.
On a general class of orthogonal learners for the estimation of heterogeneous treatment effects.
*arXiv preprint*, arXiv:2303.12687, 2023.
- Neyman (1923)
Jerzy Neyman.
On the application of probability theory to agricultural experiments.
*Annals of Agricultural Sciences*, 10:1–51, 1923.
- Nie & Wager (2021)
Xinkun Nie and Stefan Wager.
Quasi-oracle estimation of heterogeneous treatment effects.
*Biometrika*, 108(2):299–319, 2021.
- Robins (1986)
James M. Robins.
A new approach to causal inference in mortality studies with a sustained exposure period: Application to control of the healthy worker survivor effect.
*Mathematical Modelling*, 7:1393–1512, 1986.
- Robins (1994)
James M. Robins.
Correcting for non-compliance in randomized trials using structural nested mean models.
*Communications in Statistics - Theory and Methods*, 23(8):2379–2412, 1994.
- Robins (1999)
James M. Robins.
Robust estimation in sequentially ignorable missing data and causal inference models.
*Proceedings of the American Statistical Association on Bayesian Statistical Science*, pp. 6–10, 1999.
- Robins & Hernán (2009)
James M. Robins and Miguel A. Hernán.
*Estimation of the causal effects of time-varying exposures*.
Chapman & Hall/CRC handbooks of modern statistical methods. CRC Press, Boca Raton, 2009.
ISBN 9781584886587.
- Robins et al. (2000)
James M. Robins, Miguel A. Hernán, and Babette Brumback.
Marginal structural models and causal inference in epidemiology.
*Epidemiology*, 11(5):550–560, 2000.
- Rubin (1978)
Donald B. Rubin.
Bayesian inference for causal effects: The role of randomization.
*Annals of Statistics*, 6(1):34–58, 1978.
- Seedat et al. (2022)
Nabeel Seedat, Fergus Imrie, Alexis Bellot, Zhaozhi Qian, and Mihaela van der Schaar.
Continuous-time modeling of counterfactual outcomes using neural controlled differential equations.
In *ICML*, 2022.
- Shirakawa et al. (2024)
Yi Shirakawa, Toru; Li, Yulun Wu, Sky Qiu, Yuxuan Li, Mingduo Zhao, Hiroyasu Iso, and Mark van der Laan.
Longitudinal targeted minimum loss-based estimation with temporal-difference heterogeneous transformer.
In *ICML*, 2024.
- van der Laan & Gruber (2012)
Mark J. van der Laan and Susan Gruber.
Targeted minimum loss based estimation of causal effects of multiple time point interventions.
*The International Journal of Biostatistics*, 8(1), 2012.
- van der Laan & Rose (2018)
Mark J. van der Laan and Sherri Rose.
*Targeted learning in data science*.
Springer, Cham, 2018.
ISBN 978-3-319-65303-7.
- Vaswani et al. (2017)
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin.
Attention is all you need.
In *NeurIPS*, 2017.
- Wang et al. (2020)
Shirly Wang, Matthew B.A. McDermott, Geeticka Chauhan, Marzyeh Ghassemi, Michael C. Hughes, and Tristan Naumann.
MIMIC-extract: A data extraction, preprocessing, and representation pipeline for MIMIC-III.
In *CHIL*, 2020.
- Wang et al. (2025)
Xin Wang, Shengfei Lyu, Chi Luo, Xiren Zhou, and Huanhuan Chen.
Variational counterfactual intervention planning to achieve target outcomes.
In *ICML*, 2025.

## Appendix A Proofs

### A.1 Conditional average potential outcomes (CAPOs)

We split the following section into two parts: first, we derive several supporting lemmas to prove our main results ($\rightarrow$ Lemma [A.1](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem1) to [A.4](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem4)). Then, we derive the theorem from the main paper ($\rightarrow$ Theorem [A.5](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem5) and Theorem [A.6](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem6)).

#### A.1.1 Lemmas (CAPOs)

In order to prove our main theorems for CAPOs, we first introduce a series of helping lemmas.

###### Lemma A.1 .

Let

$$ $\displaystyle\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}Y_{t+\tau}+\sum_{j=t}^{t+\tau}\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})},$ (20) $$

where

$$ $\displaystyle\pi_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{P}(A_{j}=a_{j}|\bar{H}_{j}=\bar{h}_{j})$ (21) $$

is the propensity score, and

$$ $\displaystyle\mu_{t+\tau}^{\bar{a}}(\bar{h}_{t+\tau})=\mathbb{E}\Big[Y_{t:t+\tau}\;\Big|\;\bar{H}_{t+\tau}=\bar{h}_{t+\tau},A_{t+\tau}=a_{t+\tau}\Big]$ (22) $$

and

$$ $\displaystyle\mu_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{E}\Big[\mu_{j+1}^{\bar{a}}(\bar{h}_{j+1})\;\Big|\;\bar{H}_{j}=\bar{h}_{j},A_{j}=a_{j}\Big],$ (23) $$

such that

$$ $\displaystyle\mu_{t}^{\bar{a}}(\bar{h}_{t})=\mathbb{E}\Big[Y_{t:t+\tau}[a_{t:t+\tau}]\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]$ (24) $$

is the conditional average potential outcome. Then, $\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})$ is Neyman-orthogonal with respect to all nuisance functions $\eta^{\bar{a}}=\{\pi_{j}^{\bar{a}},\mu_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}$.

###### Proof.

First, $\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})$ is trivially Neyman-orthogonal with respect to $\omega_{j}^{\bar{a}}$ as it is independent of it. Second, we notice that $\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})$ is the uncentered efficient influence function of $\mathbb{E}\Big[\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big]$ (Frauen et al., 2025; van der Laan & Gruber, 2012), which is Neyman-orthogonal with respect to all nuisance functions.
∎

###### Lemma A.2 .

Let

$$ $\displaystyle\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k}),$ (25) $$

where

$$ $\displaystyle\pi_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{P}(A_{j}=a_{j}|\bar{H}_{j}=\bar{h}_{j})$ (26) $$

is the propensity score, and

$$ $\displaystyle\omega_{j}^{\bar{a}}(\bar{h}_{\ell})=p(A_{j:t+\tau}=a_{j:t+\tau}\mid\bar{H}_{\ell}=\bar{h}_{\ell})=\mathbb{E}\Big[\prod_{k=j}^{t+\tau}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{\ell}=\bar{h}_{\ell}\Big]$ (27) $$

is the weight function. Then, $\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})$ is Neyman-orthogonal with respect to all nuisance functions $\eta^{\bar{a}}=\{\pi_{j}^{\bar{a}},\mu_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}$.

###### Proof.

First, $\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})$ is trivially Neyman-orthogonal with respect to $\mu_{j}^{\bar{a}}$ as it is independent of it. Second, we show that $\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})$ is the uncentered efficient influence function of $\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]$, and hence, Neyman-orthogonal with respect to the nuisance functions $\{\pi_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}$. For this, we make use of the chain rule for pathwise derivatives (Kennedy, 2022; Luedtke, 2024). First, we compute the efficient influence function of $\omega_{t}^{\bar{a}}(\bar{h}_{t})$ via

$$ $\displaystyle\mathbbm{IF}\Big(\omega_{t}^{\bar{a}}(\bar{h}_{t})\Big)$ (28) $\displaystyle=$ $\displaystyle\mathbbm{IF}\Big(\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]\Big)$ (29) $\displaystyle=$ $\displaystyle\mathbbm{IF}\Big(\sum_{{h}_{t+1:t+\tau}}p(h_{t+1:t+\tau}\mid\bar{h}_{t})\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)$ (30) $\displaystyle=$ $\displaystyle\sum_{{h}_{t+1:t+\tau}}\Big[\underbrace{\mathbbm{IF}\Big(p(h_{t+1:t+\tau}\mid\bar{h}_{t})\Big)\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})}_{(*)}+\underbrace{p(h_{t+1:t+\tau}\mid\bar{h}_{t})\mathbbm{IF}\Big(\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)}_{(**)}\Big].$ (31) $$

For $(*)$, we have that

$$ $\displaystyle\mathbbm{IF}\Big(p(h_{t+1:t+\tau}\mid\bar{h}_{t})\Big)\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})$ (32) $\displaystyle=$ $\displaystyle\mathbbm{IF}\Big(p(\bar{h}_{t+\tau})/p(\bar{h}_{t})\Big)\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})$ (33) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\Big(\mathbbm{1}_{\{{h}_{t+1:t+\tau}={H}_{t+1:t+\tau}\}}-p({h}_{t+1:t+\tau}|\bar{h}_{t})\Big)\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j}).$ (34) $$

Further, we obtain $(**)$ via

$$ $\displaystyle p(h_{t+1:t+\tau}\mid\bar{h}_{t})\mathbbm{IF}\Big(\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)$ (35) $\displaystyle=$ $\displaystyle p(h_{t+1:t+\tau}\mid\bar{h}_{t})\sum_{j=t}^{t+\tau}\mathbbm{IF}\Big(\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)\prod_{k\neq j}\pi_{k}^{\bar{a}}(\bar{h}_{k})$ (36) $\displaystyle=$ $\displaystyle p(h_{t+1:t+\tau}\mid\bar{h}_{t})\sum_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{\bar{h}_{j}=\bar{H}_{j}\}}}{p(\bar{h}_{j})}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)\prod_{k\neq j}\pi_{k}^{\bar{a}}(\bar{h}_{k})$ (37) $\displaystyle=$ $\displaystyle p(h_{t+1:t+\tau}\mid\bar{h}_{t})\sum_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}\mathbbm{1}_{\{{h}_{t+1:j}={H}_{t+1:j}\}}}{p(\bar{h}_{j})}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)\prod_{k\neq j}\pi_{k}^{\bar{a}}(\bar{h}_{k})$ (38) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\sum_{j=t}^{t+\tau}\mathbbm{1}_{\{{h}_{t+1:j}={H}_{t+1:j}\}}p(h_{j+1:t+\tau}\mid\bar{h}_{j})\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)\prod_{k\neq j}\pi_{k}^{\bar{a}}(\bar{h}_{k})$ (39) $$

Combining both results, the efficient influence function of $\omega_{t}^{\bar{a}}(\bar{h}_{t})$ is given by

$$ $\displaystyle\mathbbm{IF}\Big(\omega_{t}^{\bar{a}}(\bar{h}_{t})\Big)$ (40) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\sum_{h_{t+1:t+\tau}}\Bigg[\Big(\mathbbm{1}_{\{{h}_{t+1:t+\tau}={H}_{t+1:t+\tau}\}}-p({h}_{t+1:t+\tau}|\bar{h}_{t})\Big)\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})$ (41) $\displaystyle+\sum_{j=t}^{t+\tau}\mathbbm{1}_{\{{h}_{t+1:j}={H}_{t+1:j}\}}p(h_{j+1:t+\tau}\mid\bar{h}_{j})\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)\prod_{k\neq j}\pi_{k}^{\bar{a}}(\bar{h}_{k})\Bigg]$ (42) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\sum_{h_{t+1:t+\tau}}\Bigg[\Big(\mathbbm{1}_{\{{h}_{t+1:t+\tau}={H}_{t+1:t+\tau}\}}-p({h}_{t+1:t+\tau}|\bar{h}_{t})\Big)\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{h}_{j})$ (43) $\displaystyle+\sum_{j=t}^{t+\tau}\mathbbm{1}_{\{{h}_{t+1:j}={H}_{t+1:j}\}}p(h_{j+1:t+\tau}\mid\bar{h}_{j})\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{h}_{j})\Big)$ (44) $\displaystyle\quad\times\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{h}_{k})\prod_{k>j}\pi_{k}^{\bar{a}}(\bar{h}_{k})\Bigg]$ (45) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\Bigg[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(H_{t+1:j},\bar{h}_{t})-\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]$ (46) $\displaystyle+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(H_{t+1:j},\bar{h}_{t})\Big)$ (47) $\displaystyle\quad\times\prod_{t\leq k<j}\pi_{k}^{\bar{a}}({H}_{t+1:k},\bar{h}_{t})\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;H_{t+1:j},\bar{H}_{t}=\bar{h}_{t}\Big]\Bigg]$ (48) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\Bigg[-\omega_{t}^{\bar{a}}(\bar{h}_{t})+\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(H_{t+1:j},\bar{h}_{t})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(H_{t+1:j},\bar{h}_{t})\Big)$ (49) $\displaystyle\quad\times\prod_{t\leq k<j}\pi_{k}^{\bar{a}}({H}_{t+1:k},\bar{h}_{t})\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;H_{t+1:j},\bar{H}_{t}=\bar{h}_{t}\Big]\Bigg].$ (50) $$

Using this result, we derive the efficient influence function of $\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]$ via

$$ $\displaystyle\mathbbm{IF}\Big(\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]\Big)$ (51) $\displaystyle=$ $\displaystyle\sum_{\bar{h}_{t}}\mathbbm{IF}\Big(p(\bar{h}_{t})\omega_{t}^{\bar{a}}(\bar{h}_{t})\Big)$ (52) $\displaystyle=$ $\displaystyle\sum_{\bar{h}_{t}}\Big[\mathbbm{IF}\Big(p(\bar{h}_{t})\Big)\omega_{t}^{\bar{a}}(\bar{h}_{t})+p(\bar{h}_{t})\mathbbm{IF}\Big(\omega_{t}^{\bar{a}}(\bar{h}_{t})\Big)\Big]$ (53) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})-\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]-\omega_{t}^{\bar{a}}(\bar{H}_{t})$ (54) $\displaystyle+\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\underbrace{\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{j}\Big]}_{=\omega_{j+1}^{\bar{a}}(\bar{H}_{j})}$ (55) $\displaystyle=$ $\displaystyle-\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]+\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau}),$ (56) $$

which concludes the proof.
∎

###### Lemma A.3 .

Let

$$ $\displaystyle\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}Y_{t+\tau}+\sum_{j=t}^{t+\tau}\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})},$ (57) $$

where

$$ $\displaystyle\pi_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{P}(A_{j}=a_{j}\mid\bar{H}_{j}=\bar{h}_{j})$ (58) $$

is the propensity score, and

$$ $\displaystyle\mu_{t+\tau}^{\bar{a}}(\bar{h}_{t+\tau})=\mathbb{E}\Big[Y_{t:t+\tau}\;\Big|\;\bar{H}_{t+\tau}=\bar{h}_{t+\tau},A_{t+\tau}=a_{t+\tau}\Big]$ (59) $$

and

$$ $\displaystyle\mu_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{E}\Big[\mu_{j+1}^{\bar{a}}(\bar{h}_{j+1})\;\Big|\;\bar{H}_{j}=\bar{h}_{j},A_{j}=a_{j}\Big],$ (60) $$

such that

$$ $\displaystyle\mu_{t}^{\bar{a}}(\bar{h}_{t})=\mathbb{E}\Big[Y_{t:t+\tau}[a_{t:t+\tau}]\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]$ (61) $$

is the conditional average potential outcome. Then, it holds that

$$ $\displaystyle\mathbb{E}\Big[\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]=\mu_{t}^{\bar{a}}(\bar{H}_{t}).$ (62) $$

###### Proof.

$$ $\displaystyle\mathbb{E}\Big[\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]$ (63) $\displaystyle=$ $\displaystyle\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}Y_{t+\tau}+\sum_{j=t}^{t+\tau}\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})}\;\Big|\;\bar{H}_{t}\Big]$ (64) $\displaystyle=$ $\displaystyle\underbrace{\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}Y_{t+\tau}\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a}}(\bar{H}_{t})}$ (65) $\displaystyle+\sum_{j=t}^{t+\tau}\mathbb{E}\Bigg[\mathbb{E}\Big[\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})}\;\Big|\;\bar{H}_{j}\Big]\;\Big|\;\bar{H}_{t}\Bigg]$ (66) $\displaystyle=$ $\displaystyle\mu_{t}^{\bar{a}}(\bar{H}_{t})+\sum_{j=t}^{t+\tau}\mathbb{E}\Bigg[\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbb{E}[\mathbbm{1}_{\{A_{j}=a_{j}\}}|\bar{H}_{j}]}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})}\;\Big|\;\bar{H}_{t}\Bigg]$ (67) $\displaystyle=$ $\displaystyle\mu_{t}^{\bar{a}}(\bar{H}_{t})+\sum_{j=t}^{t+\tau}\mathbb{E}\Bigg[\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\pi_{j}^{\bar{a}}(\bar{H}_{j})}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})}\;\Big|\;\bar{H}_{t}\Bigg]$ (68) $\displaystyle=$ $\displaystyle\mu_{t}^{\bar{a}}(\bar{H}_{t}).$ (69) $$

The DR-pseudo outcomes, which are a sub-component of our pseudo-outcomes, have been introduced in Frauen et al. (2025); van der Laan & Gruber (2012).
∎

###### Lemma A.4 .

Let

$$ $\displaystyle\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k}),$ (70) $$

where

$$ $\displaystyle\pi_{j}^{\bar{a}}(\bar{h}_{j})=\mathbb{P}(A_{j}=a_{j}|\bar{H}_{j}=\bar{h}_{j})$ (71) $$

is the propensity score, and

$$ $\displaystyle\omega_{j}^{\bar{a}}(\bar{h}_{\ell})=p(A_{j:t+\tau}=a_{j:t+\tau}\mid\bar{H}_{\ell}=\bar{h}_{\ell})=\mathbb{E}\Big[\prod_{k=j}^{t+\tau}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{\ell}=\bar{h}_{\ell}\Big]$ (72) $$

is the weight function. Then, it holds that

$$ $\displaystyle\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]=\omega_{t}^{\bar{a}}(\bar{H}_{t}).$ (73) $$

###### Proof.

$$ $\displaystyle\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]$ (74) $\displaystyle=$ $\displaystyle\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{t}\Big]$ (75) $\displaystyle=$ $\displaystyle\underbrace{\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})\;\Big|\;\bar{H}_{t}\Big]}_{=\omega_{t}^{\bar{a}}(\bar{H}_{t})}+\mathbb{E}\Big[\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{t}\Big]$ (76) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})+\sum_{j=t}^{t+\tau}\mathbb{E}\Bigg[\mathbb{E}\Big[\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{j}\Big]\;\Big|\;\bar{H}_{t}\Bigg]$ (77) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})+\sum_{j=t}^{t+\tau}\mathbb{E}\Bigg[\Big(\mathbb{E}\Big[\mathbbm{1}_{\{a_{j}=A_{j}\}}\;\Big|\;\bar{H}_{j}\Big]-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{t}\Bigg]$ (78) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})+\sum_{j=t}^{t+\tau}\mathbb{E}\Bigg[\Big(\pi_{j}^{\bar{a}}(\bar{H}_{j})-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{t}\Bigg]$ (79) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t}).$ (80) $$

∎

###### Lemma A.1 .

###### Proof.

###### Lemma A.2 .

###### Proof.

###### Lemma A.3 .

###### Proof.

###### Lemma A.4 .

###### Proof.

#### A.1.2 Theorems (CAPOs)

We now prove the CAPO version of our theorems from the main paper. For both proofs, we leverage additional helping lemmas that we derived in Supplement [A.1.1](https://arxiv.org/html/2510.19643v1#A1.SS1.SSS1).

###### Theorem A.5 (Weighted population risk (CAPO)) .

Let

$$ $\displaystyle\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\mu_{t}^{\bar{a}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a}}(\bar{H}_{t})}{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big),$ (81) $$

where

$$ $\displaystyle\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k}),$ (82) $$

and

$$ $\displaystyle\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})=\prod_{j=t}^{t+\tau}\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}Y_{t+\tau}+\sum_{j=t}^{t+\tau}\mu_{j}^{\bar{a}}\left(\bar{H}_{j}\right)\left(1-\frac{\mathbbm{1}_{\{A_{j}=a_{j}\}}}{\pi_{j}^{\bar{a}}(\bar{H}_{j})}\right)\prod_{k=t}^{j-1}\frac{\mathbbm{1}_{\{A_{k}=a_{k}\}}}{\pi_{k}^{\bar{a}}(\bar{H}_{k})},$ (83) $$

with $\bar{Z}_{t+\tau}=(\bar{H}_{t+\tau},A_{t+\tau},Y_{t+\tau})$. Then, the population risk function

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a}})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (84) $$

minimizes the oracle risk

$$ $\displaystyle\mathcal{L}^{*}(g;\eta^{\bar{a}})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg].$ (85) $$

###### Proof.

In order to show that $\mathcal{L}(g;\eta^{\bar{a}})$ and $\mathcal{L}^{*}(g;\eta^{\bar{a}})$ have the same minimizer $g$, we need to show that

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]=\mathbb{E}\Bigg[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]+C,$ (86) $$

where $C$ is some constant term that does not depend on $g$. For this, notice that

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (87) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})+\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (88) $\displaystyle=$ $\displaystyle\underbrace{\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)^{2}\Bigg]+2\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\mu_{t}^{\bar{a}}(\bar{H}_{t})\Bigg]}_{=C}$ (89) $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\Bigg]+\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg].$ (90) $$

Here, the first two terms do not depend on $g$ and are therefore constant. Next we focus on

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\Bigg]$ (91) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (92) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a}}(\bar{H}_{t})}{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (93) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[{\omega_{t}^{\bar{a}}(\bar{H}_{t})}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (94) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[{\omega_{t}^{\bar{a}}(\bar{H}_{t})}\Big(\underbrace{\mathbb{E}\Big[\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a}}(\bar{H}_{t})}-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\Bigg]$ (95) $\displaystyle=$ $\displaystyle 0,$ (96) $$

where the result we apply in Equation [95](https://arxiv.org/html/2510.19643v1#A1.E95) follows from Lemma [A.3](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem3).

Finally, we focus on the last term in Equation [90](https://arxiv.org/html/2510.19643v1#A1.E90). That is,

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (97) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\;\Big|\;\bar{H}_{t}\Bigg]\Bigg]$ (98) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\underbrace{\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\omega_{t}^{\bar{a}}(\bar{H}_{t})}\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (99) $\displaystyle=$ $\displaystyle\mathbb{E}[\omega_{t}^{\bar{a}}(\bar{H}_{t})]\mathcal{L}^{*}(g;\eta^{\bar{a}}),$ (100) $$

where the result we apply in Equation [99](https://arxiv.org/html/2510.19643v1#A1.E99) follows from Lemma [A.4](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem4).

Hence, combining Equation [90](https://arxiv.org/html/2510.19643v1#A1.E90) with Equation [96](https://arxiv.org/html/2510.19643v1#A1.E96) and Equation [100](https://arxiv.org/html/2510.19643v1#A1.E100), and multiplying with $1/\mathbb{E}[\omega_{t}^{\bar{a}}(\bar{H}_{t})]$ yields

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a}})=\mathcal{L}^{*}(g;\eta^{\bar{a}})+C,$ (101) $$

which proves the theorem.
∎

###### Theorem A.6 (Neyman-orthogonality (CAPO)) .

The weighted population risk

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a}})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (102) $$

is Neyman-orthogonal with respect to all nuisance functions $\eta^{\bar{a}}=\{\pi_{j}^{\bar{a}},\mu_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}$.

###### Proof.

In order to show Neyman-orthogonality, we first calculate the pathwise-derivative with respect to the first argument, i.e., the target parameter $g$, via

$$ $\displaystyle D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g]$ (103) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\Big[g(\bar{H}_{t})+r\{\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\}\Big]\Big)^{2}\Bigg]\Bigg|_{r=0}$ (104) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\Big[g(\bar{H}_{t})+r\{\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\}\Big]\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (105) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (106) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a}}(\bar{H}_{t})}{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})}\Big[\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big]-g(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (107) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)+\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg].$ (108) $$

Next, we compute the pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g]$ with respect to all nuisance functions $\eta^{\bar{a}}=\{\pi_{j}^{\bar{a}},\mu_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}$. When calculating the pathwise derivative of the functions $f_{t}^{\bar{a}}\in\{\mu_{t}^{\bar{a}},\gamma_{t}^{\bar{a}},\rho_{t}^{\bar{a}},\omega_{t}^{\bar{a}}\}$ with respect to $g_{j}^{\bar{a}}\in\eta^{\bar{a}}$, we use $f_{t}^{\bar{a}}(\cdot;g_{j}^{\bar{a}})$ to make our notation more explicit to highlight which $f_{t}^{\bar{a}}$ depends on the nuisance $g_{j}^{\bar{a}}$.

First, we calculate the pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g]$ with respect to the nuisances $\pi_{j}^{\bar{a}}$ for $j=t,\ldots,t+\tau$ via

$$ $\displaystyle D_{\pi_{j}^{\bar{a}}}D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g,\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}]$ (109) $\displaystyle=$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}D_{g}\mathcal{L}\Big(g;\{\mu_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}\cup\{\pi_{0}^{\bar{a}},\ldots,\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}),\ldots,\pi_{t+\tau}^{\bar{a}}\}\Big)[\hat{g}-g]\;\Big|\;_{r=0}$ (110) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (111) $\displaystyle+\omega_{t}^{\bar{a}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (112) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (113) $\displaystyle+\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)$ (114) $\displaystyle+\omega_{t}^{\bar{a}}(\bar{H}_{t})\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (115) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (116) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\Big\{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (117) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\underbrace{\mathbb{E}\Big[\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a}}(\bar{H}_{t})}-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (118) $\displaystyle=$ $\displaystyle 0,$ (119) $$

where the result we apply in Equation [113](https://arxiv.org/html/2510.19643v1#A1.E113) follows from Lemma [A.2](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem2), in Equation [115](https://arxiv.org/html/2510.19643v1#A1.E115) from Lemma [A.1](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem1), and in Equation [118](https://arxiv.org/html/2510.19643v1#A1.E118) follows from Lemma [A.3](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem3).

Next, we compute the pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g]$ with respect to the nuisances $\mu_{j}^{\bar{a}}$ for $j=t,\ldots,t+\tau$ via

$$ $\displaystyle D_{\mu_{j}^{\bar{a}}}D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g,\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}]$ (120) $\displaystyle=$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}D_{g}\mathcal{L}\Big(g;\{\pi_{j}^{\bar{a}},\omega_{j}^{\bar{a}}\}_{j=t}^{t+\tau}\cup\{\mu_{0}^{\bar{a}},\ldots,\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}),\ldots,\mu_{t+\tau}^{\bar{a}}\}\Big)[\hat{g}-g]\;\Big|\;_{r=0}$ (121) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))-g(\bar{H}_{t})\Big)$ (122) $\displaystyle+\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))-\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (123) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (124) $\displaystyle+\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}-\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (125) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (126) $\displaystyle-\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (127) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (128) $\displaystyle-\omega_{t}^{\bar{a}}(\bar{H}_{t})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (129) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\underbrace{\mathbb{E}\Big[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\omega_{t}^{\bar{a}}(\bar{H}_{t})}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (130) $\displaystyle-\omega_{t}^{\bar{a}}(\bar{H}_{t})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (131) $\displaystyle=$ $\displaystyle 0,$ (132) $$

where the result we apply in Equation [125](https://arxiv.org/html/2510.19643v1#A1.E125) follows from Lemma [A.1](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem1), and in Equation [130](https://arxiv.org/html/2510.19643v1#A1.E130) from Lemma [A.4](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem4).

Finally, we compute the pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g]$ with respect to the nuisances $\omega_{j}^{\bar{a}}$ for $j=t,\ldots,t+\tau$ via

$$ $\displaystyle D_{\omega_{j}^{\bar{a}}}D_{g}\mathcal{L}(g;\eta^{\bar{a}})[\hat{g}-g,\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}]$ (133) $\displaystyle=$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}D_{g}\mathcal{L}\Big(g;\{\pi_{j}^{\bar{a}},\mu_{j}^{\bar{a}}\}_{j=t}^{t+\tau}\cup\{\omega_{0}^{\bar{a}},\ldots,\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}),\ldots,\omega_{t+\tau}^{\bar{a}}\}\Big)[\hat{g}-g]\;\Big|\;_{r=0}$ (134) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (135) $\displaystyle+\omega_{t}^{\bar{a}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (136) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}\Big(\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (137) $\displaystyle+\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (138) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (139) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (140) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\underbrace{\mathbb{E}\Big[\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a}}(\bar{H}_{t})}-\mu_{t}^{\bar{a}}(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (141) $\displaystyle=0,$ (142) $$

where the result we apply in Equation [137](https://arxiv.org/html/2510.19643v1#A1.E137) follows from Lemma [A.2](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem2), and in Equation [141](https://arxiv.org/html/2510.19643v1#A1.E141) from Lemma [A.3](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem3).
∎

###### Theorem A.5 (Weighted population risk (CAPO)) .

###### Proof.

###### Theorem A.6 (Neyman-orthogonality (CAPO)) .

###### Proof.

### A.2 Conditional average treatment effects (CATEs)

We split the following section into two parts: first, as for the CAPOs, we derive several supporting lemmas to prove our main results ($\rightarrow$Lemmas [A.7](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem7) to [A.10](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem10)). Then, we derive the theorems from the main paper ($\rightarrow$Theorem [A.11](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem11) and Theorem [A.12](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem12)).

#### A.2.1 Lemmas (CATEs)

In order to prove our main theorems for CATEs, we first introduce a series of helping lemmas.

###### Lemma A.7 .

Let

$$ $\displaystyle\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\gamma_{t}^{\bar{b}}(\bar{Z}_{t+\tau}).$ (143) $$

Then, $\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})$ is Neyman-orthogonal with respect to all nuisance functions $\eta^{\bar{a},\bar{b}}=\eta^{\bar{a}}\cup\eta^{\bar{b}}$.

###### Proof.

The proof immediately follows from linearity of the efficient influence function and Lemma [A.1](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem1).
∎

###### Lemma A.8 .

Let

$$ $\displaystyle\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\omega_{t}^{\bar{b}}(\bar{H}_{t})+\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})\omega_{t}^{\bar{a}}(\bar{H}_{t})-\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t}).$ (144) $$

Then, $\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})$ is Neyman-orthogonal with respect to all nuisance functions $\eta^{\bar{a},\bar{b}}=\eta^{\bar{a}}\cup\eta^{\bar{b}}$.

###### Proof.

As in Lemma [A.2](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem2), we notice that $\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})$ is trivially Neyman-orthogonal with respect to $\mu_{j}^{\bar{a}}$ and $\mu_{j}^{\bar{b}}$ as it does not dependent on it. Further, we show that $\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})$ is the uncentered efficient influence function of $\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]$, and hence, Neyman-orthogonal with respect to the nuisance functions $\{\pi_{j}^{\bar{a}},\omega_{j}^{\bar{a}},\pi_{j}^{\bar{b}},\omega_{j}^{\bar{b}}\}_{j=t}^{t+\tau}$. For this, we make once again use of the chain rule for pathwise derivatives (Kennedy, 2022; Luedtke, 2024). We start with the efficient influence function of $\omega_{t}^{\bar{a}}(\bar{h}_{t})$, which is given by

$$ $\displaystyle\mathbbm{IF}\Big(\omega_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})\Big)$ (145) $\displaystyle\mathbbm{IF}\Big(\omega_{t}^{\bar{a}}(\bar{h}_{t})\omega_{t}^{\bar{b}}(\bar{h}_{t})\Big)$ (146) $\displaystyle=$ $\displaystyle\underbrace{\mathbbm{IF}\Big(\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]\Big)}_{(*)}\omega_{t}^{\bar{b}}(\bar{h}_{t})+\omega_{t}^{\bar{b}}(\bar{h}_{t})\underbrace{\mathbbm{IF}\Big(\mathbb{E}\Big[\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{b}}(\bar{H}_{j})\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]\Big)}_{(**)}$ (147) $$

For both $(*)$ and $(**)$, we can follow the derivations in Lemma [A.2](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem2), which yields

$$ $\displaystyle\mathbbm{IF}\Big(\omega_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})\Big)$ (148) $\displaystyle=$ $\displaystyle\frac{\mathbbm{1}_{\{\bar{h}_{t}=\bar{H}_{t}\}}}{p(\bar{h}_{t})}\Bigg[\Big\{-\omega_{t}^{\bar{a}}(\bar{h}_{t})+\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(H_{t+1:j},\bar{h}_{t})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(H_{t+1:j},\bar{h}_{t})\Big)$ (149) $\displaystyle\qquad\times\prod_{t\leq k<j}\pi_{k}^{\bar{a}}({H}_{t+1:k},\bar{h}_{t})\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;H_{t+1:j},\bar{H}_{t}=\bar{h}_{t}\Big]\Big\}\omega_{t}^{\bar{b}}(\bar{h}_{t})$ (150) $\displaystyle\quad+\omega_{t}^{\bar{a}}(\bar{h}_{t})\Big\{-\omega_{t}^{\bar{b}}(\bar{h}_{t})+\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{b}}(H_{t+1:j},\bar{h}_{t})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{b}}(H_{t+1:j},\bar{h}_{t})\Big)$ (151) $\displaystyle\qquad\times\prod_{t\leq k<j}\pi_{k}^{\bar{b}}({H}_{t+1:k},\bar{h}_{t})\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{b}}(\bar{H}_{k})\;\Big|\;H_{t+1:j},\bar{H}_{t}=\bar{h}_{t}\Big]\Big\}\Bigg].$ (152) $$

Finally, we derive the efficient influence function of $\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]$ via

$$ $\displaystyle\mathbbm{IF}\Big(\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]\Big)$ (153) $\displaystyle=$ $\displaystyle\sum_{\bar{h}_{t}}\mathbbm{IF}\Big(p(\bar{h}_{t})\omega_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})\Big)$ (154) $\displaystyle=$ $\displaystyle\sum_{\bar{h}_{t}}\Big[\mathbbm{IF}\Big(p(\bar{h}_{t})\Big)\omega_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})+p(\bar{h}_{t})\mathbbm{IF}\Big(\omega_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})\Big)\Big]$ (155) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]$ (156) $\displaystyle+\Big\{-\omega_{t}^{\bar{a}}(\bar{H}_{t})+\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\underbrace{\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{a}}(\bar{H}_{k})\;\Big|\;\bar{H}_{j}\Big]}_{=\omega_{j+1}^{\bar{a}}(\bar{H}_{j})}\Big\}\omega_{t}^{\bar{b}}(\bar{H}_{t})$ (157) $\displaystyle+\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big\{-\omega_{t}^{\bar{b}}(\bar{H}_{t})+\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{b}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{b}}(\bar{H}_{j})\Big)\prod_{t\leq k<j}\pi_{k}^{\bar{b}}(\bar{H}_{k})\underbrace{\mathbb{E}\Big[\prod_{k>j}\pi_{k}^{\bar{b}}(\bar{H}_{k})\;\Big|\;\bar{H}_{j}\Big]}_{=\omega_{j+1}^{\bar{b}}(\bar{H}_{j})}\Big\}$ (158) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]+\Big\{-\omega_{t}^{\bar{b}}(\bar{H}_{t})+\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})\Big\}\omega_{t}^{\bar{b}}(\bar{H}_{t})+\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big\{-\omega_{t}^{\bar{b}}(\bar{H}_{t})+\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})\Big\}$ (159) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]-2\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\omega_{t}^{\bar{b}}(\bar{H}_{t})\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})+\omega_{t}^{\bar{a}}(\bar{H}_{t})\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})$ (160) $\displaystyle=$ $\displaystyle-\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]+\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau}).$ (161) $$

∎

###### Lemma A.9 .

Let

$$ $\displaystyle\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\gamma_{t}^{\bar{b}}(\bar{Z}_{t+\tau}),$ (162) $$

for two treatment sequences $a_{t:t+\tau}$, $b_{t:t+\tau}$, and let

$$ $\displaystyle\mu_{t}^{\bar{a},\bar{b}}(\bar{h}_{t})=\mathbb{E}\Big[Y_{t:t+\tau}[a_{t:t+\tau}]-Y_{t:t+\tau}[b_{t:t+\tau}]\;\Big|\;\bar{H}_{t}=\bar{h}_{t}\Big]$ (163) $$

be the conditional average treatment effect. Then, it holds that

$$ $\displaystyle\mathbb{E}\Big[\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]=\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t}).$ (164) $$

###### Proof.

The proof immediately follows from linearity of expectations and Lemma [A.3](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem3).
∎

###### Lemma A.10 .

Let

$$ $\displaystyle\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\omega_{t}^{\bar{b}}(\bar{H}_{t})+\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})\omega_{t}^{\bar{a}}(\bar{H}_{t})-\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})$ (165) $$

for two for two treatment sequences $a_{t:t+\tau}$, $b_{t:t+\tau}$, and let

$$ $\displaystyle\omega_{j}^{\bar{a},\bar{b}}(\bar{h}_{\ell})=\omega_{j}^{\bar{a}}(\bar{h}_{\ell})+\omega_{j}^{\bar{b}}(\bar{h}_{\ell})$ (166) $$

be the weight function. Then, it holds that

$$ $\displaystyle\mathbb{E}\Big[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]=\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t}).$ (167) $$

###### Proof.

The proof follows from Lemma [A.4](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem4) via

$$ $\displaystyle\mathbb{E}\Big[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]$ (168) $\displaystyle=$ $\displaystyle\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\omega_{t}^{\bar{b}}(\bar{H}_{t})+\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})\omega_{t}^{\bar{a}}(\bar{H}_{t})-\omega_{t}^{\bar{a}\bar{b}}(\bar{H}_{t})\Big]$ (169) $\displaystyle=$ $\displaystyle\mathbb{E}\Big[\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]\omega_{t}^{\bar{b}}(\bar{H}_{t})+\mathbb{E}\Big[\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]\omega_{t}^{\bar{a}}(\bar{H}_{t})-\omega_{t}^{\bar{a}\bar{b}}(\bar{H}_{t})$ (170) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})\omega_{t}^{\bar{b}}(\bar{H}_{t})+\omega_{t}^{\bar{a}}(\bar{H}_{t})\omega_{t}^{\bar{b}}(\bar{H}_{t})-\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})$ (171) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t}).$ (172) $$

∎

###### Lemma A.7 .

###### Proof.

###### Lemma A.8 .

###### Proof.

###### Lemma A.9 .

###### Proof.

###### Lemma A.10 .

###### Proof.

#### A.2.2 Theorems (CATEs)

Finally, we can prove the CATE version of our theorems from the main paper. For both proofs, we leverage additional helping lemmas that we derived in Supplement [A.2.1](https://arxiv.org/html/2510.19643v1#A1.SS2.SSS1).

###### Theorem A.11 (Weighted population risk (CATE)) .

Let

$$ $\displaystyle\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big),$ (173) $$

where

$$ $\displaystyle\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\rho_{t}^{\bar{a}}(\bar{Z}_{t+\tau})+\rho_{t}^{\bar{b}}(\bar{Z}_{t+\tau})$ (174) $$

and

$$ $\displaystyle\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})=\gamma_{t}^{\bar{a}}(\bar{Z}_{t+\tau})-\gamma_{t}^{\bar{b}}(\bar{Z}_{t+\tau})$ (175) $$

with $\bar{Z}_{t+\tau}=(\bar{H}_{t+\tau},A_{t+\tau},Y_{t+\tau})$. Then, the population risk function

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a},\bar{b}})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (176) $$

minimizes the oracle risk

$$ $\displaystyle\mathcal{L}^{*}(g;\eta^{\bar{a},\bar{b}})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg].$ (177) $$

###### Proof.

The proof follows the exact same steps as for Theorem [A.5](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem5), where we can replace Lemma [A.3](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem3) with Lemma [A.9](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem9), and Lemma [A.4](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem4) with Lemma [A.10](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem10).

For completeness, we repeat the derivations in the following:

As in Theorem [A.5](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem5), we need to show that

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]=\mathbb{E}\Bigg[\omega_{t}^{\bar{a}}(\bar{H}_{t})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]+C,$ (178) $$

where $C$ is some constant term that does not depend on $g$. For this, notice that

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (179) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\mu_{t}^{\bar{a}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (180) $\displaystyle=$ $\displaystyle\underbrace{\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)^{2}\Bigg]+2\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Bigg]}_{=C}$ (181) $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\Bigg]+\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg].$ (182) $$

Again, the first two terms do not depend on $g$ and are therefore constant. Hence, we focus on

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\Bigg]$ (183) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (184) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (185) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[{\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (186) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[{\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}\Big(\underbrace{\mathbb{E}\Big[\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)g(\bar{H}_{t})\Bigg]$ (187) $\displaystyle=$ $\displaystyle 0,$ (188) $$

where the result in Equation [187](https://arxiv.org/html/2510.19643v1#A1.E187) follows from Lemma [A.9](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem9).

Finally, we simplify Equation [182](https://arxiv.org/html/2510.19643v1#A1.E182) via

$$ $\displaystyle\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (189) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\;\Big|\;\bar{H}_{t}\Bigg]\Bigg]$ (190) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\underbrace{\mathbb{E}\Big[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (191) $\displaystyle=$ $\displaystyle\mathbb{E}[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})]\mathcal{L}^{*}(g;\eta^{\bar{a},\bar{b}}),$ (192) $$

where the result Equation [191](https://arxiv.org/html/2510.19643v1#A1.E191) follows from Lemma [A.10](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem10).

Hence, combining Equation [182](https://arxiv.org/html/2510.19643v1#A1.E182) with Equation [188](https://arxiv.org/html/2510.19643v1#A1.E188) and Equation [192](https://arxiv.org/html/2510.19643v1#A1.E192), and multiplying with $1/\mathbb{E}[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})]$ yields

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a},\bar{b}})=\mathcal{L}^{*}(g;\eta^{\bar{a},\bar{b}})+C.$ (193) $$

∎

###### Theorem A.12 (Neyman-orthogonality (CATE)) .

The weighted population risk

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a},\bar{b}})=\frac{1}{\mathbb{E}\Big[\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]}\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)^{2}\Bigg]$ (194) $$

is Neyman-orthogonal with respect to all nuisance functions $\eta^{\bar{a},\bar{b}}=\eta^{\bar{a}}\cup\eta^{\bar{b}}$.

###### Proof.

The proof follows the proof for Theorem [A.6](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem6), where we can replace Lemma [A.1](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem1) with Lemma [A.7](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem7), and Lemma [A.2](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem2) with Lemma [A.8](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem8).

Again, for completeness, we provide the steps below.

In order to show Neyman-orthogonality, we first calculate the pathwise-derivative with respect to the target parameter $g$ via

$$ $\displaystyle D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g]$ (195) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\Big[g(\bar{H}_{t})+r\{\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\}\Big]\Big)^{2}\Bigg]\Bigg|_{r=0}$ (196) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\Big[g(\bar{H}_{t})+r\{\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\}\Big]\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (197) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-g(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (198) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})}\Big[\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big]-g(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (199) $\displaystyle=$ $\displaystyle-2\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)+\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg].$ (200) $$

Without loss of generality, we compute the pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g]$ with respect to the nuisance functions $\eta^{\bar{a}}$. The case for $\eta^{\bar{b}}$ follows completely analogously.

Again, for the pathwise derivative of the functions $f_{t}^{\bar{a},\bar{b}}\in\{\mu_{t}^{\bar{a},\bar{b}},\gamma_{t}^{\bar{a},\bar{b}},\rho_{t}^{\bar{a},\bar{b}},\omega_{t}^{\bar{a},\bar{b}}\}$ with respect to $g_{j}^{\bar{a}}\in\eta^{\bar{a}}$, we use $f_{t}^{\bar{a},\bar{b}}(\cdot;g_{j}^{\bar{a}})$ to make our notation more explicit to highlight which $f_{t}^{\bar{a},\bar{b}}$ depends on the nuisance $g_{j}^{\bar{a}}$.

The pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g]$ with respect to the nuisances $\pi_{j}^{\bar{a}}$ for $j=t,\ldots,t+\tau$ is given by

$$ $\displaystyle D_{\pi_{j}^{\bar{a}}}D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g,\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}]$ (201) $\displaystyle=$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}D_{g}\mathcal{L}\Big(g;\{\mu_{j}^{\bar{a}},\mu_{j}^{\bar{b}},\omega_{j}^{\bar{a}},\omega_{j}^{\bar{b}},\pi_{j}^{\bar{b}}\}_{j=t}^{t+\tau}\cup\{\pi_{0}^{\bar{a}},\ldots,\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}),\ldots,\pi_{t+\tau}^{\bar{a}}\}\Big)[\hat{g}-g]\;\Big|\;_{r=0}$ (202) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (203) $\displaystyle+\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (204) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (205) $\displaystyle+\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)$ (206) $\displaystyle+\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (207) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (208) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\Big\{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (209) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\pi_{j}^{\bar{a}}+r(\hat{\pi}_{j}^{\bar{a}}-\pi_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\underbrace{\mathbb{E}\Big[\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (210) $\displaystyle=$ $\displaystyle 0,$ (211) $$

where Equation [205](https://arxiv.org/html/2510.19643v1#A1.E205) follows from Lemma [A.8](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem8), in Equation [207](https://arxiv.org/html/2510.19643v1#A1.E207) from Lemma [A.1](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem1), and in Equation [210](https://arxiv.org/html/2510.19643v1#A1.E210) follows from Lemma [A.9](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem9).

The pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g]$ with respect to the nuisances $\mu_{j}^{\bar{a}}$ for $j=t,\ldots,t+\tau$ is given by

$$ $\displaystyle D_{\mu_{j}^{\bar{a}}}D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g,\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}]$ (212) $\displaystyle=$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}D_{g}\mathcal{L}\Big(g;\{\pi_{j}^{\bar{a}},\pi_{j}^{\bar{b}},\omega_{j}^{\bar{a}},\omega_{j}^{\bar{b}},\mu_{j}^{\bar{b}}\}_{j=t}^{t+\tau}\cup\{\mu_{0}^{\bar{a}},\ldots,\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}),\ldots,\mu_{t+\tau}^{\bar{a}}\}\Big)[\hat{g}-g]\;\Big|\;_{r=0}$ (213) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))-g(\bar{H}_{t})\Big)$ (214) $\displaystyle+\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (215) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (216) $\displaystyle+\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big(\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}-\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (217) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (218) $\displaystyle-\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big(\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (219) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (220) $\displaystyle-\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (221) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\underbrace{\mathbb{E}\Big[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}$ (222) $\displaystyle-\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\mu_{j}^{\bar{a}}+r(\hat{\mu}_{j}^{\bar{a}}-\mu_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (223) $\displaystyle=$ $\displaystyle 0,$ (224) $$

where Equation [217](https://arxiv.org/html/2510.19643v1#A1.E217) follows from Lemma [A.7](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem7), and Equation [222](https://arxiv.org/html/2510.19643v1#A1.E222) from Lemma [A.10](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem10).

Finally, the pathwise derivative of $D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g]$ with respect to the nuisances $\omega_{j}^{\bar{a}}$ for $j=t,\ldots,t+\tau$ is given by

$$ $\displaystyle D_{\omega_{j}^{\bar{a}}}D_{g}\mathcal{L}(g;\eta^{\bar{a},\bar{b}})[\hat{g}-g,\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}]$ (225) $\displaystyle=$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}D_{g}\mathcal{L}\Big(g;\{\pi_{j}^{\bar{a}},\pi_{j}^{\bar{b}},\mu_{j}^{\bar{a}},\mu_{j}^{\bar{b}},\omega_{j}^{\bar{b}}\}_{j=t}^{t+\tau}\cup\{\omega_{0}^{\bar{a}},\ldots,\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}),\ldots,\omega_{t+\tau}^{\bar{a}}\}\Big)[\hat{g}-g]\;\Big|\;_{r=0}$ (226) $\displaystyle\propto$ $\displaystyle\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\mathbb{E}\Bigg[\Big\{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (227) $\displaystyle+\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]\Bigg|_{r=0}$ (228) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\Big\{\underbrace{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}}_{=0}\Big(\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})-g(\bar{H}_{t})\Big)$ (229) $\displaystyle+\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big\}\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (230) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (231) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\mathbb{E}\Big[\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\;\Big|\;\bar{H}_{t}\Big]\Bigg]$ (232) $\displaystyle=$ $\displaystyle\mathbb{E}\Bigg[\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}r}\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t};\omega_{j}^{\bar{a}}+r(\hat{\omega}_{j}^{\bar{a}}-\omega_{j}^{\bar{a}}))\;\Big|\;_{r=0}\Big(\underbrace{\mathbb{E}\Big[\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t+\tau})\;\Big|\;\bar{H}_{t}\Big]}_{=\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)\Big(\hat{g}(\bar{H}_{t})-g(\bar{H}_{t})\Big)\Bigg]$ (233) $\displaystyle=0,$ (234) $$

where Equation [229](https://arxiv.org/html/2510.19643v1#A1.E229) follows from Lemma [A.8](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem8), and Equation [233](https://arxiv.org/html/2510.19643v1#A1.E233) from Lemma [A.9](https://arxiv.org/html/2510.19643v1#A1.Thmtheorem9).

∎

###### Theorem A.11 (Weighted population risk (CATE)) .

###### Proof.

###### Theorem A.12 (Neyman-orthogonality (CATE)) .

###### Proof.

### A.3 Generalizing the R-learner

Remark: *For a single-step-ahead prediction $\tau=0$ (i.e., when there is no time-varying confounding as in the static setting), the R-learner has the same overlap weights as our WO-learner for CATE.*

###### Proof.

We show how our weighted population risk function reduces for $\tau=0$ and leverage previous findings on the identity of the R-learner (Nie & Wager, 2021).

Notice that under our identifiability assumptions, for a single-step ahead prediction $\tau=0$, conditioning on the observed history (i.e., a backdoor-adjustment) is sufficient to adjust for all confounders, as *there is no time-varying confounding*.

Hence, for $\tau=0$, we can treat the observed history $\bar{H}_{t}$ as a fixed set of covariates (typically denoted as $X$), the treatment variable $A_{t}\in\{0,1\}$ as well as the intervention $\bar{a}=a_{t}\in\{0,1\}$ as a binary treatment (denoted as $A$ and $a$, respectively), and the outcome $Y_{t}$ as the instantaneous outcome (denoted as $Y$). Finally, $\bar{Z}_{t}$ summarizes all variables $(\bar{H}_{t},A_{t},Y_{t})$, which corresponds to $(X,A,Y)$ in the static setting (see Figure [6](https://arxiv.org/html/2510.19643v1#A1.F6)).

Let $\tau=0$. Then, the pseudo-outcomes and weights simplify as

$$ $\displaystyle\rho_{t}^{\bar{a}}(\bar{Z}_{t})$ (235) $\displaystyle=$ $\displaystyle\prod_{j=t}^{t+\tau}\pi_{j}^{\bar{a}}(\bar{H}_{j})+\sum_{j=t}^{t+\tau}\Big(\mathbbm{1}_{\{a_{j}=A_{j}\}}-\pi_{j}^{\bar{a}}(\bar{H}_{j})\Big)\omega_{j+1}^{\bar{a}}(\bar{H}_{j})\prod_{t\leq k<j}\pi_{k}^{\bar{a}}(\bar{H}_{k})$ (236) $\displaystyle=$ $\displaystyle\pi_{t}^{\bar{a}}(\bar{H}_{t})+\underbrace{\Big(\mathbbm{1}_{\{a_{t}=A_{t}\}}-\pi_{t}^{\bar{a}}(\bar{H}_{t})\Big)\omega_{t+1}^{\bar{a}}(\bar{H}_{t})\underbrace{\prod_{t\leq k<t}\pi_{k}^{\bar{a}}(\bar{H}_{k})}_{=0}}_{=0}$ (237) $\displaystyle=$ $\displaystyle\pi_{t}^{\bar{a}}(\bar{H}_{t}),$ (238) $$

and

$$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})=\mathbb{E}\Big[\pi_{t}^{\bar{a}}(\bar{H}_{t})\;\Big|\;\bar{H}_{t}\Big]=\pi_{t}^{\bar{a}}(\bar{H}_{t}),$ (239) $$

such that

$$ $\displaystyle\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t})$ (240) $\displaystyle=$ $\displaystyle\rho_{t}^{\bar{a}}(\bar{H}_{t})\omega_{t}^{\bar{b}}(\bar{H}_{t})+\rho_{t}^{\bar{b}}(\bar{H}_{t})\omega_{t}^{\bar{a}}(\bar{H}_{t})-\omega_{t}^{\bar{a}}(\bar{H}_{t})\omega_{t}^{\bar{b}}(\bar{H}_{t})$ (241) $\displaystyle=$ $\displaystyle\pi_{t}^{\bar{a}}(\bar{H}_{t})\pi_{t}^{\bar{b}}(\bar{H}_{t})+\pi_{t}^{\bar{a}}(\bar{H}_{t})\pi_{t}^{\bar{b}}(\bar{H}_{t})-\omega_{t}^{\bar{a}}(\bar{H}_{t})\omega_{t}^{\bar{b}}(\bar{H}_{t})$ (242) $\displaystyle=$ $\displaystyle\omega_{t}^{\bar{a}}(\bar{H}_{t})\omega_{t}^{\bar{b}}(\bar{H}_{t})$ (243) $\displaystyle=$ $\displaystyle\pi_{t}^{\bar{a}}(\bar{H}_{t})\pi_{t}^{\bar{b}}(\bar{H}_{t})$ (244) $\displaystyle=$ $\displaystyle\pi_{t}^{\bar{a}}(\bar{H}_{t})(1-\pi_{t}^{\bar{a}}(\bar{H}_{t}))$ (245) $$

and finally

$$ $\displaystyle\xi_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t})$ (246) $\displaystyle=$ $\displaystyle\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\frac{\omega_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})}{\rho_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t})}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)$ (247) $\displaystyle=$ $\displaystyle\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})+\frac{\pi_{t}^{\bar{a}}(\bar{H}_{t})(1-\pi_{t}^{\bar{a}}(\bar{H}_{t}))}{\pi_{t}^{\bar{a}}(\bar{H}_{t})(1-\pi_{t}^{\bar{a}}(\bar{H}_{t}))}\Big(\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t})-\mu_{t}^{\bar{a},\bar{b}}(\bar{H}_{t})\Big)$ (248) $\displaystyle=$ $\displaystyle\gamma_{t}^{\bar{a},\bar{b}}(\bar{Z}_{t})$ (249) $\displaystyle=$ $\displaystyle\gamma_{t}^{\bar{a},1-\bar{a}}(\bar{Z}_{t}),$ (250) $$

where $\gamma_{t}^{\bar{a},1-\bar{a}}(\bar{Z}_{t})$ simplify to the DR pseudo-outcomes for CATE in the static setting (Curth & van der Schaar, 2021; Frauen et al., 2025). As shown by Morzywolek et al. (2023) and highlighted by other works (Chernozhukov et al., 2024; Fisher, 2024), the R-learner is an *overlap-weighted DR learner* and, hence, minimizes the loss

$$ $\displaystyle\mathcal{L}(g;\eta^{a})=\frac{1}{\mathbb{E}\Big[\pi^{{a}}(X)(1-\pi^{{a}}(X))\Big]}\mathbb{E}\Big[\pi^{{a}}(X)(1-\pi^{{a}}(X))\Big(\gamma^{{a},1-{a}}(X,A,Y)-g\Big)^{2}\Big].$ (251) $$

Since for $\tau=0$, we only need one set of nuisances for CATE, i.e., $\eta^{\bar{a}}=\eta^{\bar{b}}=\eta^{\bar{a},\bar{b}}$, and it follows that Equation [251](https://arxiv.org/html/2510.19643v1#A1.E251) exactly mirrors

$$ $\displaystyle\mathcal{L}(g;\eta^{\bar{a}})=\frac{1}{\mathbb{E}\Big[\pi_{t}^{\bar{a}}(\bar{H}_{t})(1-\pi_{t}^{\bar{a}}(\bar{H}_{t}))\Big]}\mathbb{E}\Big[\pi_{t}^{\bar{a}}(\bar{H}_{t})(1-\pi_{t}^{\bar{a}}(\bar{H}_{t}))\Big(\gamma_{t}^{\bar{a},1-\bar{a}}(\bar{Z}_{t})-g\Big)^{2}\Big]$ (252) $$

in our time-varying notation.
∎

Figure: Figure 6: One-step ahead prediction. For a one-step ahead prediction $\tau=0$, there is no time-varying confounding. Hence, we can treat the observed history $\bar{H}_{t}$ as a fixed set of covariates $X$, and the single intervention and instantaneous outcomes as in the static setting. Our WO-learner for CATE then simplifies to the R-learner in the static setting.
Refer to caption: x6.png

###### Proof.

## Appendix B Details on the data-generating processes

### B.1 Synthetic data generation

We now describe the data-generating processes for the synthetic datasets
$\mathcal{D}^{\gamma}$ *(low-overlap regime)*, $\mathcal{D}^{\pi}$ *(complex propensity)*, $\mathcal{D}^{\mu}$ *(complex response function)*, and $\mathcal{D}^{N}$ *(low-sample setting)*. All of them have the following general structure:

As in Frauen et al. (2025), for each $*\in\{\gamma,\pi,\mu,N\}$, we first simulate an initial confounder $X_{0}\sim\mathcal{N}(0,1)$. Then, for time steps $t=1,\ldots,T^{*}$, we generate $d_{x}^{*}$-dimensional time-varying confounders via

$$ $\displaystyle X_{t}=(X_{t,1},\ldots,X_{t,d_{x}^{*}})=f_{x}^{*}(Y_{t-1},A_{t-1},\bar{H}_{t-1})+\varepsilon_{x}$ (253) $$

and time-varying treatments via

$$ $\displaystyle A_{t}\sim\sigma\Big(f_{a}^{*}(\bar{H}_{t})\Big),$ (254) $$

where $\sigma(\cdot)$ is the sigmoid function. The outcomes are then simulated via

$$ $\displaystyle Y_{t}=f_{y}^{*}(A_{t},\bar{H}_{t})+\epsilon_{y}.$ (255) $$

with $\varepsilon_{y}\sim\mathcal{N}(0,0.3^{2})$. For each dataset, we simulate $n^{*}$ samples for training and $1000$ samples for testing. For the test set, we always generate the ground-truth CATE of a $\tau^{*}$-step *always treat* against a $\tau^{*}$-step *never treat* intervention.

We provide the specific configurations of $f_{x}^{*},f_{a}^{*},f_{y}^{*}$, $\tau^{*}$, $d_{x}^{*}$, $T^{*}$ and $n^{*}$ below:

(1) Low-overlap regime ${\mathcal{D}^{\gamma}}$: For ${\mathcal{D}^{\gamma}}$, we set $\tau^{\gamma}=1$, $d_{x}^{\gamma}=1$, $T^{\gamma}=5$, and $n^{\gamma}=4000$. The covariates are generated via $f_{x}^{\gamma}(Y_{t-1},A_{t-1},\bar{H}_{t-1})=0.5X_{t-1}$, the treatments via $f_{a}^{\gamma}(\bar{H}_{t})=\gamma(0.5X_{t}+0.5Y_{t-1}-0.5(A_{t-1}-0.5))$, where $\gamma$ controls the overlap strength, and the outcomes via $f_{y}^{\gamma}(A_{t},\bar{H}_{t})=0.5\exp(-X_{t}^{2})(A_{t}-0.5)$. In order to decrease the overlap, we vary the overlap parameter $\gamma\in\{0.5,1.0,1.5,\ldots,6.5\}$.

(2) Complex treatment ${\mathcal{D}^{\pi}}$: For ${\mathcal{D}^{\pi}}$, we set $d_{x}^{\pi}=1$, $T^{\pi}=15$, and $n^{\pi}=4000$. The covariates are generated via $f_{x}^{\pi}(Y_{t-1},A_{t-1},\bar{H}_{t-1})=0.5X_{t-1}$, the treatments via $f_{a}^{\pi}(\bar{H}_{t})=\sin(0.5X_{t}+0.5Y_{t-1}-0.5(A_{t-1}-0.5))$, and the outcomes via $f_{y}^{\pi}(A_{t},\bar{H}_{t})=0.5\exp(-X_{t}^{2})(A_{t}-0.5)$. In order to increase the complexity of the treatment propensity, we increase the prediction horizon $\tau^{\pi}\in\{1,3,5,7\}$.

(3) Complex response ${\mathcal{D}^{\mu}}$: For ${\mathcal{D}^{\mu}}$, we set $\tau^{\mu}=1$, $T^{\mu}=15$, and $n^{\mu}=4000$. The covariates are generated via $f_{x}^{\pi}(Y_{t-1},A_{t-1},\bar{H}_{t-1})=0.5X_{t-1}$, the treatments via $f_{a}^{\pi}(\bar{H}_{t})=0.5\sum_{p=1}^{d_{x}^{\mu}}X_{t,p}/d_{x}^{\mu}+0.5Y_{t-1}-0.5(A_{t-1}-0.5)$, and the outcomes via $f_{y}^{\pi}(A_{t},\bar{H}_{t})=\exp(0.5(A_{t}-0.5)\sum_{p=1}^{d_{x}^{\mu}}\cos(X_{t-1,p})\cos(\cos(X_{t-1,p}))/d_{x}^{\mu})$. In order to increase the complexity of the response function, we increase the dimensionality $d_{x}$ of the time-varying confounders $X_{t}=(X_{t,1},\ldots,X_{t,d_{x}^{\mu}})$ via $d_{x}^{\mu}\in\{5,10,15,20,25,30,35\}$.

(4) Low-sample setting ${\mathcal{D}^{N}}$: For ${\mathcal{D}^{N}}$, we set $\tau^{N}=1$, $d_{x}^{N}=5$, and $T^{N}=5$. The covariates are generated via $f_{x}^{N}(Y_{t-1},A_{t-1},\bar{H}_{t-1})=0.5X_{t-1}$, the treatments via $f_{a}^{N}(\bar{H}_{t})=3.5(0.5\sum_{p=1}^{d_{x}^{\mu}}X_{t,p}/d_{x}^{N}+0.5Y_{t-1}-0.5(A_{t-1}-0.5))$, and the outcomes via $f_{y}^{N}(A_{t},\bar{H}_{t})=0.5\exp(-(\sum_{p=1}^{d_{x}^{N}}\cos(X_{t,p})/d_{x}^{N})^{2})(A_{t}-0.5)$. We vary the sample size for training the nuisance functions and second-stage estimators via $n^{N}\in\{8000,7000,6000,5000,4000,3000,2000\}$.

### B.2 Semi-synthetic data generation

For our semi-synthetic experiments, we employ the MIMIC-III (Wang et al., 2020) extract based on the MIMIC-III dataset (Johnson et al., 2016). Here, we use time-varying real-world covariates *heart rate, red blood cell count, sodium, mean blood pressure, systemic vascular resistance, glucose, chloride urine, glascow coma scale total, hematocrit, positive end-expiratory pressure set,* and *respiratory rate*. All measurements are aggregated at hourly levels. Further, we include *gender* and *age* as a static covariates. We summarize all covariates as $X_{t}=(X_{t,1},\ldots,X_{t,d_{x}})$. Then, we simulate treatments $A_{t}$ and outcomes $Y_{t}$ based on these covariates.

Our data generating process is designed to have time-varying confounding, has a complex propensity score, and a complex response function. Specifically, we simulate the treatments via

$$ $\displaystyle A_{t}\sim\sigma\Big(f_{a}(\bar{H}_{t})+\varepsilon_{a}\Big),$ (256) $$

with $\varepsilon_{a}\sim\mathcal{N}(0,0.2^{2})$, $f_{a}(A_{t},\bar{H}_{t})=\sin(Y_{t-1}))-A_{t-1}\sum_{p=1}^{d_{x}}\sin(X_{t,p})/d_{x}$,
and the outcomes via

$$ $\displaystyle Y_{t}=f_{y}(A_{t},\bar{H}_{t})+\varepsilon_{y}$ (257) $$

with $f_{y}(A_{t},\bar{H}_{t})=Y_{t-1}+2(A_{t}-0.5)\exp\Big(2(A_{t}-0.5)\sin(Y_{t-1})\sum_{p=1}^{d_{x}}\cos(X_{t,p})/(td_{x})\Big)$ and $\varepsilon_{y}\sim\mathcal{N}(0,0.1^{2})$.
We include trajectories of length $T=20$, and simulate $n_{\emph{train}}=1500$ samples for training. For evaluation of CATE, we again compare a $\tau$-step *always treat* against a $\tau$-step *never treat* treatment intervention sequence.

## Appendix C Implementation details

$\bullet$ Implementation details: We report implementation details for our transformer instantiation in Section [5](https://arxiv.org/html/2510.19643v1#S5). Here, we closely follow the setup by Frauen et al. (2025) (see Table [5](https://arxiv.org/html/2510.19643v1#A3.T5)):

- •
All nuisance functions and second-stage estimators can be written as regression models that take the history $\bar{H}_{t}$ as input and learn some $\delta$-step-ahead outcome $\tilde{Y}_{t+\delta}$ (e.g., for the HA-learner, $\tilde{Y}_{t+\delta}=Y_{t+\delta}$).
- •
Hence, we parametrize each regression model as $g_{\theta}(\bar{h}_{t})=g_{\theta}^{2}(g_{\theta}^{1}(\bar{h}_{t}))$, where $g_{\theta}^{1}$ is a representation function (in our main experiments: a standard transformer), and $g_{\theta}^{2}$ a read-out function (a standard multi-layer perceptron).
- •
As in (Frauen et al., 2025), we learn the propensity scores $\pi_{t+j}^{\bar{a}}$ in a joint model, whereas we learn the response functions $\mu_{t+j}^{\bar{a}}$ and the weight functions $\omega_{t+j}^{\bar{a}}$ in separate models.
- •
*Representation function $g_{\theta}^{1}(\cdot)$:* For our main experiments in Section [5](https://arxiv.org/html/2510.19643v1#S5), we use an encoder transformer (Vaswani et al., 2017) with a single transformer block and a causal mask to avoid look-ahead bias, as well as non-trainable positional encodings. The transformer block has a self-attention mechanism with $d_{\text{att}}$ attention heads and a hidden state dimension $d_{\text{hid}}$, followed by a feed-forward network with hidden layer size $d_{\text{ff}}$. The self-attention mechanism and the feed-forward network use residual connections, followed by dropout layers with dropout probability $0.1$, and post-normalization for regularization.
- •
*Read-out function $g_{\theta}^{2}(\cdot)$:* We use a simple multilayer-perceptron with one hidden layer of size $d_{\text{mlp}}$, ReLU nonlinearities, and either a linear (regression) or softmax (classification) output activation.

We summarize all parameterizations in Table [5](https://arxiv.org/html/2510.19643v1#A3.T5). To ensure a fair comparison, all nuisance models and second-stage estimators share, where appropriate, the exact same architecture and parametrization.

$\bullet$ Runtime: For each transformer-based learner, training took approximately $1.5$ minutes with $n_{\text{train}}=4000$ samples and an AMD Ryzen 7 Pro CPU and 32GB of RAM. The runtime was comparable for our WO-learner and the existing meta-learners.

Table: Table 5: Hyperparameters of our transformer instantiations for the nuisance function estimators and second-stage estimators. To ensure a fair comparison, the nuisance functions for all meta-learners share the exact same parametrization.

## Appendix D LSTM ablations

In the following, we repeat the experiment with decreasing overlap on the dataset $\mathcal{D}^{\gamma}$. Here, instantiate all meta-learners with simple LSTMs (Hochreiter & Schmidhuber, 1997). That is, we substitute the transformer instantiations for both the nuisance function and the second-stage estimator from the main experiments in Section [5](https://arxiv.org/html/2510.19643v1#S5) with LSTM architectures. As meta-learners can be instantiated with *any* neural backbone, this serves as an example that our main results are robust for different instantiations and parameterizations of the neural backbone.

Results: As for our main results in Section [5](https://arxiv.org/html/2510.19643v1#S5), our WO-learner greatly improves over all existing meta-learners for decreasing overlap. This is expected and in line with our theoretical findings, which are *agnostic* of the neural backbone. Hence, our novel WO-learner has clear advantages over existing standard meta-learners, *especially when the overlap is low* (large $\gamma$).

**Table 6: Low-overlap regime $\mathcal{D}^{\gamma}$: Reported are the average RMSEs $\pm$ standard deviation for CATE estimation, and the relative improvement of our WO-learner over the best performing baselines across different levels of overlap.**
| Overlap $\gamma$ | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| (a) HA | $0.55\pm 0.06$ | $0.51\pm 0.06$ | $0.54\pm 0.10$ | $0.52\pm 0.04$ | $0.50\pm 0.05$ | $0.52\pm 0.03$ | $0.51\pm 0.04$ | $0.50\pm 0.04$ | $0.47\pm 0.03$ | $0.52\pm 0.03$ | $0.52\pm 0.01$ | $0.53\pm 0.01$ | $0.57\pm 0.01$ |
| (b) RA | $0.30\pm 0.02$ | $0.31\pm 0.05$ | $0.31\pm 0.04$ | $0.31\pm 0.03$ | $0.29\pm 0.04$ | $0.27\pm 0.02$ | $0.29\pm 0.02$ | $0.28\pm 0.02$ | $0.27\pm 0.04$ | $0.25\pm 0.04$ | $0.27\pm 0.03$ | $0.28\pm 0.03$ | $0.27\pm 0.03$ |
| (c) IPW | $0.20\pm 0.04$ | $0.12\pm 0.06$ | $0.12\pm 0.05$ | $0.18\pm 0.08$ | $0.14\pm 0.04$ | $0.24\pm 0.07$ | $0.27\pm 0.08$ | $0.52\pm 0.35$ | $0.45\pm 0.21$ | $0.35\pm 0.12$ | $0.52\pm 0.49$ | $0.72\pm 0.04$ | $0.47\pm 0.08$ |
| (d) DR | $\mathbf{0.05\pm 0.03}$ | $\mathbf{0.04\pm 0.01}$ | $0.06\pm 0.02$ | $0.07\pm 0.03$ | $0.09\pm 0.04$ | $0.10\pm 0.03$ | $0.09\pm 0.03$ | $0.23\pm 0.19$ | $0.13\pm 0.06$ | $0.16\pm 0.05$ | $0.28\pm 0.12$ | $0.15\pm 0.04$ | $0.18\pm 0.08$ |
| (e) IVW | $0.72\pm 1.26$ | $6.14\pm 8.63$ | $0.21\pm 0.22$ | $6.24\pm 7.74$ | $3.51\pm 6.77$ | $4.98\pm 7.21$ | $9.79\pm 6.22$ | $31.71\pm 31.68$ | $138.57\pm 165.75$ | $41.99\pm 74.10$ | $61.49\pm 67.81$ | $1351.68\pm 1351.68$ | $76.66\pm 76.66$ |
| (*) WO (*ours*) | ${0.07\pm 0.01}$ | ${0.05\pm 0.03}$ | $\mathbf{0.03\pm 0.02}$ | $\mathbf{0.04\pm 0.03}$ | $\mathbf{0.04\pm 0.01}$ | $\mathbf{0.03\pm 0.01}$ | $\mathbf{0.06\pm 0.02}$ | $\mathbf{0.06\pm 0.03}$ | $\mathbf{0.08\pm 0.03}$ | $\mathbf{0.05\pm 0.03}$ | $\mathbf{0.08\pm 0.03}$ | $\mathbf{0.08\pm 0.03}$ | $\mathbf{0.08\pm 0.03}$ |
| Rel. improv. (%) | $-29.9\%$ | $-7.8\%$ | $46.0\%$ | $37.5\%$ | $54.8\%$ | $68.2\%$ | $37.0\%$ | $74.7\%$ | $36.0\%$ | $66.6\%$ | $72.6\%$ | $47.1\%$ | $54.3\%$ |

## Appendix E Real-world outcome prediction

In the following, we report the performance of our WO-learner on factual outcome prediction on real-world data.

In Table [7](https://arxiv.org/html/2510.19643v1#A5.T7), we report the performance of the propensity-based meta-learners and the HA baseline performance. For this, we use the MIMIC-III dataset (Johnson et al., 2016; Wang et al., 2020). The outcome variable of interest is diastolic blood pressure, and we consider mechanical ventilation as the treatment variable. We further use $19$ time-varying covariates such as cholesterol, respiratory rate, heart rate, and sodium, and further include gender as a static covariate for predicting the factual outcome. All measurements are aggregated at hourly levels.

Notwithstanding, we emphasize that *factual outcome prediction is not the task our method is tailored for*, nor any of the other meta-learners that perform time-varying adjustments. Specifically, for factual outcome prediction, there is no need to adjust for time-varying confounding, as there is no causal intervention on future sequences of treatments. Instead, this is purely a predictive task and can be performed with a standard regression. However, different from other learners, our method still produces reliable estimates.

**Table 7: Reported are the RMSEs for factual outcome prediction. We emphasize that *factual outcome prediction is not the task our WO-learner is tailored for*. Yet, different from the other propensity-based methods, it has very robust performance over all prediction horizons. In line with our theoretical considerations, we conclude that the HA learner has the best performance since regressing on the observed history – a simple history-adjustment – is sufficient for factual outcome prediction.**
| Prediction horizon $\tau$ | HA | IPW | DR | IVW | WO (ours) |
| --- | --- | --- | --- | --- | --- |
| $1$ | ${0.708\pm 0.022}$ | $5.627\pm 8.498$ | $1.601\pm 1.038$ | $53.200\pm 98.958$ | $1.077\pm 0.025$ |
| $2$ | ${0.774\pm 0.023}$ | $3.419\pm 4.767$ | $1.255\pm 0.305$ | $37.408\pm 36.641$ | $1.079\pm 0.022$ |
| $3$ | ${0.822\pm 0.024}$ | $3.729\pm 5.094$ | $3.978\pm 5.208$ | $21.151\pm 40.098$ | $1.087\pm 0.027$ |
| $4$ | ${0.869\pm 0.024}$ | $1.426\pm 0.674$ | $5.556\pm 8.939$ | $22.128\pm 32.211$ | $1.099\pm 0.006$ |

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: - The paper claims to estimate HTE but reports only CATE in the experiments. There is no discussion of heterogeneity across subgroups or covariate-dependent variation in effects. It may mislead readers about what “HTE” means in this context.
- Please see weaknesses.
soundness: 3
strengths: - The paper proposes a model-agnostic framework for adjusting time-varying confounders.
- The paper provides a theoretical foundation for orthogonality.
- An implementation code is available for review.
summary: The paper proposes an overlap-weighted orthogonal (WO) meta-learner to estimate time-varying heterogeneous treatment effects (HTEs) under low-overlap conditions. The framework aims to stabilize inverse-propensity-based estimators by re-weighting high-overlap regions.
weaknesses: - The claim that weighting by overlap improves estimator stability is weakly justified. By emphasizing high-overlap regions, the learner avoids extreme inverse-propensity weights but can ignore potentially outcome-informative regions. Is there a trade-off between propensity and outcome predictions?
- The motivation behind the study problem (treatment effect estimation over time) needs to be further clarified, particularly in relation to its real-world applicability. It is not evident whether the problem is testable, whether the authors have empirically validated it in real applications, or whether the underlying assumptions commonly hold in practical settings.
- The experimental validation is primarily based on synthetic data. Even in the semi-synthetic MIMIC-III experiment, both treatments and outcomes are simulated, so there is no demonstration of applicability to real observational data. The study does not evaluate whether estimating CATE over time provides practical benefits or is actually necessary or useful in real-world contexts. The study shows performance under a controlled scenario rather than demonstrating practical utility.
- The authors explicitly note that “factual outcome prediction is not the task our method is tailored for,” yet give no alternative validation criterion for real-world use. How would it be verified in practice? The authors should add a discussion and limitations on this point.
- The baselines are narrow, limited to RA, IPW, DR, and IVW  from [1]. The paper omits several relevant baselines for CATE over time, e.g., G-Net [2] and G-Transformer [3].

[1] Frauen, D., Hess, K., & Feuerriegel, S. Model-agnostic meta-learners for estimating heterogeneous treatment effects over time. In The Thirteenth International Conference on Learning Representations.

[2] Li, R., Shahn, Z., Li, J., Lu, M., Chakraborty, P., Sow, D., ... & Lehman, L. W. H. (2020). G-Net: a deep learning approach to G-computation for counterfactual outcome prediction under dynamic treatment regimes. arXiv preprint arXiv:2003.10551.

[3] Hess, Konstantin, Dennis Frauen, Valentyn Melnychuk, and Stefan Feuerriegel. "G-Transformer for Conditional Average Potential Outcome Estimation over Time." CoRR (2024).

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: - In table 4, why the IVW performance get dramatically incresed when the prediction horizon increases? In the paper [1], the performance looks normal when $\tau=2,3$.
soundness: 4
strengths: - Compared with other meta-learners in this field, the study proposes a novel method to address the challenges of other learners in the low overlap scenario, which is common in the time varying treatment assignment strategy. 
- The overlap weighted orthogonal 
- The study has conducted both theorectical and empirical analysis to demonstrate the advantages of their proposed method.
- The results demonstrate dramatical improvement of the performance in Table 2. And the performance is uniformly the best over all the sample sizes.
summary: The paper proposes an over-weighted meta-learner to estimate HTE in a very challenging time-varying scenario with a treatment sequence scenario. It also develops a new risk esimation to reduce the effects of first order bias in the estimation. The paper provides detailed theorectical analysis. The empirical results show that the proposed method achieve the best performance compared with other adjustment method.
weaknesses: - Can you please revise the writing of the problem setup? Maybe provide some examples to illustrate what the treatment looks like. Based on my understanding of your problem, the two sequences should be a list of 1 or 0, correct? i.e. $a = [1, 0, 0, 1], b = [0, 1, 1, 1]$
- One very relevant study as mentioned in the paper is the IVW method. As stated in the IVW paper, the framework is a composition of IVW and DR learner [1]. The paper provides very brief description in line 116-125. I wonder here is the IVW refers to the same model as the IVW-DR learner in the original paper or just the IVW only.  
- Another concern about the time-varying treatment strategy is that, the proposed method only considers confounders happen before $t$. But in the real-world setting, the propensity scores are usually depend on what happened during $t:t+\tau$. I know it makes the problem much harder, but it is very realistic in healthcare domain. 

[1] Frauen, D., Hess, K., & Feuerriegel, S. Model-agnostic meta-learners for estimating heterogeneous treatment effects over time. In The Thirteenth International Conference on Learning Representations.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: - Do we expect the performance of the proposed estimator to deteriorate more that the DR-learner when propensity is hard to learn, as the weights also rely on the propensity estimates?
soundness: 3
strengths: - Clear motivation and the proposed estimator addresses the problem of insufficient overlap. 
- The estimator is flexible and is robust to nuisance errors as well. 
- Performed comprehensive experiments that addresses different settings. Results demonstrates the effectiveness of the proposed method at addressing the different issues discussed in the paper.
summary: This paper proposes a new framework, Overlap-Weighted Orthogonal Learning (OWL), for estimating heterogeneous treatment effects in time varying settings where covariate overlap between treatment groups decreases exponentially with time horizon. The method incorporates the a propensity/overlap weights that up-weights samples with a higher probability to having the treatment sequence of interest. Moreover, the proposed estimator is Neyman orthogonal which makes the estimator insensitive to nuisance estimation error.
weaknesses: - While the paper showed that the proposed population risk is Neyman orthogonal, papers on mete-learners usually also provide error analysis and show that the error terms from the nuisance functions are higher order, which is not presented in the paper. 
- The main innovation is to address the problem of limited overlap, so it would be nice to have some theorems that showcase how this estimator have better behavior (e.g. variance) in those regimes.

---

confidence: 5
contribution: 2
initial_rating: 6
presentation: 3
questions: **CATE vs. CAPO Weighting:** Why is the overlap weight for CATE (Eq 10) defined as the *product* of the individual propensity weights ($\omega^{\overline{a}}\omega^{\overline{b}}$)? Why not estimate the CAPO for each arm separately (using the CAPO propensity weight $\omega^{\overline{a}}$) and then simply subtract them? What is the theoretical motivation for this specific product-based weighting for CATE?
soundness: 3
strengths: * **Problem Significance:** The paper addresses a highly relevant and challenging problem at the intersection of causal inference and time-series analysis. Estimating HTEs over time is crucial for personalized medicine, and the issue of exponentially decreasing overlap is a core, practical barrier that the paper directly confronts.
* **Methodological Novelty:** The proposed WO-learner is methodologically novel and sound. The key idea of designing a population risk function that explicitly minimizes an *overlap-weighted* oracle risk, while simultaneously enforcing Neyman-orthogonality, is the primary technical contribution.
* **Theoretical Guarantees:** The method is supported by strong theoretical guarantees. The proof that the weighted population risk is Neyman-orthogonal with respect to all nuisance functions (including the newly defined weight functions) is a non-trivial and important result.
* **Experimental Design:** The paper presents a broad set of experiments, including four synthetic datasets designed to isolate specific challenges (low overlap, complex nuisance functions, small sample size) and a semi-synthetic experiment based on real-world MIMIC-III covariates.
summary: This paper tackles the challenge of estimating HTEs in time-varying settings, where treatment sequence overlap decreases with longer horizons. The authors propose a Weighted Orthogonal (WO) meta-learner, which introduces an overlap-weighted, Neyman-orthogonal population risk function to focus learning on regions with sufficient treatment overlap. This design stabilizes estimation variance and enhances robustness to nuisance model misspecification. The method is model-agnostic and demonstrated with transformer and LSTM backbones, showing improved reliability and accuracy over existing time-varying meta-learners.
weaknesses: * **Marginal Absolute Improvement:** While the paper reports large *relative* improvements the *absolute* improvements in RMSE are often in the $10^{-2}$ magnitude. The practical significance of such a small absolute gain is questionable, especially given the significant added complexity of estimating the new WO pseudo-outcomes and weight functions.
* **Flawed Real-World Validation:** The validation on real-world data is fundamentally flawed. The authors state this is a "real-world outcome estimation" task, which is a standard *prediction* (supervised learning) task, not a *causal inference* (HTE estimation) task. Demonstrating that the WO-learner is "not worse" than baselines on a predictive task provides **no evidence** of its efficacy for its stated purpose (HTE estimation). This experiment does not support the paper's core claims and raises questions about the method's applicability.
* **Omission of Key Baselines:** The paper omits a crucial and common baseline: a standard DR learner or IPW learner with **propensity score truncation (clipping)**. This is a very common heuristic to handle low-overlap regimes. It is unclear if the WO-learner's complexity is justified over this much simpler, strong baseline.
* **Lack of Intuition for Key Formulas:** The derivation of the key components, particularly the WO pseudo-outcomes $\xi_{t}^{\circ}$ and the weighting term $\rho_{t}^{\circ}$ in Definition 4.2, lacks intuition. They appear to be complex formulas derived to make the orthogonality proof work, rather than being motivated by a clear, first-principles-based explanation. This makes the method difficult to understand and build upon.
* **Unclear Practicality of New Nuisance Function:** The new weight function $\omega_{j}^{\overline{a}}$ (Eq 11) is itself a complex expectation of products of future propensity scores. The practical challenges and stability of estimating this new, non-trivial nuisance function (especially for long horizons $\tau$) are not sufficiently discussed.
* **Missing Discussion of Failure Cases:** The paper does not adequately discuss the potential failure cases or limitations of the WO-learner. When would this method *not* be preferred? How does it behave in very high-dimensional or extremely sparse data settings? A "Limitations" section is missing.
* **Overall Impression and Presentation:** Overall, while the problem is valid and the method is technically correct, the contribution feels incremental. The solution is an orthodox application of established causal inference principles (weighting, orthogonality) to the time-series setting, but the practical gains appear marginal, and the real-world validation is not fit-for-purpose. The work lacks a "surprising" element. Furthermore, the paper's presentation is overly dense, and it notably lacks a "Conclusion" section, which weakens the paper's summary and impact.