Doubly Robust Estimation of Causal Effect on CVR with Targeted Regularization

Doubly Robust Estimation of Causal Effect on CVR with Targeted Regularization

Jiayi Dan 1 Bo Li 1 Lu Deng 2 Yong Wang 2

Abstract

1. Introduction Causal effect estimation plays a critical role in enterprise decision-making. In e-commerce and ride-hailing scenarios, platforms distribute coupons to stimulate user transactions. Existing research has extensively studied the causal effect of strategies on conversion probability, yet the stage-wise mechanism behind the conversion process has received relatively less attention. For example, high-value coupons with strict usage requirements can substantially increase users’ click probability while simultaneously reducing post-click conversion probability, which indicate potential deterioration in user experience. Industry practitioners have recognized this issue and introduced post-click conversion rate (CVR) as a core metric describing second-stage conversion efficiency. Let Y1 and Y2 denote whether a user clicks and whether conversion occurs (with Y2 = 1 implying Y1 = 1); CVR is defined as P(Y2 = 1 | Y1 = 1). A healthy policy should improve overall conversion probabilities while ensuring stage-wise efficiency does not degrade.

arXiv:2608.13461v1 [cs.LG] 13 Aug 2026

Post-click conversion rate (CVR) is a key metric in various scenarios including e-commerce and advertising, reflecting the efficiency and user experience in the second stage of the conversion process. Estimating the causal effect on CVR is therefore of great practical importance. However, directly applying existing causal inference methods to clicked samples introduces sample selection bias and increased variance due to the exclusion of non-click data. Recent studies on CVR prediction introduce “ideal loss”, which optimizes model parameters using an unbiased estimate of the loss over the full sample. Nevertheless, there is no guarantee that unbiasedness of the loss implies unbiasedness of the final estimator. We revisit this challenge from the perspective of semiparametric theory. Specifically, we develop a new doubly robust causal effect estimator for chain-structured outcomes such as CVR, and derive its theoretical properties in detail. It achieves a faster convergence rate compared to nuisance parameters estimation and is therefore more robust when using flexible nonparametric estimators, including neural networks. Based on these theoretical findings, we further design a framework based on targeted regularization to improve numerical stability and practical applicability.

Under the potential outcomes framework, let A and X denote treatment and covariates, we define the conditional and average potential outcomes of CVR as: ψa (x) = P(Y2 (a) = 1 | Y1 (a) = 1, X = x) , ψa = E[P(Y2 (a) = 1 | Y1 (a) = 1, X)] , which is defined on the entire population under the strategy. From a mechanism decomposition perspective, mediation analysis is another classic framework, aiming to decompose the total effect into direct and indirect effects. A key step is estimating cross-world expectations E[Y (a, M (a∗ ))] where M denotes the mediator. We cannot simply treat Y1 as a mediator since Y1 and Y2 are always defined under the same treatment value, and mediation analysis degenerates into standard causal inference without cross-world quantities, since E[Y (a, M (a))] ≡ E[Y (a)]. To circumvent crossworld estimation, (Robins and Greenland, 1992) proposed the controlled direct effect (CDE), which is characterized by the estimand E[E(Y (a, m) | X)]. We cannot adopt related methods for two main reasons: (1) the identifiability assumptions M ⊥⊥ Y (a, m) | A, X rarely hold as the dependency between click and conversion cannot be blocked, (2) the key

Extensive experiments on synthetic and realworld data demonstrate the effectiveness and robustness of our method. In addition, we find that naively combining loss debiasing with standard causal estimators underperforms our method, highlighting the necessity of developing the new estimator tailored to this CVR-style objective with solid theoretical guarantees. 1

Tsinghua University, Beijing, China 2 Tencent Inc., Shenzhen, China. Correspondence to: Jiayi Dan, Bo Li <danjy24@tsinghua.org.cn, libo@sem.tsinghua.edu.cn>.

1

2. Related Works

nuisance parameter E(Y2 | Y1 = 1, X = x, A = a) cannot be estimated directly. We cannot regress Y2 on Y1 , X, and A as in mediation analysis since Y2 is strictly dependent on Y1 (Y1 = 0 → Y2 = 0), and selecting samples with Y1 = 1 would introduce selection bias.

2.1. Causal Inference based on Deep Learning Deep learning has become a key tool for causal effect estimation due to its flexibility. Many studies focus on mitigating confounding bias through balanced representation learning. (Johansson et al., 2016) and (Shalit et al., 2017) first established a generalization bound incorporating empirical risk and Integral Probability Metric (IPM) distance. Building on this, several works introduce weighting strategies(Johansson et al., 2018; Hassanpour and Greiner, 2019; Assaad et al., 2021). (Wang et al., 2022) and (Kazemi and Ester, 2024) extend this paradigm to continuous treatment setting. Meanwhile, another classical neural-based causal estimation paradigm emphasizes propensity-score estimation (Shi et al., 2019; Nie et al., 2021). In particular, (Nie et al., 2021) propose a spline-based variable-coefficient neural network for continuous treatment settings, which prevents the treatment signal from being overshadowed by other high-dimensional covariates. As industry applications continue to grow, some recent studies adopt more flexible network structures(Ke et al., 2021; Zhong et al., 2022; Liu et al., 2023). Beyond standard feedforward networks, specialized architectures have also been explored, including VAE (Louizos et al., 2017), GAN (Yoon et al., 2018) and diffusion models(Sanchez and Tsaftaris, 2022). However, directly applying the above methods to causal effect estimation on CVR can only be conducted on the clicked subset, which may lead to selection bias and data sparsity issues.

The preceding discussion leads to the first challenge: selection bias. Most strategies, such as coupon allocation, are implemented prior to any click behavior, and our focus is on the full population. Therefore, directly treating Y2 as the outcome and applying existing causal estimators to samples with Y1 = 1 would introduce bias due to the distribution shift induced by click selection. (Huang et al., 2024) demonstrated that estimating treatment effect on CVR only with clicked samples will lead to incorrect conclusions. However, its proposed method applies only to RCT data and lacks sound theoretical properties. Another line of work introduces the notion of ideal loss to correct selection bias and develops related CVR prediction models (Wang et al., 2019; Zhang et al., 2020; Guo et al., 2021; Dai et al., 2022; Li et al., 2022b;a). These methods construct unbiased estimators of the loss over full samples, yet no theory directly guarantees that unbiased loss implies unbiased estimation of the target estimand. To address these limitations, we propose a new estimator for CVR causal effect with both practical applicability and theoretical guarantees. It directly targets the final estimand instead of relying on loss debiasing. Specifically, we derive the influence function and the von Mises expansion of the target estimand based on semiparametric theory. This inspires us to construct a doubly robust estimator, which remains consistent even if one of the nuisance estimators is inconsistent, and achieves desirable asymptotic properties. Leveraging the above theoretical results, we extend the idea of targeted regularization to develop a more stable estimation framework.

2.2. Doubly Robust Estimator and Targeted Regularization To address the estimation bias in the plug-in estimator, the DR learner—first proposed by (Van der Laan, 2005) and further developed by (Chernozhukov et al., 2017) and (Chernozhukov et al., 2018)—achieves doubly robustness by incorporating a bias correction term. Their theoretical work √ demonstrated that these doubly robust estimators attain nconvergence rates under appropriate conditions. Following this framework, (Nie and Wager, 2021) introduced the R-Learner, a general class of two-step algorithms for estimating treatment effects. (Kennedy, 2024) provides a comprehensive review of doubly robust estimation, with emphasis on its asymptotic properties and minimax-style efficiency bounds. To overcome the finite-sample instability of the standard doubly robust estimator based on one-step correction(Kennedy, 2024),(Shi et al., 2019) introduced targeted regularization inspired by TMLE (Van der Laan et al., 2011), which corrects estimation bias through an additional perturbation parameter, (Nie et al., 2021) further extends this practical framework to continuous-treatment setting. The above studies provide a solid theoretical foundation for establishing our new framework tailored to CVR. However,

Our contributions can be summarized as follows: (1) We revisit the problem from a semiparametric perspective. Specifically, we develop a new doubly robust estimator tailored to chain-structured outcomes, especially CVR, which is a nontrivial constructive problem, and further show its desirable theoretical properties based on the von Mises expansion. (2) Building on the above findings, we develop a practical estimation framework based on targeted-regularization to enhance applicability and empirical performance. (3) We conduct extensive experiments on synthetic, semisynthetic, and real-world data, demonstrating the strong performance and robustness of our proposed method.

2

our target estimand and nuisance parameters differ fundamentally from those in standard causal-inference problems, requiring corresponding extensions and adaptations.

notational device in theoretical analysis. Let (z1 , . . . , zn ) be i.i.d. samples drawn from P, and denote zi = (xi , ai , y1i , y2i ), where xi ∈ X ⊂ Rd is the covariate vector, and ai ∈ A ⊂ R is the treatment. Here y1i ∈ {0, 1} indicates whether a click occurs, and y2i indicates whether a conversion occurs , where y2i = 1 implies y1i = 1. We use uppercase Z, X, A, Y1 , Y2 to denote random variables corresponding to the observed lower-case ones. To ensure general applicability, we denote the treatment variable A as continuous, which naturally extends to the discrete treatment case (Nie et al., 2021).  Our target estimand is defined as: ψa (P) = E P(Y2 (a) = 1 | Y1 (a) = 1, X) ., which is a functional regarding the data-generating distribution P. For standard causal inference problems, the estimand is typically ψa = E[E(Y (a) | X)]. In our setting, A is not randomly assigned, and both Y1 and Y2 depend on A. A naive strategy is to restrict estimation to samples with Y1 = 1 and estimate E[P(Y2 (a) = 1 | X)] with existing methods. However, this introduces bias since Y1 is non-random.

2.3. Debiased CVR Prediction Sample-selection bias is a major challenge in CVR prediction, motivating many researchers to employ techniques from causal inference to address this problem. (Wang et al., 2019) first propose the “ideal loss” concept, seeking to derive an unbiased estimate of the training loss over the entire sample. Building on this idea, follow-up studies refine the estimation method to improve accuracy and reduce the variance (Zhang et al., 2020; Guo et al., 2021; Dai et al., 2022; Li et al., 2022b;a). However, this series of studies focus on CVR prediction instead of causal effect estimation. Moreover, their starting point is to estimate the loss over the entire sample, not to target the final estimand, while no theoretical results indicate that an unbiased loss necessarily leads to an unbiased final estimate. For causal-effect estimation on CVR, (Huang et al., 2024) demonstrate through real-world data analysis that restricting to clicked samples may yield misleading conclusions, and propose a multitask model for estimation of CTR and CVR treatment effects. This work further corroborates the importance of our problem, but the proposed method is only applicable to randomizedexperiment data. In addition, the approach mainly focuses on network-structure design and can be regarded as a plugin estimator, which lacks desirable theoretical properties including double robustness and consistency.

We define µ1 (x, a) = E(Y1 | X = x, A = a, µ2 (x, a) = E(Y2 | X = x, A = a), and let π(a | x) denote the treatment density given X = x (propensity score). Our estimation procedure will involve three nuisance parameters (µ1 , µ2 , π). For notational simplicity in later derivations, we write µ1 , µ2 , π and µ̂1 , µ̂2 , π̂ to refer to µ1 (x, a), µ2 (x, a), π(a | x) and µ̂1 (x, a), µ̂2 (x, a), π̂(a | x). Our analysis is based on the following assumptions:

Research gap: To the best of our knowledge, few studies have directly targeted the final estimand for CVR causal effect estimation while providing both sound theoretical guarantees and strong practical utility.

(i) Overlap. There exists a constant c > 0 such that for all x ∈ X and a ∈ A, π(a | x) ≥ c; (ii) Unconfoundedness. Covariates X block all back-door paths between treatment A and outcomes Y1 , Y2 ;

3. Problem Formulation and Notations Notation We use E to denote expectation, P ∈R P to denote true probability measure and we write P(f ) = f (z)dP(z), where P is a set of possible probability distributions. Similarly, we use Pn Rto denote the empirical measure and we write Pn (f ) = f (z)dPn (z). We denote convergence

(iii) There exists a constant c > 0 such that µ1 (x, a) ≥ c. Assumptions (i) and (ii) are standard in causal inference. Assumption (iii) is reasonable in practice since the population is finite and click probability usually has a positive lower bound, samples with µ1 = 0 have undefined CVR and are outside our scope.

d

in distribution by → and convergence in probability by p →. Xn = OP (rn ) means Xn /rn is bounded in probap bility and Xn = oP (rn ) means Xn /rn → 0.τ denotes Rademacher random variables. We denote Rademacher complexity ofP a function class F : X → R as Radn (F) = n E(supf ∈F | n1 i=1 τi f (Xi )|). Given two functions f1 , f2 : X → R, we define ∥f1 R− f2 ∥∞ = supx∈X |f1 (x) − f2 (x)| and ∥f1 − f2 ∥L2 = ( x∈X (f1 (x) − f2 (x))2 dx)1/2 . For a function class F, we define ∥F∥∞ = supf ∈F ∥f ∥∞ . an ≍ bn denotes that both an /bn and bn /an are bounded. δ(·) denotes the Dirac measure, which is used only as a

Under the assumptions above,

  P(Y2 (a) = 1, Y1 (a) = 1 | X =) ψa (P) = E P(Y1 (a) = 1 | X)     E(Y2 (a) | X) E(Y2 | X, A = a) =E =E . E(Y1 (a) | X) E(Y1 | X, A = a) We fully exploit the dependence between Y1 and Y2 , which 3

allows all nuisance parameters to be estimated using the full sample.

However, the validity of this naive strategy heavily relies on the function class of the prediction models. When using nonparametric methods such as neural networks instead of maximum √ likelihood, the nuisance estimators often fail to achieve n convergence and may introduce non-negligible estimation bias. To better understand the estimation error and motivate the method developed later, consider the Taylor expansion of µ̂µ̂21 − µµ12 :

4. Doubly Robust Estimation 4.1. Plug-in Estimator and Nuisance Parameter Estimation We begin with a naive plug-in estimator to illustrate the estimation of nuisance components and the limitations of this approach. To leverage the full dataset, the simplest strategy is to predict µ1 (x, a) and µ2 (x, a) using covariates and the treatment. The final estimate is obtained by taking their ratio: n 1 X µ̂2 (xi , a) ψ̂ plug-in = . n i=1 µ̂1 (xi , a)

µ̂2 1 µ2 µ̂2 − = − 2 (µ̂1 − µ1 ) + (µ̂2 − µ2 ) µ̂1 µ1 µ̂1 µ̂1
+ O (µ̂1 − µ1 )2 + |µ̂1 − µ1 | |µ̂2 − µ2 | . The “first-order bias” terms µ̂1 − µ1 and µ̂2 − µ2 imply that the plug-in estimator is not consistent unless all nuisance parameters are consistently estimated. The convergence rate is dominated by the slowest-converging nuisance parameter, which makes the estimator sensitive to model misspecification, especially under flexible nonparametric estimation that trades increased bias for reduced variance. Eliminating first-order bias is precisely the motivation behind doubly robust estimation (Kennedy, 2024),which we will discuss next.

This is similar to the approach for CVR prediction used in ESMM (Ma et al., 2018). For causal estimation, following (Shi et al., 2019), we extract representations related to the treatment A by predicting π(a | x) before estimating µ2 (x, a) and µ1 (x, a), enabling more accurate nuisance estimation with reduced noise. In practice, we add a small constant ϵ (e.g. 10−9 ) to µ̂1 (x, a) to ensure a strictly positive lower bound. Under continuous treatment settings, a good estimator should not be dominated by other features, and maintain continuity and smoothness. To achieve this, we adopt the varying coefficient model (Hastie and Tibshirani, 1993; Fan and Zhang, 1999; Chiang et al., 2001) to estimate µ2 (x, a) and µ1 (x, a), where the treatment variable a directly determines neural network parameters. Specifically, spline basis functions are used to parameterize weights: w(a) =

L X

4.2. The New Doubly Robust Estimator A good estimator can achieve lower bias and is robust to model misspecification, i.e., DR property (Bang and Robins, 2005; Kennedy, 2024; Chen et al., 2024). In this section, we propose a new DR estimator tailored to our target estimand, which remains consistent even if not all nuisance parameters are consistently estimated, and achieves a faster convergence than the nuisance estimators. This is a nontrivial constructive problem, and it forms the theoretical basis that supports our later model design. Specifically, we construct the estimator as follows:

αℓ ϕℓ (a),

ℓ=1

where αℓ are coefficients and ϕℓ (·) are polynomial basis functions. Therefore, µ2 (x, a) and µ1 (x, a) are continuous as long as the network activations are continuous. For the generalized propensity score π(a | x), the key difficulty is ensuring that the estimator forms a valid density for continuous treatment. Following (Nie et al., 2021), we uniformly discretize the interval [0, 1] into B grids and estimate π(·|x) on these (B + 1) grid points as πgrid (x) = sof tmax(wx) ∈ RB+1 , and then the conditional density could be given via linear interpolation, i.e., a1 a2 a1 π(a|x) = πgrid (x) + B(πgrid (x) − πgrid (x))(a − a1 ), where a1 = ⌊Ba⌋, a2 = ⌈Ba⌉.

ψ̂adr = Pn



  
µ̂2 δ(A = a) + Pn (Y2 − µ̂2 )µ̂1 − (Y1 − µ̂1 )µ̂2 µ̂1 π̂ µ̂21

:= ψa (P̂) + Pn (ϕa (P̂)). h

i where ϕa (P) = δ(A=a) µ Y − µ + µ Y − µ + 2 1 2 2 2 1 1 πµ 1

µ2 µ1 − ψa (P) is the influence function of the target estimand,

which can be viewed as the “derivative over the distribution space”(Kennedy, 2024). P̂ denotes the fitted data-generating distribution.

We estimate the nuisance parameters by minimizing the negative log-likelihood, and the overall training loss is n i 1 Xh L(µ̂1 , µ̂2 , π̂) = ℓ(y1i , µ̂1 ) + ℓ(y2i , µ̂2 ) − log π̂ . n i=1

To gain intuition about the construction and properties of this estimator, we begin with the following lemma, which can be regarded as the ”distributional Taylor expansion” of the target estimand. 4

Lemma 4.1. ψ confirms the von Mises expansion (Kennedy, 2024): Z ψa (P̂) − ψa (P) = − ϕa (P̂) d(P) + R2 (P̂, P),

networks (Chernozhukov et al., 2018; Farrell et al., 2021). In contrast, root-n consistency is a desirable property that is difficult for nonparametric estimators to achieve. Theorem 4.2 shows that ψ̂adr attains this property under mild assumptions, allowing us to obtain accurate estimates using flexible nuisance estimators. This is precisely the asympwhere totic meaning of “double robustness” (Kennedy, 2024; Chen  µ2 δ(A = a)  (Y2 −µ2 )µ1 −(Y1 −µ1 )µ2 + −ψa (P), et al., 2024): the final estimator converges faster than any ϕa (P) = πµ21 µ1 of the nuisance estimators, and is therefore robust to model misspecification. (Kernelized IF can be used to improve smoothness under continuous treatment setting(Kennedy Z et al., 2017), and it can be extended to heterogeneous causal 1 R2 (P̂, P) = (π − π̂)(µ2 − µ̂2 ) dP(x) effects estimation by pseudo-outcome regression (Kennedy, π̂ µ̂1 Z Z 2023). We do not elaborate on these standard practices in µ̂2 − (π − π̂)(µ1 − µ̂1 ) dP(x) + Rµ dP(x), detail since they are not specific to the target estimand and 2 π̂ µ̂1 is not the focus of our study.) The theoretical results above lay the foundation for constructing the models for causal effect estimation on CVR. However, the direct one-step correction approach may be unstable in practice, and can even produce estimates outside the parameter space (Kennedy, 2024). In our setting specifically, µ̂1 appears in the denominator of the correction term Pn (ϕa (P̂)). Since click-through rates are often low in practice, the correction term may become extremely large, further amplifying this instability. Furthermore, the crossfitting required by condition 1, which estimates nuisance parameters using sample-splitting across multiple folds, introduces additional complexity. To address these issues, we extend the idea of targeted regularization to our target estimand based on the findings above, and develop a more practical framework.

and 



|Rµ | ≤ C (µ̂1 − µ1 )2 + |µ̂1 − µ1 | |µ̂2 − µ2 | . Lemma 4.1 suggests how to debias the plug-in estimators, R namely by estimating the ”first order bias term” − ϕa (P) and subtracting it off, which leads directly to the construction of ψ̂adr . This ”one-step correction” approach can be viewed as a generalization of Newton methods. Intuitively, if the first-order bias is eliminated, the estimation error will be dominated by the second-order remainder R2 (P̂, P), which only involves second-order errors of the nuisance estimators, and thus yields a faster convergence rate. Formally, we analyze the asymptotic properties of ψ̂adr via the decomposition:

ψ̂adr − ψa (P) = (Pn − P){ϕa (P)} + (Pn − P){ϕa (P̂) − ϕa (P)} + R2 (P̂, P) 5. A Practical Framework Based on Targeted

Regularization

which leads to the following theorem, based on Lemma 4.1.

As discussed above, the basic one-step correction procedure consists of two stages: (1) estimating nuisance parameters (which typically involves cross-fitting), (2) predicting the correction term and adding it to the plug-in estimator. In practice, this two-stage procedure has been found to be insufficiently stable. Inspired by TMLE(Van der Laan et al., 2011), the widely used DragonNet (Shi et al., 2019) introduces targeted regularization. The idea is to force the correction term Pn (ϕa (P̂)) ≈ 0 with an extra regularizer, and recover it by learning a low-dimensional parameter ϵ. Extensive empirical results demonstrate that this approach yields a more stable and practically friendly framework. Nie et al. (Nie et al., 2021) further extend this framework to continuous treatment settings. However, the above methods are designed for standard causal estimand, and both their methodological construction and theoretical analysis largely rely on the well-studied doubly robust estimator for   E E(Y | X, A = a) . In contrast, such an established the-

Theorem 4.2. If √ 1. (Pn − P){ϕa (P̂) − ϕa (P)} = oP (1/ n); 2. ∥π̂(a|x) − π(a|x)∥ = oP (n−1/4 ); 3. ∥µ̂1 (a, x) − µ1 (a, x)∥ = oP (n−1/4 ); 4. ∥µ̂2 (a, x) − µ2 (a, x)∥ = oP (n−1/4 ), then √ ψ̂adr − ψa = (Pn − P){ϕa (P)} + oP (1/ n), and so ψ̂adr is root-n consistent, asymptotically normal by the central limit theorem and Slutsky’s theorem. Condition 1 can be satisfied by cross-fitting (Kennedy, 2024). The conversion rate oP (n−1/4 ) in conditions 2 and 3 can be satisfied by various estimators including neural 5

(iv) BKn equals the closed linear span of B-spline with equally spaced knots, fixed degree, and dimension Kn ≍ n1/6 .

oretical foundation is not available for our setting, which is a key challenge of this work. Building on the preceding theoretical results, we now extend this framework to our target estimand.   For the standard estimand ψ ′ = E E(Y | X, A = a) , the ′ influence function is δ(A=a) π(a|x) (Y − µ(x, a)) + µ(x, a) − ψ , and the targeted regularization is constructed as (Shi et al., 2019; Nie et al., 2021):

Theorem 5.1. Under the assumptions above and the loss function LT R , we have p ∥ψ̂atr − ψa ∥ =Op n−1/3 log n + r1 (n)r2 (n)
+ r1 (n)r3 (n) + r2 (n)r3 (n) + r2 (n)2 , where

2 n  1X ϵ̂(ai ) R′ (y, µ̂, π̂, ϵ̂) = . yi − µ̂i − n i=1 π̂i

∥µ̂2 − µ2 ∥∞ = Op (r3 (n)). The proof of Theorem 5.1 is in the Appendix. The main idea is to decompose ψ̂atr − ψa into three terms, among which a key component is exactly the remainder R2 (P̂, P) derived in the previous section. Once the first-order bias is controlled by targeted regularization, the convergence rate is dominated by this remainder, allowing us to leverage the results established earlier to obtain the conclusion.

which is minimized directly during the training phase, ϵ is an additional learnable parameter. The final estimator becomes: n

ψ̂atr = ψ̂aplug−in +

∥π̂ − π∥∞ = Op (r1 (n)), ∥µ̂1 − µ1 ∥∞ = Op (r2 (n)),

1 X ϵ̂(a) . n i=1 π̂i

Intuitively, it trains an additional parameter ϵ to approximate the correction term during the training phase. According to the influence function of our target estimand (as mentioned in the previous section), we adapt targeted regularization as follows:

Theorem 5.1 shows that under mild regularity conditions, ψ̂atr maintains the doubly robust property, i.e. it achieves a convergence rate exceeding the individual convergence rates of the nuisance parameter estimators, and is therefore robust to model misspecification, allowing the use of flexible nonparametric estimators, including neural networks.

n

i2 1 X h (y2i −µ̂2i ) (y1i −µ̂1i )µ̂2i ϵ̂(ai ) R(µ̂1 , µ̂2 , π̂, ϵ̂) = − − . As discussed in (Nie et al., 2021), adding the extra regular2 µ̂ π̂ µ̂ 1i i 1i n i=1 izer does not affect the consistency of nuisance parameter estimation and can substantially improve estimation stabilCombining the loss function of plug-in estimator, we define ity. The magnitude of the targeted regularization term is the final loss function as: controlled by a hyperparameter β, which allows it to be kept on a comparable scale to the main loss function. Intuitively, LT R (µ̂1 , µ̂2 , π̂, ϵ̂) = L(µ̂1 , µ̂2 , π̂) + βR(µ̂1 , µ̂2 , π̂, ϵ̂), this approach replaces the “hard” correction in the original DR estimator with a “soft” correction via regularization. In Under continuous treatment setting, optimizing ϵ(a) over practice, stability can be further enhanced by controlling the the function space of all mappings from A to R is not feasilearning rate of ϵ or by introducing additional regularization ble in practice, we follow (Nie et al., 2021) to use splines on ϵ. Moreover, this end-to-end method does not involve n {Bk }K k=1 with Kn basis function Bk (·) to approximate ϵ. cross-fitting, which leads to substantially higher efficiency in realistic settings involving complex models. Therefore, We now leverage Lemma 4.1 and Theorem 4.2 to derive we adopt this approach as the primary modeling framework. the asymptotic properties of ψ̂atr . We first state several standard and mild assumptions commonly adopted in the literaAs discussed above, obtaining the final estimator requires ture (Nie et al., 2021): estimating three nuisance parameters: π, µ , and µ , where 1

2

µ1 corresponds exactly to the click probability. A natural idea is that once π̂ and µ̂1 are obtained, one can immediately construct an estimator for the causal effect of the clickthrough rate (CTR):

(i) There exists constant c > 0 such that for any a ∈ A, x ∈ X , and π̂ ∈ U, we have 1/c ≤ π̂(a | x) ≤ c, 1/c ≤ π(a | x) ≤ c, ∥Q∥∞ ≤ c and ∥µ1 ∥∞ ≤ c,∥µ2 ∥∞ ≤ c.

ψactr = E[E(Y1 | X, A = a)] .

(ii) π, µ1 , µ2 and π̂, µ̂1 , µ̂2 have bounded second derivatives for any π̂ ∈ Q µ̂1 ∈ U1 and µ̂2 ∈ U2

Motivated by this observation, we design a multi-task model to jointly estimate the causal effects of CTR and CVR, denoted by ψ̂actr and ψ̂acvr . The overall loss function becomes:

(iii) The Rademacher complexities satisfy Radn (G), Radn (Q), Radn (U1 ), and Radn (U2 ) = O(n−1/2 ).

Lmulti−task = L1 + L2 + αLπ + β(R1 + R2 ) 6

where

L1 =

L2 =

• RQ3: Is the proposed method stable under different hyperparameter settings?

n
1X Lπ = − log π̂(ai | xi ) , n i=1 n 1 X

n i=1

• RQ4: Is it necessary to redesign the estimation framework, or can existing loss-debiasing approach developed for CVR prediction be directly combined with standard causal effect estimation methods?



− y1i log(µ̂1i ) − (1 − y1i ) log(1 − µ̂1i ) ,

n  1 X − y2i log(µ̂2i ) − (1 − y2i ) log(1 − µ̂2i ) , n i=1

6.1. Setup 6.1.1. S YNTHETIC AND S EMI - SYNTHETIC DATA G ENERATION

2 n  ϵ̂1 (ai ) 1X y1i − µ̂1i − , R1 = n i=1 π̂i R2 =

n  X

1 ϵ̂2 (ai ) y2i − µ̂2i (y1i − µ̂1i )µ̂2i − − 2 n i=1 µ̂1i µ̂1i π̂i

To obtain the true value of the target estimand, we generate synthetic datasets following (Nie et al., 2021), each dataset consists of 3,000 training samples and 1,000 test samples. Further, we reuse the 498-dimensional covariates X of the real-word dataset News to generate the assigned treatments and their corresponding outcomes following (Schwab et al., 2020; Bica et al., 2020; Nie et al., 2021), and each dataset is randomly split into a training set (67%) and a test set (33%). Hyperparameters are tuned on 20 extra synthetic datasets disjoint from the test samples. We provide details of data generation in Appendix A.4.

2 .

The final estimators are given by ψ̂actr =

n n ϵ̂1 (a) 1X 1 X µ̂2i ϵ̂2 (a) [µ̂1i + [ ], ψ̂acvr = + ]. n i=1 π̂i n i=1 µ̂1i π̂i

Figure 1 illustrates the overall model architecture. To explicitly model the dependency µ2 < µ1 , we construct the prediction of µ2 as µ̂2 = µ̂1 × µ̃2 , where µ̃2 ∈ (0, 1). Motivated by the TMLE, targeted regularization should not affect the estimation of π̂. Accordingly, gradients from the targeted regularization terms R1 and R2 are blocked from propagating to π̂, which improves the accuracy of propensity score estimation.

Table 1 reports the means and standard deviations of the observed outcomes Y1 and Y2 on both simulated and semisynthetic datasets (with one representative sample selected). As shown, the mean of Y1 is below 0.5, and overall P (Y2 = 1 | Y1 = 1) < P (Y1 = 1), which is consistent with realworld scenarios. Table 1. Summary Statistics of Outcomes

Synthetic Data Y1

Y2

Mean 0.301 0.052 Std. Dev. 0.459 0.222

Y2 | Y1 = 1 0.170 0.375

News Y1

Y2

0.347 0.080 0.476 0.272

Y2 | Y1 = 1 0.231 0.421

6.1.2. BASELINES AND S ETTINGS We compare our method with: (1) DragonNet (Shi et al., 2019): Introduces targeted regularization for causal effect estimation, requiring discretization for continuous treatments; (2) VCNet (Nie et al., 2021): Extends DragonNet to continuous treatments via varying-coefficient networks and spline functions; (3) DRNet (Schwab et al., 2020): Estimates continuous treatment effects by partitioning the treatment space while preserving treatment information through hierarchical inputs; (4) TARNet (Shalit et al., 2017): Learns a shared representation across treatment groups; (5) Causal Forest (Athey et al., 2019): Generalizes random forests to estimate causal effects; (6) ECUP (Huang et al., 2024): A recent method

Figure 1. Model architecture.

6. Experiments Our experiments address the following questions: • RQ1: For our target estimand, does the proposed method outperform existing baseline models? • RQ2: Does the targeted regularization lead to improved estimation performance? 7

focusing on treatment effect estimation for CVR, and leverages attention mechanisms to enhance the interaction between the treatment and other features; (7) DR Estimator: The basic doubly robust estimator proposed in Section 4 with cross-fitting; (8) Ours (w/o. t-reg): Our method without the designed targeted regularization, which reduces to a plug-in estimator;

validating the feasibility of jointly estimating the causal effects on CTR and CVR. • Without the designed targeted regularization, the performance of our method drops significantly, demonstrating its effectiveness and necessity. • The performance of the basic DR estimator is also competitive; however, it is inferior to our final proposed method, which is consistent with the discussion in previous sections and findings in (Shi et al., 2019; Nie et al., 2021).

Since we focus on the underlying methodological framework, our baselines exclude approaches that solely target model architecture design (e.g., (Liu et al., 2023)). Our backbone network can be replaced by arbitrary architectures; here, we unify the model capacity (e.g., the number of layers and hidden layer dimensions) across all methods for fair comparison. Some baselines require adaptation for continuous treatment, and we follow the implementation in (Nie et al., 2021).

6.2.2. H YPERPARAMETER S ENSITIVITY A NALYSIS We conduct a sensitivity analysis by varying the propensity score loss weight α and the targeted regularization weight β, and evaluate the resulting AMSE on the CVR task. As shown in the Figure 2, our method exhibits strong stability across a wide range of α values, the AMSE consistently remains substantially lower than that of the ablation study, demonstrating the robustness of our method. In contrast, Figure 3 reveals a monotonic degradation in performance as the targeted regularization weight β decreases, which further verifies the effectiveness and necessity of this component.

Each model is trained for 600 epochs. The loss weights α and β are set to 0.5 and 1, respectively. The learning rate and the number of hidden units are selected via grid search within {10−5 , 10−4 , 5 × 10−4 , 10−3 } and {8, 32, 128} (we observed that model performance is insensitive to the scale of model capacity). All other settings are kept the same as in (Nie et al., 2021). We replace the MSE loss in baseline models with cross-entropy loss since the outcomes are binary. 6.1.3. M ETRIC We use the average mean squared error (AMSE) as the primary evaluation metric, which is consistent with prior studies on causal effect estimation with continuous treatment (Nie et al., 2021).

(a) Synthetic Data S Z  s 2 1X AMSE = ψ̂ − ψa π(a) da, S s=1 A a

(b) News

Figure 2. AMSE under different settings of α.

where ψ̂as denotes the estimate obtained in the s-th simulation. 6.2. Result and Analysis 6.2.1. OVERALL P ERFORMANCE AND ABLATION STUDY Table 2 presents the AMSE of our proposed multi-task framework and baseline models on synthetic data and News, where the “CVR task” and “CTR task” correspond to the estimation of ψ̂acvr and ψ̂actr , respectively. Several key observations can be drawn from the results:

(a) Synthetic Data

(b) News

Figure 3. AMSE under different settings of β.

6.2.3. C OMPARISON WITH DIRECT LOSS DEBIASING

• Overall, our proposed method significantly outperforms all baselines on the CVR task across different datasets, demonstrating strong effectiveness. The performance on the CTR task remains desirable when additional task and regularization terms are introduced,

While our proposed method directly targets the final estimand, a well-established line of prior work that specifically addresses the sample selection bias issue in CVR prediction is based on the concept of the ideal loss (Wang et al., 2019; Zhang et al., 2020; Guo et al., 2021; Dai et al., 2022; 8

Table 2. Results on Synthetic Data and News. Reported: AMSE (mean ± standard deviation) over 20 simulations.

Synthetic Data DragonNet VCNet DRNet TARNet Causal Forest ECUP DR Estimator

News

CVR Task

CTR Task

CVR Task

CTR Task

0.01130±0.00359 0.00935±0.00287 0.01375±0.00392 0.01529±0.00320 0.01031±0.00172 0.00816±0.00209 0.00511±0.00298

0.00115±0.00032 0.00093±0.00031 0.00423±0.00173 0.00621±0.00147 0.00878±0.00080 0.00145±0.00062 0.00310±0.00153

0.00706±0.00157 0.00325±0.00184 0.01112±0.00320 0.01163±0.00224 0.00341±0.00055 0.00233±0.00057 0.00321±0.00202

0.00102±0.00025 0.00040±0.00017 0.00178±0.00083 0.00149±0.00039 0.00132±0.00014 0.00061±0.00032 0.00129±0.00076

Ours(w/o. t-reg) 0.00981±0.00346 0.00473±0.00122 0.00698±0.00249 0.00132±0.00076 Ours 0.00248±0.00087 0.00096±0.00031 0.00101±0.00052 0.00047±0.00025

Li et al., 2022b;a). The core idea of this approach is to train models using only clicked samples, while constructing an unbiased estimator of the loss over the full population, typically via reweighting, and then optimize model parameters using this debiased loss. Existing studies primarily apply this methodology to prediction problems, rather than causal effect estimation. This naturally raises a question: can the loss-debiasing idea be straightforwardly combined with standard causal inference methods? Is it necessary to construct a new estimator tailored to the estimand of this work?

R′2 = |C|−1

2 X 1  ϵ̂′ (ai ) y2i − µ̂′2i − 2 , µ̂1i π̂i i∈C

where gradients are blocked from propagating through µ̂1 in both L′2 and R′2 . The final estimator becomes: ψ̂a′cvr =

 n  1X ϵ̂′ (a) µ̂′2i + 2 . n i=1 π̂i

Table 3 compares the performance of this alternative approach with our previously proposed method on the CVR To answer this question, we propose an alternative approach task (the method and performance on the CTR task rethat combines loss debiasing with standard causal effect main unchanged). It can be observed that the alternative estimation, and compare it with our previously proposed approach consistently outperforms the plug-in estimator method. Specifically, when using standard causal estimators without targeted regularization across different datasets, and directly, selection bias mainly arises from the following two also achieves better results than most baselines reported in components: Tables 2. This is because it also leverages a targeted regularization to reduce estimation bias, and obtains an unbiased • The prediction loss for µ′2 = E(Y2 |Y1 = 1, X = estimate of the loss terms. However, the alternative approach x, A = a), which is similar to that in CVR predicperforms noticeably worse than our proposed method. The tion: key  reason is that it focuses on eliminating the estimation X biased −1 ′ ′ bias of the loss terms, rather than directly targeting the final L2 = |C| −y2i log(µ̂2i )−(1−y2i ) log(1−µ̂2i ) ; estimand. There is no theoretical guarantee that unbiased i∈C estimation of the loss terms necessarily leads to accurate • The standard targeted regularization term(Nie et al., estimation of the final estimand. Even if the ”unbiased loss” 2021): is well approximated over the entire parameter space, its 2 ′ X minimizer may still be poorly estimated. Therefore, it is necϵ̂ (ai ) Rbiased = |C|−1 y2i − µ̂′2i − 2 . 2 essary to directly construct estimation procedures tailored to π̂i i∈C the target estimand with sound theoretical properties, rather than simply combining existing methods. where C denotes the clicked set. Therefore, a straightforward solution is to get unbiased estimates of these two 6.3. Performance on Real-World Dataset terms for backpropagation. Here, we adopt the widely used and most convenient inverse propensity scoring (IPS) To further demonstrate the practical applicability of our method (Schnabel et al., 2016), and replace L2 and R2 in method, we conduct experiments and ablation studies on a our proposed method with L′2 and R′2 , respectively: large-scale real-world dataset CRITEO-UPLIFTv2 (Diemert  X 1  ′ −1 ′ ′ L2 = |C| −y2i log(µ̂2i )−(1−y2i ) log(1−µ̂2i ) ; et al., 2021). It contains approximately 13 million samples µ̂1i with 12 features, a binary treatment variable, and two binary i∈C 9

Table 3. Comparison with the Alternative Method. Reported: AMSE (mean ± standard deviation) over 20 simulations.

Model

Synthetic Data

Table 4. Performance on the CVR task (CRITEO).

AUUC

News

DragonNet TARNet T-Learner Causal Forest ECUP

Alternative Method 0.00494±0.00161 0.00323±0.00260 Ours 0.00248±0.00087 0.00101±0.00052 w/o. t-reg 0.00981±0.00346 0.00698±0.00249

QINI

0.02216 0.04483 0.02031 0.04129 0.01798 0.03467 0.00405 0.00798 0.02765 0.05253

Ours 0.03208 0.06014 Ours (w/o t-reg) 0.01017 0.01950

outcomes: visit and conversion. We randomly sample 10% of the data for our experiments, and further split it into a training, validation and a test set with a ratio of 60%, 20% and 20%.

Following previous work using this dataset (Liu et al., 2023), we compute AUUC and QINI using uplift auc score and qini auc score from sklearn, both of which are normalized by their theoretical maximum values.

Since the publicly available datasets contain only binary treatment, we adapt the varying coefficient network to separately predict nuisance parameters µ1 and µ2 for the two treatment groups, and removed the baselines specifically designed for continuous treatments. Moreover, since the true value of ψ cannot be obtained, commonly used evaluation metrics are AUUC and QINI, which require defining the estimand at the individual level. Our previous theoretical analysis are established at an averaged level, which is consistent with the settings adopted in widely used methods such as DragonNet (Shi et al., 2019) and VCNet (Nie et al., 2021). At the individual level, a natural approach is to remove the final averaging step in ψ̂ tr . In summary, due to the limitations of publicly available datasets, our method may be underestimated. The preceding experiments provide a more comprehensive and rigorous evaluation that better aligns with the settings of our work. This section serves only as a supplementary reference, primarily to further address potential concerns regarding the practical applicability of our method in real-world scenarios.

practical framework based on targeted regularization. Extensive experiments demonstrate the strong performance of the proposed method and the effectiveness of targeted regularization, as well as its advantages over the indirect loss-debiasing approach.

As shown in Table 4, our method still achieves significantly better performance in estimating the causal effect on CVR than the other models. The ablation study once again demonstrates that the targeted regularization term plays a critical role in improving the performance. The cumulative gain curves shown in Fig. 4 provide an intuitive illustration of the performance gains brought by the targeted regularization and the doubly robust design. These results further validate the effectiveness and practical utility of our method in real-world scenarios.

This paper proposes a general framework for estimating causal effects of chain-structured outcomes such as CVR. Our method can be applied to a wide range of applications, including decision-making in e-commerce and online advertising scenarios.

7. Conclusion

References

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A. Appendix

for some bounded constant C. Therefore, Z 1 (π − π̂)(µ2 − µ̂2 ) dP(x) R2 (P̂, P) = π̂ µ̂1 Z Z µ̂2 − (π − π̂)(µ1 − µ̂1 ) dP(x) + Rµ dP(x). π̂ µ̂21

A.1. Proof of Lemma 4.1 Proof. Z R2 (P̂, P) = ψa (P̂) − ψa (P) + ϕa (P̂) dP Z ( δ(A = a) h Y2 − µ̂2 (X, A) = π̂(A|X)) µ̂1 (X, A) µ̂ (X, A)(Y − µ̂ (X, A)) i

A.2. Proof of Theorem 4.2 )

µ̂2 (X, A) dP − ψa (P) Proof. By the decomposition µ̂1 (X, A) Z Z ψ̂adr −ψa (P) = (Pn −P){ϕa (P)}+(Pn −P){ϕa (P̂)−ϕa (P)}+R2 (P̂, P), µ̂2 (y1 − µ̂1 ) = p(y1 , x, a) dy1 dx − √ π̂ µ̂21 X Y1 the first term (Pn −P){ϕa (P)} is n-consistent and asympZ Z y2 − µ̂2 totically normal by the central limit theorem. It therefore + p(y2 , x, a) dy2 dx suffices to show that under π̂ µ̂1 X Y2 Z µ̂2 + dP(x) − ψa (P) ∥π̂(a|x) − π(a|x)∥ = oP (n−1/4 ), µ̂1 Z Z −µ̂2 (y1 − µ̂1 ) ∥µ̂j (x, a) − µj (x, a)∥ = oP (n−1/4 ), j = 1, 2, p(y1 | x, a) π(a | x) p(x) dy1 dx = 2 π̂ µ̂ X Y1 1 and the existence of a constant ε > 0 such that π̂(x, a) ≥ ε Z Z y2 − µ̂2 and µ̂1 (x, a) ≥ ε (with probability 1), we have R2 (P̂, P) = p(y2 | x, a) π(a | x) p(x) dy2 dx + π̂ µ̂1 −1/2 X Y2 o ). P (n Z µ̂2 + dP(x) − ψa (P) By Lemma 4.1 and an application of the Cauchy–Schwarz µ̂1 inequality, we have Z Z −µ̂2 (y1 − µ̂1 ) π p(y | x, a) dy dP(x) =
1 1 µ̂21 |R2 (P̂, P)| ≤ C1 ∥π̂ − π∥ ∥µ̂1 − µ1 ∥ + ∥µ̂2 − µ2 ∥ X π̂ Y1 Z Z   π y2 − µ̂2 + p(y2 | x, a) dy2 dP(x) + C2 ∥µ̂1 − µ1 ∥2 + ∥µ̂1 − µ1 ∥ ∥µ̂2 − µ2 ∥ . µ̂1 X π̂ Y2 Z µ̂2 where C1 , C2 < ∞. Hence, under the above oP (n−1/4 ) + dP(x) − ψa (P) µ̂1 convergence conditions, ! Z Z µ̂2 π π µ2 − µ̂2 − 2 (µ1 − µ̂1 ) dP(x) dP(x) + = R2 (P̂, P) = oP (n−1/2 ). π̂ µ̂1 π̂ µ̂1 ! Z Together with Assumption 1, which states that (Pn − µ2 µ̂2 − − dP(x). P){ϕa (P̂) − ϕa (P)} = oP (n−1/2 ), Slutsky’s theorem imµ1 µ̂1 plies ψ̂adr − ψa (P) = oP (n−1/2 ), Here p(·) denotes the corresponding probability density. and asymptotic normality follows. Define f (x, y) = y/x. Under the assumption that both µ1 and µ̂1 have positive lower bound, the function f is A.3. Proof of Theorem 5.1 twice continuously differentiable on this region. By the second-order Taylor expansion of a bivariate function with Proof. Define a Lagrange remainder, we have  .     (Y2 − µ̂2 ) (Y1 − µ̂1 )µ̂2 S ⊤ µ1 − µ̂1 S = − , ϵ̌ = P A = a P π̂n−2 A = a . n f (µ1 , µ2 ) − f (µ̂1 , µ̂2 ) = ∇f (µ̂1 , µ̂2 ) + Rµ 2 µ̂1 µ̂1 π̂n µ2 − µ̂2 −

2

1

1

µ̂21 (X, A)

+

µ̂2 1 (µ1 − µ̂1 ) + (µ2 − µ̂2 ) + Rµ , From the proof of Lemma 3 in (Nie et al., 2021), which re2 µ̂1 µ̂1 lies only on properties of spline functions and is independent where the remainder term Rµ satisfies of the target estimand, we have   p
|Rµ | ≤ C (µ̂1 − µ1 )2 + |µ̂1 − µ1 | |µ̂2 − µ2 | , ∥ϵ̂n − ϵ̌n ∥ = Op n−1/3 log n . =−

13

Let ψ̂atr = n1

Pn



ϵ̂(ai ) µ̂2 µ̂1 + π̂

n



i=1



A.4. Synthetic and Semi-synthetic Data Generation

. Then,

Synthetic data generation. We first generate covariates X ∼ Unif(0, 1) ∈ R8 , and the treatments and outcomes are generated as follows (the basic forms of all components are kept consistent with related work (Nie et al., 2021)):

 µ̂2 ε̂(a) + − ψa µ̂1 π̂   Z 1 δ(A = a)S ≤ ε̂(a) dPn (x) − P π̂ π̂  X  δ(A = a)S µ̂2 + P + − ψa π̂ µ̂1   n 1 X µ̂2 µ̂2 + −P n i=1 µ̂1 µ̂1

1X ψ̂atr − ψa = n i=1

ã | x =

10 sin(max(x1 , x2 , x3 )) + max(x3 , x4 , x5 , x8 )3 1 + (x1 + x5 )2
+ sin(0.5x3 ) 1 + exp(x4 + x8 − 0.5x3 ) + x23 + 2 sin(x4 ) + cos(x8 ) + 2x5 − 6.5 + N (0, 0.25),

a =σ(ã).

=: T1 + T2 + T3 .

where σ(·) = (1 + exp(−(·)))−1 denotes the sigmoid function.

For the first term,   1 S dPn (x) − π(a) P A=a π̂ X π̂ Z
1 dPn (x) ≤ ε̂(a) − ε̃(a) X π̂ Z   1 1 + ε̃(a) P dPn (x) − π(a) P 2 A = a π̂ X π̂ Z 1 ≲ ∥ε̂ − ε̃∥ + ε̃(a) d(Pn − P)(x) X π̂  Z µ2 − µ̂2 π̂ − π 1 + P A=a · dP(x) π̂ µ̂1 π̂ π̂ X  Z (µ1 − µ̂1 )µ̂2 π̂ − π 1 + P A=a · dP(x) π̂ µ̂1 π̂ π̂ X   p −1/3 = OP n log n + r1 (n)r2 (n) + r1 (n)r3 (n) . Z

T1 = ε̂(a)

For the second term,   δ(A = a)S µ̂2 + T2 = P − ψa π̂ µ̂1 Z = ϕa (P̂) + ψ̂a − ψa = ∥R2 (P̂, P)∥.

Z1 | x, a = cos (t − 0.5) · 2π

! 4 · max(x1 , x6 )3 t + sin(x4 ) , 1 + 2x23 2

4 · max(x1 , x7 )3 Z2 | x, a = sin (−0.5 t + 0.1) · 2π (−t2 + t + sin(x8 )), 0.5 + x33 − x3 Y1 = B(1, 0.7σ(Z1 )),

Y2 = Y1 × B(1, 0.9σ(Z2 )).

Here, B(1, p) denotes a Bernoulli random variable with mean p. Y1 and Y2 are binary random variables corresponding to whether a click occurs and whether a conversion occurs, respectively. By construction, Y2 = 0 if Y1 = 0. Semi-synthetic data generation.We first generate a set of parameters V i = U i /||U i || and i = 1, 2, 3, 4, where U i is sampled from a normal distribution N (0, 1), then: v4⊤ x 2 v3⊤ x

 a ∼ Beta 2,

(1)

 .

(2) 
 Z1 | x, t = 2 4(t − 0.5)2 sin 0.5πt

From Lemma 4.1 and the proof of Theorem 4.2, we obtain 2

×





max − 2, min 2, exp

v⊤ x 0.3(π v3⊤ x − 1) 4

!



+ 20 v1⊤ x



+ 20 v2⊤ x

,

T2 = Op r1 (n)r2 (n)+r1 (n)r3 (n)+r2 (n)r3 (n)+r2 (n) . 
 Z2 | x, t = 2 − 2(t + 0.1)2 cos 0.5πt2

Under Assumption (iii), T3 = Op (n−1/2 ).

×

Therefore,



max − 1, min 1, exp

Y1 = B 1, 0.6 σ(Z1 ) , p
∥ψ̂atr −ψ̂adr ∥ = Op n−1/3 log n+r1 (n)r2 (n)+r1 (n)r3 (n)+r2 (n)r3 (n)+r2 (n)2 .

14



v⊤ x 0.5(π v3⊤ x − 1) 4

!

Y2 = Y1 × B 1, 0.6 σ(Z2 ) .

,

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