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Doubly Robust Estimation of Causal Effect on CVR with Targeted Regularization

Jiayi Dan, Bo Li, Lu Deng, Yong Wang

arXiv:2608.13461v1cs.LG

TL;DR

CVR causal-effect estimation can suffer selection bias and variance when restricted to clicked samples, while unbiased loss estimation does not guarantee an unbiased final estimator. The paper develops a semiparametric doubly robust estimator with targeted regularization, achieving √n-consistency and strong performance in experiments.

  • Problem

    Existing CVR causal-effect methods can suffer sample-selection bias and increased variance, while unbiased loss estimation does not guarantee an unbiased final estimator.

  • Method

    The paper derives a semiparametric influence function and von Mises expansion to construct a doubly robust estimator with targeted regularization for stable CVR estimation.

  • Results

    The proposed estimator achieves √n-consistency under mild conditions, and experiments show strong performance, robustness, and advantages over indirect loss debiasing.

  • Takeaways & Limitations

    CVR causal-effect estimation benefits from directly targeting the final estimand with theoretical guarantees rather than naively debiasing losses before applying standard estimators.

  • Takeaways & Limitations

    Direct one-step correction can become unstable when click-through rates are low and may produce estimates outside the parameter space.

Abstract

from arXiv · show

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. Extensive experiments on synthetic and real-world 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. Introduction

The introduction frames CVR as a second-stage efficiency metric whose causal effect must be estimated over the full population, rather than only clicked users. It identifies click-induced selection bias and dependence as key obstacles, motivating a semiparametric doubly robust estimator with targeted regularization and empirical validation.

  • Motivation and objective: CVR measures second-stage conversion efficiency as P(Y2 = 1 | Y1 = 1), while healthy policies should improve overall conversion without degrading stage-wise efficiency.Y1 indicates clicking and Y2 conversion, with Y2 = 1 implying Y1 = 1.
  • Challenges: Mediation methods are unsuitable because click and conversion dependence can violate required assumptions, and the nuisance parameter E(Y2 | Y1 = 1, X = x, A = a) cannot be directly estimated.Regressing conversion on click, covariates, and treatment is invalid because Y2 is strictly dependent on Y1, while restricting to Y1 = 1 introduces selection bias.
  • Challenges: Applying standard causal estimators only to clicked users creates selection bias because strategies are assigned before clicks while the target concerns the full population.Click selection induces a distribution shift between clicked samples and the population on which the causal effect is defined.
  • Proposed method: The paper derives the target estimand’s influence function and von Mises expansion to construct a doubly robust estimator tailored to chain-structured outcomes, especially CVR.The estimator remains consistent when one nuisance estimator is inconsistent and directly targets the final estimand rather than relying on loss debiasing.
  • Proposed method: A targeted-regularization estimation framework is introduced to improve the method’s applicability and empirical performance.The framework builds on the estimator’s theoretical findings.
  • Experiments: Extensive experiments on synthetic, semi-synthetic, and real-world data demonstrate the proposed method’s strong performance and robustness.The evaluation spans multiple data settings to assess practical effectiveness.

2. Related Works

Prior work addresses causal-effect estimation through balanced representations, weighting, and doubly robust learners, while CVR research focuses on debiasing training loss under sample selection. However, few studies directly target the final CVR causal-effect estimand with both theoretical guarantees and practical utility.

  • Causal effect estimation: Deep-learning causal estimators mitigate confounding through balanced representations, generalization bounds using empirical risk and IPM distance, and weighting strategies.These directions include work by Johansson et al. (2016, 2017, 2018), Shalit et al. (2017), Hassanpour and Greiner (2019), and Assaad et al. (2021).
  • Doubly robust estimation: Doubly robust learners correct plug-in estimator bias and achieve √n-convergence rates under appropriate conditions.The DR learner was proposed by Van der Laan (2005) and further developed by Chernozhukov et al. (2017, 2018).
  • CVR causal effects: CVR causal-effect estimation differs fundamentally from standard causal inference because its target estimand and nuisance parameters require corresponding extensions and adaptations.This distinction motivates methods tailored to the CVR objective rather than direct reuse of standard causal estimators.
  • CVR estimation: CVR research addresses sample-selection bias through the “ideal loss,” which estimates training loss over the entire sample without bias.Follow-up studies refine this estimation method to improve accuracy and reduce variance.
  • Research gap: Few studies directly target the final CVR causal-effect estimand while providing both sound theoretical guarantees and strong practical utility.This identifies the research gap addressed by the paper.

3. Problem Formulation and Notations

The paper formulates CVR estimation with click and conversion outcomes in a chain structure, treating treatment as continuous and accounting for selection among clicked samples. It defines three nuisance parameters and states overlap, unconfoundedness, and positive click-probability assumptions supporting full-sample estimation.

  • Data structure: Each observation contains covariates, a continuous treatment, click status Y1, and conversion status Y2, with conversion implying a click.The data are represented as z_i = (x_i, a_i, y1_i, y2_i), where y1_i is binary and y2_i = 1 implies y1_i = 1.
  • Target estimand and selection bias: Restricting estimation to clicked samples is biased because click status Y1 is non-random and depends on treatment.The standard causal estimand is ψ_a = E[E(Y(a) | X)], while the naive clicked-sample strategy estimates conversion effects only among Y1 = 1.
  • Nuisance parameters: The procedure uses outcome regressions µ1 and µ2 together with the treatment density π as its three nuisance parameters.Here µ1(x, a) = E(Y1 | X = x, A = a), µ2(x, a) = E(Y2 | X = x, A = a), and π(a | x) is the propensity density.
  • Assumptions: Identification and analysis assume overlap, covariate-based unconfoundedness, and a positive lower bound µ1(x, a) ≥ c.The positive click-probability condition excludes samples with µ1 = 0, for which CVR is undefined.
  • Full-sample estimation: The method exploits dependence between Y1 and Y2 so all nuisance parameters can be estimated from the full sample.This avoids restricting nuisance estimation to clicked observations.

4. Doubly Robust Estimation

The section shows that a naive ratio plug-in estimator is vulnerable to nuisance-estimation bias, motivating a tailored doubly robust estimator that removes first-order bias and achieves root-n consistency under flexible nuisance models. It also identifies numerical instability in direct one-step correction, especially when click-through rates and the denominator are small.

  • Naive Plug-in Estimation: The naive plug-in estimator predicts µ1(x, a) and µ2(x, a) from covariates and treatment, then estimates CVR by their ratio.This strategy leverages the full dataset and resembles ESMM’s CVR prediction approach.
  • Naive Plug-in Estimation: With neural networks, first-order nuisance-estimation errors prevent plug-in consistency unless all nuisance parameters are consistently estimated.The estimator is governed by the slowest-converging nuisance parameter and is sensitive to model misspecification.
  • Doubly Robust Estimator: The doubly robust estimator remains consistent when not all nuisance parameters are consistently estimated and achieves faster convergence than the nuisance estimators.This provides robustness to model misspecification and supports flexible nonparametric nuisance estimators.
  • Doubly Robust Estimator: The proposed doubly robust estimator subtracts an estimated first-order bias through a one-step correction, leaving a second-order remainder.This construction is based on the influence function and a von Mises expansion of the target estimand.
  • Asymptotic Properties: Under mild assumptions, the doubly robust estimator is root-n consistent and asymptotically normal, with cross-fitting and oP(n^-1/4) nuisance rates supporting the result.Neural networks are among the estimators that can satisfy the required nuisance-rate conditions.
  • Practical Stability: Direct one-step correction can be numerically unstable and produce estimates outside the parameter space because low click-through rates make denominator-based correction terms extremely large.The instability specifically arises from µ1 appearing in the denominator of the correction term.

5. A Practical Framework Based on Targeted Regularization

The framework adapts targeted regularization to the CVR causal-effect estimand, embedding a soft correction in training to improve stability while retaining doubly robust theoretical guarantees. It also jointly estimates CTR and CVR using a multi-task architecture that enforces the dependency between click and conversion probabilities.

  • Targeted regularization: Targeted regularization adapts the influence-function correction for the CVR estimand and adds it to the plug-in loss during training.The regularizer trains an additional learnable parameter ϵ to approximate the correction term.
  • Theoretical properties: Theorem 5.1 establishes that the targeted-regularization estimator remains doubly robust under mild regularity conditions.Its convergence rate exceeds the individual convergence rates of nuisance-parameter estimators, supporting flexible nonparametric models, including neural networks.
  • Practical stability: The method improves numerical stability by replacing the original estimator’s hard correction with a soft regularized correction controlled by hyperparameter β.Adding the regularizer does not affect consistency of nuisance-parameter estimation, while β keeps the regularization term comparable to the main loss.
  • Multi-task estimation: A multi-task model jointly estimates causal effects of CTR and CVR after estimating the propensity and click-probability nuisance parameters.The joint design is motivated by the fact that µ1 corresponds to click probability and can support CTR estimation alongside CVR estimation.
  • Model architecture: The architecture enforces µ2 < µ1 by parameterizing the conversion prediction as ˆµ2 = ˆµ1 × ˜µ2, with ˜µ2 ∈ (0, 1).Gradients from targeted regularization are blocked from the propensity model ˆπ to improve propensity-score estimation accuracy.

6. Experiments

Experiments show that the proposed doubly robust method achieves the strongest CVR causal-effect estimation performance across synthetic, semi-synthetic, and public-data settings. Targeted regularization is crucial for performance and robustness, while loss debiasing combined with standard causal estimators remains inferior to the tailored framework.

  • Main results: Our method significantly outperforms all baselines on the CVR task across synthetic datasets and News, while maintaining desirable CTR performance.These results support jointly estimating causal effects on CTR and CVR.
  • Main results: Removing targeted regularization significantly degrades performance, demonstrating that the term is effective and necessary.The basic doubly robust estimator remains competitive but is inferior to the final proposed method.
  • Sensitivity analysis: AMSE remains substantially lower than the ablation study across a wide range of α values, showing strong stability of the proposed method.Increasing the targeted regularization weight β instead produces monotonic performance degradation.
  • Loss debiasing comparison: The loss-debiasing alternative consistently outperforms the plug-in estimator without targeted regularization and achieves better results than most baselines across datasets.Its gains are attributed to targeted regularization reducing estimation bias, but the experiments compare it against the previously proposed method rather than replacing the tailored framework.
  • Public-data evaluation: On public binary-treatment datasets evaluated with AUUC and QINI, our method significantly outperforms other models for CVR causal-effect estimation.The ablation study again shows targeted regularization is critical, while cumulative gain curves illustrate gains from targeted regularization and the doubly robust design.

7. Conclusion

The paper formulates CVR causal effect estimation from a semiparametric perspective and proposes a doubly robust estimator with theoretical guarantees. It also develops targeted regularization, whose effectiveness and advantages over indirect loss debiasing are supported by extensive experiments.

  • Conclusion: The proposed framework formulates CVR causal effect estimation from a semiparametric perspective.It combines theoretical guarantees with practical value.
  • Conclusion: The new doubly robust estimator is tailored to the target estimand and achieves √n-consistency under mild conditions.This provides the framework’s central theoretical guarantee.
  • Conclusion: Targeted regularization forms a practical framework built on the estimator’s theoretical findings.The framework is intended to support practical application of the proposed method.
  • Conclusion: Extensive experiments demonstrate strong performance, targeted regularization’s effectiveness, and advantages over indirect loss debiasing.The experiments support both the proposed method and its targeted-regularization component.

8. Impact Statement

The paper presents a general framework for estimating causal effects of chain-structured outcomes such as CVR, with applications in e-commerce and online advertising decision-making.

  • 8. Impact Statement: The framework estimates causal effects for chain-structured outcomes such as CVR and applies to decision-making in e-commerce and online advertising.It is presented as broadly applicable across these scenarios.

A. Appendix · A.1. Proof of Lemma 4.1

The proof of Lemma 4.1 analyzes the ratio function f(x, y) = y/x under positive lower-bound assumptions and uses a second-order Taylor expansion with a bounded remainder.

  • A.1. Proof of Lemma 4.1: The derivation includes integrals involving p(y1, x, a) over y1 and x.
  • A.1. Proof of Lemma 4.1: It also includes integrals involving p(y2, x, a) over y2 and x.
  • A.1. Proof of Lemma 4.1: The proof further writes integrals of the conditional densities p(y1 | x, a) and p(y2 | x, a) with respect to dP(x).
  • A.1. Proof of Lemma 4.1: The proof defines f(x, y) = y/x and assumes both µ1 and ˆµ1 have positive lower bounds.Under these conditions, f is twice continuously differentiable on the relevant region.
  • A.1. Proof of Lemma 4.1: A second-order Taylor expansion of the bivariate function f is applied with a Lagrange remainder.
  • A.1. Proof of Lemma 4.1: The proof specifies that p(·) denotes the corresponding probability density.
  • A.1. Proof of Lemma 4.1: The expansion introduces a remainder term Rµ whose behavior is analyzed separately.
  • A.1. Proof of Lemma 4.1: The remainder analysis uses a bounded constant C.

A.2. Proof of Theorem 4.2 · A.3. Proof of Theorem 5.1 · A.4. Synthetic and Semi-synthetic Data Generation

The appendices establish asymptotic normality through a decomposition, remainder-rate control, and Slutsky’s theorem, then specify synthetic and semi-synthetic data-generation procedures for chain-structured click and conversion outcomes.

  • A.2. Proof of Theorem 4.2: The leading empirical-process term is √n-consistent and asymptotically normal by the central limit theorem.
  • A.2. Proof of Theorem 4.2: Under positivity conditions for ˆπ(x, a) and ˆµ1(x, a), the second-order remainder satisfies R2(ˆP, P) = oP(n−1/2).
  • A.2. Proof of Theorem 4.2: Assumption 1 and Slutsky’s theorem yield asymptotic normality for the estimator.
  • A.3. Proof of Theorem 5.1: Theorem 5.1’s proof decomposes the relevant expression into three components, T1 + T2 + T3.
  • A.3. Proof of Theorem 5.1: Its rate expression contains log n and products of nuisance-estimation rates, including r1(n)r2(n) and r1(n)r3(n).
  • A.4. Synthetic and Semi-synthetic Data Generation: Synthetic data use X ∼Unif(0, 1) ∈R8, with treatments and outcomes generated using component forms consistent with Nie et al. (2021).
  • A.4. Synthetic and Semi-synthetic Data Generation: The synthetic construction defines Y1 as click and Y2 as conversion, with Y2 = 0 whenever Y1 = 0.
  • A.4. Synthetic and Semi-synthetic Data Generation: Semi-synthetic data generation first samples normalized parameter vectors V i = U i/||U i|| for i = 1, 2, 3, 4, with U i ∼N(0, 1).
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