Title: Density Ratio-Free Doubly Robust Proxy Causal Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Proxy causal learning

doubly robust estimation

kernel mean embeddings
Abstract: We study the problem of causal function estimation in the Proxy Causal Learning (PCL) framework, where  confounders are not observed but proxies for the confounders are available. Two main approaches have been proposed: outcome bridge-based and treatment bridge-based methods. In this work, we propose two kernel-based doubly robust estimators that combine the strengths of both approaches, and naturally handle continuous and high-dimensional variables. Our identification strategy builds on a recent density ratio-free method for treatment bridge-based PCL; furthermore, in contrast to previous approaches, it does not require indicator functions or kernel smoothing over the treatment variable. These properties make it especially well-suited for continuous or high-dimensional treatments. By using kernel mean embeddings, we propose the first density-ratio free doubly robust estimators for proxy causal learning, which have closed form solutions and strong uniform consistency guarantees. Our estimators outperform existing methods on PCL benchmarks, including a prior doubly robust method that requires both kernel smoothing and density ratio estimation.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes two kernel-based doubly robust (DR) estimators that leverage both outcome and treatment bridge functions, enabling them to handle continuous and high-dimensional covariates. Following a framework similar to Cui et al. [17], the estimators integrate the KPV/PMMR method [15] for estimating $h$ and KAP [19] for estimating $\\varphi$, thereby constructing the DR estimator. The authors establish consistency for the proposed estimator and support their approach with extensive simulation studies.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality**\n\nThe submission demonstrates technical soundness, presenting well-substantiated claims grounded in rigorous theoretical analysis and simulation-based evidence.\n\n**Clarity**\n\nThe paper is clearly written and well-organized.  \n\n**Significance**\n\nThe work addresses a topic of considerable interest, as many practical causal inference problems involve continuous treatments. The high-dimensional extension is also promising, though the theoretical justification in this context requires further clarification.\n\n**Originality**\n\nWhile the paper builds on ideas from [15], [17], and [19], the novelty of its contributions is not clearly delineated. The main methodological development appears between Line 250 (Page 6) and Line 270 (Page 7), but this section seems to offer only incremental innovation. Furthermore, some relevant literature is omitted, reducing the clarity of the paper's positioning relative to existing work.\n\nQUESTIONS:\n1. **Figure 1**\n\nThe arrow from U to W should be solid, and the arrow from Y to W should be dashed. \nConsider an example where $U \\sim N(0,1)$, $W \\sim N(0,1)$, $A \\sim N(U,1)$, and $Y \\sim N(A+W,1)$. \nThis example is compatible with Figure 1, but we find $E[U|W,A=0] = E[U|A=0]=0$, indicating that $\\ell(U) = U$ serves a counterexample of Assumption 2.5.\nThus, if $U$ and $W$ are not associated, Assumption 2.5 cannot be satisfied, which requires a solid arrow between $U$ and $W$. \nFor further justification, see Table A.1 in Tchetgen Tchetgen et al. (2024, Statistical Science), which also specifies the requirement for a direct connection between $U$ and $W$.\n\n\n2. **Theorem**\n\nWhat notion of convergence is used in the theorems?\n\nThe paper includes high-dimensional simulations, but it’s unclear whether Theorems 4.1 and 4.2 are valid in such settings. How is the curse of dimensionality addressed? Please clarify the assumptions under which these theoretical results extend to high-dimensional cases.\n\n\n3. **Reference**\n\nIn explaining Equation (2), it would be helpful to cite foundational work on stabilized weights for continuous treatments (e.g., Robins et al., 2000, Epidemiology). This would situate your methods more clearly in the literature and justify the weighting strategy.\n\n4. **Contribution**\n\nThe fundamental concept of integrating KPV/PMMR with KAP is sound. Nevertheless, the manuscript fails to clearly delineate whether this integration yields significant theoretical or methodological innovations. Notably, [19] also asserts applicability to continuous treatment and high-dimensional settings, suggesting that these aspects are not exclusive contributions of the current paper. Consequently, the paper's contribution, as articulated, appears to be primarily confined to the development of the DR property. Consequently, the paper's contribution, as articulated, appears to be primarily confined to the development of the DR property. The paper should place clearer emphasis on why this DR development is a nontrivial task.\n\n5. **Typo**\n\nLine 320, Page 8: $f_{ATE} \\rightarrow \\theta_{ATE}$\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-14"}

comments: SUMMARY:
The work  presents a way of estimating causal functions under the proxy causal learning (PCL) framework. They account for confounders by proxy of variables for treatment (Z) and outcome (W). The two estimators presented are both of the doubly  robust (DR) kernel-based type, and combine the benefits of  both Z and W bridge-based estimattors. 
The density-ratio freedom implies they they circumvent the need of the indicator finction or kernel smoothing over treatment variables.The experiments show improved performance over sysnthetic high-dimensional data.

STRENGTHS AND WEAKNESSES:
Kernel mean embeddings help with the closed-form and strongly-consistent guarantees. 

It appears that the work is an important step in measuring confounders.

Experiments are on high dim (quasi continuous)  data, but not on copious volumes.  Kernel computation may be a bottleneck in that.

QUESTIONS:
Between Z/W, one may sometimes be information-poor, and the other the more informative proxy. How does that play into the calculus?

LIMITATIONS:
I'm not able to determine those critically!
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-06-27

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses causal function estimation in the Proxy Causal Learning (PCL) framework, where confounders are unobserved but proxy variables are available. The authors propose two novel kernel-based doubly robust estimators that integrate outcome and treatment bridge methods. These estimators avoid kernel smoothing over the treatment and density ratio estimation, making them especially effective for continuous or high-dimensional treatments. Using kernel mean embeddings, the method achieves closed-form solutions and strong consistency. Empirical results on PCL benchmarks show superior performance over existing methods, including previous doubly robust approaches.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1.\tThe proposed estimators avoid kernel smoothing over the treatment or density ratio estimation, making them particularly suitable for continuous or high-dimensional treatment settings.\n2.\tBoth estimators are shown to possess the doubly robust (DR) property, which enhances their robustness to model misspecification.\n3.\tThe paper provides uniform consistency guarantees and demonstrates empirical performance, offering a compelling combination of theoretical justification and practical utility.\n\nWeaknesses:\n1.\tThe idea lacks novelty. The core idea of combining estimators from [10] and [19] to construct a new DR estimator is similar to the classical causal inference literature. Notably, the claimed benefits—such as avoiding kernel smoothing and handling high-dimensional treatments—are inherited from [19].\n2.\tBoth [10] and [19] derive convergence rates for their respective estimators. Does the proposed DR estimator—formed via their combination—achieve a similar rate? If so, is it the minimum of the two, or can it attain a faster rate under certain conditions?\n3.\tOne key advantage of DR estimators is inference. However, the paper does not analyze the asymptotic distribution of the proposed estimator, and thus it is unclear whether valid confidence intervals can be constructed.\n4.\tIn Lines 276–278, the authors mention that different stages require different samples, obtained via either data splitting or reuse. If data splitting is applied, sample inefficiency might be a concern. If reuse is employed, what are the implications for theoretical guarantees (e.g., bias or overfitting)? This trade-off requires further discussion.\n\nQUESTIONS:\n1.\tIn Supp lines 939-945, the author mention that the method of utilizing IF is locally effective. In Theorem 2.7, the author derives an identification formula that inherit the dr property. Will the proposed estimators also inherit local efficiency, or at least satisfy asymptotic normality? Given the nonregularity of continuous treatment estimators, should we expect slower-than-$\\sqrt{n}$ asymptotic normal.\n2.\tGiven that kernel-based methods are involved in parts of the estimation, can the authors clarify the computational efficiency of the proposed approach?\n\nLIMITATIONS:\nThe work does not have potential negative societal issues.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nWhen hidden factors influence a treatment's outcome, researchers can use proxy variables to estimate the true causal effect. However, standard methods require estimating complex density ratios, which can be unstable.\nThe authors developed a doubly robust method that avoids this step, leading to more stable and effective causal effect estimation with proxy variables.\n\nSpecifically, they introduces a density ratio-free, doubly robust kernel-based estimator for treatment effect estimation in the presence of unmeasured confounding, leveraging proxy variables.\n\n- Their estimator does not rely on estimating density ratios, improving robustness and practicality.\n- They employ RKHS, supporting nonparametric inference.\n- Through synthetic and real-world datasets, the proposed method outperforms standard and proximal baselines, particularly in misspecified or high-dimensional scenarios.\n\nSTRENGTHS AND WEAKNESSES:\nAssumptions 2.3 and 2.5 require that the proxies are \"sufficiently informative\" relative to the unobserved confounder, which seems resasonable to me. Though in practice it might be hard to verify. \n\n**Strengths:**\n- They developed a density ratio-free approach, which is more scalable and stable for continuous or high-dimensional treatments.\n- Theorem 2.7 shows that the proposed estimators are designed to be consistent if either the outcome bridge model or the treatment bridge model is correctly specified.\n- They also demonstrate Theorem 2.7 empirically through misspecification experiments, where the estimators recover the true causal effect even when one of the bridge functions is intentionally perturbed.\n- The paper establishes uniform consistency for both proposed estimators (Theorems 4.1 and 4.2).\n\n**Weakness:**\n- The computational cost of kernel methods may make the method scale poorly with the number of samples. Thus might be impractical for large datasets.\n- RKHS methods typically rely on kernel selection and tuning. The paper does not explore how robust the method is to kernel and hyperparameter choices.\n- The DRKPV estimator, which builds on the KPV and KAP algorithms, may use data splitting for its different regression stages, but seems less discussed? Or did I missed it?\n\nQUESTIONS:\nSee my questions in the above “Weakness” part and below: \n\n- How does the computational cost of the proposed kernel method scale with the number of samples and proxies?\n- How sensitive is the estimator’s performance to the choice of kernel function and its parameters?\n\nLIMITATIONS:\nSee the limitations also in the above “Weakness” and “Questions” part.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}