Paper: Doubly Robust Proximal Causal Learning for Continuous Treatments (ICLR-2024)

Reviewer comment (Theory):
The inference issue seems to be ignored without the analysis of asymptotical distribution for causal effect.

Author response:
**A.** Currently, we cannot obtain the asymptotic normality due to the error introduced in kernel approximation. With this error, we can show in Theorem E.9 that our estimator is $n^{2/5}$-consistent, while the asymptotic normality means $\sqrt{n}$-consistent. Besides, according to [4], since the estimand is non-regular, therefore it may not enjoy the properties of $\sqrt{n}$-consistent and asymptotically normality. We illustrate this point through an empirical study in Appendix E.5.

Reviewer comment (Theory):
The paper’s rationale for the calculation of the influence function appears to be lacking, as it directly selects a specific submodel. It might be more appropriate to refer to the function derived in this paper as the efficient influence function, given that the previous doubly robust estimator for binary treatment is efficient.

Author response:
**A.** While the DR estimator for binary treatment is shown to be the effective influence function, it is unclear whether it still holds for continuous treatments without explicit computation. To verify, we calculate the influence function.

Reviewer comment (Theory):
Although one single optimization reduces computational burden, but asymptotical normality of causal effect estimate may not hold using the proposed kernel estimation method. So please make the motivation clear about why this paper does not focus on the inference issue.

Author response:
**A.** Currently, we cannot obtain the asymptotic normality due to the error introduced in kernel approximation. With this error, we can show in Theorem E.9 that our estimator is $n^{2/5}$-consistent, while the asymptotic normality means $\sqrt{n}$-consistent. Besides, according to [4], since the estimand is non-regular, therefore it may not enjoy the properties of $\sqrt{n}$-consistent and asymptotically normality. We illustrate this point through an empirical study in Appendix E.5.

Reviewer comment (Theory):
Why does the projected residual mean squared error make sense? Does it mean \hat{q} is consistent for q0?

Author response:
**A.** The projected residual mean squared error measures how much $\hat{q}$ violate Eq. (3). This performance metric holds significant prominence in the theoretical analysis of minimax problems, as noted in references [1, 2, 3]. When the measures of the ill-posedness of inverse problems are bounded, $\hat{q}$ is consistent for $q_0$. Please refer to Remark 6.3 for details.

Reviewer comment (Theory):
This paper seems to consider the high-dimensional setting in experiments, what about theoretical properties?

Author response:
**A.** In this context, we follow the setting in [5,6], in which the term "high-dimensional" merely indicates a scenario with relatively more covariates compared to those in section 7.1.1, without implying that the number of samples is smaller than the number of features. Therefore, our theory still applies to this case.

Reviewer comment (Theory):
I believe that the statement of Theorem 4.5 is incorrect. More precisely, the influence function of $B(a)$ does not exist, meaning that $B(a)$ is not pathwise differentiable. However, $B(a; P^{\epsilon,h_{bw}})$ is pathwise differentiable, indicating that the influence function does exist. Therefore, to accurately state this, the term "lim" should be removed and the statement should be made with "for any $h_{bw} > 0$".

Author response:
While $\beta(a,P^{\varepsilon,h_{bw}})$ is pathwise differentiable, its influence function is not of our interest. Our goal is to obtain the influence function for $\beta(a)$. To obtain this, we take the gradient with respect to $\varepsilon$ and let $h$ go to 0. Such a treatment has been similarly adopted in [1].

Reviewer comment (Reproducibility):
A practical guide is needed to solve the optimization problem in Equations (8,9) and apply these methods in practice.

Author response:
Equations (8,9) are a minimax optimization problem, and we can solve it in different ways, depending on what function space class we use. For example, We can parameterize $q$ (resp. $h$) and $m$ (resp. $g$) as reproducing kernel Hilbert space (RKHS) with kernel function or neural networks. For the former, we derive their closed solutions in Appx. F. In the latter case, we can employ Generative Adversarial Networks [2]. We have appended this discussion to make it clearer.

Reviewer comment (Theory):
I believe that the statement of Theorem 4.5 is incorrect. More precisely, the influence function of $B(a)$ does not exist, meaning that $B(a)$ is not pathwise differentiable. However, $B(a; P^{\epsilon,h_{bw}})$ is pathwise differentiable, indicating that the influence function does exist. Therefore, to accurately state this, the term "lim" should be removed and the statement should be made with "for any $h_{bw} > 0$".

Author response:
While $\beta(a,P^{\varepsilon,h_{bw}})$ is pathwise differentiable, its influence function is not of our interest. Our goal is to obtain the influence function for $\beta(a)$. To obtain this, we take the gradient with respect to $\varepsilon$ and let $h$ go to 0. Such a treatment has been similarly adopted in [1].

Reviewer comment (Theory):
I understand that $q_0$ represents the maximum value for the equation in Lemma 5.1. However, I am unsure about the connection between Lemma 5.1 and Equation (8). In other words, why do we need to minimize the empirical quantity for the equation in Lemma 5.1?

Author response:
We have modified the description of Lemma 5.1 and added a paragraph to discuss its connection to Eq. (3) and Eq. (8). Simply speaking,  We first show that solving $q_0$ from data is equivalent to minimizing $\mathcal{L}_q(q;p):=\mathbb{E} [ \left( \mathcal{R} _q\left( q,p \right) \right) ^2 ]$ over $q$, where $\mathcal{L}_q(q;p)$ is equivalent to the maximization form in Lemma 5.1.