Paper: ReTaSA: A Nonparametric Functional Estimation Approach for Addressing Continuous Target Shift (ICLR-2024)

Reviewer comment (Theory):
The paper should discuss the theoretical computational complexity and practical scalability of ReTaSA, in terms of the source and target sample sizes \(n\) and \(m\), data dimension \(p\), etc. Relatedly, under "Related Work", the paper claims "empirical evidence from... our experimental studies confirms that KMM is computationally inefficient in categorical and continuous cases," but I couldn't find evidence of this in the paper.

Author response:
Thanks for pointing this out. Our method computationally faster than the KMM method is because our method only needs to solve a linear equation, while KMM needs to optimize the constraint quadratic programming problem. More specifically, the major computational cost in our method is inverting a $n\times n$ matrix, whose computational complexity is generally $\mathcal{O}(n^3)$. The KMM method proposed by Zhang et al. (2013) needs to optimize a quadratically constrained ***quadratic*** program with a $n\times n$ matrix.  We believe the computational efficiency can be attributed to the fact that our proposed method only requires a one-step linear system solving while the KMM method (as well as the added L2IWE method) needs to loop through iteration in the optimization process. **From the empirical aspect, we detail the running time comparisons in Figure S.4 in Section C.2.2 of the supplementary materials**.

Reviewer comment (Writing):
Page 2, last sentence, "the target shift assumption implies that... (1)": It would be helpful to include a few more details on the steps by which the target shift assumption implies Eq. (1). Although this can be derived using Bayes' rule, it interrupts reading and could be clarified by adding another intermediate equality in Eq. (1) without increasing length.

Author response:
Thanks for the suggestion. We have provided more details in the derivation in the revision in Equation (1).

Reviewer comment (Theory):
Page 3, just after Eq. (3), "$T$ and $T^∗$ are adjoint operators because \(\langle T\phi, \psi \rangle = \langle \phi, T^* \psi \rangle\)": It is not immediately obvious why this is the case. Since this observation is important for the remainder of the paper, a more detailed explanation or proof should be included in the main text, or a reference provided to where it can be found.

Author response:
We have refined the definition of inner products and added a few more lines of equations between equation (3) and (4) to validate this fact.

Reviewer comment (Writing):
Page 3, just after Eq. (4), "where... \(\rho(y)\) is an unknown function to be solved": This is confusing because it sounds like \(\rho = \omega\). Only later is it explained that \(\rho = \omega - 1\). This fact should be explained earlier.

Author response:
Thanks for the suggestion; we have clarified the usage of notation $\rho$ as you suggested and edited the identifiability subsection to reflect this change.

Reviewer comment (Theory):
First paragraph of Section 3: \(\eta = p_t/p_s\) is estimated as the ratio of two density estimates. There is a significant body of work on density ratio estimation showing that estimating the ratio by the ratio of two density estimates is often suboptimal, both in theory and practice ([K17, SSK10]). The paper should consider using direct density estimation methods for this step to improve performance and relax assumptions (specifically, Assumption 3).

Author response:
Thanks for providing closely connected references. Utilizing direct density estimation to yield an improved rate of consistency will be an interesting future theoretical work to pursue. In the meantime, we believe that assumption 3 in our setting is natural as the equation (1) would not be valid if violated. We have added a remark to explain connections between the assumption 3 and the equation (1).
    
Furthermore, in the revised paper, we compared two ways of computing the density ratio: directly computing the ratio of two estimated densities and the Relative unconstrained Least-Squares Importance Fitting (RuLSIF) method (Yamada et al. (2011)), which is proposed by the author of [SSK10]. The numerical study shows that these two methods have similar performance. The details of the numerical comparison can be found in Section C.2.3 of the Supplementary Materials.

Reviewer comment (Theory):
Theorem 1 bounds the \(L^2\) error of the estimated \(\rho\), but \(\rho\) is only used to re-weight the risk function to adapt to the test domain. It is not clear whether estimating \(\rho\) well in \(L^2\) distance is necessary or sufficient for this adaptation. The paper should provide a more concrete connection between estimation of \(\rho\) in \(L^2\) and test-domain performance of the new risk minimizer.

Author response:
Thanks for pointing this out. The key reason for the weight estimation is motivated by the loss relationship between training and testing datasets.
$$
\mathbb{E}_t[\ell\{f(x),y\}]=\mathbb{E}_s[\omega(y)\ell\{f(x),y\}]
$$
With a well-estimated $\rho$, we can recover the importance weight function $\omega$ accurately, which can facilitate us to learn a predictive model with respect to the test-domain distribution. As $\rho$ is a function, we use the $L^2$-distance to measure the estimation accuracy in the functional space.

Reviewer comment (Theory):
Page 1, Paragraph 2: It is unclear why the Sequential Organ Failure Assessment (SOFA) example satisfies the label shift assumption, particularly why \(P_{X|Y}\) is invariant between domains. A more convincing example would strengthen motivation. It may also be worth pointing out that label-shift assumptions appear naturally under anti-causal structural assumptions (see Section 5 of [S22]).

Author response:
Thanks for the suggestion. First, we want to clarify the relationship between label shift and biased sampling*.* Suppose $r$ is a binary random variable indicating a patient is sent to a big or small hospital (i.e., domain), and $r$ only depends on the SOFA score (i.e., the severity of the condition): $p(r|x,y)=p(r|y)$, which is equivalent to $r$ and $x$ are independent conditional on $y$. This conditional independence relationship can be rewritten as $p(x|y,r)=p(x|y)$. In other words, the conditional distribution of $p(x|y)$ is domain invariant (in this case, $y$ does not have to cause $x$), thus honoring the target shift assumption. The anti-causal setting is also an important motivation for target shift, and we have discussed it in the revision of the paper.

Reviewer comment (Theory):
The paper focuses on the non-parametric setting, making weak assumptions on the relationship between \(x\) and \(y\), but Theorem 1 suggests that performance scales poorly with dimension \(p\) of the feature \(x\). The authors could comment on whether a parametric variant of ReTaSA could be useful for high-dimensional data.

Author response:
Indeed, nonparametric methods often suffer from the curse of dimensionality. To address this issue, our suggestion is to adopt the black box shift estimation (BBSE) method proposed by Lipton et al. (2018), where the high-dimensional features $x$ are mapped to $\widehat{E}_s(y|x)$, where $\widehat{E}_s(\cdot|x)$ is a fitted predictive model. In this revision, we have adopted this approach and added numerical studies on data with many covariates. As you have suggested, it is possible to solve this problem by assuming a parametric model for the importance weight function. One of the simplest models is a polynomial model. For example, $\omega(y)=\exp(\alpha+\beta_1 y+\beta_2 y^2)$. Then, we can estimate the parameters using a semiparametric framework (we do not have to specify a parametric model for $p_s(y|x)$). Similar work has been done for the classification task (e.g., Tian et al., 2023). We want to comment that although we can address the high-dimensional using either the black-box or semiparametric methods, there is no free lunch. If we adopt the black-box method, we need a stronger identifiability assumption. After the mapping, the mapped $\widehat{E}_s(y|x)$ needs to satisfy the completeness condition, which is stronger than the current identifiability assumption. On the other hand, adapting a parametric model for the density ratio will impose model constraints and introduce model misspecification errors.

Reviewer comment (Theory):
Do the authors believe the rate in Theorem 1 is minimax optimal under Assumptions 1-5?

Author response:
Thanks for pointing out this issue. We admit that our theoretical result does not follow the minimax optimal framework. The goal of our Theorem 1 is to demonstrate the consistency of the estimated importance weight function and provide the theoretical variation. To reach the minimax optimal, it will involve more assumptions and analysis for this non-parametric approach. We would like to put it for the future investigation.

Reviewer comment (Experiments):
How robust is ReTaSA to small violations of the label-shift assumption, e.g., small changes in \(P_{X|Y}\)?

Author response:
Our proposed method relies on the target (label) shift assumption. Acknowledging the practical reality of open-world scenarios, it is inevitable that this assumption may be violated. In line with prior research, such as Sahoo et al. 2022, a common strategy to quantify these minor violations involves imposing a divergence-based constraint on $p(x|y)$. Specifically, it is posited that there exists a $\epsilon>0$ and a distance (or divergence) measure, denoted as $D$, satisfying $D(p_s(x|y), p_t(x|y))\leq\epsilon$. Distances encompass well-known metrics like $f$-divergences and Wasserstein distance. Recognizing the potential for a more comprehensive framework to characterize robustness, as outlined above, we appreciate this valuable suggestion and intend to explore it further in future research endeavors.

Reviewer comment (Theory):
Beginning of Page 4: Tikhonov regularization is added to address non-identifiability. It might be worth adding that the regularized criterion has a unique solution (i.e., \(\alpha I + T^* T\) is always invertible), especially in infinite-dimensional settings.

Author response:
Thanks for the comment; we have emphasized the uniqueness of the solution to the first-order equation in the revision.

Reviewer comment (Novelty):
Remark 1: The estimate/approximation may simply be the standard Nadaraya–Watson regression estimate of \(\mathbb{E}[\rho(y)|x]\); this could be noted.

Author response:
Thanks for the comment. Yes, the estimate can be viewed as the standard Nadaraya-Watson regression estimates, and we have modified our presentation accordingly.

Reviewer comment (Theory):
Remark 8: The sentence "the assumption fits into the regime where the dimension of the feature is smaller than the smoothness level of densities and the order ofthe generalized kernel function" is unclear. It seems to say \(\text{dim}(x) = p \leq \min\{k, \ell\} = \gamma\), but it is not clear how this follows from the previous sentence about \(\alpha\).

Author response:
Thanks for the comment, and sorry for the confusion. We rephrased the original remark 8 to the current remark 5 & 7. In remark 5, we discuss the relationship between dimension $p$ and the sample sizes $m$ and $n$. In remark 7, we discuss the relationship between dimension $p$ and regularization parameter $\alpha$, which also involves the smoothness of the generalized kernel functions.

Reviewer comment (Presentation):
Page 7, under "Evaluation Metrics": The hardware description belongs elsewhere, such as in the first paragraph of Section 5.

Author response:
Thanks for pointing this out. We have rephrased the presentation and fixed the typo.

Reviewer comment (Writing):
Page 7, under "Experimental Results": Typo — should be "performs significantly better *than* KMM-Adaptation."

Author response:
Thanks for pointing this out. We have rephrased the presentation and fixed the typo.

Reviewer comment (Presentation):
Figure 2: The lines plotted are essentially flat, so the plot illustrates little. Perhaps a larger range of smaller sample sizes should be shown.

Author response:
Thanks for raising this issue. Here, we want to clarify that the lines look flat due to the consistently significant improvement of our method compared with KMM-Adaptation and None-Adaptation methods under different sample sizes. Specifically, for our method, we list the weight MSE in the following table: MSE is about 0.45 at sample size = 100 and less than 0.2% at sample size = 1000. Thus, the improvement is more than 50% by increasing the sample size from 100 to 1000 under our method.

| Sample Size | Weight MSE |
| --- | --- |
| 100 | 0.454114 |
| 200 | 0.353673 |
| 500 | 0.235435 |
| 1000 | 0.192919 |

Reviewer comment (Presentation):
Figure 3: The sub-captions seem incorrect (both say "vs Sample Size" but the x-axis is \(\mu_t\)).

Author response:
Thanks for pointing out the typo. We have fixed it.

Reviewer comment (Presentation):
All Figures: The font size of axis labels, legends, etc., should be increased.

Author response:
Thanks for the suggestion. We have iterated all the figures in the main paper with a larger font size.

Reviewer comment (Presentation):
Definition 2 is confusing because it is not used anywhere in the paper.

Author response:
**a)**. We have moved Definition 2, which was originally reserved for generalized kernel function, to Section 3, as it has close relevance to the methodology section. **b)**. Thank you for pointing it out; we have corrected notations for the kernel function consistently throughout the paper. **c)** Thanks for the comment, and sorry for the confusion. Without loss of the generality, in the theoretical analysis, we assume that the data are standardized in a unit ball so that the mentioned condition is satisfied. We have used the generalized kernel function, which does not necessarily include symmetric kernels. We have fixed the kernel notations to be consistent throughout the paper. We refer to (Muller.1991) for more details on the generalized kernel. **d).** Thank you for the suggestion. We have modified the notations to avoid confusion.

Reviewer comment (Writing):
In Definition 2, the kernel is written as a bivariate function in contrast to Section 3's univariate form — is there a reason for this?

Author response:
**a)**. We have moved Definition 2, which was originally reserved for generalized kernel function, to Section 3, as it has close relevance to the methodology section. **b)**. Thank you for pointing it out; we have corrected notations for the kernel function consistently throughout the paper. **c)** Thanks for the comment, and sorry for the confusion. Without loss of the generality, in the theoretical analysis, we assume that the data are standardized in a unit ball so that the mentioned condition is satisfied. We have used the generalized kernel function, which does not necessarily include symmetric kernels. We have fixed the kernel notations to be consistent throughout the paper. We refer to (Muller.1991) for more details on the generalized kernel. **d).** Thank you for the suggestion. We have modified the notations to avoid confusion.

Reviewer comment (Theory):
In Definition 2, the condition \(k_h(x, y) = 0\) if \(x \notin [y - 1, y] \cap \mathcal{C}\) is unclear, especially since bivariate kernels are usually symmetric.

Author response:
**a)**. We have moved Definition 2, which was originally reserved for generalized kernel function, to Section 3, as it has close relevance to the methodology section. **b)**. Thank you for pointing it out; we have corrected notations for the kernel function consistently throughout the paper. **c)** Thanks for the comment, and sorry for the confusion. Without loss of the generality, in the theoretical analysis, we assume that the data are standardized in a unit ball so that the mentioned condition is satisfied. We have used the generalized kernel function, which does not necessarily include symmetric kernels. We have fixed the kernel notations to be consistent throughout the paper. We refer to (Muller.1991) for more details on the generalized kernel. **d).** Thank you for the suggestion. We have modified the notations to avoid confusion.

Reviewer comment (Writing):
In Definition 2, the use of \(x\) and \(y\) is confusing as it may suggest a relation to covariate \(x\) and label \(y\) discussed earlier; perhaps different variables should be used.

Author response:
**a)**. We have moved Definition 2, which was originally reserved for generalized kernel function, to Section 3, as it has close relevance to the methodology section. **b)**. Thank you for pointing it out; we have corrected notations for the kernel function consistently throughout the paper. **c)** Thanks for the comment, and sorry for the confusion. Without loss of the generality, in the theoretical analysis, we assume that the data are standardized in a unit ball so that the mentioned condition is satisfied. We have used the generalized kernel function, which does not necessarily include symmetric kernels. We have fixed the kernel notations to be consistent throughout the paper. We refer to (Muller.1991) for more details on the generalized kernel. **d).** Thank you for the suggestion. We have modified the notations to avoid confusion.

Reviewer comment (Theory):
Page 3, Eqs. (2)-(3): It is unclear why \(T\) and \(T^*\) map \(L^2\) to \(L^2\). Is this an additional implicit assumption or does it follow from their forms? Relatedly, just after Eq. (4), the implication \(\mathbb{E}_s\{\eta(x)\} = 0 \Rightarrow \eta(x) \in L^2(x)\) is not clear; why does \(\mathbb{E}_s\{\eta^2(x)\} < \infty\) hold?

Author response:
Thanks for the comment. First, this is not an additional assumption but a definition. In Eqs (2)-(3), we want to present that *$T$ and $T^{\ast}$ are two conditional expectation operators to map the functions between $L^{2}(x)$ and $L^{2}(y)$.* We believe the confusion arose from the definition of $L^2$ space we use. We have clarified the definition of $L^2$ space on page 3. Secondly, in our analysis, *we only consider the mean-zero and square-integrable function space. And $\mathbb{E}_{s}\{\eta^{2}(x)\} < \infty$ is due to the definition of square-integrable.*

Reviewer comment (Theory):
Assumption 2: While the mathematics is understood, its ramifications here are hard to grasp. More intuition or examples of \(\beta\)-regularity for a conditional expectation operator would help.

Author response:
The $\beta$-regularity space amounts to a condition of the relationship between the **smoothness** on the function $\rho(y)$ (i.e., rate of decay of its Fourier coefficients) and the **degree of ill-posedness** on the operator $T$ (i.e., rate of decay of its singular values). For example, suppose the operator is severely ill-posed (i.e., the rate of decay of the singular values of $T$ is fast), the condition says that we will also need the function $\rho(y)$ to be very smooth (rapidly decay). Otherwise, $\rho(y)$ is no longer in the $\beta$-regularity space.
    
In addition to the solution's existence (as you have mentioned: invertible), it is crucial for the Moore Penrose generalized inverse of $T$ to be **continuous**. If it is not continuous, the solution is unstable (i.e., explosive behavior). However, when we generalize a functional estimation from a parametric setting to a nonparametric one (i.e., a limit of finite dimensional problems), the Moore-Penrose generalized inverse is not continuous. Thus, there is no way we can obtain a consistent estimator because we do not have the true function $\eta$ and have to estimate it (where the estimation will introduce “perturbation” (i.e., estimation error) in the input $\eta$). After recasting the problem from $L^2(X)$ and $L^2(Y)$ to the $\beta$-regularity spaces. The ill-posed problem becomes well-posed with topologies on the $\beta$-regularity spaces. In other words, the Moore-Penrose generalized inverse is **bounded** under the new topologies, thus leading to a continuous solution. Thus making it possible to have a consistent estimation of the function $\rho(y)$.

Reviewer comment (Theory):
Assumption 5: Similar to Assumption 2, more intuition or examples would help explain when it is satisfied.

Author response:
Thanks for the comment we claim the original assumption 5 can be justified under all other previous assumptions and thus presented as a lemma. We have removed this assumption from the main paper and prove it in the supplementary.

Reviewer comment (Theory):
Assumption 4: "Letting \(\gamma = \min\{k, \ell\}\)": \(\gamma\) is never used again in this assumption — possibly an error.

Author response:
Thanks for the catch; this condition is not necessary for assumption 4, so we have deleted it.

Reviewer comment (Writing):
Remark 6: Clarify if "consistent" means "in operator norm."

Author response:
Yes, the consistency is under the operator norm. We have added an additional phrase in remark 5, which was originally remark 6, to clarify the norm. In the meantime, one can also show that it is consistent under the Hilbert-Schmidt norm. Furthermore, we have moved this assumption to the appendix and provided references for its justification.

Reviewer comment (Evaluation):
The definition of "Delta Accuracy" in the experiments is unclear; a mathematical formula should be provided along with whether 100% means perfect prediction.

Author response:
Sorry for the confusion. Here, the Delta Accuracy is defined as the percentage of improved prediction MSE compared with Non-Adaptation. Mathematically, it can be formulated as \[$\Delta$ Accuracy = (prediction MSE of None Adaptation- prediction MSE of proposed method ) /  prediction MSE of None Adaptation * 100\%\]. The larger value represents better prediction performance while 100% indeed means the perfect prediction because the prediction MSE of the proposed method is 0 then.

Reviewer comment (Theory):
Page 8, last sentence: In the described resampling procedure, if the bootstrap is used, sampling should be done with replacement — clarify the reasoning.

Author response:
Yes, we used bootstrap to obtain confidence intervals for performances.  Specifically, in each trial, we random sample 80% training data so that there is no duplicated training data in the sample trial. While across trials, the random sampling is with replacement. This bootstrap method is also valid in the literature, e.g., Sitter et al. (1992). When bootstrapping with replacement, the bootstrap sample size is the same as the original data; for bootstrapping without replacement, the bootstrap sample size is smaller than the original one (in our case, 80%). But under the independent and identically distributed (i.i.d.) assumption, both methods are valid because, essentially, both methods sample from the same empirical distribution.

Reviewer comment (Writing):
Page 9, first paragraph: The phrase "use the humidity for the importance weight estimation" is unclear — does it mean only this feature is used rather than all available features?

Author response:
Sorry for the confusion. We want to clarify that we only select the *humidity for the weight estimation. This is* because, from Meteorology study, it has been widely known that there is a causality relationship between humidity and temperature: the high temperature will lead to low humidity, which is also validated in our exploratory data analysis in the supplementary Section C.1. *However, the other features may not satisfy the target shift assumption.*

Reviewer comment (Evaluation):
Page 9, second paragraph: The claim that the method improves about 4% for the SOCR dataset is unclear, as this improvement is not visible in Table 1.

Author response:
Sorry for the confusion. Here is how the 4% comes: The KMM method has the $\Delta$ Prediction MSE as 3%, and our method has it as 6.7%. Thus, the difference is about 6.7% -3% = 3.7%, which we mean by about 4%.

Reviewer comment (Experiments):
In the experiments, only low-dimensional and small-scale datasets are used for comparison. Thus, the empirical results are not sufficient to support the success of the proposed algorithm. Considering the increasing need of learning with high-dimensional data such as images and the poor performance of kernel density approximation in high-dimensional data, the comparison in high-dimensional data is important.

Author response:
In the revision, we have demonstrated the effectiveness of our proposed method with larger datasets with higher dimensions (e.g., high dimension regression problem on UCI community and crime dataset and knowledge distillation regression problem on MNIST dataset). Due to space limits, we put the **additional numerical studies** in **Sections C.4.1 & 4.2** of the supplementary materials. Furthermore, to handle the density approximation problem with high-dimensional data, we adopted the black-box-shift-estimation method Lipton et al. (2018) proposed in our setting. Please refer to **Section C.4** for the details.

Reviewer comment (Experiments):
Determining the hyperparameter $\alpha$ for different data sets is difficult.

Author response:
Thanks for pointing this issue out. Based on Theorem 1, the regularization parameter must satisfy the condition that $\lim_{n\to\infty}n\alpha^2=\infty$. Thus, in our experiment, we proposed to choose $\alpha=n^{-1/4}$ as a rule of thumb to meet the condition for practical simplicity. In Section C.2.3 of the supplementary materials, we conducted a sensitivity analysis from which $\alpha=n^{-1/4}$ achieves good performance. We use this rule of thumb for all the numerical studies, and the results are good.

Reviewer comment (Theory):
In classical ratio estimation approaches, the estimation variance is sometimes large, e.g. $p_s(y)$ goes to zero, making the algorithm unstable. In the continuous label space, it is unclear whether this problem becomes worse. It is suggested to discuss this issue or to introduce an additional variance reduction mechanism.

Author response:
Indeed, it is problematic when $p_s(y)$ goes to zero. This is why we need the assumption that $p_s(y)$ is bounded from below by some constant $\epsilon>0$ (see Assumption 3). Also, we need the support of $p_t(y)$ to be a subset of the support of $p_s(y)$. When $p_s(y)$ goes to zero (while $p_t(y)$ is not zero), it corresponds to the out-of-distribution (OOD) setting. Although this is beyond the scope of our paper, it represents an interesting direction for future research. We discussed this issue as a future research direction in the final discussion.

Reviewer comment (Experiments):
It is suggested to test the performance of the proposed algorithm on large-scale and high-dimensional datasets.

Author response:
In this revision, we **applied our method to larger datasets with higher dimensions** in additional experiments. More specifically, we include the experiments of high dimension regression problem on the UCI community and crime dataset and knowledge distillation regression problem on the MNIST dataset in the supplementary. The proposed method demonstrated good performance. Due to space limits, the numerical study results can be found in **Sections C.4.1 & 4.2** of the supplementary materials.

Reviewer comment (Theory):
Given the importance weight function, is the estimation process sensitive to noise or outliers in the data and thus potentially leading to inaccurate estimates of the importance weight function? Or this limitation is conquered by applying the regularization technique in your paper?

Author response:
Yes, the noise and outlier data will affect the estimation performance and the regularization applied in our method can enhance our ability to robustly adjust to uncertainties in estimating the importance weight function.

Previous studies have illustrated such advantage of regularization in classification tasks; for instance, Azizzadenesheli et al. (2019) introduced a regularized version of the label shift estimator called RLLS, building upon the BBSE method of Lipton et al. (2018), who did not employ regularization. The regularized estimator RLLS demonstrated superior performance to the original non-regularized version BBSE, indicating that the regularization does help.

In the continuous continuous, regularization serves a **dual** purpose: addressing the ill-posedness problem while providing the advantage of robustness. However, unlike the classification case where regularization is optional (i.e., Lipton et al. 2018), the problem we consider becomes ill-posed due to the 'explosive behavior' in the continuous case without regularization, as detailed in our paper. So, regularization is a must-have in our case.

Reviewer comment (Theory):
Estimating a continuous importance weight function is a probabilistic process, together with the regularization process, this may inherent uncertainty in the estimates, affecting the model's reliability in certain cases. How do you solve this issue?

Author response:
The functional estimation, and indeed any estimator, is inherently contingent upon the randomness of the data sample; uncertainty becomes an inherent characteristic of the estimation. In our study, **we establish a theoretical bound for the estimator’s convergence rate in Theorem 1**, and the error bound measures the estimation uncertainty from the theoretical aspect. **Also, in our numerical evaluation, we presented all the estimation performance with confidence intervals and standard deviations, which can capture the estimation uncertainty.** 

A related aspect involves the uncertainty quantification for the continuous case. For instance, considering the randomness inherent in the estimation, how to construct a prediction set for the response. While this lies beyond the scope of our current paper, it represents an interesting direction for future research. Existing efforts have been made on uncertainty quantification in the classification case (e.g., Podkopaev and Ramdas 2021), but notably, there appears to be a gap in addressing the continuous case. We have added a discussion on future research in the paper.

Reviewer comment (Theory):
Regularization can help stabilize the estimation process, but accurate estimation of a continuous importance weight function may still require a substantial amount of data from both source and target domains. What is the sample complexity of your algorithm?

Author response:
We agree that the sample sizes play an important role in the weight estimation. Theorem 1 provides a stochastic order bound for our proposed functional estimation, in which the sample size will directly affect the convergence rate. In Equation (11), the bound is a function of $n$ (sample size of the source data) and $m$ (sample size of the target data) given other quantities (e.g., the tuning parameter $\alpha$ and the bandwidth $h$).