Strengths:

The paper is generally very well written and easy to follow. The authors excellently introduce the problem and previous methods. The manuscript also clearly motivates the problem. Specific strengths are:

1. The manuscript contains detailed and extensive theoretical results, providing strong theoretical guarantees.

2. The theoretical results are described very well. Each theorem and proposition is described in terms of its broader importance and the purpose for it being used. This is very important in theoretical papers, to allow the reader to follow without getting bogged down by extensive theory.

3. The manuscript tackles an interesting problem and the contribution is important. The method is applicable to many important settings.

Our Response: Thank you for your positive feedback. We are delighted that you found the manuscript addresses an interesting problem and the contribution is important. We also want to thank you for dedicating your time and expertise to review our paper and provide constructive comments and suggestions. We appreciate your careful review, and we will address all comments thoroughly in our revision.

Weaknesses:

My main concerns are regarding the empirical results. I'm not convinced by both the scope and outcome of the results. Specifically:

1. Both the synthetic and real data studies are very limited in scope. For an approach that is so widely useful, it is a shame that it has not been applied to more settings to showcase its usefulness. It would be insightful to find and present scenarios showing the extreme ends of performance (very good and very bad performance) of the approach and discuss why this is.

Our Response: Thank you for your comments and suggestions. In this revised version, we have carefully addressed your concerns by conducting additional experiments and data analyses to explore diverse settings.

First, we have now included an additional real-world dataset to further validate the practical utility of our methods. The analysis results are now presented in Figure 3 and discussed at the end of Section 5.

Second, we have now expanded synthetic experiments for various scenarios in Appendix G, including varying levels of noise, sample sizes, and data generation models, to provide a more comprehensive understanding of the performance of our proposed methods.

Furthermore, as ICRF comprises a component in our proposed methods (clarified at the end of Section 3.3), we have now conducted further experiments to evaluate how different implementations of ICRF influence performance of our proposed methods. Results and discussions are now included in Appendix G.3.

2. The results from the studies are not overly convincing. The method performs the same as imputation and standard boosting (which already exists) and does not provide a large benefit over the naive approach in the synthetic study. I have asked a clarifying question around this in the Questions, so this weakness can be addressed.

Our Response: Thank you for your comments, which we have now carefully addressed. In our initial version, we did not include comparison details due to space limitations. In this revision, we have added more information to compare the two algorithms, explaining their differences and similarities, in Section 3.2 for greater clarity.

Furthermore, in this revised manuscript, we have conducted additional experiments; details can be found in Appendix G. The results demonstrate the practical utility of our proposed methods.

3. I think there should be more discussion on the assumptions and limitations of the method. This would help to understand under which scenarios we might expect the method to not do well.

Our Response: To address your comments, we have now included the discussion in Section 6.

4. There is no discussion on computational cost or complexity. This could either be a theoretical complexity analysis or even just simple timing tests. Either would provide useful insights for practitioners who wish to decide on whether to apply the method to their problem.

Our Response: Thank you for highlighting the importance of discussing computational complexity. In response, in this revision we have now included the discussion in Appendix E.2. In addition, in this revision we have re-ran experiments and added computational timing comparisons for better insights into finite sample implementation. These results are now reported in Table F.1 of Appendix F.3.

Thank you for rigorous responses and updates.
However, it is still unclear for me about the benefit of employing boosting in the proposed problem setup, although the method itself becomes clear for me.
It would be the best if the performance outperformed existing methods, but it looks not true as far as reading the experimental results (although certain improvements are seen). However, it is good if there are good insights through the proposed method, for example, "this is an interesting result in the field of boosting" or "this methodology can be extended to other learning methods than boosting". How do the authors think?

Strengths:

• The paper found that, by CUT, existing L2boost algorithm can be applied to interval-censored datasets.
• There is a constraint that we need to estimate the joint distribution of $X$ and $Y$ without knowing $Y$ itself but knowing only intervals. They showed that estimating the distribution by ICRF experimentally worked well.
• Theoretical convergence rates and lower bounds of MSE (for regressions) and misclassifications (for classifications) are presented.

Our Response: Thank you for dedicating your time and expertise to review our paper and provide constructive comments and suggestions. We appreciate your careful review, and we will address all comments thoroughly in our revision.

Weaknesses:

• The procedure itself looks somewhat simple; first we apply CUT and then L2boost. If there is a difficulty or interesting results of these combinations, please emphasize.
• Perhaps, the combination of boosting and interval-censored data is the novelty? If so, please emphasize the discussion on the novelty (e.g., limitations in existing methods).

Our Response: Thank you for your comments and suggestions, which have helped us revise the paper and more clearly articulate the contributions of our work. The key innovation of our work lies in presenting a framework that extends boosting methods to handle interval-censored data — a crucial yet underexplored problem in machine learning. Standard boosting algorithms are not directly applicable in this context due to the challenges posed by interval censoring. Our approach addresses this gap by leveraging the censoring-unbiased transformation (CUT) to construct unbiased loss functions tailored for interval-censored data.

Motivated by your comment and feedback from other reviewers, we have revised the paper to include extensions of our framework to other loss functions that can be used to handle interval-censored data. We have also highlighted the uniqueness of the $L_2$ loss function employed in our current theoretical development and emphasized that establishing similar theoretical results for these extensions is not straightforward.

In response to your suggestion, we have included a discussion on possible extensions in Appendix E.3 as well as the limitations of our methods in Section 6.

Dear Reviewer VgwR:

Many thanks for your prompt response. We value your comments, suggestions, and feedback, which greatly help us sharpen the presentation of our work. With kind regards.

Strengths:

• The manuscript is clearly written and accessible, making the methodology easy to follow.
• The authors conduct a rigorous theoretical analysis of their proposed methods, assessing mean squared error (MSE), variance, and bias to establish a solid foundation for their approach.

Our Response: Thank you for your encouraging feedback. We are pleased that you found the manuscript clear and accessible, and we appreciate your recognition of the rigorous theoretical analysis. We also want to thank you for dedicating your time and expertise to review our paper and provide constructive comments and suggestions. We appreciate your careful review, and we have addressed all comments thoroughly in our revision.

Weaknesses:

• In Proposition 2, using a distinct notation for the smoother matrix would help prevent confusion with the survival function $S$

Our Response: Thank you for this suggestion. In the revision, we have now incorporated a different symbol, $\Psi$, for improved clarity.

• Adding experiments for the Cox proportional hazards model would strengthen the manuscript's applicability and support its conclusions.

Our Response: In response to your suggestion, we have included results derived from the Cox model in both data analyses and experiments. These results are now presented in Figure 3 (Section 5) and Figure G.4 (Appendix G.2), along with corresponding comments in the respective sections.

• Additional ensemble methods exist for interval-censored data, such as: Yao, W., Frydman, H., and Simonoff, J.S., 2021. An ensemble method for interval-censored time-to-event data. Biostatistics, 22(1), pp.198-213. Including comparisons with such methods would enhance the evaluation of the proposed approach.

Our Response: Thank you for this suggestion, which has now been incorporated in this revision. Specifically, additional experiment results, reported in Figure G.4 in Appendix G.2, shows that the results from the Yao et al. (2021) method are in good agreement with those produced from our proposed methods. However, the SKDT (Kendall's $\tau$) values from this method appears slightly more variable than those from our methods.

• In both synthetic and real data experiments, the proposed CUT method and the imputation approach (Bian et al. 2024a) yield comparable results, with the imputation method occasionally performing better (e.g., in real data). Could the authors further elaborate on the unique advantages of their proposed method?

Our Response: Thank you for raising this interesting point, which we have now carefully addressed. In our initial version, we did not include comparison details due to space limitations. In this revision, we have added more information to compare the two algorithms, explaining their differences and similarities in Section 3.2 for greater clarity.

• In Appendix A, the authors outline several conditions primarily related to the smoother matrix $S$. Providing relevant examples of $S$ that meet these conditions would improve clarity.

Our Response: Thank you for the suggestion. In this revision, we have now inserted the following material at the end of Appendix A.

• All theoretical results assume consistent estimation of the survival function. How does the convergence rate of the survival function estimator impact the final estimator?

Our Response: Thank you for raising this important concern. We agree that the convergence rate of the survival function estimator can affect the speed at which the estimator approaches the true value. A slower convergence rate may increase variability in the final estimator, potentially resulting in less reliable predictions, particularly in finite sample settings. To make this point clear, in this revision we have now included discussions in Section 3.3.

Questions:

• In page 3 line 108, should the survival probability be $P(Y>s)$?

Our Response: Thank you. We have now fixed this.

• In page 3 Equation (5), what is the update compared to Equation (3)?

Our Response: For greater clarity, we have revised the presentation of Equation (5) by combining it with the follow-up description originally in Equation (6).

• In traditional boosting methods, a learning rate is typically included. Is there a similar parameter in Algorithm 1?

Our Response: Thank you for this insightful question. To make this point clear, in this revision we have now included discussions in Appendix E.1.

• In Condition (C1), what does $Q_i^2$ mean? Does it denote $Q_i^\prime Q_i$ as $Q_i$ as is an eigenvector?

Our Response: Thank you very much for pointing out this error. You are correct; it was meant to be $Q_i^\top Q_i$. We have now corrected it.

Thank you for providing more detailed comments. Here, we further clarify and address the concerns you raised.

1. The differences between Bian et al. (2024a) and our L2Boost-IMP (IMP)

We would like to clarify that our L2Boost-IMP method is fundamentally different from the approach proposed by Bian et al. (2024a). The reference to Bian et al. (2024a) in the Introduction is included solely to provide context about boosting methods in diverse applications.

Both the L2Boost-IMP method and Bian et al.'s (2024a) approach share a common use of imputation to create complete datasets for applying algorithms designed for full data. However, key differences between the methods pose distinct challenges in establishing their respective theoretical guarantees, as highlighted below.

(a) different contexts:

The method proposed by Bian et al. (2024a) was developed for missing responses, where a simple missing data indicator $\Delta_i$ ($\Delta_i= 1$ if response $Y_i$ is observed, and 0 otherwise) suffices to reflect the missingness status for each subject $i$. In contrast, our L2Boost-IMP (IMP) method is tailored for interval-censored responses, which require a more complex representation. Instead of a scalar missing data indicator $\Delta_i$, a sequence of censoring indicators $\Delta_{i,j}$ is needed for each subject $i$, corresponding to different intervals indexed by $j$.

(b) different constructions of pseudo-outcomes:

Bian et al. (2024a) used the Buckley-James formulation (Buckley and James 1979) by defining a pseudo-outcome as $Y_i^* \triangleq \Delta_i Y_i+(1-\Delta_i) E(Y_i|X_i, \Delta_i=0).$ However, our approach constructs the pseudo-outcome differently, as shown in (7), which incorporates the survivor function.

(c) different implementations:

In determining the pseudo-outcome $Y_i^*$, Bian et al. (2024a) approximated the conditional expectation $E(Y_i|X_i, \Delta_i=0)$ using the Monte-Carlo method. They generated a sequence of variates $y_i^{(k)}$ from the conditional distribution $f(y_i|X_i, \Delta_i=0)$ for $k =1, \ldots, K$, which $K$ is a user-specified, and computed the average $(1/K)\sum_{k=1}^K y_i^{(k)}$. On the contrary, determining the pseudo-outcome in our approach, as outlined in (7), requires estimating the survivor function, making the implementation of our $L_2$Boost-IMP (IMP) method considerably more complex than that of Bian et al. (2024a).

(d) different properties of imputed outcome/loss function:

The pseudo-outcomes in Bian et al. (2024a) satisfy the properties $E(Y^*_i|X_i)=E(Y_i|X_i), \ \ \mbox{and thus}, \ \ E(Y^*_i)=E(Y_i).$

However, pseudo-outcomes in our approach lack this property due to the complexity introduced by the interval-censoring structure of the data.

2. differences between our proposed two methods: L2Boost-CUT (CUT) and L2Boost-IMP (IMP)

The CUT-based loss function $L_{\rm CUT}({\cal O}_i, f(X_i))$ in (9) modifies the $L_2$ loss, denoted $L(u,v)$, to ensure Proposition 1 holds. However, this adjusted loss function is quadratic in the difference between its first and second arguments, meaning it is not identical to the $L_2$ loss with its first argument imputed by the transformed response $\tilde Y_1({\cal O}_i)$ in (7).

Specifically $L_{\rm CUT}({\cal O}_i, f(X_i)) \ne L(\tilde Y_1({\cal O}_i),f(X_i)).$

This distinction underpins our earlier statement that "These two algorithms are derived from distinct perspectives in addressing interval-censored data, and they may be expected to yield different results." The $L_2$Boost-CUT method focuses on adjusting the loss function so its expectation recovers that of the original $L_2$ loss $L$, whereas the $L_2$Boost-IMP method preserves the functional form of the original loss $L$ but replaces its first argument with transformed response $\tilde Y_1({\cal O}_i)$ in (7).

Despite this difference, since the $L_2$ loss has a derivative linear in its first argument, the increment terms in both $L_2$Boost-CUT and $L_2$Boost-IMP are closely related. However, if the loss function were $L_q$ with $q\ge 3$, the development of $L_q$Boost-CUT and $L_q$Boost-IMP methods would differ more substantially. However, as noted in Appendix E.3, establishing theoretical guarantees for such cases is challenging.

For a general loss function, particularly when it is nonlinear, constructing an adjusted loss function like $L_{\rm CUT}$ ​ in (9) to ensure Proposition 1 holds is challenging, making it difficult to implement. However, using the original loss function with imputed values determined by the transformed response in (7) is a straightforward implementation, though establishing theoretical results for both cases remains difficult.

Thank you for your thoughtful question on this aspect. We will incorporate these clarifications in the next revision.

Strengths:

The article is clearly written and provides a thorough theoretical analysis of the proposed algorithm, which is really important to ground further applicative work using this method in practice.

Our Response: Thank you for your positive feedback and for recognizing the importance and thoroughness of our work. We also want to thank you for dedicating your time and expertise to review our paper and provide constructive comments and suggestions. We appreciate your careful review, and we will address all comments thoroughly in our revision.

Weaknesses:

There are several limitations of this work.

• my main concern is that the paper does not clarify the empirical improvement compared to the use of ICRF. It is slightly different indeed to use boosted trees compared to random forests (bagging), but the article fails to show a clear gain in performance, both on simulated and real data.

Our Response: Thank you for raising this concern. We would like to clarify the distinction of our work in relation to ICRF. ICRF (Cho et al., 2022) is a tree-based, nonparametric method specifically designed for estimating survival functions for interval-censored data, which differs from our objective. We focus on developing boosting methods for regression and classification tasks with interval-censored data. Specifically, we aim to construct a predictive model $f(\cdot)$ that well predicts a transformed target variable $g(Y)$, where $g(\cdot)$ can take various forms to address different tasks. These tasks include predicting the survival time, the survival status at specific time points, or any functional outcome of survival time of interest.

Our method builds upon survival function estimation as a basic step, for which ICRF is utilized as a component of our framework. Therefore, our work is not designed to improve ICRF but rather to leverage it within a broader algorithm to achieve new predictive objectives. Consequently, the goal of our paper is distinct from ICRF’s, and direct numerical comparisons between the two would not be meaningful. For greater clarity, in this revision we have included some discussions at the end of Section 3.3 to address your comment, as well as a related comment from another reviewer.

• there should be more baselines, like Cox models, even if they are designed for right-censored data rather than interval data. There are very few baselines included in the benchmark presented here. It's unclear whether the I method is similar to ICRF or a novel method. If not, ICRF should be included as a baseline to demonstrate the contributions of this work clearly.

Our Response: Thank you for your feedback. Regarding your comment on ICRF, as explained in the response to the previous bullet point, the I method (now referred to as the L2Boost-IMP method, or IMP for short) is not similar to ICRF. Instead, ICRF constitutes a component of the I method and is used specifically for estimating the survivor function.

In response to your suggestion regarding the use of the Cox model, we have included additional results derived from it in the revised manuscript. Specifically, we now present results for both the data analysis and experimental settings involving the Cox model. They now appear in Figure 3 in Section 5, as well as Figure G.4 in Appendix G.2.

• I don’t think there is any mention that the code to use L2Boost-CUT or L2Boost-Impute will be made available, is it?

Our Response: Thank you for your comment. We plan to make the code publicly available after acceptance and will post it on GitHub at that time. In this revision, we have added a note after Proposition 1 in Section 3.2 to inform readers of this intention.

Questions:

• are there any additional performance metrics that could be used? like the c-index (or a variation of it)?

Our Response: Thank you for your thoughtful suggestion regarding additional performance metrics. To address this, we have incorporated Kendall’s $\tau$ to evaluate the concordance between the estimator and its target. Kendall’s $\tau$ quantifies the ordinal association between two variables by comparing concordant and discordant pairs. This metric captures both the consistency of the predicted ranking with the true ranking and the extent of any discrepancies between them. Alongside the metrics considered in the initial version, we now present experimental results using Kendall’s $\tau$ in Figure 1, as well as Figures G.1. G.2, G.4, and G. 5 in Appendix G. Additionally, we have included the discussions in Appendix F.1, Section 5, and Appendix G.1.

Strengths:

• The paper is mathematically well written. The notation is consistent and precise. Definitions and proofs are provided in the Appendix.
• The theoretical results are novel, clearly structured and formulated. The implications of each theoretical result are well discussed and framed within the literature context if relevant.
• The proposed algorithms are experimentally tested on the synthetic dataset under various scenarios. The method is further applied to real-life dataset.

Our Response: Thank you for the positive feedback. We are glad you recognize the novelty of the theoretical results and the mathematical rigor of the paper. We also want to thank you for dedicating your time and expertise to review our paper and provide constructive comments and suggestions. We appreciate your careful review, and we have carefully addressed all comments thoroughly in our revision.

Weaknesses:

1. The underlying part of the model is estimating survival function. It would be interesting to include the experiment with different survival function estimators to assess how sensitive the method is to the potential biases of the estimator of the survival times as this part of the model is not properly covered by presented theory.

2. It is not clear how sensitive the proposed method is to the noise in the underlying data.

3. Authors did not provide much insights into how well the algorithm scales to the larger datasets.

Our Response: In this revision, we have carefully addressed each of your comments to improve upon the weakness of our initial submission. Specifically, in response to your suggestion about survivor function estimation, we have now conducted additional experiments and reported the results in Figure G.5, with detailed comments provided in Appendix G.3. Regarding your comments on the sensitivity of our methods to noise level and data size, we have thoroughly addressed them. Details are provided in the following bullet points, corresponding to your individual questions.

Questions:

1. What are the scaling capabilities of the models to the large datasets? (provided datasets are of order of magnitude $10^3$? It would be good to see experiments for varying sizes of the dataset.

Our Response: Thank you for raising this important question. To address your concern, in this revision we have conducted additional experiments using a larger dataset with $n=10^3$ to evaluate the scalability of our methods. Results for these experiments have now been summarized in Figure G.1 in Appendix G.1.

2. The imputation algorithm and the CUT boosting in the presented experiments (especially Figure 3, E.3) seems to produce similar results. Can you provide more insights into why that is?

Our Response: Thank you for raising this interesting point, which we have now carefully addressed. In our initial version, we did not include comparison details due to space limitations. In this revision, we have added more information to compare the two algorithms, explaining their differences and similarities. Specifically, we have inserted the following material in Section 3.2 for greater clarity.

3. It would be great to see performance for other real-life datasets with already known benchmarks to demonstrate that the presented method performs well against other well explored survival dataset problems.

Our Response: Thank you for this suggestion. We agree that benchmarking the proposed methods on multiple survival datasets would demonstrate the broader applicability of our approaches. In response to your suggestion, we have included another dataset -- the Bangkok HIV data -- in this revision and analyzed it using the proposed methods. The results are now presented alongside the initial analysis of the Signal Tandmobiel dataset in Figure 3 in Section 5 (Here, results of implementing the Cox model are also included in response to other reviewers' suggestion).

4. For the synthetic dataset, e.g. scenario 1, how does the performance changes based on the amount of the noise in the data, so when you change the variance in the error term, e.g. instead of only 0.25 variance for normal, when you increase the noise for to 0.5, 1, 1.5?

Our Response: Thank you for the suggestion. In this revision, we have incorporated your feedback by conducting additional experiments to extend the initial analysis with $\sigma=0.25$ to include $\sigma=0.5$, $\sigma=1$, and $\sigma=1.5$. The results are summarized in Figure G.4, together with some comments presented in Appendix G.2.

Dear reviewers:

Thank you all for your continued feedback on our rebuttal and for your helpful suggestions. In addition to providing our further responses to your additional feedback individually, we have revised the manuscript once again to incorporate your comments for greater clarity. The changes made in response to your initial feedback are marked in red, while the new revisions are highlighted in blue for ease of your reference.

We deeply appreciate your time and valuable expertise, which have significantly improved the presentation of our work. We believe this revised version represents a considerable improvement over the initial draft for greater clarity.

Thank you once again for your support.

With warm regards