Step-Controlled DPO: Leveraging Stepwise Errors for Enhancing Mathematical Reasoning of Language Models

Q3: This work is quite similar to another work "Step-dpo: Step-wise preference optimization for long-chain reasoning of llms", but the differences between these papers are not clearly demonstrated. The authors need to elaborate more about the comparison.

A3: Thank you for your valuable suggestion. Step-DPO [1] is a work concurrent with ours, which was also submitted to ICLR 2025 (https://openreview.net/forum?id=H5FUVj0vMd). The approach in [1] uses GPT-4 for erroneous step localization, which is less cost-effective. In contrast, our method incorporates stepwise information into the DPO process by constructing positive and negative pairs that differ after a specific step.

To compare Step-DPO with our method, we trained MetaMath-Mistral-7B using the dataset and code of Step-DPO. As shown in the table below, our method outperforms Step-DPO on most datasets without relying on any stronger LLMs (e.g., GPT-4), demonstrating the effectiveness of our approach. We have addressed the differences between our work and the concurrent Step-DPO in the related work section (line 485, blue font) and added the experimental results in Section 4.2.

[1] Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs

Q4: Line 466 misses a citation.

A4: Thank you for pointing this out. We have corrected this typo in the revised paper. Additionally, we have thoroughly proofread the entire paper again to ensure that any other typos or errors have been addressed. We sincerely appreciate your valuable feedback.

Thank you for reviewing our work and providing thoughtful suggestions. Your valuable feedback has been instrumental in improving our research and presentation. We address your concerns as follows:

Q1: The paper mentions SCDPO works well on both CoT and Code-integrated solutions, but the results do not demonstrate this comparison. The author needs to add a table that highlights the comparison between these two different types of solutions.

A1: Thank you so much for the suggestion. We present the comparison in Table 2 of the paper, which has been expanded to include results on out-of-domain datasets. MetaMath-Mistral-7B uses CoT solutions, while Mistral-7B-Ours and MathCoder-Mistral-7B utilize Code-Integrated solutions. To better compare these two types of solutions, we evaluated these models on five out-of-domain datasets: OCW, Hungarian, Mathematics, SVAMP, and Simuleq. The results are shown below. Code-Integrated solutions outperform CoT solutions on benchmarks involving complex computations, such as MATH, OCW, and Mathematics. On less computation-intensive datasets, such as GSM8K, SVAMP, and Simuleq, the performance of the two types of solutions is similar.

Q2: This work lacks the experiments to show generalization ability of the method. The author could do an evaluation on out-of-distribution data, comparing the results with the in-distribution data.

A2: Thank you for the suggestion. Firstly, as shown in Table 1 and the table below, we tested our models on 10 datasets, among which GSM8K, MATH, and APE210K are used for training and are considered in-distribution. The other 7 are out-of-distribution datasets. OCW Courses is a dataset of challenging university-level problems. The Hungarian National Exam consists of held-out high school national final math exams, making it an excellent benchmark for testing the model’s generalizability. Mathematics is a dataset focused on solving complex equations, while SVAMP and Simuleq feature math problems that are worded differently from those in GSM8K. These datasets cover a broad range of domains and problem types.

Secondly, we also tested SCDPO’s effect on three different SFT models with out-distribution datasets, as shown in the table in A1. We have added these new tests to Table 2 in the revised paper. These results can effectively demonstrate the generalization ability of the method.

Dear Reviewer 1hBn,

We hope this message finds you well. As the author-reviewer discussion period progresses, we respectfully seek your confirmation on the adequacy of our rebuttal in addressing the concerns raised in your review.

We really appreciate the substantial time and effort you have committed to reviewing our work and are grateful for the additional insights. Your comments have been very helpful in refining our project.

Thank you once again for your valuable perspectives. We eagerly await your further guidance.

Sincerely,
Authors

Thank you for reviewing our work and for your thoughtful suggestions, which have greatly helped improve our project. We address your concerns below:

Q1: The concept of a “step” in the reasoning chain is not clearly defined in the paper. It would be more clear if the authors could provide a more precise explanation of how they segment steps within a reasoning chain.

A1: Thank you for your suggestion. In CoT-format solutions, the reasoning steps are separated by “\n”. In Code-Integrated format solutions, the model generates reasoning steps in the following order: a natural language reasoning step, a code snippet, the output of the code execution, and then repeats this cycle. Each block begins with <|text|>, <|code|> or <|execution|> and ends with <|end_of_text|>. We consider each text block as a reasoning step, and each code snippet paired with its execution output is also regarded as a reasoning step.

Q2: Potential positional bias in error generation. The approach uses an ascending temperature strategy to create erroneous data, which may unintentionally introduce a bias where errors occur more frequently in the latter half of the reasoning chain. It would be helpful if the authors could investigate whether the distribution of errors indeed skews toward the end. Since errors cannot be automatically located within the chain, manually sampling and inspecting some dispreferred solutions could provide insights into any potential biases.

A2: Thank you for your suggestion. We sampled 100 SCDPO-rejected solutions and manually inspected them. Our findings indicate that while not all errors occur at the beginning of the generation of erroneous steps, most occur early in the process, as earlier steps often make decisions that significantly influence the correctness of the solution. We present the distribution of the step index where the first actual error occurs in the 100 SCDPO-rejected solutions, alongside the distribution of the index where the generation of erroneous steps begins. As shown in the table below, the two distributions are closely related, demonstrating that the starting point of error generation has a strong impact on the step index of the actual error. In most cases, the step index of the actual error is close to the starting point of error generation. This analysis has been added to Appendix E of the revised paper.

Q3: (Following up on Weakness 2) Could the authors provide an analysis of the distribution of the dispreferred data? Specifically:

(a) What is the distribution of the step (referenced in Line 138) that is used as the starting point for generating errors in the resulting dispreferred data?

(b) What is the distribution of the "actual error" location within the dispreferred reasoning chains?

A3: We present (a) and (b) in the table below. (a) corresponds to the “starting point distribution”, while (b) corresponds to the “actual error distribution”. As shown in the table, the two distributions are closely related, with the starting point of error generation having a strong influence on the location of the actual error.

Q4: Line 467: There appears to be an unlinked reference.

A4: Thank you for pointing out this typo. We have corrected it in the revised paper. Additionally, we have thoroughly proofread the entire paper again to address any other potential errors. We sincerely appreciate your attention to detail and your valuable feedback.

A4 (continued): We also tested manipulating the distribution of the step index where error generation begins. Specifically, we trained the Mistral-7B-SFT model with SCDPO data where the starting index was either less than or equal to 4, or greater than 4. As shown in the table below, limiting the error starting point index decreases performance compared to not imposing any such limitation. We have added this result to Appendix E of the revised paper.

Q5: Mathematical reasoning is one of reasoning types. It would interesting to see if it can expand to other reasoning tasks as well.

A5: Thank you for the valuable suggestion. In this paper, we mostly focus on mathematical problem-solving tasks as it is a representative reasoning task and can be easily verified. We will explore the effect of SCDPO on other reasoning tasks such as code generation, theorem proving, and other STEM subjects in future works. We have added this to the Limitation and Future Work Section of our paper.

We test our 20B models on GPQA Diamond, a challenging multiple-choice Q&A dataset featuring difficult questions in biology, physics, and chemistry. As shown in the table below, SCDPO also improves performance on GPQA, demonstrating the potential of our method for reasoning tasks in subjects beyond mathematics.

Q6: There are many "-" in table 1. Any reason?

A6: The performance of the baselines is taken from the reports in previous works ([1], [2], [3], [4], [5]). However, some datasets were not included in these works. To make the comparison in Table 1 clearer, we conducted our own evaluation of several important open-source baseline models on these previously unreported datasets. The results are as follows, and we have added them to Table 1 of the revised paper.

[1] Xu, Yifan, et al. "ChatGLM-Math: Improving Math Problem-Solving in Large Language Models with a Self-Critique Pipeline." arXiv preprint arXiv:2404.02893 (2024). [2] Gou, Zhibin, et al. "Tora: A tool-integrated reasoning agent for mathematical problem solving." arXiv preprint arXiv:2309.17452 (2023). [3] Wang, Ke, et al. "Mathcoder: Seamless code integration in llms for enhanced mathematical reasoning." arXiv preprint arXiv:2310.03731 (2023). [4] Yue, Xiang, et al. "Mammoth: Building math generalist models through hybrid instruction tuning." arXiv preprint arXiv:2309.05653 (2023). [5] Lu, Zimu, et al. "Mathgenie: Generating synthetic data with question back-translation for enhancing mathematical reasoning of llms." arXiv preprint arXiv:2402.16352 (2024). Experiment: testing

Example 1:

Question: There are 20 dolphins in the aqua park. One-fourth of the dolphins are fully trained. Two-third of the remaining dolphins are currently in training and the rest will be trained next month. How many dolphins will be trained next month?

Correct Steps:

To solve this problem, let's proceed step by step:

Step 1: Find the number of fully trained dolphins To find out how many dolphins are fully trained, we can calculate one-fourth of the 20 dolphins in the aqua park.

The formula for calculating one-fourth of a number $n$ is:

$$\text{Fully trained dolphins} = \frac{n}{4}$$

In this case, $n = 20$.

Let's calculate.

# Given number of total dolphins in the aqua park
total_dolphins = 20

# Calculate the number of fully trained dolphins
fully_trained_dolphins = total_dolphins // 4
fully_trained_dolphins

5

The number of fully trained dolphins in the aqua park is 5.

Step 2: Calculate the number of dolphins still to be trained The number of still-to-be-trained dolphins can be found by subtracting the number of fully trained dolphins from the total number of dolphins:

$$\text{Dolphins still to be trained} = \text{Total dolphins} - \text{Fully trained dolphins}$$

Let's calculate this.

# Calculate the number of dolphins that are still to be trained
still_to_be_trained_dolphins = total_dolphins - fully_trained_dolphins
still_to_be_trained_dolphins

15

The number of dolphins that are still to be trained is 15.

Step 3: Find the number of dolphins currently in training To find out how many dolphins are currently in training, we take two-thirds of the number of still-to-be-trained dolphins.

The formula for this is:

$$\text{Dolphins in training} = \frac{2}{3} \times \text{Dolphins still to be trained}$$

Let's calculate this next.

Erroneous Steps:

# Calculate the number of dolphins that are currently in training
dolphins_in_training = 2 * fully_trained_dolphins // 3
dolphins_in_training

3

The number of dolphins that are currently in training is 3.

Step 4: Calculate the number of dolphins that will be trained next month The number of dolphins that will be trained next month can be calculated by subtracting the number of dolphins currently in training from the number of remaining dolphins that are still to be trained:

$$\text{Dolphins for next month} = \text{Still to be trained dolphins} - \text{Dolphins in training}$$

Let's perform this calculation.

# Calculate the number of dolphins that will be trained next month
dolphins_next_month = still_to_be_trained_dolphins - dolphins_in_training
dolphins_next_month

12

$$\boxed{12}$$

The number of dolphins that will be trained next month is $\boxed{12}$.

Example 2:

Question: A cookfire burns three logs every hour. It was built with six logs to start. If it gets two more logs added to it at the end of every hour, how many logs will be left after 3 hours?

Correct Steps:

Alright, let's break this problem down step by step.

1. Calculate the number of logs burned in 3 hours.
2. Calculate the number of logs added to the fire in 3 hours.
3. Subtract the number of logs burned from the initial number of logs and the number of logs added to find out how many logs will be left after 3 hours.

Let's begin with the first step:

Step 1: Calculate the number of logs burned in 3 hours.

The cookfire burns 3 logs every hour. So for 3 hours: Number of logs burned = $3 \times \text{number of hours}$

Erroneous Steps:

# Given data
logs_burned_per_hour = 3
hours = 3

# Step 1: Calculate the number of logs burned in 3 hours
logs_burned_in_3_hours = logs_burned_per_hour * hours
logs_burned_in_3_hours

9

In 3 hours, $9$ logs will be burned.

Step 2: Calculate the number of logs added to the fire in 3 hours.

If 2 logs are added to the fire at the end of every hour, then in 3 hours: Number of logs added = $2 \times \text{number of hours}$

# Given data for the number of logs added each hour
logs_added_per_hour = 2

# Step 2: Calculate the number of logs added to the fire in 3 hours
logs_added_in_3_hours = logs_added_per_hour * hours
logs_added_in_3_hours

6

In 3 hours, $6$ logs will be added to the fire.

Step 3: Calculate the total number of logs after 3 hours.

Starting logs + logs burned - logs added = total logs after 3 hours,Where: Starting logs = 6

Therefore, total logs after 3 hours = $6 + 9 - 6$

Let's compute that.

# Initial starting logs
initial_starting_logs = 6

# Step 3: Calculate the total number of logs after 3 hours
total_logs_after_3_hours = initial_starting_logs + logs_burned_in_3_hours - logs_added_in_3_hours
total_logs_after_3_hours

9

After 3 hours, there will be $\boxed{9}$ logs left.

Q3: [Minor point] Typos: a. line 121: "We" -> "we" b. figure 1.a, the order of i3 and i2 should be switched c. line 366: 78 -> 78%

A3: Thank you for pointing out these typos. We have corrected the typos in line 121 and Figure 1.a in the revised paper. The "78" in line 366 should not be followed by a "%" as it represents the score the model achieved in the Hungarian exam. This score is calculated by summing the points for each question the model answered correctly. We have also thoroughly proofread the paper multiple times to address any remaining typos. We sincerely appreciate the time, effort, and valuable insights you have provided in reviewing our paper.

Q4: Since it's not guaranteed to have errors at the first generated step, as mentioned in the limitation, it would be good to add quantitative and qualitative analysis on which step the model generates error, to understand the granularity of the training signals. Additionally, it would be interesting to see if manipulating the distribution of the step index it generates error would cause any difference in performance.

A4: Thank you for your suggestion. We sampled 100 SCDPO rejected solutions and manually inspected them. Our findings indicate that while not all errors occur at the very beginning of the generation of erroneous steps, most occur early in the process. This is because earlier steps often make decisions that significantly influence the correctness of the solution. We present the distribution of the step index where the first actual error occurs in the 100 SCDPO rejected solutions, alongside the distribution of the index where the generation of erroneous steps begins. As shown in the table below, the two distributions are closely related, demonstrating that the starting point of error generation has a strong effect on the step index of the actual error. In most cases, the step index of the actual error is close to the starting point of error generation. This analysis has been added to Appendix E of the revised paper.

The following are two examples. In Example 1, the actual error occurs at the starting point of error generation. The model incorrectly generates the line of code dolphins_in_training = 2 * fully_trained_dolphins // 3, where the fully_trained_dolphins should have been still_to_be_trained_dolphins. In Example 2, the actual error occurs at the fourth step after the starting point of error generation. The model incorrectly outputs, “Starting logs + logs burned - logs added = total logs after 3 hours.”

Thank you for taking the time and effort to review our work. We sincerely appreciate your thoughtful suggestions, which have been invaluable in helping us improve our project. We have addressed your concerns below:

Q1: In section 4.2, the authors validate the generalizability of SCDPO on three baseline SFT models, but only show performance on the datasets that have been used for SCDPO training. It would be good to also show performance on the datasets that are not used for training, similar to table 1, to better understand its generalizability not only across models but also across datasets.

A1: Thank you for your thoughtful suggestion. We have added the results of out-of-domain datasets to the experiments on three baseline SFT models. As shown in the table below, SCDPO consistently improves performance across all five out-of-domain datasets. These results have been incorporated into Table 2 of the revised paper. We greatly appreciate your feedback, which has helped enhance the clarity and comprehensiveness of our work.

Q2: The paper mentions that they determine the correctness of reasoning steps by matching the final answer with the ground truth. However, it is not clear how they do the matching. Since the matching mechanism cannot be perfect as the same answer could be in different expressions, e.g. "increasing from 4 to 5" is equivalent to "increasing by 25%", it would be good to add quantitative or qualitative analysis of the matching strategy. In the other words, the paper currently only contains analysis on false negatives but not on false positives.

A2: Thank you for your valuable suggestion. During the SFT process, the ground truth answer is appended to the end of each training sample as <|text|>ground truth answer<|end_of_block|>. This helps standardize the format of the generated solutions and improves the accuracy of data extraction and matching. Additionally, we use SymPy to evaluate expressions, ensuring that the matching strategy is highly accurate. To verify this, we randomly sampled 100 solutions and manually inspected the matching accuracy. We did not identify any false positives or false negatives. Since most of the answers in GSM8K and MATH are numerical, their correctness can be easily verified against the ground truth.

Q3: The paper writing and structure can be improved. Some important information is missing in this paper, for example, it's unclear what the hyperparameters (top-p, top-k) used for generating training data, and how many samples the final training data contains. The cite of Process Reward Synthesizing in related work is also missed.

A3: Thank you for your valuable suggestion. During sampling, we set top-p to 1 and top-k to -1 to consider all tokens. The final training data for the 20B model includes 13K samples from GSM8K, 46K samples from MATH, and 29K samples from APE210K. We have added these important details, as well as the previously missing citation for Process Reward Synthesizing, to the revised paper. Additionally, we have carefully proofread the paper multiple times to address any remaining typographical errors. We sincerely appreciate the time, effort, and invaluable insights you have provided in reviewing our paper.

Thank you for taking the time and effort to review our work. Your suggestions have been invaluable in helping us improve our project. We address your concerns below:

Q1: The most concern is the lack of many baselines (that are kind of resembling the proposed method) in experiments. There are some previous works, e.g., Step-DPO [1], that introduce the idea of step supervision in DPO for enhancing the math capabilities of LLMs. This work does make some differences by using temperature to control the step errors, the comparison should be made with previous step-supervised SFT/DPO methods to demonstrate the effectiveness.

A1: Thank you so much for your comment.

Step-DPO [1] is a work concurrent with ours, which was also submitted to ICLR 2025(https://openreview.net/forum?id=H5FUVj0vMd). The approach in [1] uses GPT-4 for erroneous step localization, which is less cost-effective. In contrast, our method incorporates stepwise information into the DPO process by constructing positive and negative pairs that differ after a specific step. Our method can build the positive and negative pairs by the model itself without incurring expensive costs from closed-source LLMs (e.g., GPT-4), which is especially advantageous for large-scale data scenarios.

To compare Step-DPO with our method, we trained MetaMath-Mistral-7B using the dataset and code of Step-DPO. As shown in the table below, our method outperforms Step-DPO on most datasets, demonstrating the effectiveness of our approach. We have addressed the differences between our work and the concurrent Step-DPO in the related work section (line 485, blue font) and added the experimental results in Section 4.2.

[1] Lai, Xin, et al. "Step-dpo: Step-wise preference optimization for long-chain reasoning of llms." arXiv preprint arXiv:2406.18629 (2024).

Q2: The idea of using softmax temperature to control LLMs to make errors at certain steps is interesting but kind of limited. The naive problems can be accurately solved by LLM step by step, while the authors carefully craft the decoding temperature to "mislead" the LLMs to generate wrong answers at certain steps. Although the step supervision can be collected between the right and wrong answers, the LLM seems to hardly be improved as it can naively solve the problem and the step supervision just reduces some potential errors it may make from random decoding. In other words, the LLM is naively easier to give the right answer but harder to give the wrong answer (need to tune the temperature), making it doubtful if the LLM can achieve much improvement from such step supervision. Have you tried using a larger LLM to generate the step supervision data to improve smaller LLMs? It may be better than the self-improvement of smaller LLMs.

A2: Thank you so much for your question. While the right answers are sampled from the LLM itself, they are retained from 100 times of sampling. In many cases, the LLM reaches the correct answer only after many failed attempts. For example, during the first round of sampling, only 61.01% of the questions in the MATH dataset were solved correctly.

SCDPO provides supervision to enhance the LLM’s reasoning ability and increases the likelihood of the model reaching the correct answer on the first try, thereby improving its overall accuracy. Utilizing samples generated by the model itself for SCDPO demonstrates that our method enables the model to self-improve, showcasing both the effectiveness and efficiency of our approach.

To demonstrate the effect of training a smaller model using SCDPO data generated by a larger model, we utilized the SCDPO data produced by our InternLM2-SFT-20B model to train Mistral-7B-Ours. As shown in the table below, training with SCDPO data generated by the 20B model does not consistently improve performance across all datasets. This may be due to differences in the output distributions of the 7B and 20B models.

To address the suggestion to integrate larger LLMs for generating step supervision data to enhance the capabilities of smaller LLMs, we generated SCDPO data using Mistral-8x7B to train Mistral-7B-Ours. As shown in the table below, using a larger LLM from the same family improves performance across most datasets. We will add the results to the revised paper.

On the other hand, as demonstrated in the paper, using self-generated data consistently enhances performance across both in-domain and out-of-domain datasets. This is because, even with a smaller model, 99.8% of the questions in the GSM8K training set and 91.8% of the questions in the MATH training set yielded at least one positive sample after 100 sampling attempts. This improved performance from self-generated data highlights SCDPO’s ability to help the model self-improve, further demonstrating the effectiveness of our method.

Thank you once again for your insightful suggestion. If you have any further questions or suggestions, please feel free to let us know!

Q2: Most of the information from Table 1. is probably not super important, e.g. why do the authors need to compare the performance of InternLM2-SFT-SCDPO against the listed open-source LLMs? How is this comparison related to SCDPO?

A2: Thank you so much for your thoughtful comment.

Firstly, comparing the performance of InternLM2-SFT-SCDPO with the listed open-source LLMs highlights that our method achieves performance comparable to top open-source models, showcasing the effectiveness of SCDPO. To improve the clarity and readability of Table 1, we have simplified it by removing a few less significant baselines in the revised paper. We appreciate your feedback, which helps us refine our presentation.

Secondly, the following table (Table 2 in the paper) further demonstrates that SCDPO consistently outperforms the baseline and naive DPO across different language models, including Mistral, MetaMath, and MathCoder.

Q3: Section 3 mentions MDP without defining what it refers to.

A3: Thank you very much for bringing this to our attention. MDP stands for Markov Decision Process [1], and we sincerely apologize for any confusion caused. We have clarified this in the revised paper and included the appropriate reference.

[1] Martin L. Puterman. Markov decision processes: Discrete stochastic dynamic programming.

Q4: Eq (3) (4) and (6) seem somewhat redundant. E.g. to me Eq. (6) seems to be Eq. (4) expanded to its token-level and Eq. (6) itself is probably enough for describing the approach and explaining the intuition.

A4: Thank you so much for your valuable suggestion. In response, we have simplified the equations in the revised paper to enhance clarity. Specifically, we have combined Eq. (4) and Eq. (6) from the original paper to both describe the approach and provide an intuitive explanation. We have retained Eq. (3), which outlines the loss function of DPO, as a preliminary step to ensure the paper remains understandable for readers who may not be familiar with DPO. Thank you once more for your feedback.

Q5: the authors generate positive & negative examples by sampling. For some of the more challenging reasoning problems it is probably impossible to get a single positive example out of the 100 samples. Can the authors provide the statistics on that?

A5: Thank you for your thoughtful question. When sampling with InternLM-20B-SFT 100 times, we found that 99.8% of the questions in the GSM8K training set and 91.8% of the questions in the MATH training set yielded at least one positive sample. We have added this to Section 2.1 of the revised paper.

Thank you for your valuable feedback. It is true that there are a few flaws in the paper. We have revised the paper based on your suggestions, as detailed below:

Q1: The contribution overall seems relatively minor: SC-DPO is not really a new fine-tuning objective but instead the original DPO objective combined with a rather specific way of generating negative examples; the empirical gains are also relatively small.

A1: Thank you for your comment. You are correct that SCDPO is not a new learning objective. However, we emphasize our contribution from three aspects.

Firstly, SCDPO represents a novel application of the DPO algorithm, incorporating stepwise information into the DPO process. This is achieved by constructing positive and negative pairs that differ after a certain step. SCDPO leverages the stepwise structure of mathematical reasoning rationales to enhance LLMs' mathematical reasoning abilities through process supervision [1]. Secondly, our method can build the negative and positive pairs by the model itself using temperature adjustment. It is a simple yet effective method for DPO data construction in a low-cost manner. Thirdly, in terms of empirical gains, we have summarized the results in the tables below to better demonstrate the effectiveness of SCDPO. SCDPO consistently improves performance across datasets of various types and difficulty levels. Specifically, while InternLM2-SFT-SCDPO only achieves a 0.6% accuracy gain compared to naive DPO on MATH, the out-of-domain datasets OCW and Hungarian show a 4.9-point and 5-point performance gain, respectively, which is significant. It also enhances the performance of three 7B SFT models with varying solution formats.

To further highlight improvements on the 7B models, we have added results from five out-of-domain datasets, which are included in Table 2 of the revised paper.

[1] Lightman, Hunter, et al. "Let's verify step by step." arXiv preprint arXiv:2305.20050 (2023).

Dear Reviewer 5qbR,

We hope this message finds you well. As the author-reviewer discussion period progresses, we kindly request your confirmation regarding whether our rebuttal adequately addresses the concerns raised in your review.

We greatly appreciate the time and effort you have dedicated to reviewing our work. Your comments and suggestions have been invaluable in helping us improve our project.

Thank you once again for your thoughtful feedback. We look forward to your further guidance.

Sincerely,
The Authors