Breaking Determinism: Fuzzy Modeling of Sequential Recommendation Using Discrete State Space Diffusion Model

First of all, thank you very much for reading our work carefully and for your valuable comments and suggestions, from which we have greatly benefited. Next, I will explain and respond to the shortcomings you have pointed out, and we will make corresponding revisions in the official version of our work based on your reminders.

W1: Regarding a clearer description of our algorithm, we think your suggestion is very reasonable. To avoid making the theoretical part too lengthy within the main discussion, we placed the description of discrete diffusion separately in Section 3. However, this approach may not facilitate readers' understanding of the overall workflow. We add our algorithm flowchart below and will incorporate it into the official version of our paper based on your suggestion.

Training of DDSR.

Input:

• Historical Interaction Sequence of user $u$: $v_{1:n-1} = c_{1:n-1;1:m}$ ;
• Target item: $v_n = c_{n;1:m}$;
• Transition matricest: $Q_t$;
• Approximator: $f_{\theta}(\cdot)$.

Output:

• Well-trained Approximator: $f_{\theta}(\cdot)$.

Procedure:

1. while not converged do
2.   Sample Diffusion Time: $t \sim [0,1,...,T]$;
3.   Calculate $t$-step transition probability: $\quad\overline{\boldsymbol{Q}}_t=\boldsymbol{Q}_1\boldsymbol{Q}_2\ldots\boldsymbol{Q}_t$;
4.   Convert $c_{n;1:m}$ to one-hot encoding $x_{n;1:m}^0$;
5.   Obtain the discrete state $x_{n;1:m}^t$ after $t$ steps by Equation (2), thereby obtain the 'fuzzy set' $c_{1:n-1;1:m}^{t}$;
6.   Modeling $c_{2:n;1:m}$ based on 'fuzzy sets' through Equation (5);
7.   Take gradient descent step on $\nabla$ $L_{CE}$ ($\hat{c}_{2:n;1:m}$, $c_{2:n;1:m}$).

Sampling of DDSR.

Input:

• Historical Sequence: $v_{1:n-1} = c_{1:n-1;1:m}$
• Well-trained Approximator: $f_{\theta}(\cdot)$
• Sampling Steps: $T$.

Output:

• Predicted Target Item: $v_{n}$

1. Let $x_T$ = $c_{1:n-1;1:m}$;

2. Let t = T;

3. while $t>0$ do

4.   Use the trained $f_{\theta}(\cdot)$ to obtain predictions $\widetilde{x}_{0}$ with $x_t$ and $t$ as inputs;

5.   Substitute $\widetilde{x}_{0}$ into equation (7) to obtain the distribution of $t-1$ step;

6. end while

7. $\widetilde{v}_{n}$ = $x_0$[-1;1:m];

8. if the same code project exists: $v_n$ = $\widetilde{v}_{n}$;

  else: $v_n$ is the project in the space closest to $\widetilde{v}_{n}$.

W2: Regarding the issues with the experimental section, we greatly appreciate your constructive suggestions. We recognize that our experimental setup was not comprehensive, so we have added a study on the impact of codebook length on recommendation performance. We will present the following experimental results on dataset Scientific in the form of bar charts in the paper:

W3: Regarding the comparison of actual runtime, we have compared the GPU memory usage and runtime of the UniSRec, DiffuRec, and our DDSR model on a single A100 GPU. The specific results are as follows:

Firstly, we sincerely appreciate your careful review of our work and the extremely valuable suggestions you have made! In response to the issues you have pointed out, we will improve our research and provide detailed answers to each of your questions, hoping to clearly address your concerns.

W1: Regarding the concern you raised about the insufficient baseline comparison with traditional sequence recommendation models, we indeed acknowledge that there is a gap in our work in this area. During our research, we primarily focused on comparing with other text-based and generative model-based sequential recommendation approaches, which may have led to an inadequate evaluation of traditional models. To address this, we have added experiments with the following three traditional sequential recommendation models:

1. HSTU[1]: A novel sequence recommendation architecture introduced by Meta in May this year. HSTU utilizes a hierarchical sequence transduction framework, incorporating a modified attention mechanism and a micro-batching algorithm (M-FALCON) for efficient handling of high-cardinality and dynamic recommendation data.

2. Mamba4Rec[2]: A model proposed by Liu et al. in June this year, based on the latest Mamba design, which includes a series of sequence modeling techniques.

3. FDSA[3]: The feature-level deep self-attention network introduced by Zhang et al., which integrates various heterogeneous features of items into a feature sequence with different weights through a fundamental mechanism.

Here are our experimental results:

Among these, we have integrated HSTU into our framework, which based on RecBole, as its original source code did not include a test set partition. Thank you for your suggestion; we will enhance the baseline comparisons in the formal version of the paper.

[1] Zhai J, Liao L, Liu X, et al. Actions speak louder than words: Trillion-parameter sequential transducers for generative recommendations[J]. arXiv preprint arXiv:2402.17152, 2024.

[2] Liu C, Lin J, Wang J, et al. Mamba4rec: Towards efficient sequential recommendation with selective state space models[J]. arXiv preprint arXiv:2403.03900, 2024.

[3] Zhang T, Zhao P, Liu Y, et al. Feature-level deeper self-attention network for sequential recommendation[C]//IJCAI. 2019: 4320-4326.

Q1&W2: Thank you for pointing that out. We will address these omissions in the final version of the paper. Below are the actual runtime cost comparison results of our DDSR model with UniSRec and DiffuRec:

Regarding the performance variations with different codebook lengths that you mentioned, we have added experiments here. Taking the Scientific dataset as an example, we explore how codebooks obtained through quantization methods affect the recommendation performance at various lengths：

For the Scientific dataset alone, a codebook length of 64 achieved best results most of the time. We regret not having made sufficient attempts before.

Q1: During our experiments, we ran each model five times and took the average as the final result to validate the significance of improvements. We failed to mention this point in the implementation details section of the paper, which was an oversight. We are very sorry for this and will include this information in the final version of the paper.

If you believe that our response has resolved the issues raised and alleviated your concerns, we kindly request that you consider adjusting the score accordingly. If you still have any doubts or suggestions, we earnestly ask you to point them out to further enhance our work. Once again, thank you for your thoughtful comments and consideration.

Dear Reviewer,

I hope this message finds you well. I am writing to inquire about the progress of my rebuttal submission. I understand that the review process can be quite involved, but I would appreciate any updates you could provide regarding the status of the review.

Thank you very much for your time and effort!

Best regards！

We sincerely appreciate the valuable time you have dedicated to our work and the encouragement you have given us! Thank you for thoroughly reading our paper and pointing out some issues, which indeed arose due to our oversight; for this, we deeply apologize. In the final version, we will carefully review and correct each issue you have raised and perform a comprehensive proofreading from start to finish to ensure that such problems do not recur.

Regarding the limitations you've mentioned, specifically the need to discuss improvements in computational complexity, we indeed recognize the importance of this issue and plan to address it in the final version of our paper. We believe that there are two feasible approaches for improvement: through model architecture and coding strategies. For instance, within the model architecture, one could employ linear attention mechanisms in Transformers as a substitute, redesigning the computation of $K$ and $V$ matrices to reduce the time complexity of the attention mechanism from $O(n^2d)$ to $O(nd^2)$. Additionally, we have taken note of the recently proposed mamba model, which maintains linear complexity while demonstrating robust performance. From a coding perspective, exploring ways to convey sufficient information with shorter codebook lengths seems to be an effective strategy. For example, in our research, the codebook lengths generated by RQ-VAE are significantly shorter than those required by traditional quantization methods, yet maintaining effectiveness on this basis remains a critical challenge.

Lastly, we greatly appreciate the valuable suggestions you have provided.

[1]Gu A, Dao T. Mamba: Linear-time sequence modeling with selective state spaces[J]. arXiv preprint arXiv:2312.00752, 2023.

[2]Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pappas, and François Fleuret. “Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention”. In: International Conference on Machine Learning. PMLR. 2020, pp. 5156–5165.

Firstly, we appreciate the valuable time you have invested in our work and the constructive suggestions you have offered; we will rigorously revise our paper based on your feedback. We hope the following explanations will somewhat alleviate your concerns.

W1: We find your suggestion to relocate Section 2 to the end of the paper quite insightful, and we plan to adopt this adjustment in the final version. Regarding the idea of combining Sections 3 and 4, we need to explain that the original intention of discussing Section 3 before Section 4 separately was to separate the theoretical and model sections. This was aimed at preventing the model section from becoming too lengthy, which could negatively affect the reader's experience. However, your advice has highlighted a potential compromise in clarity that we had not fully appreciated. To enhance our discussion, we will introduce an algorithm flowchart below as an initial step. In the formal version, we will thoughtfully consider how to integrate your feedback. Currently, we are contemplating including Sections 3.1 and 3.2 within the methods part of Chapter 4 and positioning Section 3.3 just before the experimental segment, to provide theoretical backing for our proposed DDSR model. We would appreciate your thoughts on whether this revised structure seems appropriate. Here is the algorithm flowchart we provided (after generating discrete semantic encoding).

Training of DDSR.

Input:

• Historical Interaction Sequence of user $u$: $v_{1:n-1} = c_{1:n-1;1:m}$ ;
• Target item: $v_n = c_{n;1:m}$;
• Transition matricest: $Q_t$;
• Approximator: $f_{\theta}(\cdot)$.

Output:

• Well-trained Approximator: $f_{\theta}(\cdot)$.

Procedure:

1. while not converged do
2.   Sample Diffusion Time: $t \sim [0,1,...,T]$;
3.   Calculate $t$-step transition probability: $\quad\overline{\boldsymbol{Q}}_t=\boldsymbol{Q}_1\boldsymbol{Q}_2\ldots\boldsymbol{Q}_t$;
4.   Convert $c_{n;1:m}$ to one-hot encoding $x_{n;1:m}^0$;
5.   Obtain the discrete state $x_{n;1:m}^t$ after $t$ steps by Equation (2), thereby obtain the 'fuzzy set' $c_{1:n-1;1:m}^{t}$;
6.   Modeling $c_{2:n;1:m}$ based on 'fuzzy sets' through Equation (5);
7.   Take gradient descent step on $\nabla$ $L_{CE}$ ($\hat{c}_{2:n;1:m}$, $c_{2:n;1:m}$).

Sampling of DDSR.

Input:

• Historical Sequence: $v_{1:n-1} = c_{1:n-1;1:m}$
• Well-trained Approximator: $f_{\theta}(\cdot)$
• Sampling Steps: $T$.

Output:

• Predicted Target Item: $v_{n}$

1. Let $x_T$ = $c_{1:n-1;1:m}$;

2. Let t = T;

3. while $t>0$ do

4.   Use the trained $f_{\theta}(\cdot)$ to obtain predictions $\widetilde{x}_{0}$ with $x_t$ and $t$ as inputs;

5.   Substitute $\widetilde{x}_{0}$ into equation (7) to obtain the distribution of $t-1$ step;

6. end while

7. $\widetilde{v}_{n}$ = $x_0$[-1;1:m];

8. if the same code project exists: $v_n$ = $\widetilde{v}_{n}$;

  else: $v_n$ is the project in the space closest to $\widetilde{v}_{n}$.

W2: I apologize for the lack of clarity in our description and hope that the following explanation will address your concerns about the technical innovations. Unlike traditional diffusion models that add noise to features, we use discrete diffusion to induce structured transformations among discrete nodes, such as semantic labels, within a discrete state space using transition matrices. This approach replaces the introduction of noise in conventional diffusion models and avoids the deviation of representations towards meaningless directions. The semantic space itself is inherently meaningful, and the discrete diffusion framework allows us to design controllable transition schemes, such as importance-based transitions, which are a major source of performance improvement. We also note that traditional diffusion models add noise to embeddings, which often exacerbates training difficulties in highly sparse scenarios like sequence recommendation. For instance, in our experiments, we observed that DiffuRec required over three hundred training epochs to converge on most datasets. Our work is also the first attempt to enhance diffusion models with textual information in the context of recommendation.

Q1: We introduce the concept of fuzzy sets because diffusion models are inherently designed for generative tasks, which fundamentally differ in form from autoregressive tasks like sequence recommendation, where the $n$th item is predicted using the previous $n-1$ items. By employing fuzzy modeling to redefine the diffusion process, we ensure that the diffusion can be theoretically applied to the target embeddings but also enhances adaptability without compromising the robust theoretical support inherent to diffusion models. As for discrete diffusion, it represents a form of fuzzy modeling in our work. The specific reasons for using discrete diffusion are presented in W1.

Thank you for your thorough review. If you feel our clarifications have indeed alleviated any doubts, we kindly ask you to consider adjusting the score accordingly. Should you have any further questions or require additional clarification, please do not hesitate to point them out.

Dear Reviewer,

I hope this message finds you well. I am writing to inquire about the progress of my rebuttal submission. I understand that the review process can be quite involved, but I would appreciate any updates you could provide regarding the status of the review.

Thank you very much for your time and effort!

Best regards！