Paper: Decoupled Marked Temporal Point Process using Neural Ordinary Differential Equations (ICLR-2024)

Reviewer comment (Experiments):
The authors insist that an advantage of the proposed method (Dec-ODE) over conventional methods (e.g., Yang et al., 2020) is that Dec-ODE can compute multiple integrations simultaneously [Section 3.2]. But Dec-ODE demands to solve ODEs to compute integrations, which is generally more time-consuming than Monte Carlo integrations needed in conventional methods. In Experiment, there are no comparative analysis about computation time (sec/iter), and the authors’ insistence about computation efficiency remains to be verified.

Author response:
- **Is Dec-ODE computationally more efficient than others?** The reviewer raised a good point and we are sorry for the misleading text in Section 3.2. What we intended to argue was that computation with Neural ODEs is heavy by its nature but we mitigated the issue with the characteristics of the differential equation within our proposed framework, simultaneously solving the multi-dimensional differential equation with respect to the hidden state $h_t$. The actual computation time per epoch is given in the comment to all reviewers, where Dec-ODE takes comparable or less time than the SOTA approach. Please see the revised text in the updated manuscript highlighted in yellow starting from the bottom of page 4. We would like to make it clear that faster computation is not the advantage of our model (nor we argue it as a contribution in the introduction), but we proposed an efficient learning of a flexible model which would have taken much longer computation without our proposed optimization scheme. However, we still want to mention that numerical computation of integrals often leads to more accurate results than MCMC, which may lead to faster convergence even if sec/iter may be slower. Moreover, other methods that involve transformer do not consider the temporal nature (e.g., Markovian) of the events and need to perform integration iteratively whenever a new event is introduced.

Reviewer comment (Experiments):
RMTPP is a standard benchmark model in the literature, and should be included in comparative experiments. Otherwise, the authors need to mention the reason for not including it.

Author response:
- **Why not compare with RMTPP?** We appreciate the reviewer for pointing us to the RMTPP as a fundamental baseline. We reported the results only from the latest literature so we missed it, but we are more than happy to include RMTPP results in our paper. The results from RMTPP under our experimental setting are given in the table below which is now updated in the revised paper. The results from RMTPP compared with Dec-ODE can be found in the table below. The implementation provided by the author of RMTPP was employed, with minor adjustments made to accommodate length-varying sequences and present the results in a consistent format. The hyper parameters were found using a grid search from the list of options that can be found in RMTPP Section 6.4 paragraph 5. In most cases RMTPP showed a good performance in the time prediction task with best RMSE on StackOverflow data, and its NLL in MIMIC dataset showed a better result compared to Dec-ODE.

Reviewer comment (Reproducibility):
The explanation of the experiment setup seems to be inadequate for reproducibility. The details of neural networks (e.g., activation functions) and the machine spec used in the experiments should be shown.

Author response:
- **Implementation details?** We feel sorry that we missed these important implementation descriptions. We utilized simple 3-layer Multi Layer Perceptrons (MLPs) for implementing the Neural ODE $\gamma()$ with linear activations. Specifically, softplus normalization on the weights is used for parameterizing $\gamma (\theta)$ as in (Chen et al., ICLR 2021). For decoders, 3-layer MLPs with ReLU activations have been used. All experiments were performed with a single NVIDIA RTX A6000.

Reviewer comment (Theory):
The detailed equation about how to compute the derivative of the likelihood regarding model parameter is not shown.

Author response:
- **Details on deriving the derivative of the likelihood?** The derivative of the likelihood regarding the model parameter is calculated using the adjoint sensitivity method in (Chen et al. NeurIPS 2018), which is extensively discussed in the referenced paper. For detailed description and implementation please refer to the Neural ODEs (Chen et al. NeurIPS 2018) and the public repository (https://github.com/rtqichen/torchdiffeq).

Reviewer comment (Evaluation):
Discussions about the limitation of the proposed model are not in the paper.

Author response:
- **Limitations?**  A major limitation is that we assume that each $\mu(t)$ is independent from other events. Even though such an approach is beneficial for interpretability, it may be limited in learning inter-event relationships. Also, even though Dec-ODE shows comparable training time, it requires longer training time for an iteration than most conventional methods due to the flexible but complex nature of ODEs. However, notice that the actual computation time per epoch is quite reasonable and even better than ANHP (shown in the common question) given our proposed training scheme. Finally, the experimented version of the Dec-ODE can only express a self-excitatory effect, where each event only increases the intensity but not decreases, which makes the model less flexible. Actually, if we design the model to both increase and decrease the intensity simultaneously, the performance becomes better as we have shown in the Appendix A.1. However, in this case, the parallel training scheme in Section 4.3  cannot be adopted as it violates eq. (16) in the paper, and therefore we had to leave it in the Appendix.

Reviewer comment (Theory):
[2nd paragraph in Section 2] The authors explain that the modeling of intensity function is preferred over the pdf \( f^*(t) \) due to the complex dynamics of \( f^*(t) \), but the complex dynamics depending on the past events is not limited to \( f^*(t) \).

Author response:
- **Technical errors?** We have corrected all the technical issues addressed by the reviewer and went through the paper again to check for other errors if there were any. These errors can be easily corrected in the text and do not affect the soundness of our model. We really appreciate the reviewer for going through the paper thoroughly to improve its quality. 
   - Complex dynamics depending on the past events is not limited to $f^*(t)$?: Our objective was to underscore the challenge of modeling the probability density function $f^*(t)$ with a parametric model. As the intricate nature of the dynamics involved is quite diverse, describing these temporal variations through intensity functions allows for a more flexible modeling, often achieved with nonparametric methods.

Reviewer comment (Theory):
[2nd paragraph in Section 2] \( \int_0^{\infty} f^*(s) ds = 0 \) is correctly \( \int_0^{\infty} f^*(s) ds = 1 \).

Author response:
- **Technical errors?** We have corrected all the technical issues addressed by the reviewer and went through the paper again to check for other errors if there were any. These errors can be easily corrected in the text and do not affect the soundness of our model. We really appreciate the reviewer for going through the paper thoroughly to improve its quality. 
   - In 2nd paragraph in Section 2, $\int f^*(t) = 0$?: It is now $\int_0 ^{\infty} f^*(t) = 1$.

Reviewer comment (Writing):
[Eq. (9)] The definition of \( f_{\theta}(h,t) \) is not found.

Author response:
- **Technical errors?** We have corrected all the technical issues addressed by the reviewer and went through the paper again to check for other errors if there were any. These errors can be easily corrected in the text and do not affect the soundness of our model. We really appreciate the reviewer for going through the paper thoroughly to improve its quality. 
   - In Eq. (9), the definition of $f_\theta(ℎ,t)$ is not found?: It should be $\gamma (\bf{h_t}, t, \mathbf{e_i} ;\theta) $ and not $f_\theta ()$, where $\mathbf{e_i}$ is a vector of  events corresponding to $\mathbf{h_t}$.

Reviewer comment (Experiments):
No simulation studies are conducted to demonstrate the estimation accuracy of the proposed algorithm when the data-generating process is assumed in the paper.

Author response:
We sincerely thank the reviewer for highly valuing the ideas and presentations given in our paper. 

- **No simulation studies?**: We are currently performing a simulation study with synthetically generated data from Neural Hwakes Process (NeurIPS 2017). We will update the results soon, which will also be discussed in the paper (or in the Appendix). Regarding the computing time, the comparisons on the real-world data are given in the general response at the top of this rebuttal.

Reviewer comment (Experiments):
Some comparisons with existing methods are desirable to provide better ideas of the advantages and limitations of the proposed method, in terms of estimation accuracy, predictive accuracy, and computing times.

Author response:
We have performed a simulation study on the Hawkes process following the similar procedure from the Neural Hawkes Process (NeurIPS 2017). In order to obtain the truth intensity function we randomly sampled the parameters for the kernel function of the multivariate Hawkes process, and we simulated synthetic data using the tick library (https://x-datainitiative.github.io/tick/modules/hawkes.html). 

We have updated the figures that illustrate the results from the experiment in the Appendix D.3. In Figure 8 (a) - (d) THP struggles from simulating the flexible dynamics of the intensity function $\lambda_g$, whereas ANHP and Dec-ODE show similar dynamics to the ground truth intensity. The table below also demonstrates that Dec-ODE shows better results than THP and ANHP in all metrics. The result also aligns with the results in Table 1 of the main paper, which suggests that Dec-ODE can simulate reliable MTPPs compared to other state of the art methods. The Figure 8 (e) illustrates $\mu(t)$ that compose $\lambda_g(t)$ from Figure 8 (d), which shows that the ground intensity $\lambda_g(t)$ can be reconstructed from individually constructed trajectory, i.e. a MTPP can be modeled from decoupled information.

|         | RMSE       | ACC       | NLL        |
|:---------:|:------------:|:-----------:|:------------:|
| THP     | 0.7734     | 27.48     | 3.0121     |
| ANHP    | 0.6540 | 29.31     | 0.2804     |
| Dec-ODE | **0.6398**     | **29.35** | **0.2730** |