A Decoupled Learning Framework for Neural Marked Temporal Point Process

We really appreciate all the valuable feedback you've given us. We hope our response below can help sort out your concerns.

Q1: The decoupled model can have fewer parameters because each single model is only responsible for one event type. For historical events, some of them have influences on event type $i$, some of them have influences on event type $j$, some of them have influences on event type $k$, and so forth. In our decoupled framework, we have a complete EEHD model for each event type $i$. And in each dedicated model, the encoder only needs to encode the historical events that have influences on event type $i$ and ignore other historical events. While for the standard model, all dynamics of historical events are encoded into one single hidden vector. As a result, larger hidden vector, encoder and decoder are required. We have provided the experimental results when the encoder-decoder parameters of the decoupled networks are equivalent to those of the shared networks in Table A4 and A5 of the updated submission.

Q2: Maybe there exists a misunderstanding. In Figure 3, we are showing the clustering effects of our decoupled framework. That is, if two event types have similar influences on event type $i$, then their embedding in the EEHD model of event type $i$ should be close. We have a detailed explanation for the meaning of Figure 3 in our reply to reviewer Aaqp (“Q1”). If we still miss some details, don't hesitate to let us know.

We know you are expecting us to draw plots like Figure 3 in reference [1]. We understand how to plot the kernels for their model, i.e., kernel $q$ in Equation 8 of their paper, but to be honest, we don’t know how the authors plot these kernels for other models like THP (Figure 3 of their paper) as the authors provided inadequate details in Section 7 of their paper. THP (or SAHP, etc.) doesn't model the triggering kernel of pairs of event types, but the model in reference [1] does, i.e., the $q$ in Equation 8 of their paper. Since THP doesn't model that, neither does our decoupled model Dec-THP. (Our framework only decouple the learning of THP but doesn't alter the architecture of the original model.)

Q3: Beyond IFL and THP, I have provided more results in Table A3 in the appendix of the updated submission.

Q4: Like our method, the two models in reference [1] and [2] are also generalized Hawkes process and are not standard EEHD models. The difference is that our proposed method is a framework and not limited to a specific encoder and decoder while these two benchmarks both specifically use a kernel followed by a summation as the Encoder (Equation 6 in reference [1] and Equation 4 in reference [2]), like the standard Hawkes process. Actually, we have benchmarked this kind of method (ODETPP in Table 2 of our paper). Here's the performance of the two models in reference [1] and [2].

GSHP is better than SPRITE as SPRITE captures only time decaying effects while GSHP further captures local effects (see Figure 2 in reference [1]). Both of them underperform our model because of their less expressive encoder. (To obtain the total influence of historical events, they simply add up the influence of each historical event like the standard Hawkes process.)

Q5: Maybe there exists a misunderstanding. Actually, we have no assumptions. In our framework, the occurrence of an event is dependent rather than independent of other events. As shown in Figure 2, when predicting event type $a$, we will evaluate which kind of historical events have influences on event type $a$, i.e., events painted with yellow color $(k_2=2,t_2)$ and $(k_4=K,t_4)$, and use a tailored encoder to encode them into the hidden vector. Finally, we use a tailored decoder to decode the future dynamism of event type $a$. For the discussion regarding Theorem 2, kindly refer to our reply to reviewer Aaqp (“W”).

[1] Yamac et al. (2023), "Hawkes Process with Flexible Triggering Kernels", MLHC.

[2] Pan et al. (2021), "Self-adaptable point processes with nonparametric time decays", NeurIPS.

Thank you for your valuable feedback, we are hopeful that our upcoming response will effectively tackle your concerns.

Q1: Thank you for your suggestion. When proving the proposed framework is no weaker in expressive power than the standard learning framework, our idea is to construct a decoupled model that can produce the same output with the given standard model. That’s why we said “let $NN_k$ in Equation 6 equal to that in Equation 5”, where $NN_k$ in Equation 6 is the decoder of our framework and $NN_k$ in Equation 5 is the decoder of the standard framework. ($NN_k$ is the decoder to decode the future dynamics of event type $k$.)

Q2: For the first question in Q2, the reviewer please refer to our reply to reviewer Aaqp (“W”). For the second question about “which model is more advantageous to use in which specific situations”, we have additionally conducted ablation study to see if it is necessary to let the embedding, encoder and decoder be type-specific like we do in our framework. Table A6 of the updated submission givens positive answers in most datasets. In our framework, each event type is treated by a tailored EEHD model, which can therefore invoke more event types that may be under-represented in the standard model, as shown in Figure A1, A2, A3, and A4. For the third question, we here introduce two commonly used techniques we can think about to mitigate the issue of frequency bias of the standard framework.

One method is to normalize the loss. Specifically, we first count the frequency of each event type in the training data, denoted as $\alpha_i$, where $i$ is a specific event type and $\sum_{i=1}^{K} \alpha_i=1$. Then we multiply their reciprocals with the training objective, i.e., $\log \mathcal{L}(\mathrm{S})=\sum_{l=1}^{L} \log \frac{1}{\alpha_{k_l}} \lambda_{k_l}(t_l) - \int_{0}^{T} \sum_{k=1}^{K} \frac{1}{\alpha_{k}} \lambda_{k}(t) dt$. We retrain the model using this objective and report the performance below. (the number in the bracket is the entropy of the frequency distribution of event types and the frequency distribution is reported in Figure A5 of the updated submission.)

Another method is to directly normalize the outputs. We don’t retrain the model and directly do the normalization during the inference stage based on IFL and THP. That is, we consider the learned intensity function $\lambda_i(t)$ as $\frac{1}{\alpha_{i}}\lambda_i(t)$. The results of this variant are summarized below.

We see the two normalization methods both work not very well. For the first method, the main reason is that there always exists several event types that dominate the training loss and the training of other event types is actually not converged, as shown in Figure 4. For the second method that needn't retrain, the reason for the failure mainly comes from the propagation of errors: the normalization factor may don’t fit in the learned intensity function.

Q3: We have ever considered this issue when preparing this submission, though we still set the hyperparameters equally in our work (Line 352). As you said, it's actually for simplicity. Otherwise, we have to list a lot of hyperparameter settings in our paper (ICEWS has 201 event types). Setting tailored hyperparameters could improve performance in principle and the optimal hyperparameters can be automatically searched by program. Intuitively, if there are more event types that have influences on event type $i$, then the size of the EEHD model of event type $i$ should be bigger to capture its dynamics. We mainly search the size of the embedding vector, layers of encoder, size of hidden vector, and the learning rate. For Dec-IFL, their ranges to seach are {1,2,3,4,8}, {1,2}, {4,8,16,32}, and {0.0005,0.001,005,0.01}, respectively; For Dec-THP, their ranges to seach are {4,8,16,32}, {3,4,5,6}, {4,8,16,32}, and {0.0005,0.001,005,0.01}, respectively. The optimal results are reported below.

Almost surely, employing tailored hyperparameters can make the performance better as using unified hyperparameters is just a special case.

Q9: There is no additional operation to compute the influence matrix, instead, it is simply the local embedding itself. We have provided a detailed introduction to Figure 3 in our reply to reviewer Aaqp ("Q1"). If we still miss some details, don't hesitate to let us know.

Q10: The total parameter counts of our model is the number of event types multiplies the parameter counts of one single EEHD model. For example, Dec-IFL has 22 $\times$ 1K= 22K parameters over dataset SOflow and Dec-THP has 22 $\times$ 6K = 0.132M parameters. What if traditional models use as many parameters as that used in our individual EEHD model and vice versa? Table A4 and A5 answers this. Please note that each individual EEHD model in our framework can have smaller parameter scales because of decoupled dynamics for each event type. For the EEHD model of event type $k$, it only needs to encode the historical events that have influences on $k$ and ignore others. (For historical events, some of them have influences on event type $i$, some of them have influences on event type $j$, some of them have influences on event type $k$, and so forth.)

Q4&Q8&Q11: Thank you for your suggestion, we will do that for readers' convenience. We are not very understanding Q8, can you kindly provide more details?

We deeply value the feedback you have given, which is extremely useful. We anticipate that our subsequent response will satisfactorily address your concerns and provide the answers you seek.

Q1 and Q2: To see the imbalance of event types and failure of the standard model, the reviewer please refer to Table 3, Figure A1, A2, A3, A4 and A5 (Figure A5 is newly added in the updated submission), where we see some event types occur less than 100 times while some event types occur more than 20000 times, and our method can invoke more event types than the standard model. In Table 3, Dec-IFL outperforms IFL for 10 event types on dataset SOflow, whereas IFL surpasses Dec-IFL for only 4 event types; Dec-THP outperforms THP for 12 event types on dataset SOflow, whereas THP surpasses Dec-THP for only 5 event types. Similar conclusions can be drawn over datasets MOOC (97 event types) and ICEWS (201 event types) as shown in Figure A1-A4.

Q3: Thank you for your suggestion. Firstly, it should be noted that “common embedding and encoder but event specific decoder” is exactly the standard model. That is, both the standard method and our method use event specific decoder (see $NN_k$ in Equation 5 and 6). Secondly, kindly note that there is no such model that use common embedding, encoder and decoder as it could not output different intensity functions for different event types. Hence, except for the standard model and our model, there are a total of five kinds of variants, as shown in Table A6. The results show that it performs better to let the encoder be type-specific to encode the unique dynamics of the corresponding event type and that it performs worse to let the decoder be shared across event types.

Q5: From our understanding, the standard Hawkes process (Equation 11) has no global shared component. $\alpha_{i,j}$ is dedicated to build the intensity function of event type $i$, encoded by encoder $\phi_i$, which is the same as our case: the local embedding $z_m^i(j)$ is dedicated to build the intensity function of event type $i$, encoded by encoder $Encoder_i$. We are not very understanding your point “global shared” as $\alpha_{i,j}$ is only used in encoder $\phi_i$ to produce $\lambda_i(t)$ and is not used (shared) in other encoders. If we have overlooked anything, we would greatly appreciate it if you could kindly inform us.

Q6: As shown in Line 312 of our paper, each dataset is split into training/validation/testing set according the number of event sequences, with each part accounting for 60%/20%/20%, respectively. Taking dataset SOflow, which has 6633 event sequences (see Table 1), as an example, we randomly select 6633*0.6=3979 event sequences as the training data, 6633*0.2=1327 event sequences as the validation data, and 6633*0.2=1327 event sequences as the testing data.

Q7:The proposed method shows no advantage over the standard model when the sequence length is very short, but this doesn’t mean small sequence length is a problem for our proposed method. (The results of our method and the standard one in this case are similar.) Our method shows no improvement when the sequence length is very short because the hidden vector in standard model can well summarize the very short historical events. Of course, our method can also summarize the historical events well. Another finding on dataset MIMIC, which has 715 sequences in total (Table 1), is that there are 564 sequences that repeat the same event type. That is, in these sequences, the event type of the next event is always the same as the event type of historical events. In the remaining 151 sequences, there are 128 sequences looking like this (ignoring the event times): (sequence 1) [1, 0, 0, 0], (sequence 2) [13, 1, 1, 1, 1], (sequence 3) [11, 1, 1, 1, 1], etc. As you can see, there are only 2 event types in each sequence and one of them tends to repeat. These kind of simple patterns, together with the too short length to provide enough information about the dynamics, makes our method show no advantage.

Thank you for the response and clarifications. The newly added baseline results in Table A.6 are helpful to get a better picture of the proposed method. However, the limited technical novelty of the proposed approach (trivial modification of an existing method), the contentious philosophy of complete decoupling over parameter sharing, and the inability to convincingly demonstrate the effectiveness on events with low frequency (Table 3) are still some drawbacks associated with the paper.

We here answer the questions you raised.

Q1: To clarify this, let’s start with some important background (they can also be found in Line 110 of our paper). In standard neural MTPPs, the embedding layer assigns each event type a vector representation and the learned representations can naturally group similar event types according to the spirit of Word2vec. However, this kind of embedding can not reflect the dependencies between different event types. For example, which event types have similar influences on the given event type $k$? Their embedding layer has size $K\times d$, where $K$ is the number of event types and $d$ is the embedding dimension.

As a comparison, we create $K$ vector spaces and in each, an event type will have a vector representation, which we call local embedding. That is, we have $K$ EEHD models and in each model, the embedding layer has size $K \times d$. If we consider the $K$ EEHD models as a system, then the embedding layer of this system has size $K \times K \times d$. (Note that our embedding dimension $d$ can be very small because of the decoupled dynamics for each event type, see our response for W1.) Using our method, an event type can have different vector representations (embeddings) in different EEHD models, indicating that it can have different influences on different event types. In EEHD models of event type $k$, if event type $a$ and event type $b$ have close embeddings, then event type $a$ and event type $b$ have similar influences on event type $k$ according to the spirit of Word2vec. We illustrate an example in Figure 2, where the embedding of event type 2, event type 5, event type $K-2$ and event type $K$ are close (all painted yellow) in the EEHD model of event type $a$, indicating that these four event types have similar influences on event type $a$. But in the EEHD model of event type $b$, it is event type 1, event type 5 and event type $K-1$ that have similar influences on event type $b$.

By now, we can start talking about Figure 3, which illustrates the relationships between different event types, just like Figure 2. Specifically, we conduct experiments on four datasets generated by Hawkes Process (Equation 11), namely Haw1, Haw2, Haw3, and Haw4, where the ground truth influences among event types are known. In Figure 4, the upper four matrices present the configurations of parameters $\alpha_{i,j}$ for the four Hawkes datasets. The parameters $\beta_{i,j}$ are uniformly set to 2.5 across all instances and all datasets. The parameter $\alpha_{i,j}$ quantifies the magnitude of influence that event type $j$ exerts on event type $i$, analogous to the embedding of event type $j$ in the EEHD model of event type $i$. Recall that the embedding dimension for Dec-IFL is set to 1 (see the caption and Table A1), thus the embedding of our whole system has size $K\times K\times d$=$K\times K \times 1$=$K\times K$. We present them by the lower four matrices in Figure 3 (rounded to one decimal place), where each row corresponds to an EEHD model. The element in the $i^{th}$ row and $j^{th}$ is the embedding of event type $j$ in the EEHD model of event type $i$.

Next, we provide an example to analyze the results. For dataset Haw4, according to the ground truth influence parameters $\alpha_{1,2}=1.5$, $\alpha_{1,3}=1.5$ and $\alpha_{1,4}=1.5$, event type 2, event type 3, and event type 4 all exert influences on event type 1. This kind of relationship is well reflected by their embeddings in the EEHD model of event type 1. The embedding of event type 2, event type 3 and event type 4 in the EEHD model of event type 1 are -0.4, -0.8, and -0.8, respectively. They have close embeddings! Conversely, event types that have no influence on event type 1 are also clustered together. The embeddings of event type 1 and event type 5 in the EEHD model of event type 1 are 0.2 and 0.3. For more results, the reviewer can kindly refer to Figure A6 and A7 in the appendix, where we illustrate the embeddings learned by Dec-THP on dataset ICEWS.

Q2; NLL is the negative log-likelihood, and the log-likelihood of one sequence is calculated in Equation 7. In testing data, we have multiple sequences and we report the averaged NLL per sequence. In Table 2, Table A4 and Table A5, we report the results on the testing data as we are reporting the predictive performance. (We have updated Table A4 and Table A5 in the updated submission.) In Figure 4, we show the training (convergence) process as the epoch increases, and thus we are reporting the NLL results on the validation data.

We greatly appreciate the valuable feedback you have provided. We hope that our following response will adequately address your concerns and answer your questions.

W: As you said, the theorem basically guarantees that one can always find a parameterization of the standard model that matches that of the proposed model and vice versa. In the proof, we see that the proposed model only needs to copy the embedding, encoder and decoder to match the standard model, which is very easy to realize. Conversely, it requires the universal approximation theory for the standard model to produce the same outputs as that of our proposed model. In this process, the standard model should carefully encode and decode all historical events as any of them could be of significance for one specific event type. Practically, however, the training tends to favor those most common events as they dominate the loss. Moreover, by joint training, we see the component loss of the standard model usually cannot be converged, as shown in Figure 4. RNNs vs. Transformers present a similar case to us. Although theoretically, there exists an RNN that can achieve the same performance as a given Transformer (both of them are universal approximators for sequence), it is always hard for the RNN to find the desired parameters in practice.

W1: We utilize a very small embedding dimension as well as a history vector dimension because each individual model in our framework only needs to process one event type. Doing so, it accelerates the training speed (Table 4), which is one of the contributions of our study. Of course, if you prefer, you can set a very large embedding dimension, history vector dimension, encoder, and decoder. However, this may not improve the model's performance because the performance has already been saturated, as demonstrated in Tables A4 and A5 of the updated submission. In these two tables, we see the decoupled models work well with small dimensions and increasing $d$ does not provide significant gains. We use the Adam optimizer to update the model parameters, with the weight_decay set to 1e-5 and the learning rate set to 0.001.

W2: As mentioned in Section 2.4 of our paper, the $K$ decoupled models are trained in parallel. Specifically, we have 4 machines equipped with 8 GPUs (24GB) and 80 CPU cores. And for the four real-world and four synthetic datasets we used in our work, one machine is already enough as it occupies very little GPU memory for one single EEHD model (Table 4). For each EEHD model, we calculate the average training time per epoch, i.e., the time elapsed from the first epoch to the last epoch divided by the number of epochs. We report the slowest training time per epoch among the $K$ individual EEHD models as the training time (per epoch) of the entire model.

W3&W4: We have provided the error bars in Table A3 of the updated version of our submission. In Line 466 of our paper, we stated that to our knowledge, we are the first to address the issue of frequency bias. If we have overlooked anything, we would greatly appreciate it if you could kindly inform us.