Intensity-Free Learning of Temporal Point Processes
Authors: Oleksandr Shchur
Marin Biloš
Stephan Günnemann
Venue:  (2020)
Keywords: Temporal point process
neural density estimation

Abstract: Temporal point processes are the dominant paradigm for modeling sequences of events happening at irregular intervals. The standard way of learning in such models is by estimating the conditional intensity function.  However, parameterizing the intensity function usually incurs several trade-offs. We show how to overcome the limitations of intensity-based approaches by directly modeling the conditional distribution of inter-event times.  We draw on the literature on normalizing flows to design models that are flexible and efficient. We additionally propose a simple mixture model that matches the flexibility of flow-based models, but also permits sampling and computing moments in closed form.  The proposed models achieve state-of-the-art performance in standard prediction tasks and are suitable for novel applications, such as learning sequence embeddings and imputing missing data.

Published as a conference paper at ICLR 2020
INTENSITY-FREE LEARNING OF TEMPORAL POINT PROCESSES
Oleksandr Shchur," Marin Biloš; Stephan Günnemann Technical University of Munich, Germany {shchur, bilos, guennemann] in. tum. de
ABSTRACT
Temporal point processes are the dominant paradigm for modeling sequences of events happening at irregular intervals. The standard way of learning in such mod- els is by estimating the conditional intensity function. However, parameterizing the intensity function usually incurs several trade-offs. We show how to overcome the limitations of intensity-based approaches by directly modeling the conditional distribution of inter-event times. We draw on the literature on normalizing flows to design models that are flexible and efficient. We additionally propose simple mixture model that matches the flcxibility of flow-based models, but also permits sampling and computing moments in closed form. The proposed models achieve state-of- sfate-ofthhe-ae art performance in standard prediction tasks and are suitable for novel applications, such learning sequence embeddings and imputing missing data.
INTRODUCTION
Visits to hospitals, purchases in e-commerce systems, financial transactions, posts in social media various forms of human activity can be represented as discrete events happening at irregular intervals. The framework of temporal point processes is a natural choice for modeling such data. By combining temporal point process models with deep learning, we can design algorithms able to learn complex behavior from real-world data.
Designing such models, however, usually involves trade-offs along the following dimensions: flexi- bility (can the model approximate any distribution?), efficiency (can the likelihood function be eval- uated in closed form?), and ease of use (is sampling and computing summary statistics easy?). Existing methods (Du et al., Mei & Eisner 2017| Omi al. 2019) that are defined in terms of the conditional intensity function typically fall short in at least one of these categories.
Instead of modeling the intensity function, suggest treating the problem of learning in temporal point processes as an instance of conditional density estimation. By using tools from neural density estimation (Bishop 1994 Rezende & Mohamed 2015), we develop methods that have all of the above properties. To summarize, our contributions are the following:
We connect the fields of temporal point processes and neural density estimation. We show how normalizing flows can be used to define flexible and theoretically sound models for learning in temporal point processes.
We propose simple mixture model that performs par with the state-of oft-----thhhheee the-art methods. Thanks its simplicity, the model permits closed-form sampling and moment computation.
We show through wide range of experiments how the proposed models can be used for prediction, conditional generation, sequence embedding and training with missing data.
BACKGROUND
Definition. A temporal point process (TPP) is random process whose realizations consist of a quence of strictly increasing arrival times T = {t,,.., J A TPP can equivalently be represented
"Equal contribution
Code and datasets available http://t.p///gghhhh.ecc/////////

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Exponential intensity
Neural Hawkes
Normalizing Flows
Mixture Distribution
Fully NN
Closed-form likelihood
x
Flexible
X
/
/
Closed-form E[r]
X
X
x
X
Closed-for sampling
X
X
X
Table 1: Comparison of neural temporal point process models that encode history with an RNN.
as a sequence of strictly positive inter-event times т; - 4; 14-1 E RR+. Representations in terms of will use them interchangeably throughout the paper. The traditional way of specifying the dependency of the next arrival time on the history H. = {ts E T tt < t} using the conditional intensity function (t) == A(t Here, the symbol reminds us of dependence on H4. Given the conditional intensity function, we can obtain the conditional proba- bility density function (PDF) of the time т; until the next event by integration (Rasmussen, 2011) as
1 and 7; isomorphic
p" (T+) :- p( Hr,) (4- T.) exp (- (= s)ds).
Learning temporal point processes. Conditional intensity functions provide a convenient way to specify point processes with a simple predefined behavior, such as self-exciting (Hawkes 1971 and self-correcting (Isham & Westcott, 1979) processes. Intensity parametrization is also commonly used when learning model from the data: Given parametric intensity function (t) and quence of observations T, the parameters 0 can be estimated by maximizing the log-likelihood:
0- arg max Σ; log pa (T&) arg maxo Σ log Ag(tz) So {g(s)ds).
The main challenge of such intensity- approaches lies in choosing a good parametric form for (t). This usually involves the following trade-off:: For a "simple" intensity function (Du et al., 2016 Huang et al. al. 2019), the integral (T;r:) := J X (f=-1+)dds has closed form, which makes the log-nnklthood easy to compute. However, such models usually have limited expressiveness. A more sophisticated intensity function (Mei & Eisner 2017) can better capture the dynamics of the system, but computing log-likelihood will require approximating the integral using Monte Carlo.
Recently, Omi et al. (2099) proposed fully neural network intensity funetion (FullyNN) - a flexi- ble, yet computationally tractable model for TPPs. The key idea of their approach is to model the cumulative conditional intensity function A (T.) using a neural network, which allows to efficiently compute the log-likeling likeliiooddd. Still, in its current state, the model has downsides: it doesn't 't define valid PDF, sampling expensive, and the expectation cannot be computed closed form
This work. We show that the drawbacks of the existing approaches be remedied by looking the problem of learning in TPPs from different angle. Instead of modeling the conditional intensity (t), suggest to directly learn the conditional distribution p" (r). Modeling distributions with neural networks is well-researched topic, that, surprisingly. is not usually discussed in the context of TPPs. By adopting this alternative point of view, we able develop new theoretically sound and effective methods (Section 3 as well as better understand the existing approaches (Section 4..
3 MODELS
We develop several approaches for modeling the distribution of inter-event times. First, we assume for simplicity that each inter-event time is conditionally independent of the history, given the model parameters (that is, p" (Tz) = p(T,)) In Section 3.1 we show how state-of-ther art neural den- sity estimation methods based on normalizing flows can be used to model p(T.). Then in Section 3.2 we propose a simple mixture model that can match the performance of the more sophisticated flow- based models, while also addressing some of their shortcomings. Finally, discuss how make p(T)) depend on the history He, in Section 3.3
A detailed discussion of the FullyNN model follows in Section and Appendix C
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3.1 MODELING P(T) WITH NORMALIZING FLOWS
The core idea of normalizing flows TTbak & Tumerr 2013 Rezende & Mohamed 2013) is to define flexible probability distribution by transforming a simple one. Assume that 2 has PDF (2)) Let =(2) for some differentiable invertible transformation g Z + X (where 2. C R We can obtain the PDF p(x) of x using the change of variables formula as p(x) = q0 (x)) By 9 (z)]
By stacking multiple transformations 91...., 9Ms we obtain an expressive probability distribution p(z). To draw sample x ~ p(z). we need to draw ~ g(2) and compute the forward transforma- tion x = (gM ..... gi (=). the inverse transformation = = t 9M (x) and compute q(2). Modern normalizing flows architectures parametrize the transformations using extremely flexible functions fo. such as polyno- mials (Jaini et al. 2019) or neural networks (Krueger et al. 2018). The flexibility of these functions comes at cost while the inverse fg exists, it typically doesn't have closed form. That is, if we use such a function to define one direction of the transformation in a flow model, the other di- rection can only be approximated numerically using iterative root-fim poot-finding methods (Ho et al., 2 In this work, we don't t consider invertible normalizing flows based on dimension splitting, such as RealNVP (Dinh et al. 2017), since they
get the density an arbitrary point æ; it is necessary to evaluate
applicable ID data.
In the context of TPPs, our goal is to model the distribution p(T) of inter-event times. In order to be able to learn the parameters of p(r) using maximum likelihood, need to be able to evaluate the density any point T. For this need to define the inverse transformation g First, we set 2M 9M (T) log T to convert a positive
(9r -9
R+ into 2M R. Then, we stack multiple layers of parametric functions fe IR + R that can approximate any transformation. We consider two choices for fo: deep sigmoidal flow (DSF) from Krueger al. (2018) and sum-of- squares (SOS) polynomial flow from Jaini et al. (0199)
KR R ap.kag. ptg+l p (1)
DDS (2) = =
Σ Wio
x
pk
SOS
(x) = 00
8%
++
where a, w, 8, la are the transformation parameters, K is the number of components, R is the poly- (=1+)>>> We denote the two variants of the model based on fDr
nomial degree, and (zx) and f.50s' building blocks as DSFlow and SOSFlow respectively. Finally, after stacking multiple 9m = Jo... we apply a sigmoid transformation 91 = a to convert 22 into 21 E (0,
For both models, we can evaluate the inverse transformations oo... o <ww model can be efficiently trained via maximum likelihood. The density p(T) defined by either DS- Flow or SOSFlow model is extremely flexible and can approximate any distribution (Section 3.4). However, for some use cases, this is not sufficient. For example, we may be interested in the expected time until the next event, [T] In this case, flow-based models are not optimal, since for them E, [T] does not in general have closed form. Moreover, the forward transformation (gM o... - 91) cannot be computed in closed form since the functions DSF and fSOS cannot be inverted analytically. Therefore, sampling from p(T) is also problematic and requires iterative root finding.
which means the
This raises the question: Can we design a model for p(T) that is as expressive as the flwebased models, but in which sampling and computing moments is easy and can be done in closed form?
3.2 MODELING P(T) WITH MIXTURE DISTRIBUTIONS
Model definition. While mixture models are commonly used for clustering, they can also be used for density estimation. Mixtures work especially well in low dimensions (McLachlan & Peel] 2004). which is the case TPPs, where we model the distribution of one-dimensional inter-event times 7. Since the inter-event times are positive, we choose to use a mixture of log-normal distributions to model p(T). The PDF of a log- normal mixture is defined as
K
(log 11x
p(T W. M1
Wk
TSk V 2T
exp
2s
(2)
All definitions can be extended to RD for D > 1. We consider the one-dimensional case since our goal is model the distribution of inter-event times 7 R..
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Seq. emb. e Metadata
qr (2(
0:
Vor, bg
91
21
91
History h;
9m-1
RNN
p(rr 0,)
mm
gra(2m)
9M
2M
T
p(r)
f
f--2
f
f +
Figure 1: Model architecture. Parameters of p" (T, generated based on the conditional information C,.
Figure 2: Normalizing flows define
flexible distribution via transformations.
where the mixture weights, μ the mixture means, and are the standard deviations. Because of its simplicity, the log-normal mixture model has a number attractive properties.
Moments. Since each component k has finite mean, the mean of the entire distribution can be computed as [x[r] E. We exp(μi + /2) i.e., a weighted average of component means, Higher moments can be computed based the moments of each component (Frühhwirh-Schhatte) 2006).
Sampling. While flow-based models from Section 3.1 require iterative root-finding algorithms to generate samples, sampling from mixture model can be done in closed form:
~ Catogorical w)
E ~~ Normal(O, 1)
T= =>xp(ss=>> - +
where z is one-hot vector of size K. In some applications, such as reinforcement learning (Upad- hyay et al. 2018), might be interested in computing gradients of the samples w.r.t. the model parameters. The samples T drawn using the procedure above are differentiable with respect to the means fr and scales 8. By using the Gumbel-softmax trick (Jang al. 2017) when sampling z, can obtain gradients w.r.t. all the model parameters (Appendix D.6p. Such reparametrization gra- dients have lower variance and are easier implement than the score function estimators typically used in other works (Mohamed et al. 2019). Other flexible models (such multi-layer flow models from Section[..] do not permit sampling through reparametrization, and thus are not well-suited for the above- mentionen scenario. In Section 5.4 we show how reparametrization sampling can also be used to train with missing data by performing imputation on the fly.
3.3 INCORPORATING THE CONDITIONAL INFORMATION
History. A crucial feature of temporal point processes is that the time = (t: 1411 until the next event may be influenced by all the events that happemed before. standard way of capturing this dependency is to process the event history H4, with recurrent neural network (RNN) and embed it into a exeedddimensionll vector h; E R (Du et al. 2016)..
Conditioning additional features. The distribution of the time until the next event might de- pend on factors other than the history. For instance, distribution of arrival times of customers in a restaurant depends on the day of the week. As another example, we are modeling user behavior in an online system, we can obtain a different distribution (r) for each user by conditioning on their metadata. We denote such side information marks [Rasmussen] 2011). since (a) the metadata may be shared for the entire sequence and (b) y; only influences the distribution p" (Ts(9:), not the objective function.
vector y:- Such information is different from
In some scenarios, we might be interested in learning from multiple event sequences. In such case, we can assign each sequence Tj a learnable sequence embedding vector ej, By optimizing ejs the model can learn to distinguish between sequences that come from different distributions. The learned embeddings can then be used for visualization, clustering or other downstream tasks.
Obtaining the parameters. We model the conditional dependence of the distribution p" (Tx) all of the above factors the following way. The history embedding h,, metadata y, and sequence em- bedding e, concatenated into context vector c; /h.l/v//le,). Then, obtain the parameters of the distribution p" (Fr) as an affine function c, For example, for the mixture model we have ba)
w, = softmax Vw + bw)
8; =
#4 = Vuc + by (3)
where the softmax and exp transformations are applied to enforce the constraints on the distribution parameters, and (V.: Vz. Vus bw, b, bn} learnable parameters. Such model resembles the
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mixture density network architecture (Bishop 1994). We obtain the parameters of the flow-based models in a similar way (see Appendix D).
The whole process is illustrated in Figure
|
3.4 DISCUSSION
Universal approximation. The SOSFlow and DSFlow models can approximate any probability density on R arbitrarily well (Jaini et al. 2019 Theorem 3), (Krueger et al. [2018] Theorem 4). turns out, a mixture model has the same universal approximation (UA) property.
Theorem (DaSGupta 2008 Theorem 33.2), Let p(z) be a continuous density on R. If q(z) is any density on I. and is also continuous, then, given E >> and a compact set R. there exist number of components K E N, mixture coefficients w E A AK-1, locations μ E B. aand scales $ E T such that for the mixture distribution p(x) = -E= Wk (===))))))))))0))))) it holds supses (m(x) - -(x)) <.
This results shows that, principle, the mixture distribution is as expressive the dowebased models. Since we are modeling the conditional density, we additionally need to assume for all of the above models that the RNN can encode all the relevant information into the history embedding h;- This can be accomplished by invoking the universal approximation theorems for RNNs (Siegelmann Sontag 1992. Schäfer Zimmermann 2006).
Note that this result, like other UA theorems of this kind (Cybenko 1989, Daniels & Velikova 2010), does not provide any practical guarantees on the obtained approximation quality, and doesn't t say how to learn the model parameters. Still, UA intuitively seems like desirable property of a distribution. This intuition supported by experimental results. In Section 5.1 we show that models with the UA property consistently outperform the less flexible ones.
Interestingly. Theorem does not make any assumptions about the form of the base density q(x). This means could well use mixture of distribution other than log-normal. However, other popular distributions on R_ have drawbacks: log-logistic does not always have defined moments and gamma distribution doesn't permit straightforward sampling with reparametrization.
Intensity function. For both flow-based and mixture models, the conditional cumulative distribution readily available. This means we can easily compute the respective intensity functions (see Appendix A). However, we should still ask whether we lose anything by modeling p" (7) instead of X (t). The main arguments in favor of modeling the intensity function in traditional models (e.g. self-exciting process) are that it's intuitive, easy to specify and reusable (Upadhyay Rodriguez. 2019).
function (CDF) (7) and the PDF p" (T(
"Intensity function is intuitive, while the conditional density is not." While it's 's true that in simple models (e.g. in self-exciting or self-correcting processes) the dependence of X (t) on the history is intuitive and interpretable, modern RNN-based intensity functions (as in Du et al. (2016); Mei & Eisner (0mi): et al. (2019)) cannot be easily understood by humans. In this sense, our proposed models intuitive and interpretable as other existing intensity-basee neural network models.
(t) is easy to specify, since only has to be positive. On the other hand, p" (T) must integrate to one." - As we saw, by using either normalizing flows or mixture distribution, we automatically enforce that the PDF integrates to one, without sacrificing the flexibility of our model.
"Reusability: If we merge two independent point processes with intensitites (t) and Ag(t). the merged process has intensity X (t) = (t) + (t).)"" An equivalent result exists for the CDFs F (T( and F (7) of the two independent processes. The CDF of the merged process is obtained as F (T) (F) (T) (derivation in Appendix A
As just showed, modeling (T) instead of X* (t) does not impose any limitation our approach. Moreover, a mixture distribution flexible, easy to sample from and has well-defined moments, which favorably compares it to other intensity-based deep learning models.
RELATED WORK
Neural temporal point processes. Fitting simple TPP models (c.g. self-exciting (Hawkes, 1971) or self-correcting [Isham & Westcott 1979) processes) to real-world data may lead to poor results because of model misspecification. Multiple recent works address this issue by proposing more
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flexible neural-netwook-baee point process models. These neural models are usually defined in terms of the conditional intensity function. For example, Mei & Eisner (2017) propose a novel RNN architecture that can model sophisticated intensity functions. This flexibility comes at the cost of inability to evaluate the likelihood in closed form, and thus requiring Monte Carlo integration.
Du al. 2016) suggest using an RNN encode the event history into vector h4- The history embedding h; is then used to define the conditional intensity, for example, using the constant in- tensity model X² exp(v² h; + b) (Li et al., 2018 Huang al., 2019) or the more flexible exponential intensity model (4:) - exp(w((; 11-1) v² h4 + 6) (Du et al. 2016, Upadhyay al. 2018). By considering the conditional distribution p" (T) of the two models. we can better understand their properties, Constant intensity corresponds to an exponential distribution, and expo- nential intensity corresponds to Gompertz distribution (see Appendix B). Clearly, these unimodal distributions cannot match the flexibility of mixture model (as can be seen in Figure K).
Omi et al. (2019) introduce a flexible fully neural network (FullyNN) intensity model, where they model the cumulative intensity function (T) with neural net. The function A converts into an exponentially distributed random variable with unit rate (Rasmussen) [201I) similarly to how normalizing flows model p" (r) by converting into random variable with simple distribution. However, due to a suboptimal choice of the network architecture, the PDF of the FullyNN model does not integrate to I, and the model assigns non-zero probability to negative inter-event times (see Appendix C). In contrast, SOSFlow and DSFlow always define a valid PDF on R Moreover, similar to other bow-based models, sampling from the FullyNN model requires iterative root finding.
Several works used mixtures of kernels to parametrize the conditional intensity function (Taddy let al.) 2012 Tabibian et al., 2017 Okawa et al. 2019). Such models can only capture self-exciting influence from past events. Moreover, these models do not permit computing expectation and draw- ing samples in closed form. Recently, Billos al. (2099) and Türkmen al. (2019) proposed neural models for learning marked TPPs. These models focus on event type prediction and share the lim- itations of other neural intensity-based approaches. Other recent works consider alternatives to the maximum likelihood objective for training TPPs. Examples include noise- contrasttee estimation (Guo et al.) 2018), Wasserstein distance (Xiao et al. [017]] [018]] Yan et al.) 2018), and reinforce- ment learning (Li et al., 20188] [Upadhyay et al. 2018). This line of research is orthogonal to our contribution, and the models proposed in our work can be combined with the above-mentioned training procedures.
Neural density estimation. There exist two popular paradigms for learning flexible probability dis- tributions using neural networks: In mixture density networks (Bishop) 1994), neural net directly produces the distribution parameters; in normalizing flows (Tabak& & Turner 2013 Rezende & Mo- h 2015), obtain complex distribution by transforming simple one. Both mixture models [Schuster] 2000 Eirola & Lendasse] 2013 Graves 2013) and normalizing flows (Oord et al.) 2016) Ziegler & Rush 2019) have been applied for modeling sequential data. However, surprisingly, none of the existing works make the connection and consider these approaches in the context of TPPs.
5 EXPERIMENTS
We evaluate the proposed models on the established task of event time prediction (with and without marks) in Sections and 5.2 the remaining experiments, we show how the log-normal mixture model can be used for incorporating extra conditional information, training with missing data and learning sequence embeddings. We use 6 real-world datasets containing event data from various domains: Wikipedia (article edits). MOOC (user interaction with online course system), Reddit (posts in social media) (Kumar et al., 2019). Stack Overflow (badges received by users), LastFM (music playback) (Du | and Yelp (check-ins to restaurants). We also generate 5 synthetic datasets (Poisson, Renewal, Self-correcting, Hawkes Hawkes2), as described in im/Omi et al. 2019). Detailed descriptions and summary statistics of all the datasets are provided in Appendix
5.1 EVENT TIME PREDICTION USING HISTORY
Setup. We consider two normalizing flow models, SOSFlow and DSFlow (Equation [ well a normall mixture model (Equation D. denoted as LogNormMix. As baselines, we consider RMTPP (i.e. Gompertz distribution / exponential intensity from Du al. (2016) and FullyNN
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Suaddnttoed HEL
Ligi
Standentiondted
4.5
Fullyit
3.0
Loggter
RAITPP
0.0
Yelp
Reddit
La
MOOC
Stack Overflow
Wikipedia
Reddit
MOOC
Figure 3: NLL loss for event time prediction without marks (left) and with marks (right). NLL of each model is standardized by subtracting the score of LogNormMix. Lower score is better. Despite its simplicity, LogNormMix consistently achieves excellent loss values.
model by Omi et al. (2019). Additionally, we use a single log-normal distribution (denoted Log- Normal) to highlight the benefits of the mixture model. For all models, an RNN encodes the history into a vector ht, The parameters of p (T) are then obtained using h; (Equatiinn 3). We exclude the NeuralHawkes model from our comparison, since it is known to be inferior to RMTPP in time prediction (Me & Eisner 2017). and, unlike other models, doesn't 't have a dlosed-form likelihood.
Each dataset consists of multiple sequences of event times. The task is to predict the time тı until the next event given the history Hr, For each dataset, we use 60% of the sequences for training, 20% for validation and 20% for testing. We train all models by minimizing the negative log-likelihood (NLL) of the inter-event times in the training set. To ensure a fair comparison, we try multiple hyperparameter configurations for each model and select the best configuration using the validation set. Finally, we report the NLL loss of each model on the test set. All results are averaged over 10 train/validationn/estt splits. Details about the implementation, training process and hyperparameter ranges are provided in Appendix D For each real-world dataset, we report the difference between the NLL loss of each method and the LogNormMix model (Figure 3 We report the differences. since scores of all models can be shifted arbitrarily by scaling the data. Absolute scores (not differences) in tabular format, as well as results for synthetic datasets are provided Appendix F.
Simple unimodal distributions (Gompertz/RMTPP, LogNormal) are always dominated by the more flexible models with the universal approximation property (LogNormMix, DSFlow, SOS- Flow, FullyNN). Among the simple models, LogNormal provides a much better fit to the data than RMTPP/Gompertz. The distribution of inter-event times in real-world data often has heavy tails, and the Gompertz distributions fails to capture this behavior. We observe that the two proposed models, LogNormMix and DSFlow consistently achieve the best loss values.
Results.
5.2 LEARNING WITH MARKS
Setup. We apply the models for learning in marked temporal point processes. Marks are known to improve performance of simpler models (Du et al. (016)) we want to establish whether our proposed models work well in this setting. We use the same setup as the previous section, except for two differences. The RNN takes tuple (T;, m,) input at each time step, where m. is the mark. Moreover, the loss function now includes a term for predicting the next mark: [(0)
- E, [log P (Tx) + (logp)(m.)) (implementation details in Appendix F.2
Results. Figure 3 (right) shows the time NLL loss (i.c. - -e log (T,)) for Reddit and MOOC datasets. Log shows dominant performance in the marked case, just like in the previous experiment. Like before, we provide the results in tabular format, as well as report the marks NLL
loss in Appendix F
5.3 LEARNING WITH ADDITIONAL CONDITIONAL INFORMATION
Setup. We investigate whether the additional conditional information (Section 3.3) can improve performance of the model. In the Yelp dataset, the task is predict the time 7 until the next check-in for given restaurant. We postulate that the distribution p" (r) is different, depending on whether it's a weekday and whether it's an evening hour, and encode this information as a vector y:- We consider 4 variants of the MogNorMMix model, that either
don't use y, and the history embedding h
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Observod
Mcan
Sampled
sampling
12
Mean imputation
Model
NLL
Samping reparam.
1.0
No imputation
1.01 = 0.20
Mean imputation
0.81 0.27
Sampling with
0.4
0.36 0.05
reparametrization
Figure 4: By sampling the missing values from p" (T) during training, LogNormMix learns the true underlying data distribution. Other imputation strategies overfit the partially observed sequence.
Figure 5 shows the test set loss for 4 variants of the model. We see that additional condi- tional information boosts performance of the LogNormMix model, regardless of whether the history embedding is used.
Results.
5.4 MISSING DATA IMPUTATION
In practical scenarios, one often has to deal with missing data. For example, we may know that records were not kept for period of time, or that the data is unusable for some reason. Since TPPs are a generative model, they provide a principled way to handle the missing data through imputation.
We are given several sequences generated Hawkes process, where some parts are known to be missing. We consider strategies for learning from such partially observed sequence: (a) ignore the gaps, maximize log-likelihood of observed inter-event times (b) fill the gaps with the average r estimated from observed data, maximize log-likelihone of observed data, and (c) fill the gaps with samples generated by the model, maximize the expected log-likelihood of the observed points. The setup is demonstrated in Figure + Note that in case (c) the expected value depends on the parameters the distribution, hence we need to perform sampling with reparametrization to optimize such loss. A more detailed description of the setup is given in Appendix F44
Setup.
Results. The model variants are trained on the partially-observed sequence. Figure shows the NLL of the fully observed sequence (not seen by any model training time) produced by cach strategy. We see that strategies (a) and (b) overfit the partially observed sequence. In contrast, strategy (c) generalizes and learns the true underlying distribution. The ability of the LogNormMix model to draw samples with reparametrization was crucial to enable such training procedure.
Buth h 4.y,
M
Polsson
campenonce
Renewal
0.115
Hawkes1
Sell-comecting
0.10
Hawkes2
6.05
0.00
Yelp
t SNE component
Figure 5: Conditional informa- tion improves performance.
Figure 6: Sequences generated Figure 7: Sequence embeddings
based on different embeddings.
learned by the model.
5.5 SEQUENCE EMBEDDING
Different sequences the dataset might generated by different processes, and exhibit different distribution of inter-event times. We can "help" the model distinguish between them by assigning trainable embedding vector e; to each sequence j in the dataset. seems intuitive that embedding vectors learned this way should capture some notion of similarity between sequences.
Learned sequence embeddings. We learn a sequence embedding for each of the sequences in the synthetic datasets (along with other model parameters). We visualize the learned embeddings using t-SNE (Maaten & Hinton 2008) in Figure colored by the true class. As we see, the model learns to differentiate between sequences from different distributions completely unsupervised way.
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Generation. We fit the LogNormMix model to two sequences (from self-correcting and renewal processes), and, respectively, leam two embedding vectors Esc and eav- After training, we gen- erate 3 sequences from the model, using esc, 1/2(esc + eRN) and eax sequence embeddings. Additionally, we plot the learned conditional intensity function of our model for each generated se- quence (F The model learns to map the sequence embeddings very different distributions.
CONCLUSIONS
We use tools from neural density estimation to design new models for learning in TPPs. We show that a simple mixture model competitive with state-of-the-art normalizing flows methods, as well convincingly outperforms other existing approaches. By looking at learning in TPPs from dif- ferent perspective, we were able to address the shortcomings of existing intensity-basee such as insufficient flexibility, lack of closed-form likelihoods and inability generate samples analytically. We hope this alternative viewpoint will inspire new developments in the field of TPPs.
approaches,
ACKNOWLEDGMENTS
This research was supported by the German Federal Ministry of Education and Research (BMBF), grant 01IS 1803BB and the Software Campus Project Decp-RENT. The authors of this work take full responsibilitiie for its content.
REFERENCES
Marin Biloš, Bertrand Charpentier, and Stephan Günnemann. Uncertainty on asynchronous time event prediction. In Advances in Neural Information Processing Systems. 2019.
Eli Bingham, Jonathan P. Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis Karaletsos, Rohit Singh, Paul Szerlip, Paul Horsfall, and Noah D. Goodman. Pyro: Deep Univer- sal Probabilistic Programming. Journal of Machine Learning Research, 2018.
Christopher M Bishop. Mixture density networks. 1994.
Oscar Celma. Music recommendation. In Music recommendation and discovery, pp. 43-85. Springer, 2010.
George Cybenko. Approximation by superpositions sigmoidal function. Mathematics of control, signals and systems, 2(4), 1989.
Hennie Daniels and Marina Velikova. Monotone and partially monotone neural networks. 1EEE Transactions on Neural Networks, 21(6))906-913 2010.
Anirban DasGupta. Asymptotic theory of statistics and probability. Springer Science & Business Media, 2008.
Laurent Dinh, Jascha Sohl-Dickstein. and Samy Bengio. Density estimation using Real NVP. In- ternational Conference on Learning Representations, 2017.
Nan Du, Hanjun Dai, Rakshit Trivedi, Utkarsh Upadhyay, Manuel Gomez-Rodriguee and Le Song. Recurrent marked temporal point processes: Embedding event history vector. AnACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2016.
Emil Eirola and Amaury Lendasse. Gaussian mixture models for time series modelling, forecast- ing, and interpolation. In International Symposium Intelligent Data Analysis, pp. 162-173. Springer, 2013.
Sylvia Seahwwrtt-Schhattee Business Media, 2006.
Finite mixture and Markov switching models. Springer Science &
Will Grathwohl, Dami Choi, Yuhuai Wu, Geoffrey Roeder, and David Duvenaud. Backpropagation through the void: Optimizing control variates for black-box gradient estimation. International Conference Learning Representations, 2018

---

Published as a conference paper at ICLR 2020
Alex Graves. Generating sequences with recurrent neural networks. wrXiv:1308.08500 2013.
arXiv preprint
Ruocheng Guo, Jundong Li, and Huan Liu. INITIATOR: Noise-contrastive estimation for marked temporal point process. In International Joint Conference on Artificial Intelligence, 2018.
Alan G Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83-90, 1971.
Jonathan Ho, Xi Chen, Aravind Srinivas, Yan Duan, and Pieter Abbeel. Flow++: Improving flow- based generative models with variational dequantization and architecture design. International Conference on Machine Learning. 2019.
Hengguang Huang, Hao Wang, and Brian Mak. Recurrent poisson process unit for speech recogni- tion. In AAAI Conference on Artificial Intelligence, 2019.
Valerie Isham and Mark Westcott. A self-correcting point process. Stochastic Processes and Their Applications, 8(3):335-347, 1979.
Priyank Jaini, Kira A. Selby, and Yaoliang Yu. Sum-of-squares polynomial flow. In International Conference on Machine Learning, 2019.
Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with Gumbel-softmax. International Conference Learning Representations, 2017.
Diederik Kingma and Jimmy Ba. Adam: method for stochastic optimization. The International
Conference on Learning Representations, 2015.
David Krueger, Chin-Wei Huang, Alexandre Lacoste, and Aaron Courville. Neural autoregressive flows. In International Conference on Machine Learning, 2018.
Srijan Kumar, Xikun Zhang, and Jure Leskovec. Predicting dynamic embedding trajectory in tem- poral interaction networks. In ACM SIGKDD international conference on Knowledge discovery and data mining, 2019.
Shuang Li, Shuai Xiao, Shixiang Zhu, Nan Du, Yao Xie, and Le Song. Learning temporal point processes via reinforcement learning. In Advances in Neural Information Processing Systems, 2018.
Laurens van der Maaten and Geoffrey Hinton. learning research, 9Nov)):557922605 2008
Visualizing data using t-sne. Journal of machine
Geoffrey McLachlan and David Peel. Finite mixture models. John Wiley & Sons, 2004.
Hongyuan Mei and Jason M Eisner. The neural hawkes process: A neurally elffmodulating multi- variate point process. In Advances in Neural Information Processing Systems, 2017.
Shakir Mohamed, Mihaela Rosca, Michael Figurnov, and Andriy Mnih. Monte carlo gradient esti- mation machine learning. arXiv preprint arXiv::906.0 10652, 2019.
Maya Okawa, Tomoharu Iwata, Takeshi Kurashima, Yusuke Tanaka, Hiroyuki Toda, and Naonori Ueda. Deep mixture point processes: Spatio-temporal event prediction with rich contextual infor- mation. 2019.
Takahiro Omi, Naonori Ueda, and Kazuyuki Aihara. Fully neural network based model for general temporal point processes. 2019.
Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbremner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint 16Xiv:1609.00 2016.
Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. In NIPS Autodiff Workshop, 2017.
10

---

Published as a conference paper at ICLR 2020
Jakob Gulddahl Rasmussen. Temporal point processes: the conditional intensity function. Lecture Notes, Jan, 2011.
Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. In- ternational Conference on Machine Learning, 2015.
Anton Maximilian Schäfer and Hans Georg Zimmermann. Recurrent neural networks are universal approximators. In International Conference on Artificial Neural Networks, pp. 632-640. Springer, 2006.
Mike Schuster. Better generative models for sequential data problems: Bidirectional recurrent mix- ture density networks. In Advances in Neural Information Processing Systems, 2000.
Hava T Siegelmann and Eduardo D Sontag. On the computational power of neural nets. In Pro- ceedings of the fifth annual workshop on Computational learning theory, pp. 440-449. ACM, 1992.
Esteban TTaba and Cristina V Turner. A family of nonparametric density estimation algorithms. Communications on Pure and Applied Mathematics, 66(2):145-164
2013. 2013.
Behzad Tabibian, Isabel Valera, Mehrdad Farajtabar, Le Song, Bernhard Schölkopf. and Manuel Gomez- Rodriiuz. Distilling information reliability and source trustworthiness from digital traces. In International Conference World Wide Web, pp. 847-855 2017.
Matthew A Taddy, Athanasios Kottas, al. Mixture modeling for marked poisson processes. Bayesian Analysis, 7(2), 2012.
George Tucker, Andriy Mnih, Chris Maddison, John Lawson, and Jascha Sohl-Dickstein. Rebar: Low-variance, unbiased gradient estimates for discrete latent variable models. In Advances in Neural Information Processing Systems, 2017.
Ali Caner Türkmen, Yuyang Wang, and Alexander J Smola. Fastpoint: Scalable deep point processes. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, 2019.
Utkarsh Upadhyay and Manuel Gomez Rodriguez. Temporal point processes. Lecture notes for Human Centered ML. January 2019, URL http://courses mpi sws. org. org/
[cmml-ww1@/lectures//Pp pddf
Utkarsh Upadhyay, Abir De, and Manuel Gomez Rodriguez. Deep reinforceeme learning of marked temporal point processes. In Advances in Neural Information Processing Systems, 2018.
Andreas Wienke. Frailty models in survival analysis. Chapman and Hall/CRC, 2010.
Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, ((3-4):229-255) 1992.
Shuai Xiao, Mehrdad Farajtabar, Xiaojing Ye, Junchi Yan, Le Song, and Hongyuan Zha. Wasserstein learning of deep generative point process models. In Advances in Neural Information Processing Systems, 2017.
Shuai Xiao, hongteng Xu, Junchi Yan, Mehrdad Farajtabar, Xiaokang Yang, Le Song, and Hongyuan Zha. Learning conditional generative models for temporal point processes. AAAI Conference Artificial Intelligence, 2018.
Junchi Yan. Xin Liu, Liangliang Shi, Changsheng Li, and Hongyuan Zha. Improving maximum likelihood estimation of temporal point process via discriminative and adversarial learning. In International Joint Conference on Artificial Intelligence, 2018.
Zachary M Ziegler and Alexander Rush. Latent normalizing flows for discrete sequences. Inter- national Conference Machine Learning, 2019.

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A INTENSITY FUNCTION OF FLOW AND MIXTURE MODELS
CDF and conditional intensity function of proposed models. The cumulative distribution func- tion (CDF) of normalizing flow model can be obtained in the following way. If z has CDF Q(2)
and T = g(z), then the CDF F(T) is obtained as
F(T)
(r)
Since for both SOSFlow and DSFlow we can evaluate g in closed form, F(r) is easy to compute.
For the log-normal mixture model, CDF is by definition equal
K
log
F(r) = Σ wy
Sk
where ((-) is the CDF of standard normal distribution.
Given the conditional PDF and CDF, we can compute the conditional intensity X (t) and the cumu- lative intensity A (r) for each model
(1) p" (1 4-11
(T)T)) = L (t s)ds - log(1 F* (T:))
Ft (t- 1;-1)
where 6;-1 the arrival time of most recent event before (Rasmussen) 2011).
Merging two independent processes. We replicate the setup from Upadhyay & Rodriguez (2019) and consider what happens if we merge two independent TPPs with intensity functions X (L) and (t(t) (and respectively, cumulative intensity functions (T) and (T(r)) According to UUpadhyay & Rodriguez (019). the intensity function the new process is (t = (t( + (t))). Therefore, the cumulative intensity function of the process
(T) L' Xt s)ds
Ai sdds I' >g(ts-1 + s)ds
J0
= M[() + Ag(T)
Using the previous result, we can obtain the CDF of the merged process as
F*(r) exp(-A"(r))
exp(-A](r) - AT(T))
-eexplog((1 - (((()o + log 11 Fj(T))
(1+ (7) - - FT(T))
The PDF of the merged process is obtained by simply differentiating the CDF w.r.t.
This means that by using either normalizing flows or mixture distributions, and thus directly model- ing PDF / CDF, we are not losing any benefits of the intensity parametrization.
B DISCUSSION OF CONSTANT & EXPONENTIAL INTENSITY MODELS
Constant intensity model as exponential distribution. The conditional intensity function of the constant intensity model (Upadhyay et al. 2018) defined as (t,) exp(x" h, b). where h, E # is the history embedding produced by an RNN, and E IR is learnable parameter. By setting e = exp(e (v u h + b), it's easy see that the PDF of the constant intensity model p" p²(r) = cexp(-c) corresponds to exponential distribution.
Exponential intensity model as Gompertz distribution. (Wienke 2010) is defined as
PDF a Gompertz distribution
p(T|a, (3) exp (T (8r-3 exp(Bt) +
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Constant intensity
Exponential intensity
Log-normal mixture
2.0
0:5
0
0, a= 0.5
1.5
--1.0.-1
0, a:
2.2
2-2.0b-1
1.
1.0
0.5
0.0
Figure 8: Different choices for modeling p(r): exponential distribution (left), Gompertz distribution (center), log-normal mixture (right). Mixture distribution can approximate any density while being tractable and easy to sample from.
for a, 3 > 0. The two parameters and 3 define its shape and rate, respectively. For any choice of its parameters, Gompertz distribution is unimodal and light-tailed. 'he mean of the Gompertz distribution can be computed as B[r] = 3/2 exp Ei(- where Ei(z) = [ exp(-u)/v du the exponential integral function (that can be approximated numerically).
The conditional intensity function the exponential intensity model (Du et al. 2016 is defined as (4, exp(wti (ttt (4-1) v" h, b), where h2 R is the history embedding produced by RNN, and v E _" b R, w E R4 are learnable parameters. By defining d = v" h; + b, we obtain the PDF of the exponential intensity model (Du al. 2016 Equation 12) as
1 p(r|wr d) = exp WT + d exp(wr d) + exp(d)
1
By setting a exp(d) and B = w we see that the exponential intensity model is equivalent to Gompertz distribution.
Discussion. Tigure&sshove densities that can be represented by exponential and Gompertz distribu- tions. Even though the history embedding h, produced by an RNN may capture rich information, the resulting distribution p" (T;) for both models has very limited flexibility, is unimodal and light-tailed. In contrast, flow-based or mixture model significantly more flexible and can approximate any density.
C DISCUSSION OF THE FULLYNN MODEL
Summary The main idea of the approach by Omi et al. 2019 is to model the integrated condi- tional intensity function
A (-))) S. X- + s)ds
using a feedforward neural network with non-negative weights
AT(T) == f(r) softplus( (3)) tanh(W tanh tanh(W (()) b(2)) + b(0)
(4)
where T()) = Vh + ((0) h E R is the history embedding, ( E DD11 W E RDxD
(3) R non-negative weight matrices, and V RD ((0) RD, ((2) E RD
((3) are the remaining model parameters.
FullyNN normalizing flow Let Exponential(1), that is
F(z) = 1 exp(-2)
p(z) = exp( -2)
We can view f R: R transformation that maps
2 f=) - (=f
We the change of variables formula to obtain the conditional CDF and PDF of
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Alternatively, obtain the conditional intensity
0
0
(² dr (T) Or f(T)
and use the fact that p pT(Ti) = X (t,-1 + exp (- So x- +
Both approaches lead to the same conclusion
F" (T) 1 exp-(((r((())
p"(r) -xp()())())))))- o.
However, the first approach also provides intuition on how to draw samples f from the resulting distribution p" (r) an approach known the inverse method (Rasmussen) 2011
1. Sample ~ Exponential(1)
for (using e.g. bisection method) 2. Obtain by solving [(T)
Similarly to other flow-based models, sampling from the FullyNN model cannot be done exactly and requires a numerical approximation.
Shortcomings of the FullyNN model
The PDF defined by the FullyNN model doesn't integrate to
By definition of the CDF, the condition that the PDF integrates to is equivalent to lim, (T) = 1, which in turn equivalent to lim (T) co. However, be-
cause of saturation of tanh activations (i.e. suP=ER [aah(x)] = 1) in Equation
lim (T) lim f(r) < softplus
Σ
(3) 1
6() oc
Therefore, the PDF doesn't integrate to
2. The FullyNN model assigns non-zero amount of probability mass the (-o0, 0) interval, which violates the assumption that inter-event times are strictly positive.
Since the inter-event times T assumed to be strictly positive almost surely, it must hold that Prob(T < 0) = F (0) = O, or cquivalently A (0) = 0. However, we can see that (0) = f(0)= = softplus (M (3) tanh( (2) tanh(b(1), + 6(2) + b(8) > 0 which means that the FullyNN model permits negative inter-event times.
D IMPLEMENTATION DETAILS
D.I SHARED ARCHITECTURE
We implement SOSFlow, DSFlow and LogNormMix, together with baselines: RMTPP (Gompertz distribution), exponential distribution and a FullyNN model. All of them share the same pipeline, from the data preprocessing the parameter tuning and model selection, differing only in the way we calculate p" (T). This way we ensure a fair evaluation. Our implementation uses Pytorch B
From arival times 1 calculate the inter-event times t; large values, RNN takes log-tran teggrrrefformed and centered inter-event time and produces h; E R In case we have marks, we additionally input m; the index of the mark class from which we get mark embedding vector m., In some experiments we use extra conditional information, such as metadata y: and sequence embedding ej, where j the index of the sequence.
f,-1. Since they can contain very
As illustrated in Section 3.3tionn333wrm we we generate the parameters 0 of the distribution p" (Tx) from [h: |ey] using an affine layer. apply a transformation of the parameters to enforce the constraints, necessary.
All decoders are implemented using a common framework relying on normalizing flows. By deñi- --1 ing the base distribution g(z) and the inverse transformation (0) (o.-o... .... g we can evaluate the PDF p² (7) at any т, which allows us to train with maximum likelihood (Section 3.1
https://pytorcch.ogg Paszke et al. 2017
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D.2 LOG-NORMAL MIXTURE
The log-normal mixture distribution is defined in Equation 2
We generate the parameters of the dis- tribution E R le E R E IR (subject to Ex wg wk 2 0 and s& 0), using affine
transformation (Equation 3 flow model
The log-normal mixture is equivalent to the following normalizing
21 GaussianMixturr((ur u.,
22 = 021 +
exp(22)
By using the affine transformation 22 = α21 + b before the exp transformation, we obtain better initialization, and thus faster convergence. This similar to the batch normalization flow layer
(Dinh et al. 2017), except that - & Σi log Ti and = > (log - b) are estimated using the entire dataset, not using batches.
Forward direction samples a value from a Gaussian mixture, applies an affine transformation and applies exp. In the bacward direction we apply log-transformation to an observed data, center it with an affine layer and compute density under the Gaussian mixture.
D.3 BASELINES
We implement FullyNN model (Omi al., 2019) described in Appendix C using the official implementation as a reference" The model uses feed-forward neural network with non-negative weights (enforced by clipping values at after every gradient step). Output of the network a cumu-
lative intensity function A (T) from which we can easily get intensity function X (T) derivative w.r.t. using automatic differentiation Pytorch. We get the PDF as (T) (T) exp(-A>(7))
We implement RMTPP Gompertz distribution (Du et al. 2016 and the exponential distribution (Upadhyay et al., 2018) models as described in Appendix
All of the above methods define the distribution p² (T). Since the inter-event times may come at very different scales, apply linear scaling 7 ar, where N Σ is estimated from the data. This ensures a good initialization for all models and speeds up training.
D.4 DEEP SIGMOIDAL FLOW
single layer of DSFlow model is defined as
K
#DSP fo (x) = o
Σ
x
14
Wko
with parameters 0 {w E RK /4 RKK E R (subject Eσ Wk 1, uy 2 0 and 8k O).
We obtain the parameters of each layer using Equation
We define p(T) through the inverse transformation
(gT .... 9Ml) 1 described in Section 3.1
2M = 9 (T) = log
2m = 9m (2m+1) 50 (zm+1)
21 = 0(22)
21 92(22) Uniform(0,
We use the the batch normalization flow layer (Dinh et al. 2017) between every pair of consecutive
layers, which significantly speeds up convergence.
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D.5 SUM-OF-SQUARES POLYNOMIAL FLOW
single layer of SOSFlow model is defined as
K n ΣΣΣ 4
Soss (x) ao
ptg+1
There are constraints the polynomial coefficients R(R+I)XK, We obtain similarly to Equation
as a = Vac + ba, where is the context vector.
We define p(T) by through the inverse transformation (91 ....o -9M ), described in Section 3.1
RM (T) llog
2m = Im (Zm+1)
PSOS (Zm+1)
=0(22) 21 ~ (21) = Uniform (0, (0,
Same for DSFlow, the the batch normalization flow layer between every pair of consecu- tive layers. When implementing SOSFlow, we used Pyrd for reference.
D.6 REPARAMETRIZATION SAMPLING
Using log-normal mixture model allows to sample with reparametrization which proves to be useful, e.g. when imputing missing data 5..4 In a score function estimator (Williams 1992) given random variable æ (z), where 9 are parameters, we can compute VgLxr=po(t) f(z)
Ez>np(a) (f Vo log po (=))]- This is unbiased estimator of the gradients but it often suffers from high variance. If the function f is differentiable, we can obtain an alternative estimator using the reparametrization trick:
can compute V Vol===po(=)] [x)] = B,-g(c) of (go Such reparametrization estimator typically has lower variance than the score function estimator (Mohamed al.) 2019). In both cases, we estimate the expectation using Monte Carlo.
g(t), x go))). Thanks this reparametrization,
To sample with reparametrization from the mixture model we use the Straight-Throon Through Gumbel Estimator (Jang et al. 2017). We first obtain relaxed sample z" softmax (log + (T), where each o; sampled from a Gumbel distribution with zero mean and unit scale, and T is the temperature parameter. Finally, we get one-hot sample onehot(arg max, =t). While discrete z is used in the forward pass, during the backward pass the gradients will flow through the differentiable z².
The gradients obtained by the Straight-Throw Through Gumbel Estimator are slightly biased, which in prac- tice doesn't t have significant effect on the model's performance. There exist alternatives (Tucker et al. 2017; Grathwohl et al. 2018) that provide unbiased gradients, but are more expensive to compute.
E DATASET STATISTICS
E. SYNTHETIC DATA
Synthetic data is generated according to Omi et al. (2019) using well known point processes. We sample 64 sequences for each process, each sequence containing 1024 events.
Poisson. Conditional intensity function for a homogeneous (or stationary) Poisson point process is given (c) = 1. Constant intensity corresponds exponential distribution.
Renewal. A stationary process defined by a log-normal probability density function p(T), where we set the parameters to be μ 1.0 and 6.0. Sequences appear clustered.
httts::/pyro.at / Bingham
2018
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Dataset name
Number of sequences
Number events
LastFM
929
1268385
Reddit
10000
672350
Stack Overflow
6633
480414
MOOC
7047
396633
Wikipedia
1000
157471
Yelp
300
215146
Table 2: Dataset statistics.
Self-correcting. Unlike the previous two, this point process depends on the history and is defindd by a conditional intensity function X (t) exp(t Ei.<i 1). After every new event the intensity suddenly drops, inhibiting the future points. The resulting point patterns appear regular.
Hawkes. We use a self- point process with a conditional intensity function given as (t)
u+ En ز j=1 3 cxp(-3y( tt)). As per Omi et al. (019)), we create two different datasets: Hawkesl with M 1, μ = 0.02, aj = 0.8 and 31 = 1.0; and Hawkes2 with M = 2, = 0.2,
0n 0.4, 31 1.0, 02 0.4 and 83 20. For the imputation experiment use Hawkes 1 to generate the data and remove some of the events.
E.2 CRAL--WORLA DATA
In addition we use real-world datasets that are described bellow. Table 2 shows their summary. All datasets have large amount of unique sequences and the number of events per sequence varies lot. Using marked temporal point processes to predict the type of an event is feasible for some datasets (e.g. when the number classes low), and is meaningless for other.
LastEM The dataset contains sequences of songs that selected users listen over time. Artists are used an event type.
Reddit. On this social network website users submit posts to subreddits. In the dataset, most active subreddits are selected, and posts from the most active users on those subreddits are recodered. Each sequence corresponds to a list of submissions user makes. The data contains 984 unique subreddits that we use as classes in mark prediction.
Stack Overflow Users of a question ciaation-ansee answering website get rewards (called badges) over time for participation. sequence contains list of rewards for each user. Only the most active users are selected and only those badges that users can get more than once.
MOOC Contains the interaction of students with an online course system. interaction is an event and can be of various types (97 unique types), e.g. watching video, solving quiz etc.
Wikipedia A sequence corresponds to edits of Wikipedia page. The dataset contains most edited pages and users that have activity (number of edits) above certain threshold.
3
Yelp We use the data from the review forum and consider the reviews for the 300 most visited restaurants in Toronto. Each restaurant then has corresponding sequence of reviews over time.
F ADDITIONAL DISCUSSION OF THE EXPERIMENTS
F.1 EVENT TIME PREDICTION USING HISTORYY
Detailed setup. Each dataset consists of multiple sequences of inter-event times. We consider 10 train/validation/test splits of the sequences (of sizes 60%/20% 20%). train all model parameters by minimizing the negative log- likelihood (NLL) of the training sequences, defined Lisme (0)
Celma 2010
https: htttst///itttub..mmm///riiuuuuu
Kumar 2019
preprocessed according to Du et al 2016
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Exponential
RMTPP
NeuraNee
LagNorrMix
0.04
0.02
24
120
72
120
24
96
120
72
120
Figure 9: Models learn different conditional distribution p(T(H( on Yelp dataset. Since check-ins occur during the opening hours, true distribution of the next check-in resembles the one on the right.
E log Po(PP)TT) After splitting the data into the 3 sets, we break down long training sequences into sequences of length at most 128. Optimization is performed using Adam (Kingma & Ba 2015) with learning rate 10 We perform training using mini-batches of 64 sequences. We train for up to 2000 epochs (1 epoch 1 full pass through all the training sequences). For all models, we compute the validation loss at every epoch. there no improvement for 100 epochs, we stop optimization and revert to the model parameters with the lowest validation loss.
.V
We select hyperparameter configuration for each model that achieves the lowest average loss on the validation set. For each model, we consider different values of L2 regularization strength C' E {0, 10-5, 10-3). Additionally, for SOSFlow we tune number of transformation layers M (1,2,3) 3} and for DSFlow M {1, 2, 3, 5, 10). We have chosen the values of K such that the mixture model has approximately the same number of parameters as a l-layer DSFlow or a L-layer FullyNN model. More specifically, we set K 64 for LogMormMix, DSFlow and FullyNN. We found all these models to be rather robust to the choice of K, as can be seen in Table LogNormMix, For SOSFlow we used K = 4 and /t resulting in polynomial of degree 7 (per each layer). Higher values of Rt led to unstable training, even when using batch normalization.
Additional discussion. In this experiment, we only condition the distribution p" (Ti) on the history embedding ht We don't 't learn sequence embeddings e, since they can only be learned for the training sequences, and not fore the validation/test sets.
There are two important aspects related to the NLL loss values that we report. First, the absolute loss values can be arbitrarily shifted by rescaling the data. Assume, that we have distribution p(r) that models the distribution of т. Now assume that we are interested in the distribution q(x) of at (for 0). Using the change of variables formula, we obtain log g(x) log p(T) + log a. This means that by simply scaling the data we can arbitrarily offset the log-likelihood score that we obtain. Therefore, the absolute values of of the (negative) log-likelihood L for different models are of little interest all that matters are the differences between them.
The loss values are dependent on the train/val/test split. Assume that model I achieves loss values C = {1.0, 3.0} on two train/val/test splits, and model 2 achieves L2 {2.0, 4.0} on the same splits. If we first aggregate the scores and report the average L1 2.0 + 1.0, L2 3.0 + 1.0, it may seem that the difference between the two models is not significant. However, if we first compute the differences and then aggregate (L2- L1 1.0 + 0.0 we see different picture. Therefore, we use the latter strategy in Figure 3 For completeness, we also report the numbers obtained using the first
strategy in Table
As baseline, we also considered the constant intensity/ / exponential distribution model (Upadhyay al., 2018). However, we excluded the results for it from Figure since it consistently achieved the worst loss values and had high variance. We still include the results for the constant intensity model Table 4 We also performed all the experiments on the synthetic datasets (Appendix [E.T]. The results are shown in Table 5 together with NLL scores under the true model. We see that LogNormMix and DSFlow, besides achieving the best results, recover the true distribution.
Finally, in Figure 9 we plot the conditional distribution p() with models trained on Yelp dataset. The events represent check-ins into a specific restaurant. Since check-ins mostly happen during the opening hours, the inter-event time is likely to be the same day (Oh), next day (24h), the day after (48h), etc. LogNormMix can fully recover this behavior from data while others either cannot learn multimodal distributions (e.g. RMTPP) struggle to capture it (e.g. FullyNN).
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K
8
16
32
64
Reddit
10.239
10.208
10.189
10.185
10.191
10.192
LastFM
-2.828
-2.879
-2.881
-2,880
-2.877
-28860
MOOC
6,246
6.053
6.055
6.055
6.050
5.660
Stack Overflow
14.461
14.438
14.435
14.435
14.436
14.428
Wikipedia
8.399
8.389
8.385
8.384
8.384
8.386
Yelp
13.169
13.103
13.058
13.045
13.032
13.024
Poisson
1.006
0.992
0.991
0.991
0.990
0.991
Renewal
0.256
0.254
0.254
0.254
0.256
0.259
Self-correcting
0.831
0.785
0.782
0.783
0.784
0.784
Hawkesl
0.530
0.523
0.532
0.532
0.523
0.523
Hawkes2
0.036
0.026
0.024
0.024
0.026
0.024
Table 3: Performance of LogNorMMix model for different numbers K mixture components.
Roddit
LastFM
MOOC
Stack Overflow
Wikipedia
Yelp
LogNormMix
10.19 + 0.078
-2.88 i 0.147
6,03 i 0.092
14.44 0.013
8.39 0.079
13.02 1 0,070
DSFlow
10.20 + 0.074
-2.88 + 0.148
6.03 - 0.090
14.44 + 0.019
8,40 0.090
13.09 = 0.065
SOSFlow
10.27 1 0.106
0.1133
6.27 1 0.058
14.47
0.049
8.44 0.122
13.21 0.068
FullyNN
10.23 + 0.072
0.179
1.855..152
14.45 -.45+00.11
8,40 0.086
13.04 0.073
LogNormal
10.38 0.077
2.600
0.013
8.52 0.078
13.44 0.074
RMTPP
10.88 + 0.293
-1.30
10.65 0.023
10,02 + 0.085
13.36 0.056
Exponential
11.07 + 0.070
-1.28 + 0.152
10.64 = 0.026
18.48 +.48833.2 3.257
10.03 + 0.083
13.78 + 1.250
Table 4: Time prediction test NLL on real-world data.
F.2 LEARNING WITH MARKS
Detailed setup. We use the same setup as in Section F.1 except two differences. For learning in marked temporal point process, we mimic the architecture from Du et al. (2016). The RNN takes a tuple (T,, m;) as input at each time step, where is the mark. Moreover, the loss function now includes term for predicting the next mark: Lrotet (0) - [logP](T:) log ((gg))(m)))) (m:)).
The next mark my at time 4 is predicted using a categorical distribution p" (mz). The distribution is parametrized by the vector πy, where Tye is the probability event m, = We obtain π., using the history embedding h passed through a feedforward neural network
softmax
tanh (1) h, + b(4) + 6
where (1) V
(2) Di
b are the parameters of the neural network.
Additional discussion. In Figure (right) we reported the differences in time NLL between dif- Σ log (G Table we additionally provide the total NLL
ferent models Ltime (0) (f) - Σ [log (T.) + log py (m.)) averaged over multiple splits.
Using marks as input to the RNN improves time prediction quality for all the models. However, since assume that the marks are conditionally independent of the time given the history (as was done in earlier works), all models have similar mark prediction accuracy.
F.3 LEARNING WITH ADDITIONAL CONDITIONAL INFORMATION
Detailed setup. In the Yelp dataset, the task to predict the time т; until the next customer check- in, given the history of check-ins up until the current time f--1- We want to verify intuition that the distribution pa (T:) depends on the current time 44 For example, (7x) might be different depending whether it's weekday and it's evening hour. Unfortunately, model that processes the history with an RNN cannot easily obtain this information. Therefore, we provide this information directly context vector y: when modeling p" (T,).
The first entry context vector y. E (0, 1)2 indicates whether the previous event f;-1 took place on weekday weekend, and the second entry indicates whether 1--1 in the 5PM-11PM time
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Published as a conference paper at ICLR 2020
Poisson
Renewal
Self-correcting
Hawkes1
Hawkes2
True model
0.999
0.254
0.757
0.453
-0.043
LogNormMix
0.99
0.006
0.25 + 0.010
0.78 + 0.003
0.52 + 0.047
0.02 0.049
DSFlow
0.99 + 0.006
0.25 + 0.010
0.78 + 0.002
0.52 = 0.047
0.02 = 0.050
SOSFlow
1.00 0.013
0.25 + 0.010
0.88 + 0.011
0.59 0.056
0.06 0.046
FullyNN
1.00 + 0.006
0.78 + 0.004
0.55 0.047
0.06 = 0.047
LogNormal
1.08 0.008
0.25 0.010
1.03 + 0.006
0.55 + 0.047
0.06 0.049
RMTPP
0.99 + 0.006
1.01 + 0.023
0.78 + 0.003
0.74 + 0.057
0.69 + 0.058
Exponential
0.99 0.006
1.00 + 0.023
0.94 + 0.002
0.74 + 0.055
0.69 0.054
Table 5: Time prediction test NLL synthetic data.
Time NLL
Total NLL
Mark accuracy
Reddit
MOOC
Reddit
MOOC
Reddit
MOOC
LormorMMix
10.28 0.066
5.75 + 0.040
12.40 + 0.094
7.58 0.047
0.62++0.14
0.45=00.03
DSFlow
10.28 0.073
5.78 + 0.067
12.39 + 0.064
7.52 0.074
0.62+0.013
0.45=0.004
SOSFlow
10.35 0.106
6.06 0.084
12.49 + 0.158
7.78 0.107
0.62:10.13
0.469
FullyNN
10.41 0.079
6.22 i 0.224
12.51 + 0.094
7.93 0.230
0.63::0.013
0.46
LogNormal
10.42 0.076
6.38 + 0.019
12.51 + 0.080
8.11 0.026
0,62:10.013
0.42=0.005
RMTPP
11.15 0.061
10.29 + 0.209
13.26 + 0.085
12.14 0.220
0.62::0.14
0.41=0.006
Table 6: Time and total NLL and mark accuracy when learning marked TPP.
window. To each of the four possibilities we assign learnable 64-dimensional embedding vector: The distribution of p" (T:) until the next event depends on the embedding vector of the time stamp t--1 of the most recent event.
F.4 MISSING DATA IMPUTATION
Detailed setup. The dataset for the experiment is generated as two step process: 1) We generate a sequence of 100 events from the model used for Hawkesl dataset (Appendix [E.]] resulting in sequence of arrival times (ti.... ). 2) We choose random i; and remove all the events that fall inside the interval [4, [i4a] where selected such that the interval length is approximately /N/3.
We consider three strategies for learning with missing data (shown in Figure 4 (left)):
a) No imputation.
The missing block spans the time interval [4, 4+4)- We simply ignore the t,+* - t4.
missing data, i.c. training objective Ltme will include an inter-event time
b) Mean imputation. We estimate the average inter-event time / from the observed data, and impute events at times {4+ + - for n E N, such that 4, + ni < 4.+4.. These imputed events are fed into the history-encoding RNN, but are not part of the training objective.
c) Sampling. The RNN encodes the history up to and including t, and produces h; that use to define the distribution p" (2) We draw a sample 7, (6mp) form this distribution and feed it into the RNN. We keep repeating this procedure until the samples get past the point 4++4- The imputed inter-event times 7
((mp) are affecting the hidden state the RNN (thus
influencing the likelihood of future observed inter-event times 7. (obs)).
We sample multiple such sequences in order approximate the expected log-likelihood of the observed inter-event times (imp)) Σ log (T(((())) Since this objective in-
cludes an expectation that depends on p", we make use of reparametrization sampling to obtain the gradients w.r.t. the distribution parameters (Mohamed et al. 2019).
F.5 SEQUENCE EMBEDDING
Detailed setup. When learning sequence embeddings, we train the model as described in Appendix
F1 besides one difference. First, we pre-train the sequence embeddings e, by disabling the his-
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Published as a conference paper at ICLR 2020
tory embedding h; and optimizing minimize N Σ; log po( h.).
Σ, log Afterwards, enable the history and
In Figure the top row shows samples generated using esc, embedding of a self-correcting se- quence, the bottom row was generated using esc, embedding of a renewal sequence, and the middle row was generated using 1/2(esc + ERN), an average of the two embeddings.
21


Reviews:
[{"review_id": "SyxzOArRFB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper describes a simple yet effective technique for learning temporal point processes using a mixture of log-normal densities whose parameters are estimated with neural networks that also adds conditional information. The method is shown to perform better than more recent techniques for density estimation such as different versions of normalising flows. Experiments were reported on 6 datasets, comparing the approach against flow models and assessing the benefits of adding extra conditional information, performance with missing data, and benefits of sequence embeddings.\n\nThe paper makes an important point which is also my own experience when working with relatively low dimensional problems; simpler neural density estimation approaches such as MDNs usually perform similarly or even better than models using normalising flows. The task here is on learning temporal point processes which have important applications in social networks, criminality studies, disease modelling, etc, but are relatively unpopular within the machine learning community. The paper gives some motivation but I think the authors could elaborate further on the huge number of applications and potential for significant impact from these models. Apart from this the paper is well written and structured, and easy to follow. \n\nThere are not many theoretical innovations as the main contribution is a combination of several well known techniques such as MDNs and RNNs, applied to the specific temporal point process formulation. The main lesson learnt though is that these simpler techniques can perform surprisingly well. With that said, the paper would benefit from discussing the following points:\n\n1) Mixture models are effective in low dimensional problems but require the manual specification of the number of components. How was this done in the experiments and how sensitive the performance is to this parameter?\n\n2) The paper discusses several problems with normalising flows, but in particular the computational cost involved in generating samples or evaluating the density. This is true for some variations of NFs but not for all. For example RealNVP and the recent Neural Spline Flows are efficient in both, sample generation and density evaluation. With this in mind, the paper would benefit from further comparisons to these approaches. Another interesting comparison would be with autoregressive flows, such as inverse autoregressive flow or masked autoregressive flow. They can both  capture sequences and should be able to model inter-event times, instead of an RNN. \n\n  ", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "SJgwtqv3YH", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a new paradigm for learning models for point processes which circumvents the need to explicitly model the conditional intensity. This utilizes recent work on Normalizing flows and upends a long-standing paradigm. The paper is a true tour-de-force and the authors make a very convincing case for why instead of modelling conditional intensity, one could (and should) model the distribution of times directly.\n\nThe authors do an extensive literature review and place their work in the context very well. The writing is very lucid and the paper was a pleasure to read. The illustrations are also helpful and the extensive experiments complement the discussions very well. The appendix is also very easy to read and has been judiciously separated.\n\nThe two primary axes they justify their model is by first comparing it against conditional intensity-based methods in general and show that:\n 1. Their approach has universal approximation property (thanks to a result from DasGupta (2008)).\n 2. The likelihood function can be evaluated efficiently (thanks to it being a mixture of log-normals).\n 3. The expected next action time can be evaluated efficiently.\n 4. Sampling from their model is also easy. \n\nThen they show that all related work fails either one or more of these conditions. Finally, they also address the common concerns of why point processes were historically always taught using intensity functions. A bit of historical context here is that in absence of modern computers and when Cox-Hazard models still needed to be worked out by hand, the conditional intensity functions indeed were much simpler to handle and afforded desirable properties. However, with modern approaches towards modelling probability distributions, those reasons are no longer valid. Further, some of the recent research which has built on the conditional intensity functions has lost much of the advantages anyway because of the use of deep neural networks. Hence, at this point, one might as well move directly to modelling the probability (as the authors have done) instead of taking a detour via the intensity functions. The experimental results for prediction seem to justify taking this route as modelling intensity function does not seem to out-perform the intensity-free models (more on the metrics later). Finally, the authors also show that their model, thanks to its generative nature, can also be used for several ancillary tasks: Sequence embedding, missing data imputation and learning with conditional information. The provided code is easy to read through and execute.\n\nHence, I do not have any hesitation in recommending the paper for publication.\n\nAs a side-note, it is notable that in (Tabibian, 2016), a mixture model is proposed, but I believe it is not for the final probability but instead for the intensity itself to make the learning problem tractable.\n\n\nSome ways of improving the paper:\n\n - Theorem 1 can be made more rigorous by making the role of parameters like 'K' more explicit. e.g. does the theorem imply that there 'exist' K, \\mu_k, \\s_k, or does it say that for any K, \\mu_k, and \\s_k, such 'w_k' can be found, etc.?\n - Using NLL for comparing models in the experiment results is a bit unusual. Could a different metric (e.g. MAE) be employed?\n - A key discussion missing in the paper is that of the complexity of training the model. RMTPP, for example, can be trained very efficiently while Neural-Hawkes is difficult to train due to a Monte-Carlo sampling embedded in the calculation of the likelihood. Empirical results will also add to better placing the Intensity free method against other methods.\n\nCitations:\n\n Tabibian, Behzad, et al. \"Distilling information reliability and source trustworthiness from digital traces.\" Proceedings of the 26th International Conference on World Wide Web. International World Wide Web Conferences Steering Committee, 2017.", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "BJlCvHrG9H", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes to directly model the (conditional) inter-event intervals in a temporal point process, and demonstrates two different ways of parametrizing this distribution, one via normalizing flows and another via a log-normal mixture model. To increase the expressiveness of the resulting model, the parameters of these conditional distributions are made to be dependent on histories and additional input features through a RNN network, updating at discrete event occurrences.\n\nThe paper is very well written and easy to follow. I also like the fact that it is probably among the first of those trying to integrate neural networks into TPPs to look at directly modeling inter-event intervals, which offers a different perspective and potentially also opens doors for many new methods to come.\n\nI have just three comments/questions.\n\n1. The log-normal mixture model has a hyper-parameter K. Similarly, DSFlow also has K, and SOSFlow has K and R. How are these hyper-parameters selected? I don't seem to find any explanation in the paper (not even in appendix F.1)?\n\n2. To better demonstrate that a more complicated (e.g. multi-modal) inter-event interval distribution is necessary and can really help with data modeling, I'd be interested to see e.g. those different interval distributions (learnt from different models) being plotted against each other (sth. similar to Figure 8, but with actual learnt distributions), and preferably with some meaningful explanations as to e.g. how the additional modes capture or reflect what we know about the data.\n\n3. Even though the current paper mainly focuses on inter-event interval prediction, I think it's still helpful to also report the  model's prediction accuracy on marks in a MTPP. The \"Total NLL\" in Table 5 is one step towards that, but a separate performance metric on mark prediction alone would have been even clearer.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This submission proposes a new paradigm for modelling temporal point processes by using deep learning to learn to mix log-normal distributions in order to directly model the conditional distribution of event time intervals themselves.

Strengths of the paper:
-Introduces a new modelling paradigm that can lead to further research in this direction, for an important problem.
-Extensive experimentation validates the approach quantitatively.
-Easy to read.

Weaknesses:
-Several reviewers wanted more details on how the mixing parameter K was tuned. This was adequately addressed during the discussion period.

The reviewer consensus was to accept this submission.

Decision: Accept (Spotlight)