Meta Temporal Point Processes
Authors: Wonho Bae
Mohamed Osama Ahmed
Frederick Tung
Gabriel L. Oliveira
Venue: ICLR 2023 poster (2023)
Keywords: Temporal Point Process
Asynchronous Time Series
Meta-learning

Abstract: A temporal point process (TPP) is a stochastic process where its realization is a sequence of discrete events in time. Recent work in TPPs model the process using a neural network in a supervised learning framework, where a training set is a collection of all the sequences. In this work, we propose to train TPPs in a meta learning framework, where each sequence is treated as a different task, via a novel framing of TPPs as neural processes (NPs). We introduce context sets to model TPPs as an instantiation of NPs. Motivated by attentive NP, we also introduce local history matching to help learn more informative features. We demonstrate the potential of the proposed method on popular public benchmark datasets and tasks, and compare with state-of-the-art TPP methods.

Published as a conference paper at ICLR 2023
META TEMPORAL POINT PROCESSES
Wonho Bae
Mohamed Osama Ahmed Borealis Al mohamed. o a.meeeebrrrrlll com
University of British Columbia & Borealis AI whbaeêcs ubc.ca
Frederick Tung
Gabriel L. Oliveira
Borealis AI
Borealis AI
olo com
ABSTRACT
A temporal point process (TPP) is a stochastic process where its realization is a sequence of discrete events in time. Recent work in TPPs model the process using a neural network in a supervised learning framework, where a training set is a collection of all the sequences. In this work, we propose to train TPPs in meta learning framework, where each sequence is treated as a different task, via a novel framing of TPPs as neural processes (NPs). We introduce context sets model TPPs as an instantiation of NPs. Motivated by attentive NP, we also introduce local history matching help learn more informative features. We demonstrate the potential of the proposed method on popular public benchmark datasets and tasks, and compare with state-of-the-ar TPP methods.
INTRODUCTION
With the advancement of deep learning, there has been growing interest modeling temporal point processes (TPPs) using neural networks. Although the community has developed many innovations how neural TPPs encode the history of past events (Bilos et al. 2021 or how they decode these representations into predictions of the next event (Shchur et al. [0200] Lin et al.) (022). the general training framework for TPPs has been supervised learning where a model is trained on a collection of all the available sequences. However, supervised learning is susceptible to overfitting, especially in high noise environments, and generalization to new tasks can be challenging.
In recent years, meta learning has emerged adapt or generalize well on new tasks, which resembles how humans can learn new skills from a few examples. Inspired by this, end, we treat each sequence as a "task", since is realization of a stochastic process with its own characteristies. For instance, consider the pickup times of taxis in city. The dynamics of these event sequences are governed by many factors such as location, weather and the routine of taxi driver, which implies the pattern of each sequence other. Under the supervised learning framework, a trained model tends to capture the patterns seen training sequences well, but easily breaks on unseen patterns.
alternative to supervised learning it aims to
propose to train TPPs in meta learning framework. To this
be significantly different from each
As the goal of modeling TPPs is to estimate the true probability distribution of the next event time given previous event times, we employ Neural Processes (NPs), family of the model-based meta learning with stochasticity, to explain TPPs. In this work, we formulate neural TPPs as NPs by satisfying some conditions of NPs, and term as Meta TPP. Inspired by the literature in NP, we further propose the Meta TPP with cross-attention module, which we refer to as Attentive TPP. We demonstrate the strong potential of the proposed method through extensive experiments.
Our contributions can be summarized as follows.
To the best of our knowledge, this is the first work that formulates the TPP problem in meta learning framework, opening up a new research direction in neural TPPs.
Inspired by the NP literature, we present a conditional meta TPP formulation, followed by latent path extension, culminating with our proposed Attentive TPP model.

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Published as a conference paper at ICLR 2023
The experimental results show that our proposed Attentive TPP model achieves state-of- the-art results on four widely used TPP benchmark datasets, and is more successful in capturing periodic patterns on three additional datasets compared to previous methods.
We demonstrate that our meta learning TPP approach can be more robust in practical de- ployment scenarios such as noisy sequences and distribution drift.
2
PRELIMINARIES
Neural processes. A general form of optimization objective in supervised learning is defined as,
f" = arg max E B--(DD)
Σ [u.N)B
lo x)
(1)
12 -=== for imput z and label y. and B denotes a mini- matni-bbtchhhe set of (x, y) data pairs. Here, the goal is to learn a model ff parameterized by 0 that maps æ to as fo : x + y-
where D := (()))))))))))
In recent years, meta learning has emerged as an alternative to supervised learning it aims to adapt or generalize well on new tasks (Santoro et al., 2016), which resembles how humans learn new skills from few examples. In meta learning,
define meta dataset, set of different tasks, as M == {D(0) I.M. Here, ( is a dataset of i-th task consisting of a context and target set as D : C. The objective of meta learning is then defined
0- arg max E BBp(((
Σ [C,TEBD
log po (JT 1r (x
where Bn denotes match set of tasks. Also, My and V' represent inputs and labels of target set, respectively. Unlike supervised learning, the goal is to learn a mapping from x to y given C: more formally, C)
y. Although meta learning is a powerful framework to learn fast adaption to new tasks, it does not provide uncertainty for its predictions, which is becoming more important in modern machine learning literature metric to measure the reliability of a model.
To take the uncertainty into account for meta learning, Neural processes (NPs) have been proposed (Garnelo al. 2018h a). Instead of finding point estimators done in regular meta learning models, NPs learn a probability distribution of a label given an input x and context set C: po (z. C). In this work, we frame TPPs meta learning instead of supervised learning, for the first time. To this end, we employ NPs to incorporate the stochastic nature of TPPs. In Section 3.1 we propose simple modification of TPPs connect them NPs, which enables us employ rich line of works in NPs to TPPs as described in Section 3.2 and Section 3.3
Neural temporal point processes. TPPs are stochastic processes where their realizations are se- quences of discrete events in time. In notations, a collection of event time sequences is defined as D = (8( I!ol where (8) (7(()) (i) (() and L, denotes the length of i-th sequence. The history of studying TPPs started decades ago (Daeyy & Vere-Jones] 2003), but in this work, we focus on neural TPPs where TPPs are modeled using neural networks (Shchur al., 202T). ). As described in Figure h. general form of neural TPPs consists of an encoder, which takes sequence of previ- ous event times and outputs history embedding, and a decoder which takes the history embedding and outputs probability distribution of the time when the next event happens.
Previous works of neural TPPs are regressisyyy modeled in supervised learning framework. More formally, the objective of neural TPPs are defined as,
m Σ E=
0" mnx E (D)(D)
(3) Po (T411 (
where B ~ p(D) denotes batn-b of event time sequences. To frame TPPs NPs, need to define a target input and context set shown in Equation (2). from an event time history Tgi, which will be described in the following section.
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Amtesiinn Mask
Mask
1
Ti
f2
Encoder
Decoder
Encoder
T+i
(a) Neural TPP
Decoder
Weight
Atiention
Sharing
Misk
F2
22
Cross Attention
12
Encoder
Encoder
Decoder
Y--1
12-1
(b) Conditional Meta TPP
(c) Attentive TPP
Figure 1: Overall architectures of TPP models.
META TEMPORAL POINT PROCESS AND ITS VARIANTS
3.1 TEMPORAL POINT PROCESSES NEURAL PROCESSES
To frame TPPs as NPs, we treat each event time sequence as a task for meta learning, which intuitively makes sense since each sequence realization of a stochastic process. For instance, the transaction times of different account holders are very different from each other due to many factors including an account holder's financial status and characteristics.
With the new definition of tasks, we define a target input and context set for a conditional probability distribution of meta learning shown in Equation (2). using previous event times Tşt. There are many ways to define them but a target input and context set need to be semantically aligned since the target input will an element of the context set for the next event time prediction. Hence, we define target input for 7 as the latest "local history" 7- 1a where k: is the window size of the local history. Similarly, context set for 71+1 is defined as C := {Tr- )==) Here, if t k S O, we include event times from Ti. With Transformer structure, it is easy to efficiently compute the feature embeddings for the context set C. Figure Tp shows schematic of the Conditional Meta TPP with mask (shaded) used for an example case of 5 event times with a local history window size of k = 3. Then, the feature embedding Fi contains information of 71-++: With the notations for target inputs and context sets, we propose the objective of TPPs in meta learning framework as,
B -
f" = arg max E BB-p(D)
Σ
log Pogro(Ti T- cf
(4)
l=
(1) Note that have only target label 744 to predict per event unlike the general meta learning objective in Equation 2 where usually IT| > 1. It is because TPP models in general trained
to predict the next event time. Modeling TPPs to predict multiple future event times would be an interesting future work, but it is out of scope of this work.
X7 == (m:)l7 and )' = 177 a set of target inputs and labels, respectively, and be arbitrary permutation of a set. To design NP models, it required to satisfy the following two conditions.
Requirements for neural processes.
Condition 3.1 (Consistency
target set). probability distribution pe is consistent consistent under permutation: PD C), = l (XT)), C)C, and marginalization:
Po(yi-m C) = poly):n Xy, C) dyjm+inn for any positive integer
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Condition 3.2 (Permutation invariance over context set). po()'r (r, po (())
According to Kolmogorov extension theorem (Oksendal 2013). a collection of dintes-ameniin distributions is defined stochastic process if condition 3.1 is satisfied. In NP literature, condition 3.1 is satisfied through factorization: it assumes target labels are independent to each other given a larget input and a context set C, in other words, po (),,
I1!" po(ys I - Mec. C) Ro p n! [ xi, C) (Dubois et al. 2020). This assumption can be unrealistic if target labels are strongly dependent to previous target inputs and labels even after context representations are ob- served. It is. however, not necessary to assume factorization to make TPPs as NPs. As previously mentioned, we only care about predicting the next event time, which means [}' 1. When set contains only one element, its permutation is always itself. More formally, the consistency un- der permutation of Condition 3.1 in TPPs: pø (T M-K+1dd Ci) = po (1+11) [TT C.), is satisfied since (({++1}) = { and x ((TI-*+1a)) = { Also, the marginal- ization under permutation Condition 3.1 is satisfied as marginalization is not applicable for Po(T++1 T Ci) since the target label set T1+1 contains only one element.
Recall that NP models learn a probability distribution of a target label pe given a target input and context set. For computational efficiency (to make inference O[IC] + 177) time), the feature repre- sentation of should be invariant to the size of the context set, for which Condition 3.2 is required. To satisfy Condition 3.2, average-poo all the context features rı, ........ PI-1 to generate the global feature for a task G as shown in Figure [Tb and term as conditional Meta TPP following the terminology used in the NP literature. Each context feature rı, r2,".. : -T-1 represents a feature from a transformer encoder such as Transformer Hawkes Processes (THP), that encodes the corre- sponding local history of the context set C. For instance, r; contains information of T. To make r; only encode the subset of previous event times 1 (instead of the whole previous event times TsJ), we mask out events that are outside of the local history window using an attention mask as shown in Figure 1(b) and (c), which is different from regular attention mask shown in Figure (a). Using the permutation invariant feature G not only satisfies Condition 3.2, but also lets the de- coder approximate the probability distribution of a target label given both target input and context set instead of just a target input. Now that we satisfy both requirements with a new architectural design, we can treat TPPs as NPs.
Implementation. It can be expensive to compute the individual context feature r; for all 1 S t << I from each element of the context set Tt-k++1:t E Cr: the time complexity of computing all the context features for a sequence is OLL)) Instead of passing each element of a context set, using the Transformer architecture (Vaswani et al,, 2017), context features at once, of which time complexity is OkLL) where & is the window size of a local history. To this end, we employ the THP the encoder. Please refer to Zuo al. (0200) for details.
simply pass the event times to obtain all the
3.2 META TEMPORAL POINT PROCESS
In the NP literature, NPs are generally modeled as latent variable models. Instead of using the deterministic global feature G as an input to the decoder (Garnelo et al. (2018)). latent variable is sampled from a probability distribution e.g. multi-variate Gaussian, using parameters inferred from the global feature G (Garnelo et al. 2018b). As intractable to compute the log-likelihood for a latent variable model, amortized variational inference (VT) can be used to approximate inference.
In the setting of TPPs, the evidence lower bound (ELBO) of variational inference with an inference network (= Cz) can be derived as,
log TT-441 Cz) = log f po(Ti T- z)po(= Cz)dz
(5)
> E. [logpo(T<a (oo0o/t411---ae 7- =)] K L(poo CL) [=]))]
(6)
po T- KLppo Cz.) po (c))
where N denotes the number of samples of 2 ~ p0 (Cz) for Monte-Care approximation. Here, po(2 Ozz) is the posterior given the context at the last (L-th) event, which contains all the events of the sequence s (it is accessible in training time). Minimizing K L--iveeree between po C,.) and (a Ci) is to make the global latent variable inferred from C₁ be similar to the latent
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variable of sequence from Cz, in training time. To sample 2, we use the reparameterization trick as 2 = μ a @ e where ~ [0, I] as described in the latent path Figure D In inference, approximate evaluation metrics such as negative log- lilkkiihhodd or root mean squared error using Monte-Carke samples. But, as we do not have access to C₁. at I-th event when < I., we use from po(= Ci). The detailed description of the evaluation metrics provided in Appendix C
An advantage of a latent variable model is that it captures stochasticity of functions, which can be particularly beneficial to model TPPs since TPPs are stochastic processes. Experiments in Sec- tion 5.4 demonstrate it indeed helps to model TPPs over the deterministic case. In particular, it is robust noises (Section (..2). We term the latent variable model as Meta TPP throughout the paper.
Discussion. The existing TPP models treat all event sequences as realization of the same process whereas the Meta TPP treats each sequence as realization of distinct stochastic process. We achieve this by conditioning on the global latent feature that captures task-specific characteristics. For to be task specifi,,,,,,, it has to be distinct for different sequences but similar throughout different events / E [1, L 1J within the same sequence. It is natural for the global features to be distinct by sequence but we need further guidance to make the global feature shared across all the event times in sequence. Due to the permutation invariance constraint implemented in average-pooling, cannot be very different at different event time: adding some addition context feature ry will not change G as well as much. In addition, the KL-divergence between po Ccz and (2 |Cr) further enhances the task-specific characteristics of 2. We provide more detailed discussion in Appendix B
3.3 ATTENTIVE TEMPORAL. POINT PROCESS
Early works in NPs suffered from the underfitting problem. To alleviate this, Kim et al. (2019) proposed AttentiveNP, which explicitly attends the elements in a context set to obtain better fea- ture for target inputs. Inspired by this, add cross-attention module that considers the sim- ilarity between the feature of a target input and previous event times as described Figure [ Given the local history (context) features r1, P2......- at l-th time step. the key-query- pairs K E R1-1xD :q E IRIDD, and V E RR for the cross-attention, are computed using their corresponding projection weights WK E DRDx WQ E RDxD as,
K = R. WK. T T Wo, V = R where R =
(8)
Here, K corresponds to [k1, k 1=11"" in Figure De. The feature of i-th attention head h, are then computed follows,
h4 Softmar(q KT /VDD V.
With W E HDDD and some fully connected layers denoted as FC, r E RRID is computed as,
ri FC [h1, he. .., hu] W.
(10)
Finally, the decoder takes the concatenated feature of 22 and r/ as input infer a distribution.
In the TPP setting, it is common that there are multiple periodic patterns in the underlying stochastic process. The cross-attention module provides an inductive bias model that the repeating event subsequences should have similar features. Our experiments in Section 5.2 demonstrate that the explicit attention helps to model TPPs in general, especially when there are periodic patterns.
Decoder. The decoder takes the concatenated feature of the global latent feature 2, target input feature ri that encodes T1- and attention feature r' from the attention module. For the Meta TPP (without the attention module), the decoder takes as input the concatenated feature of = and ri- Here, 2, Tı, and r' are all D-dimensional vectors. The decoder consists of two fully connected layers, and the input and hidden dimension of the decoder layers are either 2D or 3D depending whether we use the feature from the attention module r'.
The decoder outputs the parameters of the probability distribution of the next event time or Po(T++1 Tt-b+1:1s 2.). Inspired by the intensity- TPP (Shchur et al., 2020), we use a mix- ture of lor-normal distributions model the probability distribution. Formally, for I [1, - 1), T41 ~ Log oormmppp (1 0 w)+1) 1) where 44+1 are the mixture means, 0 are the stan- dard deviations, and w1+1 the mixture weights.
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4 RELATED WORK
Neural temporal point processes. Neural temporal point processes (NTPPs) have been proposed to capture complex dynamics of stochastic processes in time. They are derived from traditional
temporal point processes (Hawkes) 1971 Isham & Westcoti 1979 Daley Vere-Jones 2003).
Models based on RNNs are proposed by (Du et al., 2016) and Mee & Eisnerr 2017) to improve
NTPPs by constructing continuous-tima RNNs. More recent works use Transformers to capture
long-term dependency (Kumar et al. 2019 /hhng et al. [2200] Zuo et al. 2020) Yang et al.) 2022]
Zhang et al. 2022). (Omi et al. 2019 Shehur et al. [2020] Sharma et al. 2021 propose intensity-free
NTPPs to directly model the conditional distribution of event times.
Omi et al. 2019) propose to model a cumulative intensity with a neural network. But, it suffers from problems that the probability density is not normalised and negative event times receives non-zero probabilities. Alternatively, (Scccur et al. (2020) suggest modelling conditional probability density by log-normal mixtures. Transformer- based models like Zuo et al. (2020); Zhang et al.) (2020) propose to leverages the self-attention mechanism to capture long-term dependencies. Another class of TPP methods called Neural Flows (Biloš al., 2021). proposed to model temporal dynamics with ordinary differential equations learned by neural networks. Unlike the previous TPP methods, frame TPPs as meta learning (not supervised learning) for the first time.
Neural processes. Meta learning is a learning framework that aims to adapt or generalize well on new tasks. There are three approaches in meta learning: metric-based (Koch et al. 2015 Vinyals it al. 2016 Sung et al. [2018 Snell et al., 2017). model-based (Santoro et al. 201 Munkhdalai & Yu 2017 Grant et al., 2018) and aptimization-basee (Finn et al.] 2017 2018 Nichol et al. 2018).
Neural processes (NPs) is the model-based meta learning with stochasticity. Garnelo et al. (2018a) propose conditional neural process as new formulation approximate stochastic process us- ing neural network architecture. It succeeds the advantage of Gaussian Processes (GPs) as it can estimate the uncertainty of its predictions, without having expensive inference time. Garnelo et al. (0188) generalize conditional neural process by adding latent variables, which are approximated using variational inference. Although NPs can adapt to new tasks quickly without requiring much computation, it suffers from underfitting problem. To alleviate it, Kim al. (2019) propose cross- attention module, which explicitly attends the elements in the context set to obtain better representa- tions for the elements in the target set. As another way to address the underfitting problem, Gordon itt al. (2020) propose a set convolutional layer under the assumption of translation equivariance of inputs and outputs, which expanded the latent variable counterpart in FFonn et al. (2020).
Transformer NP (Nguyen & Grover 2022 is the most relevant work to ours. Although it also models event sequences, focuses on modeling regular time series: discrete and regularly-spacee time inputs with corresponding label values. TPPs are different as they are continuous and irregularly- spaced time sequences not necessarily with corresponding label values.
EXPERIMENTS
5.1 EXPERIMENT SETTING
Datasets. To compare the effectiveness of models, we conduct experiments on 4 popular benchmark datasets Stack Overflow, Mooc, Reddit, and Wiki, and datasets with strong periodic patterns we introduce - Sinusoidal wave, Uber, and NYC Taxi. Please refer to Appendix H for details.
Metrics. We use the root mean squared error (RMSE) as the main metric along with the negative log-likelihood (NLL.) as a reference since NLL can go arbitrary low if probability density is placed mostly on the ground truth event time. RMSE may not a good metric, either, if one ignores stochastic components of TPPs and directly trains a bascline on the ground truth event times to obtain point estimations of event times (Shchur et al.) 2021). We train all the methods on NLL and obtain RMSE in test time not abuse RMSE scores, keeping stochastic components of TPPs. For marked TPP datasets, we extend the proposed method to make class predictions, and report accuracy. For details about marked cases, please refer to Appendix G
Baselines. We use intensity- TPP (Shchur et al. 2020). Neural flow (Bilos et al. 202 and Transformer Hawkes Processes (THP) (Zuo et al. 2020) as baselines. For intensity-free TPP and

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Stack Overflow
Mooc
Roddit
Wiki
Methods
RMSE
NLL
Ace
RMSE
NLL
Acc
RMSE
NLL
Ace
RMSE
NLL
Acc
3.64
3.66
0.43
0.31
0.94
0.40
0.18
1.09
0.60
0.60
7.76
0.26
Intensityffree
(0.26)
(0.02)
(0.005)
(0.006)
(0.03)
(0.004)
(0.006)
(0.04)
(0.008)
(0.05)
(0.40)
(0.03)
0.47
0.43
0.30
0.32
1.30
0.60
0.56
11.55
0.05
Neural flow
(0.006)
(0.02)
(0.04)
(0.04)
(0.33)
(0.07)
(0.05)
(2.22)
(0.01)
1.68
3.28
0.46
0.18
0.13
0.38
0.26
1.20
0.60
0.17
6.25
0.23
THP+
(0.16)
(0.02)
(0.004)
(0.005)
(0.02)
(0.004)
(0.005)
(0.04)
(0.007)
(0.02)
(0.39)
(0.03)
1.15
2.64
0.46
0.16
-0.72
0.36
0.11
0.03
0.60
0.15
6.25
0.25
Attentive TPP
(0.02)
(0.02)
(0.004)
(0.004)
(0.02)
(0.003)
(0.002)
(0.04)
(0.007)
(0.01)
(0.38)
(0.03)
Table 1: Comparison of the Attentive TPP to the tfute-of-the-arr methods on a bootstrapped test sets.
Sinusoidal
Uber
NYC Taxi
Methods
RMSE
NLL
RMSE
NLL
RMSE
NLL
Intensity-free
1.29 (0.08) 0.88 (0.02) 51.23 (2.89) 4.46 (0.02)
46.59 (26.16) 2.06 (0.07)
Neural flow
1.13 (0.07) 0.99 (0.02)
THP
1.72 (0.10) 0.84 (0.02) 90.25 (4.53) 3.63 (0.03) 10.31 (0.47) 2.00 (0.01)
Attentive TPP (Ours)
1.45 (0.11) 0.66 (0.02) 22.11 (1.94) 2.89 (0.04) 8.92 (0.42) 2.00 (0.009)
Table 2: Experiment results on bootstrapped test sets with strong periodic patterns.
neural flow, we add the survival time of the last event to NLL and fix some bugs specified in their public repositories. THP and its variants are originally based on intensity: they predict intensities from which log-like loglliiel and expectation of event times are computed. It is, however, computa- tionally expensive to compute them as requires to compute integrals: especially, to compute the expected event times, it requires to compute double integrals, which can be quite expensive and complex to compute even with thinning algorithms described in Mei & Fisner (2017). To work around it without losing performance, we add the mixture of log-normal distribution proposed in (Shchur et al., 2020) as the decoder, and we call it THP For fair comparison, we fix the number of parameters of the models in between 50K and 60K except the last fully-connected layer for class predictions since it depends on the number of classes
Hyperparameters. We grid-search every combination of dataset and method for learning rate E {0.01, 0.001, 0.0001, 0.00001} and weight decay E {0.01, 0.001, 0.0001, 0.00001} for fair com- parison. We bootstrap for 200 times test sets obtain the mean and standard deviation (in parentheses) for the metrics in Figure 2a and Table 1-3 following Yang et al. (2022). All the other hyperparameters are fixed throughout the experiments, and are reported in Appendix [ Our imple- mentation is publicly available at
/tt.alatsiiimaiaaaaaaaaaaaaaaaaa / tpp
5.2 EXPERIMENT RESULTS
In this section, we begin by comparing our attentive meta temporal point process (denoted as Atten- tive TPP) with state-of-the-art supervised TPP methods on popular benchmarks. We then inves- tigate how Attentive TPP captures periodic patterns, and show how Attentive TPP can be used to impute missing events in noisy sequences. Finally, we consider robustness under distribution drift.
Comparison with state-of-the-art methods. Table summarizes our comparison of Attentive TPP with -ath--ofthhe-rt baselines intensity-free (Shchur et al. 2020), neural flow (Biloš et al. 2021 and THP+ (uoo et al. 2020) on the Stack Overflow, Mooc, Reddit, and Wiki benchmarks. THP generally performs better than the intensity-fres and neural flow baselines. Attentive TPP further improves over THP+ on all datasets and metrics except for mark accuracy on Mooc and Wiki.
Neuual flow results on Uber, NYC Taxi and Stack Overflow (in Table datasets are missing because the official implementation runs into NaN values for long event sequences in inversion step.
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THP
TPP
Meta
140
120
40
100
RMSB
30
RMSE
80
10
30
40
50
60
70
Jan-Feb Mar-ApeMay-Jon J Jul-Aug Sep-Oct Nox-Dec
Drop ratio (%)
Target domains
(a) Imputation with different drop rates
(h) Distribution drift in NYC Taxi dataset
Figure 2: Experiment results on imputation and distribution drift.
Reddit
Uber
Reddit
Attention
Latent
RMSE
NLL
Acc
RMSE
NLL
Methods
# Params
RMSE
NI.L
Acc
x
x
0.16
0.92
0.59
63.71
3.68
THP
113K
0.26
1.19
0.60
x
0.13
-0.39
0.61
63,35
170K
0.29
0.79
0.59
0.29
47.91
226K
0.28
1.44
0.57
0.12
0.07
0.60
21.87
AtnTPP
222K
Table Comparison of the variants of Meta TPPs
Table 4: Comparison of diff. model size
Periodic patterns. As previously mentioned Section 3.3 the cross-attention module is designed to capture periodic patterns by matching the local history of the current event to the local histories of the previous event times, in addition to alleviating the underfitting problem. To validate the effec- tiveness of the cross-attention, we experiment the datasets with strong periodicity Sinusoidal, Uber, and NYC Taxi (please refer to Appendix H for details). Table 2 shows that the Attentive TPP generally outperforms the state-of-the-ar methods, except for RMSE on Sinusoidal. To investigate the behavior of the cross-attention, we provide an example in Figure 3a where we highlight 15 (out of 64) the most attended local history indices (in red) to predict the target event (in green) in a se- quence from Sinusoidal. The dotted grey lines represent the start and end of periods. We can observe that the attention refers to the local histories with similar patterns more than the recent ones.
5.3 APPLICATIONS
Imputation. We study the robustness of Meta and Attentive TPP to noise by randomly dropping events, simulating partial observability in noisy environment, and measuring imputation perfor- mance. For the experiment, we drop n percentage of all the event times drawn independently at random per sequence the Sinusoidal putation performance of THP+, Meta TPP, and Attentive TPP, in terms of RMSE. As the drop ratio increases, RMSE increases for all three models but the gap exponentially increases. Given that the performance gap between three models on 'next event' predictions is not as large (mean RMSE THP*: 1.72, Meta TPP: 1.49, Attentive TPP: 1.45), the results shown in Figure 2a imply that the Meta and Attentive TPP are significantly more robust to the noise coming from partial observability.
dataset. In Figure 2a, report the bootstrapped im-
Distribution drift. Distribution drift occurs when the distribution observed during training becomes misaligned with the distribution during deployment due to changes in the underlying patterns over time. This is common deployment challenge in real-world systems. Figure 2b shows how THP+ and Meta TPP models trained on the January-February data of the NYC Taxi dataset generalize to subsequent months. Both models show decrease in performance, suggesting the presence of non- stationary or seasonal patterns in the data that are not captured in the training months; however, Meta TPP is comparatively more robust across all oft-of-domain settings. is also worth mentioning that although the Attentive TPP generally performs better than Meta TPP in the conventional experimen- tal setting, it is not the case for distribution drifts. We conjecture it is because the cross- cross-attention is designed to alleviate the underfitting problem, which results in being less robust distribution drift.
8

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Published as a conference paper at ICLR 2023
Altwatitve TPP
Targetx
Most attended events
Target event
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(a) Self-attention in THP and proposed cross-attention
(b) Visualization of test predictions vs. targets
Figure 3: Qualitative analysis on the cross-attention and prediction results.
5.4 ABLATION STUDIES
Meta TPP and its variants. In Table we compare the proposed Meta TPP and its variants on Reddit, and Uber datasets. The result shows that both cross-attention and latent variable path generally help to improve the performance. When they are combined (resulting in the Attentive TPP). it generally performs the best in terms of both RMSE and NLL.
Different model sizes. Both latent path and cross-attention components introduce additional learn- able parameters (for the Reddit dataset with 984 classes, THP+: 113K, Meta TPP: 126K, and Atten- tive TPP: 209K parameters). We provide an ablation study with varying number of model parameters for the THP baseline validate the performance improvement does not come from the increased number of model parameters. In Table we increase the number of model parameters for THP+ on Reddit along with the result of the Attentive TPP. The result shows that the larger model does not necessarily help to improve the performance: times improves but may hurt RMSE as in the case of Table The significant improvement in performance of our proposed method shows the importance of providing effective inductive bias.
the number of parameters increases, NLL some-
Parameter sharing. In Attentive NP, the encoders of the latent path and attention path are separated to provide different features. However, it can significantly increase computational overhead, and for this reason, we share the weights for the encoders. As an ablation study, we provide the performance with and without sharing the weights for the encoders of the Attentive TPP on the Stack Overflow dataset. Although the number parameters of 'with sharing' is 50% less than 'without sharing' ('without sharing': : 86K vs. 'with sharing': : 136K), it performs better than 'without sharing' (RMSE / 'without sharing': 1.08/ 3.20 vs. 'with sharing": 1.03 2.81).
Visualization of event time predictions. In TPP literature, the evaluation relies only the RMSE and NLL metrics. It is, however, often hard to measure how practically useful a trained TPP model is. To qualitatively evaluate TPP models, we convert an event time sequence into time series sequence: we count the number of event times falling into each bin (in Figure 3b each bin is a day). Figure 3b shows how close overall predictions of the Attentive TPP and THP (in red) are the ground truth event times (in blue). In the figure, we can see that the Attentive TPP's predictions closely align with the targets whereas the predictions of the THP+ are off at some regions. Note that as the y-axis represents bin counts, even slight off from the ground truth implies large values in terms of RMSE.
6 CONCLUSION
Previously, neural temporal point processes (TPPs) train neural networks in a supervised learning framework. Although performing well on event sequences similar to a training set, they are sus- ceptible to overfitting, and may not generalize well on sequences with unseen patterns. To alleviate this, we proposed novel framework for TPPs using neural processes (NPs), We proposed the Meta TPP where TPP is formulated as NP, and further developed the Attentive TPP using a cross-attention module, which forces a model to use similar features for repeating local history patterns. Our exper- iments demonstrate that the proposed models outperform strong state-of of-te-of-theaant thethh-art baselines on several event sequence datasets, effectively capture periodic patterns, and increase robustness to noise and distribution drift. We believe this work opens up new research direction to meta learning for TPPs.

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Published as a conference paper at ICLR 2023
REPRODUCIBILITY STATEMENT
Hyperparameters and implementation details are available Section 5.1 and Appendix The code for the baselines and the proposed Meta TPP is provided in meta-tpp repo Our code based publicly available official intensity-free and neural flow code.
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Reviews:
[{"review_id": "goHZxksFeab", "reviewer": "Reviewer_Jm5S", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "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": "The paper proposes a meta-learning framework for temporal point processes (TPPs). The paper points out a limitation of the existing TPP models - treating all event sequences in the dataset as realization of the same process. The proposed meta-learning approach addresses this limitation and leads to improved performance compared to other neural TPP models, as shown by the experiments. More specifically, the proposed model uses a cross-attention encoder (based on [attentive neural process](https://arxiv.org/abs/1901.05761)) to extract the necessary information for predicting the tiem of the next event in the sequence.", "summary_of_the_review": "The proposed approach is well-motivated and achieves strong empirical results. However, some important aspects of the model are not described very clearly.", "strength_and_weaknesses": "- Strong performance: The proposed approach shows a consistent improvement over the baselines in terms of event prediction capabilities.\n- Extensive empirical evaluation: The experiments cover a large selection of datasets and tasks. These include both standard results on predictive performance (event time prediction, log-likelihood), as well as experiments designed to shed light on the properties of different methods (effect of missing events, distribution shifts on performance; effect of model size)\n\n\nWeaknesses:\n- Clarity: Some aspects of the proposed approach are hard to understand or are not described in full detail. The paper would benefit greatly if these are clarified in the revised version.\n    - It would be helpful to explain the mapping between the variables $\\mathcal{X}, \\mathcal{Y}$ in the meta-learning problem definition and $\\tau_i$ in the TPP problem definiton early in the paper, and then stick to the latter notation. Currently, section 3.1 is somewhat hard to follow with the switches between two notations.\n    - How are the context features $r$ defined for TPP models?\n    - Most notation in figure 1 ($r$, $G$) is only introduced a few pages later in the paper, making it hard to interpret the figure the first time it's encountered.\n    - What is the interpretation of the global feature $G$ and the latent variable $z$ in the context of TPP modeling (section 3.2)?\n    - Experiments in table 1 include accuracy scores (probably, for mark prediction?), but the rest of the paper only discusses unmarked TPPs.\n    - $G$ denotes the context set in the beginning of page 4 instead of the $\\mathcal{C}$ notation.\n    - How is the NLL computed for the proposed Attentive TPP model? Equation 5 seems to imply that the NLL is intractlible, and only ELBO is available.\n    - How exactly are the histograms in figure 3 obtained? As far as I understand, the models produce a one-step-ahead prediction of the inter-event times. How are these converted to the histograms? Do we predict the next inter-event time $\\tau_{i+1}$ and then feed in the actually observed value of $\\tau_{i+1}$ to then generate the prediction for $\\tau_{i+2}$, etc.?\n\n----\nPost-rebuttal update: The authors have addressed all of my concerns and updated the paper accordingly. I have raised my score to reflect this.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Sy2qtIXlUSE", "reviewer": "Reviewer_dU5L", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "This work propose to train TPP in a meta learning framework and treat each sequence as a different task via the neural process. The work defines the context dataset, target input and output for TPP model. Also, the work introduces the cross-attention module to introduce local history matching to learn more informative features. \n", "summary_of_the_review": "See strength and weakness. ", "strength_and_weaknesses": "Strength: Learning TPP in a meta learning framework is interesing and the submission provide the design of the model and training procedure. In general, this paper is easy to follow and I did not find any obvious technical errors despite of some notation problems. \n\nWeakness: The submission provide a detailed description of the encoder part and the training procedure, but almost no description of the decoder. For example, what is the structure of the decoder? How to implement the decoder? If the output of the decoder is the distribution of the next event time or inter-event interval, what distribution did you use? and how did you parameterize the distribution with your decoder? All questions above are not answered in the submission. Furthermore, the decoder takes a random latent variable Z as input, does that mean the output distribution of the decoder is also random? If so, this kind of design contradicts with the TPP model that when the historical information is fixed, the distribution of the next event time or inter-event interval should be determinnistic. \n\nThe design of experiments is also problematic. As stated in the introduction, the goal of meta learning is to improve generalization. To make the experiments convincing, you should demonstrate the meta-learning TPP is better than existing methods on generalization, e.g., training log-likelihood v.s. test log-likelihood. The curret experiment section even did not describe how to split traning and test dataset; and looks more like the submission proposed a more expressive model with lower RMSE or NLL in Table 1,2,3. In conclusion, the experiments did not convince the statement made in the introduction. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "RzMNqD3-kV", "reviewer": "Reviewer_YAR9", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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": "The authors suggest viewing the problem of predicting the next event for temporal point process models as a meta-learning problem. In particular, they advocate a neural process framework, whereby windows of previous event times act as context and target input sets. A cross-attention architecture is suggested to increase modeling capacity that does not scale quadratically in the number of events. The method seems to outperform supervised point process models empirically. \n\n\n----\nUpdate following authors' responses:\nThe presentation and clarity has been improved in the updated manuscript. The authors have also included additional experimental work that improved the experimental validation of the proposed method. I have increased my score to a weak accept.\n\n---", "summary_of_the_review": "Casting predictions with point process as a neural process is new, as far as I am aware. I feel there is scope to improve the presentation and clarity and I would consider increasing my score if this were addressed.", "strength_and_weaknesses": "Strengths:\n- The meta-learning framework is new for point process models, as far as I am aware. \n- The method appears to perform better than point process models trained in a supervised manner.\n\nWeaknesses:\n- I feel that the presentation could be improved at some points (see below). \n\nActionable feedback:\n- The equations (5) to (7) are a bit unclear to me. Why are we using a variational lower bound based on an expectation with respect to $p_{\\theta}(z|C_{L})$? Is this still a valid lower bound if we take (as I understand you do, while dropping the KL term?) using samples from $p_{\\theta}(z|C_{l})$?\n- How are the context features $r_1, r_2,$ … computed/encoded?\n- How exactly does z enter the decoder. What are the priors for z given different context sets? \n\nComments:\n- Can this be extended to multi-dimensional/marked point process setting where there is not just a single task to predict?\n- I was wondering if the presence of periodic patterns in the data somewhat makes the required permutation invariance over a context set a constraint that might decrease the forecasting performance?\n- Can this method be applied to meta-learning short sequences, e.g., setups like in Xie et al. Meta Learning with Relational Information for Short Sequences, Neuirps 2019?\n\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "1ULe8hGG8Z", "reviewer": "Reviewer_AxMo", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "In this work, the authors propose to use a meta learning way to train neural processes aiming at one-step prediction of event time of point processes. The proposed TPPs employ a latent variable layer and attention mechanism to improve the prediction performance. The method was demonstrated with benchmark datasets.", "summary_of_the_review": "This work uses a meta learning to train variant TPP aiming at one-step prediction of event time of point processes. The demonstration with a variety of benchmark datasets shows competitive performance against other SOTA methods. There are a few key concepts and details need to be clarified to make the work a better understandable and solid one.", "strength_and_weaknesses": "Strength:\nUse meta learning training to overcome the generalization issue of supervised learning.\nThe latent variable layer incorporates uncertainty.\nThe attention takes into account the similarity between inputs.\n\nWeaknesses:\nThe meta objective (4) turns out a special case of supervised learning (3). The distinction is blurry.\nThe writing needs to be improved.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster