Meta Learning with Relational Information for Short Sequences

09/04/2019 ∙ by Yujia Xie, et al. ∙ Georgia Institute of Technology Florida Atlantic University 22

This paper proposes a new meta-learning method -- named HARMLESS (HAwkes Relational Meta LEarning method for Short Sequences) for learning heterogeneous point process models from short event sequence data along with a relational network. Specifically, we propose a hierarchical Bayesian mixture Hawkes process model, which naturally incorporates the relational information among sequences into point process modeling. Compared with existing methods, our model can capture the underlying mixed-community patterns of the relational network, which simultaneously encourages knowledge sharing among sequences and facilitates adaptive learning for each individual sequence. We further propose an efficient stochastic variational meta expectation maximization algorithm that can scale to large problems. Numerical experiments on both synthetic and real data show that HARMLESS outperforms existing methods in terms of predicting the future events.



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1 Introduction

Event sequence data naturally arises in analyzing the temporal behavior of real world subjects (Cleeremans and McClelland, 1991). These sequences often contain rich information, which can predict the future evolution of the subjects. For example, the timestamps of tweets of a twitter user reflect his activeness and certain state of mind, and can be used to show when he will tweet next time (Kobayashi and Lambiotte, 2016). The job hopping history of a person usually suggests when he will hop next time (Xu et al., 2017b). Unlike usual sequential data such as text data, event sequences are always asynchronous and tend to be noisy (Ross et al., 1996). Therefore specialized algorithms are needed to learn from such data.

In this paper, we are interested in short sequences, a type of sequence data that commonly appears in many real-world applications. Such data is usually short for two possible reasons. One is that the event sequences are short in nature, such as the job hopping history. Another is the observation window is narrow. For example, we are interested in the criminal incidents of an area after a specific regulation is published. Moreover, this kind of data usually appears as a collection of sequences, such as the timestamps of many user’s tweets. Our goal is to extract information that can predict the occurrence of future events from a large collection of such short sequences.

Many existing literature considers medium-length or long sequences. They first model a sequence as a parametric point process, e.g., Poisson process, Hawkes process or their neural variants, and apply maximum likelihood estimation to find the optimal parameters

(Ogata, 1999; Rasmussen, 2013). However, for short sequences, their lengths are insufficient for reliable inference. One remedy is that we treat the collection of short sequences as independent identically distributed realizations of the same point process, since many subjects, e.g., Twitter users, often share similar behaviors. This makes the inference manageable. However, the learned pattern can be highly biased against certain individuals, especially the non-mainstream users, since this method ignores the heterogeneity within the collection.

An alternative is to recast the problem as a multitask learning problem (Zhang and Yang, 2017) – we target at multi-sequence analysis for multi-subjects. For each sequence, we consider a point process model that slightly deviates from a common point process model, i.e., , where is the common model that captures the main effect, is the model for the -th sequence, and is the relatively small deviation. Such an assumption that there exists a universal common model cross all subjects, however, is still strong, since the subjects’ patterns can differ dramatically. For example, the job hopping history of a software engineer and a human resource manager should have distinct characteristics. Furthermore, such method ignores the relationship of the subjects that usually can be revealed by side information. For example, a social network often shows community pattern (Girvan and Newman, 2002) – across the communities the variation of the subjects is large, while within the communities the variation is small. The connections in the social network, such as "follow" or retweet relationship in Twitter data, can provide us valuable information to identify such community pattern, but the aforementioned methods do not take into account such understanding to help analyzing subjects’ behavior.

To this end, we propose a HAwkes Relational Meta LEarning method for Short Sequence (HARMLESS), which can adaptively learn from a collection of short sequence. More specifically, in a social network, each user often has multiple identities (Airoldi et al., 2008). For example, a Twitter user can be both a military fan and a tech fan. Both his tweet history and social connections are based on his identities. Motivated by above facts, we model each sequence as a hierarchical Bayesian mixture of Hawkes processes – the weights of each Hawkes process are determined jointly by the hidden pattern of sequences and the relational information, e.g., social graphs.

We then propose a variational meta expectation maximization algorithm to efficiently perform inference. Different from existing fully bayesian inference methods

(Box and Tiao, 2011; Rasmussen, 2013; Xu and Zha, 2017), we make no assumption on the prior distribution of the parameters of Hawkes process. Instead, when inferring for the Hawkes process parameters of the same identity for all the subjects, we perform a model-agnostic adaptation from a common model for this identity (Finn et al. (2017), see section 3 for more details). This is more flexible since it does not restrict to a specific form. We apply HARMLESS to both synthetic and real short event sequences, and achieve competitive performance.


: Throughout the paper, the unbold letters denote vectors or scalars, while the bold letters denote the corresponding matrices or sequences. We refer the

-th entry of vector as . We refer the -th subject as subject .

2 Preliminaries

We briefly introduce Hawkes Process and Model-Agnostic Meta Learning.

Hawkes processes (Hawkes, 1971) is a doubly stochastic temporal point process with conditional intensity function defined as

where , is the nonnegative impact function with parameter , is the base intensity, and are the timestamps of the events occurring in a time interval . Function indicates how past events affect current intensity. Existing works usually use pre-specified impact functions in parametric form, e.g., the exponential function in Rasmussen (2013); Zhou et al. (2013) and the power-law function in Zhao et al. (2015).

Hawkes process captures an important property of real-world events – self-exciting, i.e., the past events always increase the chance of arrivals of new events. For example, selling a significant quantity of a stock can precipitate a trading flurry. As a result, Hawkes process has been widely used in many areas, e.g., behavior analysis (Yang and Zha, 2013; Luo et al., 2015), financial analysis (Bacry et al., 2012), and social network analysis (Blundell et al., 2012; Zhou et al., 2013).

Model-Agnostic Meta Learning (MAML, Finn et al., 2017) considers a set of tasks , where each of the tasks only contains a very small amount of data which is not enough to train a model. We want to exploit the shared structure of the tasks, to obtain models that can perform well on each of the tasks. Specifically, MAML seeks to train a common model for all tasks. From optimization perspective, MAML solves the following problem,


where is an operator,

is the loss function of task

, is the parameter of the common model, and is the step size. Here, represents one or a small number of gradient update of . For example, in cases of one gradient step, we take . This optimization problem aims to find the common model that is expected to produce maximally effective behavior on that task after performing update .

Solving (1) using gradient descent involves computing the Hessian matrices, which is computationally prohibitive. To alleviate the computational burden, First Order MAML (FOMAML) (Finn et al., 2017) and Reptile (Nichol et al., 2018) are then proposed. FOMAML drops the second order term in the gradient of (1

). Reptile further simplifies the computation by relaxing the original update with Hessian as a multi-step stochastic gradient descent updates. All three algorithms can be written in the form of (

1) with operator defined differently for different methods. Due to space limit, we defer the definition of to Appendix B.

3 HAwkes Relational Meta LEarning for Short Sequences (HARMLESS)

Figure 1: Probabilisitic graph of the suggested model. The yellow nodes are parameters, white nodes are latent variables, and the gray nodes are observed variables. The solid arrows represent probabilistic mapping, while the hollow arrows represent the deterministic mapping.

We next introduce the meta learning method for analyzing short sequences. Suppose we are given a collection of sequences . We also know some extra relational information about the subjects. For example, in social networks, we can have information on who is friend of whom; in criminal data, we have the locations of the crimes, and crimes happen near each other often have Granger causality. Such relational information can be described as a graph , where is the node set, is the edge set. Denote its adjacency matrix as .

Such social graphs often exhibit community patterns (Girvan and Newman, 2002; Xie et al., 2013). Within the communities the variation of subjects are small, while across the communities the variation is large. Moreover, the communities are overlapping with each other, i.e., each subject may belong to multiple communities and thus have multiple identities. The behaviors of the subject is based on the identities. Motivated by this observation, we first assign each subject a sum-to-one identity proportion vector , whose

-th entry represents the probability of subject

having the -th identity. In this way, we associate each subject with multiple identities rather than a single identity so that its different aspects is captured, which is more natural and flexible.

For the -th identity of subject , we adopt Hawkes process to model the timestamps of the associated events. Denote the conditional intensity function of as . For a Hawkes process , the likelihood (Laub et al., 2015) of a sequence to appear in time interval is


Here, the parameter is adapted from a common model with parameter using a relatively small model-agnostic adaptation, which we will elaborate in next section.

The identity of the -th subject is then a combination of the identities with identity proportion , and the models for individual sequences are essentially mixtures of Hawkes process models. Denote . The likelihood for the -th sequence is


Moreover, the connections of the subjects are also based on their identities. More specifically, for each connection to happen, one subject needs to approach another subject , where the identities of subjects are based on respectively. Based on this observation, we adopt a Mixed Membership stochastic Blockmodel (MMB) (Airoldi et al., 2008) to model the connections of the subjects. For each subjects pair , denote the identity of subject when subject approaches subject

as random variable

, and the identity of subject when is approached by as . The probability of represent the -th identity is , and the probability of represent the -th identity is . The probability of whether subject and have a connection is then a function dependent on this two identities - the random variable representing the existence of connection

follows Bernoulli distribution with parameter

, where is a learnable parameter.

Generative process: The above model can be summarized as the following generative process.

  • For each node ,

    • Draw a dimensional identity proportion vector .

    • Sample the -th sequence from the mixture of Hawkes processes described in (3).

  • For each pair of nodes and ,

    • Draw identity indicator for the initiator

    • Draw identity indicator for the receiver

    • Sample whether there is an edge between and , .

Here, the observed variables are and . The parameters are , , and . The latent variables are , , and . The graph model is shown in Figure 1.

4 Variational Meta Expectation Maximization

We now introduce our variational meta expectation maximization algorithm. This algorithm incorporates model-agnostic adaptation into variational expectation maximization. In the rest of the paper, we denote , , .

To ease the computation we add one more latent variable . For the -th sequence, we sample . We regard as a Hawkes process with parameter . Note that this is equivalent to the mixture of Hawkes process described in previous section, since . This can ease the computation because now the update for has close form.

Variational E step. The goal is to find an approximation of the following posterior distribution

We aim to find a distribution that minimizes the Kullback-Leibler (KL) divergence to the above posterior distribution. This can be achieved by maximizing the Evidence Lower BOund (ELBO, Blei et al., 2017),


where is a properly chosen distribution space. We adopt as the mean-field variational family, i.e.,


is the Probability Density Function (PDF) of

, is the Probability Mass Function (PMF) of , is the PMF of , is the PMF of , and , , , are variational parameters. By some derivation (see Appendix C for detail), the updates for the variational parameters for solving problem (4) are


where , and is the digamma function.

Meta inference for and . Recall that the Hawkes parameter of the -th identity of subject is . Instead of specifying that is sampled from a prior distribution, we adapt the -th common model to sequence using MAML-type updates,


Since MAML-type algorithms only perform one or few updates from the common model, the adapted individual models with parameter within one community is close to each other, which meets our expectation that the within-community variation should be small.

The gradient descent step on the log-likelihood of can then be written as


where is the step size. In this algorithm, we only need to estimate the common models with parameter , instead of all individual models. After we obtain , the individual models can be easily obtained from Equation (9).

M step. We perform maximum likelihood estimation to and , The updates are as follows,


where is the step size. The detailed derivation can be found in Appendix C.

Algorithm. We perform updates (5)-(8), (10)-(12) iteratively until convergence. Note that the updates can also be implemented in stochastic fashion – at each iteration, we sample a mini-batch of sequences, and update their associated parameters (Hoffman et al., 2013).

5 Experiments

We first briefly introduce oue experiment settings.

Impact function. Following Rasmussen (2013); Zhou et al. (2013), we choose exponential impact function . The conditional intensity function is


where and are parameters. Note that each Hawkes process model only contains three parameters, , , and . This is because we target at short sequence. To avoid overfitting, each individual models cannot have too many parameters.

Regularized likelihood function. Substitute Eq. (13) into Eq. (2), we have

To keep the parameters non-negative, in practice we replace with a regularized log-likelihood in update (10),


where is the parameter of the -th Hawkes process of the -th identity, is a regularization coefficient.

Evaluation metric. We hold out the last timestamp of each sequence, and split the hold-out timestamps into a validation set and a test set. Another option to do validation and test on event sequence data is to hold out the last two timestamps – we first use the former ones to do validation, then train a new model together with the validation timestamps, and finally report the test result based on the later ones. However, this is not suitable here. This is because the sequences we adopt for experiments are usually very short, sometimes even no more than 5 events in one sequence. As a result, the models trained without or with validation timestamps, e.g., using 3 or 4 timestamps, can be significantly different, which makes the validation procedure very unreliable.

We report the Log-Likelihood (LL) of the test set. More specifically, for each sequence and parameter , the likelihood of next arrival is

The reported score is the averaged over subjects. More details can be found in Appendix D.

To estimate of the variance of the estimated log-likelihood, we adopt a multi-split procedure for evaluation. First, we train

candidate models with different hyper-parameters. Then we repeat the following procedure for times: 1). Randomly split a validation set and a test set; 2). Pick a model with highest log-likelihood on the validation set from the candidate models; 3). Compute the log-likelihood on the test set. Accordingly, we obtain

estimates of the log-likelihood. We then report the mean and standard error of the


Baselines. We adopt four baselines as follows.

MLE-Sep: We consider each sequence as a realization of an individual Hawkes process. We perform Maximum Likelihood Estimation (MLE) on each sequence separately, and obtain models for sequences.

MLE-Com: We consider all sequences as realizations of the same Hawkes process and learn a common model by MLE.

DMHP (Xu and Zha, 2017): We model sequences as a mixture of Hawkes processes with a Dirichlet distribution as the prior distribution of the mixtures.

MTL: We perform multi-task learning as described in Section 1. More specifically, we adopt Hawkes process model for and . Denote the parameters of and as and , respectively. We solve

where is the norm regularizer of to promote the difference between and to be small, is a tuning parameter, and is the function defined in Eq. (14).

Parameter Tuning. The detailed tuning procedure and detailed settings of each experiment can be found in Appendix E.

  Ground Truth               
Table 1: Visualizations of identities by HARMLESS(MAML).

5.1 Synthetic Data

Data generation. We generate a dataset of 50 nodes with communities. For each community, we generate Hawkes meta parameters

using the following uniform distributions:

We set , i.e., the entries of is all one. Then for the -th node, the identity proportion is sampled from and the membership indicator from the corresponding categorical distribution . Based on , we then generate the Hawkes parameters by adding small perturbation to :

The sequence is then sampled based on Hawkes process with parameter in time interval . To ease the tuning we normalize the sequences by dividing by the largest timestamp. We set , for any , and . We sample the graph edges based on . Denote . The generated graphs are visualized in the second column of Table 1.

Visualization of communities. We visualize the communities learned by HARMLESS (MAML) in Table 1. Denote as the number of communities specified in HARMLESS. We adopt colors corresponding to the communities in the graph. The color of each node shown in the Table 1 is the linear combinations of the RGB values of the colors weighted by identity proportions .

HARMLESS produces reasonable identities even if is mis-specified. If , some of the communities would merge. If , some of the communities would split.

Figure 2: Plot of synthetic data.

Benefit of joint training. To validate the benefit of joint training on graphs and sequences, we compare HARMLESS result with a two step procedure: We first train an MMB model and obtain the identities, and train HARMLESS (MAML) with fixed identities. In Figure 2 we plot the obtained log-likelihood with respect to .

HARMLESS (MAML) consistently achieves larger log-likelihood than the two step procedure. This suggests joint training of graphs and the sequences indeed improve the prediction of future events.

Log-likelihood with respect to . We also include the results of the baselines and HARMLESS (FOMAML) in Figure 2. The performance of HARMLESS is consistently better than the baselines. Besides, we find the performance HARMLESS (Reptile) is very dependent on the dataset. For this synthetic dataset, Reptile cannot perform well.

Dataset 911-Calls LinkedIn MathOverflow StackOverflow
HARMLESS (Reptile)
Table 2: Log-likelihood of real datasets.

5.2 Real Data

We adopt four real datasets.

911-Calls dataset: The 911-Calls dataset111Data is provided by contains emergency phone call records of fire, traffic and other emergencies for Montgomery County, PA. The county is divided into disjoint areas, each of which has a unique ZIP Code. For each area, the timestamps of emergency phone calls in this area are recorded as an event sequence. We consider each area as a subject, and two subjects are connected if they are adjoint. We finally obtain subjects and connections among them. The average length of the sequences is .

LinkedIn dataset: The LinkedIn dataset (Xu et al., 2017b) contains job hopping records of the users. For each user, her/his check-in timestamps corresponding to different companies are recorded as an event sequence. We consider each user as a subject, and two subjects are connected if the difference in timestamps of two user joined the same company is less than 2 weeks. After removing the singleton subjects, we have subjects and connections among them. The average length of the sequences is .

MathOverflow dataset: The MathOverflow dataset (Paranjape et al., 2017) contains records of the users posting and answering math questions. We adopt the records from May 2, 2014 to March 6, 2016. For each user, her/his timestamps of answering questions are recorded as an event sequence. We consider each user as a subject, and two subjects are connected if one user answers another user’s question. After removing the singleton subjects, we have subjects and connections among them. The average length of the sequences is .

StackOverflow dataset: StackOverflow is a question and answer site similar to MathOverflow. We adopt the records from November 8, 2015 to December 1, 2015. We construct the sequences and graphs in the same way as MathOverflow. After removing the singleton subjects, we have users and connections among them. The average length of the sequences is .

Result: The log-likelihood is summarized in Table 2

. Note due to Markov chain Monte Carlo is needed for


, we cannot get reasonable result for large dataset, i.e., StackOverflow. HARMLESS performs consistently better than the baselines. Since the standard error of the results of 911-Calls dataset are large, we also performed a paired t test. The test shows the difference in log-likelihood between

MLE-Com, i.e., best of the baselines, and HARMLESS (FOMAML), i.e., best of HARMLESS series, is statistically significant (with value).

Method Log-Likelihood
HARMLESS (Reptile)
Remove inner heterogeneity ()
Remove inner heterogeneity ()
Remove grouping (MAML)
Remove grouping (FOMAML)
Remove grouping (Reptile)
Remove graph (MAML)
Remove graph (FOMAML)
Remove graph (Reptile)
Table 3: Results of ablation study.

5.3 Ablation Study

We then perform ablation study using LinkedIn dataset. Three sets of ablation study are considered here:
Remove inner heterogeneity: We model each community of sequences using the same parameters, i.e., we set .
Remove grouping: We set , so that the whole graph is one community. This equivalent to apply the MAML-type algorithms on the sequences directly.
Remove graph: We do not consider the graph information, i.e., we remove , , and from the panel in Figure 1.

The results in Table 3 suggest that MAML-type adaptation, graph information, and using multiple identities all contribute to the good performance of HARMLESS.

6 Discussions

The setting of meta learning. The goal of conventional settings of meta learning is to train a model on a set of tasks, so that it can quickly adapt to a new task with only few training samples. Therefore, people divide the tasks into meta training set and meta test set, where each of the task contains a training set and a test set. The meta model is trained on the meta training set, aiming to minimize the test errors, and validated on the meta test set (Vinyals et al., 2016; Santoro et al., 2016)

. This setting is designed for supervised learning or reinforcement learning tasks that has accuracy or reward as a clear evaluation metric. Extracting information from the event sequences, however, is essentially an unsupervised learning task. Therefore, we do not separate meta training set and meta test set. Instead, we pull the collection of tasks together, and aim to extract shared information of the collection to help the training of models on individual tasks. Here, each short sequence is a task. We exploit the shared pattern of the collection of the sequences to obtain the models for individual sequences.

Community Pattern. The target of Mixed Membership stochastic Blockmodels (MMB) is to identify the communities in a social graph, e.g., the classes in a school. However, real social graphs cannot always be viewed as Erdős-Rényi (ER) graphs assumed by MMB. As argued in Karrer and Newman (2011), for real-world networks, MMB tends to assign nodes with similar degrees to same communities, which is different from the popular interpretation of the community pattern. This property, however, is actually very helpful in our case. As an example, Twitter users that are more active tend to have similar behavior: They tend to make more connections and post tweets more frequently. In contrast, users with very different node degrees often have the tweets histories of different characteristics, and thus should be assigned to different identities. Such property of MMB allows the identities in HARMLESS to represent this non-traditional community patterns in non-ER graphs, i.e., it assigns subjects with various activeness to different communities.

Mixture of Hawkes processes. Many existing works adopt mixture of Hawkes process to model sequences that are generated from complicated mechanisms (Yang and Zha, 2013; Li and Zha, 2013; Xu and Zha, 2017). Those works are different from HARMLESS since they do not consider the hierarchical heterogeneity of the sequences, and do not consider the relational information.

Variants of Hawkes process. Some attempts have been made to further enhance the flexibility of Hawkes processes. For example, the time-dependent Hawkes process (TiDeH) in Kobayashi and Lambiotte (2016)

and the neural network-based Hawkes process (N-SM-MPP) in

Mei and Eisner (2017)

learn very flexible Hawkes processes with complicated intensity functions. Those models usually have more parameters than vanilla Hawkes processes. For longer sequences, HARMLESS can also be naturally extended to TiDeHs or N-SM-MPP. However, this work focuses on short sequences. These methods are not useful here, since they have too many degrees of freedom.


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Appendix A Related Works

Hawkes Process Hawkes process has long been used to model event sequences (Hawkes, 1971), such as earthquake aftershock sequences (Ogata, 1999), financial transactions (Bauwens and Hautsch, 2009), and events on social networks (Fox et al., 2016; Farajtabar et al., 2017). Its variant, mixture of Hawkes processes model, has also been proved effective in many area (Yang and Zha, 2013; Li and Zha, 2013; Xu and Zha, 2017). In most cases, the learning methodology is variational inference or maximum likelihood estimation (Rasmussen, 2013; Zhou et al., 2013; Zhao et al., 2015). Other possible methods includes least-squares-based method (Eichler et al., 2017), Wiener-Hopf-based methods (Bacry et al., 2012), and cumulants-based methods (Achab et al., 2017).

Instead of predefine an impact function here, some non-parametric methods use discretization or kernel-estimation when learning models (Reynaud-Bouret et al., 2010; Zhou et al., 2013; Hansen et al., 2015). Those methods usually target small datasets, and do not need a good scalability. Recently, some attempts have been made to further enhance the flexibility of Hawkes processes. The time-dependent Hawkes process (TiDeH) in Kobayashi and Lambiotte (2016) and the neural network-based Hawkes process in Mei and Eisner (2017) learn very flexible Hawkes processes with complicated intensity functions. Those methods usually target very long and multi-dimensional sequences, instead of short sequences.

Existing works targeting short sequences is usually in specific cases (Xu et al., 2017a, b), such as the data is censored. However, there is no work targeting general short sequences as we do here.

There are lines of research that involves both point processes and graphs. One is using point process to find the latent graph (Blundell et al., 2012; Linderman and Adams, 2014; Tran et al., 2015). Another one is considering the interaction of the nodes as point process and use it to construct a dynamic graph, instead of the event happens on nodes as we consider here (Farajtabar et al., 2016; Zarezade et al., 2017; Trivedi et al., 2018). These works have vary different aims from our work.

Meta Learning Meta learning has been studied since last century (Bengio et al., 1990; Chalmers, 1991). Some works focus on learning the hyperparameters, such as learning rates or initial conditions (Maclaurin et al., 2015). Some works aim to learn a metric so that a simple K nearest neighbors can perform well under such a metric (Koch et al., 2015; Vinyals et al., 2016; Sung et al., 2018; Snell et al., 2017). Some works design specific deep neural networks so that the information of different tasks are memorized and thus the model can easily generalize to new tasks (Santoro et al., 2016; Munkhdalai and Yu, 2017; Ravi and Larochelle, 2016).

Model-Agnostic Meta Learning (MAML) method (Finn et al., 2017) opens another line of research, i.e., it designs an optimization scheme so that the model can fast adapt to new tasks. Reptile (Nichol and Schulman, 2018), a variant of MAML, is proposed to simplify the computation of MAML. None of those works, however, considers the relational information between tasks like our method, which is critical in modeling short sequences.

One interesting line of follow-up works of MAML is connecting MAML with Bayesian inference (Finn et al., 2018; Ravi and Beatson, 2018; Grant et al., 2018). Since HARMLESS combines a Bayesian model with MAML, it has the potential to be rewritten into a pure Bayesian model that has better quantification of uncertainty. We left this for future work.

Appendix B Definition of Operator

As we mentioned earlier,

is the loss function for MAML, FOMAML, and Reptile algorithm with different definition of the operator .

For simplicity, here we define the operator of one gradient step. The cases of few gradient steps can be defined analogously.

For MAML, is defined as .

For First Order MAML (FOMAML), is also defined as . The difference is that the output of the operator just a value, not a function of , i.e., when we solve the gradient of , the gradient does not back-propagate into .

For Reptile, the algorithm of reptile is as follows Nichol and Schulman (2018).

  while not converged do
     Sample task with loss ;
     , where is the number of SGD steps;
     Do the update ;
  end while
Algorithm 1 Reptile

From the algorithm we can see, operator is defined as . Similar as FOMAML, computing the gradient also does not back-propagate into .

Appendix C Derivation of Variational EM

Preparation After adding latent variable

, the joint distribution is


Note that in this section we represent as one-hot vector, while in the main paper we use scalar representing the identities.

The posterior distribution is defined as

We aim to find a distribution , such that the Kullback-Leibler (KL) divergence between the above posterior distribution and is minimized. This can be achieved by maximize the Evidence Lower BOund (ELBO),

Variational family We adopt the mean-field variational family, i.e.,

We pick as PDF of , as PDF of , as PDF of , as PDF of .

Update for Again, our goal is to maximize

Now we focus on , and treat , and as given. We want to maximize

Take the derivative,

Substitute the expressions of the distributions, after some derivation we get the update for as


Update for Similarly, we have

Take the derivative,

After some derivation, we have


where is the digamma function.

Update for and The derivation of update for and is very similar to the update for , so we will not elaborate on that. Readers who are interested might also refer to Airoldi et al. (2008). The updates are


Update for We update using gradient ascent. We first pick the terms that is relevant to ,

So the gradient ascent update is,


Update for and From Airoldi et al. (2008), we have the update for and as follows


Appendix D Derivation of Evaluation Metric

In this section, we give more details on the evaluate metrics. Specifically, we show how to compute the NLL of the test set. Given a sequence , we would like to predict the timestamp of . Here, we use the probability of the arrival at time and no arrival in given history before