Semi-supervised Learning for Marked Temporal Point Processes

Initially, research in the field of Temporal Point Processes (TPP) focused primarily on Self-Exciting Hawkes (1971) and Self-Correcting Isham and Westcott (1979) processes. While such techniques demonstrated good performance, they were deemed to have limited flexibility limiting their usage in real-world scenarios. A significant breakthrough in the TPP domain happened in the last decade with the advent of deep learning by the usage of neural-networks for modelling the underlying conditional function. Traditionally, TPPs were used to only model the time occurrence of an event, however, later they were also used to model and predict the corresponding type/class of the event called the marker. Such techniques which focused on predicting the event and corresponding type were termed as modeling Marked Temporal Point Processes (MTPP) Du et al. (2016). Recently, Recurrent Neural Networks (RNN) have also been used to model the Marked Temporal Point Processes. For example, techniques such as the Recurrent Marked Temporal Point Processes Du et al. (2016) and Neural Hawkes Process Mei and Eisner (2016) utilize the RNN module as the backbone architecture. Similarly, a fully neural network based model has also been proposed for general TPPs Omi et al. (2019) to model the cumulative intensity function with a neural network. Further, recently, techniques have also been proposed to learn robust representations under the constraint of missing observations as well Gupta et al. (2021). Most of the techniques mentioned above follow the supervised learning approach which often require large amount of labeled training data.

Broadly, machine learning algorithms can be categorized into the following three categories: (i) supervised learning, (ii) unsupervised learning, and (iii) semi-supervised learning Zhu (2005). Semi-supervised learning is the branch of machine learning which attempts to utilize both labeled and unlabeled data during training. Often, the goal is to enhance the performance by incorporating the benefits of both kinds of data. Over the past decades, various algorithms for semi-supervised learning have been proposed. They are broadly classified into two categories: (i) inductive learning and (ii) transductive learning Chapelle et al. (2009). In inductive learning based semi-supervised approaches, the goal is to infer the correct mapping from $X$ (data points which are labeled) to $Y$ (corresponding labels) in the form of a classifier, whereas in transductive learning based semi-supervised techniques the goal is to infer the correct labels for the unlabeled data points only. Inductive learning is further divided into three categories namely unsupervised processing, wrapper methods, and intrinsically semi-supervised learning whereas transductive learning is mostly used in graph-based algorithms like graph construction, graph weighting and graph inference Van Engelen and Hoos (2020).

The proposed SSL-MTPP model extends the current body of literature by focusing on the intersection of the above two fields of research. Specifically, current TPP models follow supervised learning approach which often require large amount of labeled data. In real world scenarios, such data is often not easily available. In order to overcome the limitation of limited labeled data availability, this research proposes a novel SSL-MTPP algorithm which follows the inductive approach of semi-supervised learning and incorporates labeled and unlabeled data for learning an marker prediction model.

Figure 3 presents the architecture of the proposed algorithm consisting of a labeled and an unlabeled path. The labeled path takes as input of a given set of sequences ($\mathcal{S}$) consisting of $n$ sequence pairs ($x_{i},y_{i}$) having the event time information ($x_{i}$) and the marker information ($y_{i}$). It utilizes a Recurrent Neural Network (RNN) architecture for learning an embedding capturing the behavior of the marker and the time occurrence in the given sequences. The embedding is used by two independent modules for (i) marker prediction and (ii) time prediction of the next event. On the other hand, the unlabeled path utilizes an RNN Encoder-Decoder model Cho et al. (2014) for learning an embedding for the time sequence only. The learned embedding represents the time sequence and is used to supplement the marker-time embedding for a robust representation. Mathematical formulation and explanation of the different steps and models involved along with each loss component is as follows:

The proposed SSL-MTPP model has been implemented in the PyTorch environment Paszke et al. (2019). SSL-MTPP algorithm utilizes the RMTPP architecture Du et al. (2016) as the base model. The supervised branch contains a five layer Long Short-Term Memory (LSTM) module Hochreiter and Schmidhuber (1997), while the unsupervised branch contains a two layer RNN encoder and decoder. The marker and event prediction modules contain two dense layers, respectively. Dropout Hinton et al. (2012) has also been applied as a regularizer after the RNN layer. The weight value ($\lambda$ in Equation 4) has been set to $0.1$. The model has been trained for 100 epochs with a learning rate of 0.01 using the Adam optimizer Kingma and Ba (2014). Training has been performed with a batch-size of $1024$ sequences on an NVIDIA Quadro RTX6000 GPU.

Experiments have been performed on the Retweet dataset Zhao et al. (2015), which is formed through the Seismic dataset. The dataset contains multiple sequences of retweets, where each sequence corresponds to information regarding the retweets on a particular tweet. Each sequence contains information regarding the event time (retweet) and the corresponding marker information. Here, the marker refers to type of user (based on the number of followers) who has retweeted. Three categories of marker are provided: (i) normal user, (ii) influencer user, and (iii) celebrity user. The marker information is defined based on the number of followers (degree) of a given user: (i) degree lower than the median (normal user), (ii) degree higher or equal to the median but less than the $95^{th}$ percentile (influencer user), and (iii) degree higher or equal to the $95^{th}$ percentile (celebrity user). The dataset consists of over two million events with an average sequence length of 209 events.The dataset has imbalance class distribution with 50.6% event marker are normal users,45% events marker are influencer users,while only 4.4% events marker are celebrity users. For experiments, data pertaining to 1.4M events was used for training, while the remaining 60K events formed the test set.

Table 1 presents the six protocols used to evaluate the proposed SSL-MTPP algorithm. The training set (consisting of 1.4M events) is split into a labeled set and an unlabeled set for each protocol. The labeled set contains data varying from 10K (P-1) to 0.7M (P-6), while the remaining data forms the unlabeled set. For experiments, the proposed SSL-MTPP algorithm is trained for each protocol, and comparison has been performed with the baseline supervised MTPP model trained in the traditional supervised manner with labeled data only.

Table 2 presents the results obtained on the Retweet dataset. The proposed SSL-MTPP algorithm has been evaluated on different protocols containing varying amount of labeled data (Table 1). In the literature, most of the research has focused on reporting the Macro-F1 and Micro-F1 performance metrics. As part of this research, we observe that these might not be the most appropriate metrics to judge a model’s performance, especially under the scenario of imbalanced per-class data. To this effect, along with the Macro-F1 and Micro-F1 values, we also report the average precision of each model.

As can be observed from Table 2, limited variation is observed for the Macro-F1 and Micro-F1 values for the proposed and native supervised MTPP model. For the SSL-MTPP model, the Micro-F1 values lie in the range of $59.22\%-59.59\%$, regardless of the amount of labeled data, and the Macro-F1 values lie in the range of $39.48\%-40.77\%$, thus demonstrating limited variation. On the other hand, the average precision lies in the range of $39.10\%-69.49\%$ with varying labeled data. Similar behavior is observed for the native supervised MTPP model as well, where limited variations are observed for the Macro-F1 and Micro-F1 values, while higher range is observed for the average precision metric. The consistent behavior across the two models suggest average precision to be a better metric for comparing performance in the setup of imbalanced testing data across classes.

Figure 4 presents the average precision (%) of the proposed SSL-MTPP model and the native supervised MTPP model for six protocols. Across the protocols, the proposed SSL-MTPP model demonstrates improved performance as compared to the baseline model. In P-1, where only 10K labeled data is available, the SSL-MTPP model obtains an average precision of 39.10%, presenting an improvement over the baseline model (38.84%). Larger improvements are observed for P-2 to P-5, where at least 20K labeled data is available for training. For example, in P-4, the SSL-MTPP model obtains an average precision of 68.35% demonstrating an improvement of almost 24% as compared to the baseline model. Improvement is also observed for the Macro-F1 and Micro-F1 values across protocols. Further, relatively less improvement is seen when 0.7M labeled data is available for training (P-6), where the proposed SSL-MTPP model achieves an average precision of 69.49% (as compared to 66.74% of the baseline model), thus suggesting higher improvement when limited training data is available for training. The above behavior appears intuitive in nature as well, since as the amount of labeled data increases, the baseline model is able to learn better, thus reducing the gap in improvement.

Experiments have also been performed to analyze the effect of the weight parameter ($\lambda$ in Equation 4) for understanding the effect of the fusion of the supervised and unsupervised representations. Figure 5 presents the average precision obtained on P-3 (30K labeled training data) with varying $\lambda$ values. Best performance is obtained with a value of $0.1$, while a drop in performance is seen with a very small ($0.001$) and a very large ($10$) value. A small value reduces the contribution of the unsupervised representation, while a very large value offsets the contribution of the supervised embedding thus resulting in a drop in performance.