\begin{table*}
\centering
\caption{Results for Zero-shot setup for proposed method HyperHawkes under different setups against the proposed baselines. The table considers both the variants HyperHawkes-FNN and HyperHawkes-FNN-RNN as proposed methodology for ZSL}
\label{Tab:zsl}
\tabcolsep=0.05cm
\begin{tabular}{c|c|cccc|cccc} 
\hline\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Experimental\\Setup\end{tabular}}      & \multirow{2}{*}{Dataset} & \multicolumn{4}{c|}{MNLL}                                                                                                                                                                       & \multicolumn{4}{c}{MAE}                                                                                                                                                                         \\ 
\cline{3-10}
                                                                                  &                          & FNHP    & \begin{tabular}[c]{@{}c@{}}FNHP-\\Descriptor\end{tabular} & \begin{tabular}[c]{@{}c@{}}HyperHawkes-\\FNN\end{tabular} & \begin{tabular}[c]{@{}c@{}}HyperHawkes-\\FNN-RNN\end{tabular} & FNHP   & \begin{tabular}[c]{@{}c@{}}FNHP-\\Descriptor\end{tabular} & \begin{tabular}[c]{@{}c@{}}HyperHawkes-\\FNN\end{tabular} & \begin{tabular}[c]{@{}c@{}}HyperHawkes-\\FNN-RNN\end{tabular}  \\ 
\hline
\multirow{2}{*}{Zero-shot}                                                        & Yelp                     & -4.2023 & -3.3334                                                   & \textbf{\textbf{-5.7045}}                                 & -5.4934                                                       & 0.0025 & 0.0015                                                    & \textbf{0.0013}                                           & 0.0014                                                         \\
                                                                                  & Meme                     & -3.9011 & -2.9421                                                   & \textbf{\textbf{-5.4339}}                                 & -5.3403                                                       & 0.0056 & 0.0072                                                    & 0.0036                                                    & \textbf{0.0018}                                                \\ 
\hline\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Generalized\\zero-shot\end{tabular}}   & Yelp                     & -4.9279 & -4.2075                                                   & -5.2473                                                   & \textbf{\textbf{-5.6869}}                                     & 0.0025 & 0.0016                                                    & 0.0021                                                    & \textbf{0.0014}                                                \\
                                                                                  & Meme                     & -4.2735 & -4.0250                                                   & -5.2475                                                   & \textbf{\textbf{-5.2922}}                                     & 0.0036 & 0.0033                                                    & 0.0030                                                    & \textbf{0.0026}                                                \\ 
\hline\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Standard\\event modeling\end{tabular}} & Yelp                     & -4.2700 & -3.8543                                                   & -4.9030                                                   & \textbf{\textbf{-4.9475}}                                     & 0.0047 & 0.0042                                                    & \textbf{0.0024}                                           & 0.0025                                                         \\
                                                                                  & Meme                     & -4.6991 & -2.9633                                                   & -2.5325                                                   & \textbf{-4.8796}                                                      & 0.0046 & 0.0072                                                    & 0.0078                                                    & \textbf{0.0027}                                                \\
\hline\hline
\end{tabular}
\end{table*}
\paragraph{\textbf{Continual Learning:}} aims to create a learning paradigm which is able to model a stream of tasks while avoiding catastrophic forgetting. Different techniques \cite{kirkpatrick2017overcoming,li2017learning,lopez2017gradient,von2019continual}  have been proposed in this regard by consolidating knowledge either in various spaces like data, weight or meta space.
\cite{von2019continual} performs continual learning through hypernetwork which learns task conditioned weights of base model. 
\paragraph{
\textbf{Hypernetworks:}} They have been introduced as a metanetwork which can generate weights for another network~\cite{ha2017hypernetworks}. It is used for various tasks like meta learning \cite{zhao2020meta}, neural architecture search \cite{zoph2016neural}, natural language understanding~\cite{he2022hyperprompt} etc.

\begin{equation}
\label{eq:log_lik_nhp}
\begin{split}
& \log p(\{t^i_j\}_{j=1}^{n^i} ; W^i) = \sum_{j=1}^{n^i} \log p(t^i_j|\mathcal{H}^i_j;W^i) = \\
& \sum_{j=1}^{n^i} \big( \log (\frac{ \partial}{\partial \tau^i_j} \Phi(\tau^i_j |\boldsymbol{h}^i_{j-1}; W^i) )
 - \Phi(\tau^i_j|\boldsymbol{h}^i_{j-1}; W^i) \big) 
 \end{split}
\end{equation}

\begin{itemize}
\item \textbf{HyperHawkes-FNN:} Hypernetwork is considered only for the FNN  modeling the cumulative hazard function.  
\item \textbf{HyperHawkes-FNN-RNN:} This variant uses two separate hypernetworks, one to model  the RNN modeling the history and the second to model the FNN.
\end{itemize}

\subsection{HyperHawkes for Zero-shot modeling}
In this section  we discuss how we employ \textit{HyperHawkes} for zero-shot event modeling. Our goal is to train the model on seen sequences $\mathcal{D}^S$ with event sequences $\mathcal{T}^s$ and task descriptors $\mathbf{d}^s$ and predict event times of unseen sequences in $\mathcal{D}^U$ given a task descriptor $\mathbf{d}^u$. We employ \textit{HyperHawkes} for performing zero-shot learning on event sequences. The central idea of the proposed approach is to predict the parameters for the neural Hawkes process for the unseen task $\mathbf{d}^u$. This is achieved using hypernetwork which considers the sequence descriptor as input and parameters for neural Hawkes process as output as discussed in the previous section. Consequently, we can get parameters for RNN ($W_r^u$) using Equation \ref{eq:hrnn} and FFN ($W_t^u$)  using Equation \ref{eq:hfnn} and use them to model the cumulative hazard function for an unseen sequence. These parameters can then be used for predicting  events in the sequence $\mathcal{T}^u$. 

\begin{equation}
\label{eq:log_lik_nhp_cl}
\begin{split}
&\sum_{j=1}^{n^s}  -\log p(t^s_j|\mathcal{H}^s_j;(f_r(\mathbf{d}^s;\theta_{fr}), 
f_t(\mathbf{d}^s;\theta_{ft}))) \\
 & +\frac{\beta}{s-1}\sum_{c=1}^{s-1}\biggl(\parallel f_r(\mathbf{d}^c;\theta_{fr})-f_r(\mathbf{d}^c; \bar{\theta}_{fr})\!\!\parallel^2 \\
& + \parallel f_t(\mathbf{d}^c; \theta_{ft}) - f_t(\mathbf{d}^c; \bar{\theta}_{ft}) \parallel^2 \biggr)
 \end{split}
\end{equation}

\section{Experiments}
\subsection{Datasets}
Due to paucity of standard datasets for event modeling tasks which contain meta descriptions as well, we use the following two datasets: \textbf{1)Yelp:}\footnote[1]{https://www.kaggle.com/datasets/yelp-dataset/}: This is a dataset comprising of business information and their check-in information. Each business is associated with 82 attributes like \textit{Wheelchair Accessible, Accepts Insurance, By Appointment Only, Business Category, Business Timings} etc. Also, they are associated with latitude-longitude pairs. Moreover, these businesses are associated with a fine-grained category. For higher granularity, we convert them into 22 broad categories using the hierarchy mentioned in their website \footnote[2]{https://www.yelp.com/developers/documentation/v3/all\_category\_list}. Using these attributes, we create a vector of length 1229 representing a business. This vector acts as a descriptor of the business. We select businesses with more than 5000 check-ins. For continual learning, we have considered another sample of dataset consisting of business with more than 10k checkins, hence considering 26 sequences of business. This is done to reduce the number of sequences for better visualization of performance of each sequence. \textbf{2) Meme:}\footnote[3]{https://snap.stanford.edu/data/memetracker9.html}: This dataset\cite{leskovec2009meme} tracks the popular phrases and quotes which appear appear most frequently over time in news media and blogs. Each meme is associated with the content and timestamps when they were quoted in the media. We select top 200 english phrases and doc2vec representation of the meme content is considered as the descriptor of length 100. We have considered memes from April, 2009 with an average number of events as 970. 

\subsection{Experimental Setup}
We consider these experimental setups to evaluate the performance of our model - 
\textbf{1) Zero-Shot:} Training is done on seen sequences and testing is done on unseen sequences.  \textbf{2) Generalized Zero-Shot:} In this, testing is done by randomly sampling 20\% of events from seen sequences and unseen sequences.  \textbf{3) Standard Event Modeling:} Training is done on the first 70\% of the events from all sequences. Testing is done on the last 20\% events for unseen sequences. 
Mean negative log-likelihood (MNLL) and mean absolute error (MAE) are considered as evaluation metrics for both zero-shot and continual learning setup. Lower MNLL and MAE indicates better performance.

\section{Results and Analysis}
\subsection{Zero-Shot Learning}
Results for zero-shot setup are presented in Table \ref{Tab:zsl}. A better model is expected to have lower MNLL and MAE. We can observe that the proposed method HyperHawkes performs better than the baselines in terms of both evaluation metrics MNLL and MAE. We can observe that zero-shot setup, which consists of predictive performance for unseen sequences, exhibits significantly lower MNLL and MAE for HyperHawkes-FNN as compared to FNHP and FNHP-Descriptor. Also, HyperHawkes-FNN-RNN yields lower MAE for Meme dataset, however, MNLL for both the variants of the proposed methods is close. Comparing the results of generalized zero-shot setup where we consider instances from seen sequences as well as unseen sequences, we can observe that HyperHawkes-FNN-RNN performs better for both the datasets. The final section of the table discusses results for standard event modeling where we test on the last 20\% of the events of unseen sequences. HyperHawkes-FNN-RNN performs well here as well except for MAE for Yelp dataset. We can also observe that HyperHawkes-FNN-RNN performs slightly better for generalized zero-shot and standard event modeling setup. Moreover, it is interesting to note that inclusion of sequence descriptors within FNHP in FNHP-Descriptor has not helped much in prediction of unseen sequences. In fact, in various cases, it has performed worse than FNHP itself. This confirms the necessity of an event sequence specific framework which can predict better for unseen sequences as well. Therefore, our results indicate that the proposed variants of HyperHawkes perform consistently and significantly better than the baselines for both the datasets.  
{"value": "-4.2023", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "-3.3334", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "-5.7045", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "-5.4934", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "0.0025", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "0.0015", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "0.0013", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "0.0014", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "-3.9011", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "-2.9421", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "-5.4339", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "-5.3403", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "0.0056", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "0.0072", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "0.0036", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "0.0018", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "-4.9279", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "-4.2075", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "-5.2473", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "-5.6869", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "0.0025", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "0.0016", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "0.0021", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "0.0014", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "-4.2735", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "-4.0250", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "-5.2475", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "-5.2922", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "0.0036", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "0.0033", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "0.0030", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "0.0026", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Generalized zero-shot": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "-4.2700", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "-3.8543", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "-4.9030", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "-4.9475", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "0.0047", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "0.0042", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "0.0024", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "0.0025", "type": "Result", "training data/set": "Yelp", "test data/set": "Yelp", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "-4.6991", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "-2.9633", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "-2.5325", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "-4.8796", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MNLL", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}} {"value": "0.0046", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP", "model settings": {"xx": "yy"}} {"value": "0.0072", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "FNHP-Descriptor", "model settings": {"xx": "yy"}} {"value": "0.0078", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN", "model settings": {"xx": "yy"}} {"value": "0.0027", "type": "Result", "training data/set": "Meme", "test data/set": "Meme", "task": ["event modeling", "time-to-event modeling"], "metric": "MAE", "experimental settings": {"Standard event modeling": "true"}, "model": "HyperHawkes-FNN-RNN", "model settings": {"xx": "yy"}}