Multiclass Classiﬁcation for Hawkes Processes
Abstract
We investigate the multiclass classiﬁcation problem where the features are event sequences.
More precisely, the data are assumed to be generated by a mixture of simple linear Hawkes processes.
In this new setting, the classes are discriminated by various triggering kernels.
A challenge is then to build an efﬁcient classiﬁcation procedure.
We derive the optimal Bayes rule and provide a two-step estimation procedure of the Bayes classiﬁer.
In the ﬁrst step, the weights of the mixture are estimated; in the second step, an empirical risk minimization procedure is performed to estimate the parameters of the Hawkes processes.
We establish the consistency of the resulting procedure and derive rates of convergence.
Finally, the numerical properties of the data-driven algorithm are illustrated through a simulation study where the triggering kernels are assumed to belong to the popular parametric exponential family.
It highlights the accuracy and the robustness of the proposed algorithm.
In particular, even if the underlying kernels are misspeciﬁed, the procedure exhibits good performance.
INTRODUCTION
A crucial challenge in multiclass learning is to provide algorithms designed to handle temporal data.
In the present paper, we tackle the multiclass classiﬁcation problem where the features are time event sequences.
More precisely, we assume that the data come from a mixture of Hawkes processes and we focus on the classiﬁcation per trajectory (and not per event).
In neuroscience we can consider event sequences as recorded spike trains on several neurons from different populations (healthy or sick subjects, for instance).
The goal is then to predict the status (healthy or not) of a new subject from the associated recording [Lambert et al., 2018]
Hawkes processes, originally introduced in [Hawkes, 1971], are proposed to model tricky event sequences where the past events inﬂuence the future events.
Hawkes processes arise in a wide variety of ﬁelds, ranging from neuroscience to ﬁnance.
In mathematical ﬁnance, see e.g. [Bacry et al., 2015] for a complete review; in the social network literature, see e.g. [Lukasik et al., 2016] and [Qu and Lemhadri, 2021].
In neuroscience, Hawkes processes have a statistical interest for modeling neuron spike occurrences, see e.g. [Hansen et al., 2015], [Ditlevsen and Löcherbach, 2017], [Foschi, 2020].
Seminal work for Hawkes process properties is [Brémaud and Massoulié, 1996].
Furthermore, there are numerous statistical methods of inference for Hawkes processes.
For instance, one can cite [Hansen et al., 2015], [Bacry and
Muzy, 2016] and more recently [Bacry et al., 2020], or in a
Bayesian framework, [Rasmussen, 2013].
Besides, [Favetto, 2019] focuses on parameter estimation for Hawkes processes from repeated observations in the context of electricity market modeling.
However, the aim of the paper is a multiclass classiﬁcation task and not the parameter inference.
To the best of our knowledge, except the paper of [Lukasik et al., 2016], there is no work which deals with supervised classiﬁcation for
Hawkes processes.
In [Lukasik et al., 2016], the authors propose to use multivariate Hawkes processes for classifying sequences of temporal textual data, with an application to rumours coming from Twitter datasets.
They highlight that a model based on Hawkes processes is a competitive approach which takes into account the temporal dynamic of the data.
But, they do not provide any theoretical properties.
More recently, Dutta et al.
[2020], Tondulkar et al.
Ram and Srijith [2018] focus on the question of time classiﬁcation.
Indeed, for classical Twitter example from PHEME dataset, used to do rumor stance classiﬁcation, the models impose a label on each tweet (each time).
The classiﬁcation setting, with temporal and textual data, is thus a bit different from our framework.
Besides, as in [Lukasik et al., 2016], the authors do not provide theoretical properties to support their procedures.
In this work, we observe repeatedly jump times coming from the mixture of Hawkes processes, on a ﬁxed time interval [0 , T ] .
The classes are characterized by different triggering kernels.
We ﬁrst formally deﬁne the model and provide the explicit form of the Bayes classiﬁer in Section 2.
The expression of the Bayes classiﬁer suggests to consider a plug-in approach to estimate the optimal predictor.
Sectionis devoted to the deﬁnition of plug-in type classiﬁer and the study of its properties.
We show how the misclassiﬁcation error, for any plug-in predictor is linked to the estimation error of the process parameters.
We propose in Sectiona two-step procedure to build a plug-in type classiﬁer.
A ﬁrst step is dedicated to the estimation of the weights of the mixture.
In a second step the parameters of the process are estimated through an empirical risk minimization procedure by using similar ideas as in [Denis et al., 2020].
The resulting algorithm beneﬁts from the attractive properties of the empirical risk minimizer: it is computationally efﬁcient and offers good theoretical properties.
In particular, under mild assumptions, we show that the proposed procedure performs as well as the Bayes classiﬁer.
Section 5 illustrates the performance and the robustness of the method in the case where the triggering kernels are assumed to belong to the parametric exponential family.
Finally, a discussion which highlights some directions for future works is proposed in
Section 6.
GENERAL FRAMEWORK
Section 2.1 introduces the considered model, some notation and explains the objective of the paper.
In Section 2.2, we provide an explicit formula of the optimal predictor.
2.1 STATISTICAL SETTING
Let Y a random variable which takes its values in Y = { 1 , . .
, K } , with K ≥ 2 , representing the label of the observations.
The distribution of Y is denoted by p ∗ =
( p ∗ k ) k ∈Y and is unknown.
We assume that the observations come from a mixture N of simple linear Hawkes processes observed on the time interval [0 , T ] .
Precisely, conditionally on Y , N is a simple linear Hawkes process.
The number of points that lie in [0 , t ] is denoted by N t and the corresponding counting process is ( N t )
≤ t ≤ T .
The jump times of N are denoted T 1 , . .
, T N T .
The ﬁltration (or history) at time t − is denoted F t − and contains all the necessary information for generating the next point of N .
Conditional intensity The intensity of the process N at time t ≥ 0 , with respect to the ﬁltration ( F t ) t ≥ 0 , is deﬁned as
[Equation]
where the ﬁrst term µ ∗ > 0 is the baseline, or exogenous intensity, and the second term is a weighted sum over past events.
For each class k ∈ Y , the function h ∗ k is the triggering kernel which is nonnegative and supported on R + .
Besides, both parameters µ ∗ and h ∗ =
( h ∗ 1 , . .
, h ∗ K ) are assumed to be unknown.
Note that the baseline intensity is assumed to be common to all classes.
This assumption is notwithstanding consistent according to the neuronal experimental setting described in Section 1.
Indeed, if the spike trains are recorded on the same type of neurons ( e.g. neurons which play the same role), it seems relevant to assume that the exogenous intensity is homogeneous between the classes.
Objective Given a sequence T T = { T 1 , . .
, T N T } of observed jump times of N over the ﬁxed interval [0 , T ] , the goal is then to build a predictor, namely a classiﬁer g , a measurable function such that g ( T T ) is a prediction of the associated label Y .
The performance of a classiﬁer g is then measured through its misclassiﬁcation risk
R ( g )
[Equation]
In the following, we denote by G the set of classiﬁers.
2.2 BAYES RULE
The unknown minimizer of R over G is the so-called Bayes classiﬁer, denoted by g ∗ , and is characterized by
[Equation]
with π ∗ k ( T T ) =
P ( Y = k |T T ) .
The following proposition gives the expression of the conditional probabilities π ∗ k
and then provides a closed form of the Bayes classiﬁer.
Proposition 2.1.
Let T ≥ 0 .
For each k ∈ Y , we deﬁne,
[Equation]
Therefore, the sequence of conditional probabilities satisﬁes
[Equation]
where F ∗ =
( F ∗ 1 , . .
, F ∗ K ) and φ p ∗ k : ( x 1 , . . .
, x K ) (cid:55)→
[Equation]
Kj =1 p ∗ j e x j are softmax functions.
Note that conditionally on the event Y =
k , F ∗ k
( T T ) is the likelihood function of the sequence T T .
Proposition 2.
highlights the dependencies of the optimal Bayes classiﬁer w.r.t.
the unknown parameters.
In the following, for a given classiﬁer g ∈ G , we deﬁne its excess risk as
[Equation]
PLUG-IN TYPE CLASSIFIER
We ﬁrst introduce assumptions related to the model in Section 3.1 and then deﬁne a set of classiﬁers which relies on the plug-in principle in Section 3.2.
Finally, the main properties of the plug-in classiﬁer are provided in Section 3.3.
3.1 ASSUMPTIONS
We ﬁrst make the following assumptions on the triggering kernels.
Assumption 3.1 (Stability condition) .
For each k ∈ Y , h k : R + → R + is bounded and satisﬁes (cid:82) h k ( t ) d t < 1 .
Assumption 3.2.
There exist 0 < µ 0 < µ 1 such that µ 0 ≤ µ ∗ ≤ µ 1 .
Assumption 3.3.
There exists a positive constant p 0 such that min( p ∗ ) > p 0 .
Assumption 3.1 guarantees that N T admits ﬁnite exponential moments, that is, there exists a > 0 such that E [exp( a | N T | )] < ∞ , see for instance [Roueff et al., 2016].
In particular the exponential and power-law kernels satisfy this assumption (with additional assumptions on the corresponding parameters).
Assumption 3.2 is a technical assumption and Assumption 3.3 ensures that all the components of the mixture occur with non-zero probability.
Let us denote the following subset of probability weights
[Equation]
3.2 DEFINITIONS
In this section, we present the construction of the plug-in type classiﬁers.
First we introduce a set H of nonnegative functions supported on R + .
For a K -tuple h
= ( h 1 , . .
, h K ) in H K , we associate p a vector of probability weights and a baseline intensity µ > 0 .
For each k ∈ Y , we then deﬁne λ k ( t ) =
λ ( µ,h k ) ( t ) =
µ + (cid:88)
T i <
t h k ( t − T i ) , t ∈ [0 , T ] .
Hence, the random functions ( λ k ) k
=1 ,...,K are approximations of the conditional intensities λ ∗ k deﬁned by (1).
Besides, similarly with the deﬁnition (2) of F ∗ k ( T T ) , we deﬁne
[Equation]
We also consider
[Equation]
with the φ p k ’s deﬁned in the same manner of the φ p ∗ k ’s given in Proposition 2.1.
Finally, we denote π p ,µ, h ( . )
= (cid:16) π k p ,µ, h ( . ) (cid:17) k ∈Y and π := π p ,µ, h .
A plug-in type classiﬁer g π is naturally deﬁned as
[Equation]
3.3 PROPERTIES
In this section, we establish important properties of plug-in type classiﬁers.
For a vector of functions h ∈ H K , let us denote the supremum norm
[Equation]
We introduce for a positive constant A the following set
[Equation]
and the set of probabilities
[Equation]
The ﬁrst result is a key step to obtain the consistency of the classiﬁcation procedure presented in Section 4.
Proposition 3.4.
Let us consider π and π (cid:48) two vectors functions belonging to the set Π deﬁned by (5) with respective parameters ( p , µ, h ) , and ( p (cid:48) , µ (cid:48) , h (cid:48) ) .
Grant Assumptions 3.1, 3.2, 3.3, the following holds
[Equation]
, where C is a constant depending on K , T , h ∗ , µ 0 , µ 1 , pand A .
Proposition 3.4 provides a bound on L 1 -distance between two elements of the set Π .
It shows that this distance is bounded by the distance between the corresponding parameters of the associated models.
From this result, for a plug-in type classiﬁer g , we can easily deduce a bound of its excess risk.
Corollary 3.5.
For all π = π p ,µ, h ∈ Π , we have that
[Equation]
where C is a constant depending on K , T , h ∗ , µ 0 , µ 1 , pand A .
An important consequence of this result is that a plug-in type classiﬁer which relies on consistent estimators of p ∗ , µ ∗ and h ∗ is then consistent w.r.t.
misclassiﬁcation risk.
CLASSIFICATION PROCEDURE
This section is devoted to the presentation and the study of the proposed data-driven procedure that mimics the Bayes classiﬁer.
Our estimation method is then presented in Section 4.1 and theoretical guarantees of the procedure are derived in Section 4.2.
4.1 ESTIMATION STRATEGY
Based on the results of Section 3, we propose an hybrid classiﬁcation procedure which involves both plug-in and empirical risk minimization (E.R.M.) principles.
To this end, we introduce a learning sample D n = { ( T iT , Y i ) , i = 1 , . . . , n } , which consists of n independent copies of ( T T , Y ) .
We propose a two-steps procedure.
In a ﬁrst step, we estimate the vector p ∗ by its empirical counterpart (cid:98) p .
The second step relies on the empirical risk minimization over a suitable set.
In view of the results obtained in Section 3.3, we introduce the following approximation of the set Π :
[Equation]
and the corresponding set of classiﬁers:
[Equation]
Since g ∗ is the minimizer of the misclassiﬁcation risk, a natural estimator of g ∗ would be the empirical risk minimizer over the family G (cid:98) Π
[Equation]
Nevertheless, as a solution of non convex minimization problem, it is known that this estimator is computationally intractable.
Convexiﬁcation To avoid computational issues, it is then natural to replace the classical 0-1 loss with a convex surrogate (see [Zhang, 2004]).
Let us denote the scores functions set:
[Equation]
As convex surrogate, we consider the square loss and then deﬁne for a score function f , the following risk measure
[Equation]
with Z k = 2 1 { Y = k } − 1 .
The choice of the square loss as a convex surrogate is motivated by the fact that, if we deﬁne g ( · ) = argmax k ∈Y f k ( · ) , then
[Equation]
with f ∗ k ( T T ) =
2 π ∗ k ( T T ) − 1 which satisﬁes f ∗ ∈ argmin
[Equation]
R ( f ) .
Hence, consistent procedure w.r.t.
to the L 2risk involves consistent classiﬁcation procedure w.r.t.
the misclassiﬁcation risk.
Resulting estimator As suggested by the form of the optimal score function f ∗ , we then consider the set of scores functions
[Equation]
, and then consider the empirical risk minimizer over (cid:98) F :
[Equation]
f ∈
[Equation]
with
[Equation]
Finally, the resulting classiﬁer (cid:98) g is the plug-in type classiﬁer associated to (cid:98) f deﬁned as
[Equation]
Note that, in order to reduce the computational burden, we have chosen to not introduce the estimation of the probability weights p ∗ in the minimization problem given in
Equation (8).
Nevertheless it remains a possible strategy.
In the next section, we establish rates of convergence of our classiﬁcation procedure.
4.2 RATES OF CONVERGENCE
The study of the statistical performance of (cid:98) g deﬁned by (10) relies on the following assumption.
Assumption 4.1.
Let ε > 0 , we assume that there exists a ε -net
H ε ⊂ H KA , w.r.t. sup-norm (cid:107) · (cid:107) ∞ ,T
such that
[Equation]
where C ε is the number of elements of H ε , d ≥ 1 and C is a positive constant which does not depend on ε .
Theorem 4.2.
Grant Assumptions 3.1, 3.2 and 3.3 and Assumption 4.1.
If h ∗ ∈ H KA , the following holds
[Equation]
where C > 0 depends on K , T , h ∗ , µ 0 , µ 1 , p 0 and A .
Theorem 4.2 establishes that, when n goes to inﬁnity, the proposed classiﬁcation procedure is consistent provided that h ∗ belongs to H KA .
If h ∗ does not belong to H KA , a classical additional bias term appears.
We also have to note that Theorem 4.2 applies for a broad class of functions H .
In particular, Assumption 4.1 covers the case where H is a bounded linear subspace of functions.
Let ( ψ j ) j ≥ 1 an orthonormal basis such that the basis functions are uniformly bounded and then we consider for θ 0 >
[Equation]
as Laguerre basis for example.
Another important example is the parametric exponential family
[Equation]
with β 0 > 0 .
Finally, it is possible to obtain better rate of convergence when the estimation of the probability weights and the estimation of ( µ ∗ , h ∗ ) are performed on two different independent datasets, this is the purpose of the next paragraph.
Alternative strategy Hereafter, we consider an alternative strategy.
First, we split the dataset D n into two independent samples D 1 n and D 2 n .
Fore sake of simplicity, we assume that n is even and that the two datasets D 1 n and
D 2 n have same size n/ 2 .
Based on D 1 n , we estimate p ∗ , and based on D 2 n we estimate f ∗ .
The resulting classiﬁer (cid:98) g satisﬁes the following theorem.
Theorem 4.3.
Grant Assumptions 3.1, 3.2, 3.3 and 4.1.
If h ∗ ∈ H KA , we have
[Equation]
with C > 0 a numerical constant.
Therefore, the classiﬁer (cid:98) g achieves parametric rate of convergence up to a logarithmic term.
Note that from practical point of view, the splitting of the sample does not affect the performance of the classiﬁer ˆ g .
Therefore, we do not consider this strategy in the numerical section.
4.3 COMMENTS
In this section we make comments about the proposed procedure.
Parameter µ Contrary to the parameter p 0 , the estimation procedure requires the knowledge of µ 0 and µ 1 .
This assumption is important to obtain the consistency property.
However, we shall show in Section 5 that the procedure has good performance if we only assume that µ ∗ > 0 .
Estimation of the weights p ∗
For the estimation of the mixture weights, another approach is to include the estimation of p ∗ in the minimization procedure.
In this case, the rate of convergence of the classiﬁcation procedure is the same as the one provided in Theorem 4.3.
However, we do not consider this approach since it signiﬁcantly increases the computational cost of the procedure, especially if the number of classes is large.
Other approach Another strategy is possible motivated by Proposition 3.4.
For example, assuming that the triggering kernels belong to the exponential kernel family, then classical estimators of the parameters can be used.
Therefore, with these estimators we can compute a plug-in type classiﬁer.
For this task, the methods implemented in the tick library as Maximum Likelihood or Least-Squares estimator can be used.
In the next section we illustrate this strategy with the Least-Squares estimator.
NUMERICAL EXPERIMENTS
In this section, we present numerical experiments to illustrate the performance of the procedure described in Section 4.1 and refer to the resulting algorithm as ERM .
We focus on the case where the set H is the parametric exponential family.
Then our method is compared to the plug-in strategy presented in Section 4.3 which is referred as PI .
We also include a comparison with Long Short-Term
Memory ( LSTM ) algorithm.
Indeed, these recurrent neural networks are used in time series forecasting and are a natural solution to study time dependency in data.
The main numerical limitation is that the user needs to choose a length for the data whereas in the case of point processes the length is different for each sequence.
This length has consequently to be chosen large enough to not lose information on the test sample.
We use the tensorflow.keras library of Python with tuning parameters calibrated as follows: batch_size=10 , epochs=100 and learning_rate=0.01 .
The details of the implementation of the ERM estimator are given in Section 5.1.
Then, we describe the experimental setting in Section 5.2 and discuss the obtained results in Section 5.3.
The source code we used to perform the experiments can be found at https://github.com/ charlottedion/HawkesClassification .
5.1 IMPLEMENTATION
We present the implementation of our classiﬁcation procedure in the case where the set of kernel functions H is the parametric exponential family deﬁned as H [Equation]
[Equation]
(cid:55)→ αβ exp( − βt ) , 0 < α < 1 , β > 0 } .
We deﬁne for α, β ∈ R the function h α,
[Equation]
t ) , where expit denotes the inverse-logit function.
Then, we can write H as H =
{ t (cid:55)→ h
α,β ( t ) , α, β ∈ R } .
For α and β in R K , we denote by h α , β the corresponding function of H K .
Therefore the set (cid:98) Π deﬁned in Equation (6) can be rewritten as
[Equation]
Hence the minimization step is performed w.r.t.
µ , α , and β .
Note that the formulation of the above set ˆΠ shows that the optimization part of our classiﬁcation procedure does not require any constraint on the parameters.
The minimization is performed with the Python function minimize with argument method BFGS .
Algorithm 1 sums up the main steps of the procedure.
Algorithm 1 Classiﬁcation algorithm
Input: T , D n , and new observation T n +
Estimate p ∗ on D n
Solve the minimization problem (8) based on D n
Compute (cid:98) g
the resulting classiﬁer (10)
Compute ˆ Y n +
= (cid:98) g ( T n +1 )
Output: Predicted label ˆ Y n +
For the procedure PI , we use the tick function
HawkesExpKern with argument gofit = least-squares for the parameter inference.
5.2 EXPERIMENTAL SETTING
We consider K = 2 or K = 3 classes in the following.
We propose two different models for the experiments that we refer to as Model 1 and Model 2.
For Model 1, we consider the case where the triggering kernel belongs to the parametric exponential family.
For Model 2, we investigate a more general form for the kernels (see below).
We set the baseline intensity µ = 1 .
We use the library tick to generate the sequence of jump times of the Hawkes processes. [Figure]
Synthetic data
The label Y is drawn from a uniform distribution on { 1 , . .
, K } .
Conditionally on Y , we simulate the jump times according to Model 1 and Model 2 which are deﬁned as follows:
Model exponential kernels h ( t ) = αβ exp ( − βt ) , with ( α, β ) = (0 . 7 , 1 . 3) for class Y = 1 , (0 . 2 , 3) for class
Y = 2 , and if K = 3 , (0 . 5 , 5) for class Y = 3 .
Model 2 interpolation function kernels with parameters ( a, b, c ) :
h
[Equation]
t
[Equation] [Equation] [ 0 , t ≥with for ( a, b, c ) = (0 . 2 , 0 . 8 , 0 . 2) for class Y = 1 , (0 . 1 , 0 . 4 , 0 . 2) for Y = 2 , and if K = 3 , (0 . 8 , 0 .
3 , 0 . 7) for class Y = 3 .
As an illustration, Figure 1 displays the considered kernels for both models.
We can see from this ﬁgure that for Modelthe kernel of the class Y = 1 seems to be different of the kernels of the classes Y = 2 and Y = 3 which are more closed.
Hence, it should be easy to discriminate between observations from class Y = 1 and observations from class
Y ∈ { 2 , 3 } .
On the contrary, observations from class Y =and class Y = 3 would be overlapped.
Similar comments can be made for Model 2 with observations from class Y ∈ { 1 , 2 } and observations from class Y = 3 .
We also investigate the role of parameter T on the difﬁculty of classiﬁcation problem.
To this end, Figure 2 displays the error rate of the Bayes classiﬁer as a function of T for
Model 1 and K = 3 .
This error quickly decreases from 0 .to 0 .
05 as T goes from 10 to 40 .
In the following, we shall give results for T = 20 .
[Table] [Table] [Figure] Simulation scheme In order to assess the performance of our procedure, we evaluate the misclassiﬁcation risk of the Bayes classiﬁer, ERM , PI , and LSTM through MonteCarlo repetitions.
More precisely, for n ∈ { 100 , 1000 } and n test = 1000 , we repeat independently 50 times the following steps:
1. simulate two datasets D n and D n test , 2.
from D n compute the classiﬁer (cid:98) g , and 3.
based on D n test , compute the empirical error rate of the three classiﬁers.
The obtained results are presented in Table 1 for n =and Table 2 for n = 1000 .
Note that, for ERM algorithm, the following initial guess for the optimization step is considered: µ = 0 .
5 , α = 1 and β = 1 for all classes.
5.3 RESULTS
From the obtained results, we make several comments.
For
Model 1, the ERM and PI procedures achieve similar performance to the Bayes classiﬁer for n ∈ { 100 , 1000 } , and
K ∈ { 2 , 3 } .
Similar comments can be made for Model
[Figure] and K = 2 . For K = 3 , the difference between the error rates of the Bayes classiﬁer and the ERM and PI classiﬁers is larger.
Let us notice that, when n increases the error rate of ERM is closer to the error rate of the Bayes classiﬁer.
Finally, we can see that the LSTM algorithm has the worst performance in almost every scenario except for model 1 and
K
= 2 .
Hence, our classiﬁer ERM is competitive to classify event sequences and can be recommended for future works.
Let us notice that our procedure also outputs estimations of the parameters ( µ, α, β ) .
Although the estimation task is not our main purpose, it is interesting to evaluate the accuracy of the obtained estimators.
Figure 3 displays a visual description of the obtained estimates for n ∈ { 100 , 1000 } for Model 1 with observations coming from the class Y = 1 .
Again, we can see the impact of the parameter n .
For n = 1000 , the estimation of the three parameters are clearly better than for n = 100 .
Furthermore, for n = 1000 , the resulting estimates are quite good.
DISCUSSION
We investigate the multiclass classiﬁcation setting where the features come from a mixture of simple linear Hawkes processes.
In this framework, we derive the optimal predictor and provide a classiﬁcation procedure tailored to this problem.
The resulting algorithm relies on both plug-in and empirical risk minimization principles.
We establish theoretical guarantees and illustrate the good performance of the method through a numerical study.
In future works, we plan to extend our classiﬁcation procedure to the case where the observations come from a mixture of multidimensional Hawkes processes.
Indeed, in neuroscience, the modeling of multivariate neuron spike data is used for taking into account potential interactions between neurons (see e.g. [Hansen et al., 2015], [Donnet et al., 2020]).
Hence, it should capture the interactions between neurons.
In this framework, a challenge is to take into account the high dimension of the space of parameters.
For example, by considering exponential kernels, plug-in type classiﬁer should beneﬁt from algorithm as ADM4 which is adapted for high dimensional setting [Bacry et al., 2020].
Another possible development is the case of nonlinear
Hawkes process.
A few works focus on this subject, see e.g. [Brémaud and Massoulié, 1996], [Lemonnier and Vayatis, 2014], [Costa et al., 2020].
This allows us to consider kernels which can take negative values to model an inhibitory behaviour.
The proposed algorithm should remains efﬁcient.
Nevertheless, it will be trickier to establish rates of convergence.
Finally, we could also extend our method to a model with a common time-inhomogeneous baseline.
This idea is considered in many applications (see e.g. [Li et al., 2017]) and could be an improvement of the present algorithm.