FaDIn: Fast Discretized Inference for Hawkes Processes with General Parametric Kernels
Authors: Guillaume Staerman
Cédric Allain
Alexandre Gramfort
Thomas Moreau
Venue: Submitted to ICLR 2023 (2023)
Keywords: Hawkes processes
Neuroscience

Abstract: Temporal point processes (TPP) are a natural tool for modeling event-based data. Among all TPP models, Hawkes processes have proven to be the most widely used, mainly due to their simplicity and computational ease when considering exponential or non-parametric kernels. Although non-parametric kernels are an option, such models require large datasets. While exponential kernels are more data efficient and relevant for certain applications where events immediately trigger more events, they are ill-suited for applications where latencies need to be estimated, such as in neuroscience. This work aims to offer an efficient solution to TPP inference using general parametric kernels with finite support. The developed solution consists of a fast L2 gradient-based solver leveraging a discretized version of the events. After supporting the use of discretization theoretically, the statistical and computational efficiency of the novel approach is demonstrated through various numerical experiments. Finally, the effectiveness of the method is evaluated by modeling the occurrence of stimuli-induced patterns from brain signals recorded with magnetoencephalography (MEG). Given the use of general parametric kernels, results show that the proposed approach leads to a more plausible estimation of pattern latency compared to the state-of-the-art.


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FADIN: FAST DISCRETIZED INFERENCE FOR HAWKES PROCESSES WITH GENERAL PARAMETRIC KERNELS
Anonymous authors Paper under double-blind review
ABSTRACT
Temporal point processes (TPP) are a natural tool for modeling event-based data. Among all TPP models, Hawkes processes have proven to be the most widely used, mainly due to their adequate modeling for various applications. in par- ticular when considering exponential or non non-paraeetric kemels. Although non- parametric kernels are an option, such models require large datasets. While expo- nential kernels are more data efficient and relevant for certain applications where events immediately trigger more events, they are ill-suited for applications where latencies need to be estimated, such as in neuroscience. This work aims to offer an efficient solution to TPP inference using general parametric kernels with finite support. The developed solution consists of fast L. rradient-based solver lever- aging a discretized version of the events. After supporting the use discretization theoretically, the statistical and computational efficiency of the novel approach is demonstrated through various numerical experiments. Finally, the effectiveness of the method is evaluated by modeling the occurrence of induued patterns from brain signals recorded with magnetoenccehaalggaphh (MEG). Given the use of general parametric kernels, results show that the proposed approach leads to a more plausible estimation of pattern latency compared to the late-of-thhe-rt
INTRODUCTION
The statistical framework of Temporal Point Processes (TPPs; see egg., Daley & Vere-Jones 2003) is well adapted for modeling event-based data. It offers principled way to predict the rate of events as a function of time and the previous events" history. TPPs are historically used to model intervals be- tween events, such as in renewal theory, which studies the sequence of intervals between successive replacements of a component susceptible to failure. TPPs find many applications in neuroscience, in particular, to model single-cell recordings and neural spike trains (Truccolo et al., 2005; Okatan et al., 2005; Kim et al., 2011; Rad & Paninski, 2011), occasionally associated with spatial statis- ties (Pillow et al., 2008) or network models (Galves Löcherbach, 2015). In the machine learning community, there is growing interest in these statistical tools (Bompaire, 2019; Shchur et al., 2020; Mei et al., 2020). Multivariate Hawkes processes (MHP; Hawkes 1971) are likely the most popular, as they model interactions between each univariate process. They also have the peculiarity that a process can be self-exciting, meaning that a past event will increase the probability of having an- other event in the future on the same process. The conditional intensity function is the key quantity for TPPs. With MHP, it is composed of a baseline parameter and kernels. It describes the probability of occurrence of an event depending on time. The kernel function represents how processes influ- ence each other or themselves. The most commonly used inference method to obtain the baseline and the kernel parameters of MHP is the maximum likelihood (MLE: see e.g., Daley Vere-Jones, 2007 or Lewis & Mohler, 201 I). One alternative and often overlooked estimation criterion is the least squares & error, inspired by the theory of empirical risk minimization (Reynaud- -ourret & Rivoirard, 2010; Hansen et al., 2015; Bacry al., 2020).
A key feature of MHP modeling is the choice of kernels. Non-parametric and parametric kernels are the two possibilities. In the non-parametric setting, kernel functions are approximated by histograms (Lewis& & Mohler, 2011; Lemonnier & Vayatis, 2014), by a linear combination of pre-defined func- tions (Zhou et al., 2013a; Xu et al., 2016), by functions lying in RKHS (Yang et al., 2017) or, alter- natively, by neural networks (Mei & Eisner, 2017: Shchur et al., 2019; Pan et al., 2021). In addition to the frequentist approach, many Bayesian approaches, such as Gibbs sampling (Ishwaran & James,

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2001) or (stochastic) variational inference (Hoffman et al., 2013), have been adapted to MHP in par- ticular to fit non- paraamerrercc kernels. Bayesian methods also rely on the modelling of the kernel by histograms (e.g.. Donnet et al., 2020) by linear combination of pre-defined functions (e.g. Lin- derman & Adams, 2). These approaches are designed whether in continuous-time (Rasmussen, 2013; Zhang et al., 2018; Donnet et al., 2020; Sulem al., 2021) or in discrete-time (Mohler al., 2013; Linderman & Adams, 2015; Zhang et al., 2018; Browning et al., 2022). These functions allow great flexibility for the shape of the kernel, yet this comes at the risk of poor estimation of it when only small amount of data is available (Xu et al., 2017). Another approach to estimate the intensity function is to consider kernels parametrized by 17. Although it can introduce a potential bias by as- suming particular shape for kernels, this approach has several benefits. First, it reduces inference difficulties, as 7j is typically lower dimensional compared to non-parametric kernels. Moreover, for kernels satisfying the Markov property (Bacry et al., 2015), computing the conditional intensity function is linear in the total number of timestamps/cvents. The most popular kernel belonging to this family is the exponential kernel (Ogata, 1981). It is defined by l (a, 7) + ar exp( where a and y are the scaling and the decay parameters, respectively (Veen & Schoenberg, 2008; Zhou et al., 2013b). However, pointed out by Lemonnier & Vayatis (2014), the maximum likeli- hood estimator for MHP with exponential kernels is efficient only if the decay 7 is fixed. Thus, only the scaling parameter is usually inferred. This implies that the hyperparameterr y must be chosen in advance, usually by using a grid search, random search, or Bayesian optimization. This leads to a computational burden when the dimension of the MHP is high. The second option is to define r de- cay parameter common to all kernels, which results in a loss of expressiveness of the model. In both cases, the relevance of the exponential kernel relies on the choice of the decay parameter, which may not be adapted to the data (Hall & Willett, 2016). For more general parametric kernels which do not verify the Markov property, the inference procedure with both MLE or & loss scales poorly as they have quadratic computational scaling with the number of events, making their use limited in practice (see e.g. Bompaire, 2019, Chapter 1). These limitations for parametric and non-pan porammmmettiic kernels prevent their usage in some applications, pointed out by Carreira (2021) in finance Allain et al. (2021) in neuroscience. A strong motivation for this work is also neuroscience applications.
The quantitative analysis of electrophysiological signals such electroencephalograpyy (EEG) or magnetoenncephallgrrahyy (MEG) is a challenging modern neuroscience research topic (Cohen, 2014). By giving non-invasive way to record human neural activity with high temporal res- olution, EEG and MEG offer a unique opportunity to study cognitive processes as triggered by controlled stimulation (Baillet, 2017). Convolutional dictionary learning (CDL) an unsupervised algorithm that has recently been proposed to study M/EEG signals (Jas et al., 2017; Dupré la Tour et al., 2018). It consists in extracting patterns of interest in M/EEG signals. It learns a combination of time-invariant patterns- called atomy- and their activation function to reconstruct the signal sparsely. However, while CDL recovers the local structure of signals, it does not provide any global information, such as interactions between patterns or how their activations are affected by stimuli. Atoms typically correspond to transient bursts of neural activity (Sherman et al., 2016) or artifacts non-invasive electro-
such as eye blinks or heartbeats. By offering event-based based perspective magnetic brain signals, CDL makes Hawkes processes amenable to M/EEG- based studies. Given the estimated events, one important goal is then to uncover potential temporal dependencies between external stimuli presented to the subject and the appearance of the atoms in the data. More precisely, one is interested in statistically quantifying such dependencies, e.g., by estimating the mean and variance of the neural response latency following a stimulus. In Allain et al. (2021), the authors ad- dress this precise problem. Their approach is based on an EM algorithm and Truncated Gaussian kernel, which can cope with only few brain data, as opposed to non- non-parametric kernels, which are more data hungry. Beyond neuroscience, Carreira (2021) use a likelihood based approach using exponential kernels to model order book events. Their approach use high-frequenc trading data, taking account of latency at hand in the proposed loss.
This paper proposes new inference method - named FaDIn to estimate any parametric kernels for Hawkes processes. Our approach based on two key features. First, we use finite-support kernels and discretization applied to the ERM-inspired least-squares loss. Second, we propose to employ some precomputations that significantly reduce the computational cost. We then show that the implicit bias induced by the discretization procedure is negligible compared to the statistical error. Further, we highlight the efficiency of FaDIn in computation and statistical estimation over the non-parametric approach. Finally, we demonstrate the benefit of using general kernel with MEG
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data. The flexibility of FaDIn allows to model neural response to external stimuli with much better-adapted kernel than the existing method derived Allain et al. (2021).
FAST DISCRETIZED INFERENCE FOR HAWKES PROCESSES (FADIn)
2.1 HAWKES PROCESSES
Given stopping time T' R. and an observation period []. temporal point process (TPP) is stochastic process whose realization consists of a set of distinct timestamps Fr = (In, in E [0. T} occurring in continuous time. The behavior of a TPP is fully characterized by its intensity function that corresponds to the expected infinitesimal rate at which events are occurring at time E [o, T] The values of this function may depend on time (e.g., inhomogeneous Poisson processes) or rely on past events such as self-exciting processes (see Daley & Vere-Jones 2003 for an excellent account of TPP). For the latter, the occurrence of one event will modify the probability of having a new event in the near future. The conditional intensity function À : [0, T] + R+ have the following form:
P (Netde N, +1
(t)
lim dt
dt
where N, = En21 Lantt is the counting process associated to the PP. Among this family, Mul- tivariate Hawkes processes (MHP; Hawkes, 1971) model the interactions of p E N. self-exciting
TPPs. Given p sets of timestamps J/ = -th " [0, (0,T)) -,....P. P each process
described by the following intensity function:
E S.00 de ()) dNg 5=1
(e(1) = pr +
where μ4 is the baseline parameter, N₂ I.. N?" the associated multivariate counting process and diy [0, 7] + R4 the excitation function - called kernel representing the influence of j-th process' past events onto i-th process' future events. From an inference perspective, the goal is to estimate the baseline and kernels associated with the MHP from the data. In this paper, we focus on the ERM-inspired least squares loss. Assuming class of parametric kernel parametrized by n, the objective is to find parameters that minimize (see e.g., Eq. (I.2) in Bompaire, 2019, Chapter I):
P Σ NT
% ds ds -
C(0, (0,
Σ > ( ) K,
where Ny- Ei= N is the total number of timestamps, and where we denote by 0 (H, 17)- Interestingly, when used with an exponential kernel, this loss benefits from some precomputations of complexity O(Nr), making the subsequent iterative optimization procedure independent of Nr- This computational ease is the main advantage of the loss L over the log-likelihood function. How- ever, when using a general parametric kernel, these precomputations require O((NT)²) operations killing the computational benefit of the lz loss L over the log-liieldoood. It is worth noting that this loss differs from the quadratic error minimized between the counting processes and the integral of the intensity function, used in Wang et al. (2016); Eichler et al. (2017) and Xu et al. (2018).
2.2 FADIN
The approach we propose in this paper fills the need for general parametric kernels in many appli- cations. We provide a computationally and statistically efficient solver coined FaDIn that works with many parametric kernels using gradient- gradient-based algorithms. Precisely, it relies on the three key ideas: (i) the use of parametric finite-support kernels, (ii) discretization of the time interval [0, T). and (iii) precomputations allowing an efficient optimization procedure detailed below.
Finite support kernels core bottleneck for MLE or & estimation of parametric kernels is the need to compute the intensity function for all events. For general kernels, the intensity function usually requires ((N))² operations, which makes it intractable for long-time length processes. To make this computation more efficient, consider finite support kernels. Using finite support
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kernel amounts to setting limit in time for the influence of past event on the intensity, i.e. Vt & [0, W., (t)(t) = O, where W denotes the length of the kernel's support. This assumption matches applications where event camnot have influence far in the future, such in neuroscience (Krumin et al., 2010; Fichler et al., 2017; Allain et al., 2021) or high-frequency trading (Bacry et al., 2015; Carreira, 2021). The intensity function (3) can then be reformulated as convolution between the kernel Øū and the sum of Dirac functions 2 = Ee Eng de; located at the event occurrences "
P
x(11) 14 +
Pig 2,)), Σ /=1
EE (0.)
As diy has finite support, the intensity can be computed efficiently with this formula. Indeed, only events in the interval [t W. need to be considered. This usually amounts to a fraction of the events of the full process.
Discretization To make these computations even more efficient, we propose to rely on discretized processes. Most Hawkes processes estimation procedures involve continuous paradigm to min- imize (2) or its log-likelihood counterpart. Discretization has been investigated so far for non- parametric kernels (Kirchner, 2016; Kirchner & Bercher, 2018; Kurisu, 2018). The discretization of a TPP consists in projecting each event t on a regular grid G = {0, A, 2A,..., GA), where G = - We refer to as the stepsize of the discretization. Here L-] denotes the floor function. Let J be the set of projected timestamps of F/. on the grid G. The intensity function of the i-th process of our discretized MHP is defined as:
[s] H. Σ
Σ E
0ig(s -
Pmm) = +
[=]y]| 7), SE [0
(3)
( [+)
W where L = [ denotes the number of points on the discretized support, +t) (sy(s is the kernel value on the grid and 2i[s] = # { - SA I } } denotes the number of events projected on the grid timestamp From now and throughout the rest of the paper, we denote ( R_ R+ as a function while + represents the discrete vector o R4 Compared to the continuous formulation, the intensity function can be computed more efficiently as one can rely on discrete convolutions, whose worst case complexity scales as ON. can also be further accelerated using Fast Fourier Transform when Nr is large. Another benefit of the discretization is that for kernel whose values are costly to compute, at most values need to be computed. This can have a strong computational impact when N₂ >> L as all values can be precomputed and stored.
While discretization improves the computational efficiency, also introduces a bias the computa- tion of the intensity function and, thus potentially, in estimating the kernel parameters. The impact of the discrctization on the estimation considered Scction 2.3 and Section 1. Note that this bias is similar to the one incurred by quantizing the kernel histograms for pon-aaametric estimators.
Loss and precomputations FaDIn aims at minimizing the discretized &g loss, which approximates the integral on the left part of (2) by a sum on the grid & after projecting timestamps of Fr on it. boils down to optimizing the following loss:
Lg 00
(T)
A
-
2
X
2
(4)
NT
A
sE|D.G]
To find the parameters of the intensity function 0, FaDIn minimizes Lg using first-order gradient- based algorithm. The computational bottleneck of the proposed algorithm thus the computation of the gradient VLg regarding parameters 0. Using the discretized knite-suupon kernel, this gradient can be computed using convolution, giving the same computational complexity as the computation of the intensity function L).
However, gradient computation can still be too expensive for long processes with many events to get reasonable inference times. Using the least squares error of the process (4), can further reduce
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the complexity of computing the gradient by precomputing some constants D,(T: G). V (T; and Φ (T; +T) that do not depend on the parameter Indeed, by developing and rearranging the terms in (4), obtains:
G 35|s -T r)
Nr Lg (0. -1IM
24
Σ
im1
jm
(T(O)
P
P
P
IH
Σ 2/1s
7 2 71
2
N T/S +
Σ 2j
7
A
==
b (T:
The term V VT,K(T, r': G) dominates the computational cost of our precomputations. It requires O operations for each tuples (T, and (j, k). Thus, it has total complexity of O L²G) and the bottleneck of the precomputation phase. For any m E {1.....Pl p), the gradient of the loss w.r.t. the baseline parameter given by:
0CG pm 2NT 24 NT 0pm
L
m (T:
For any tuple (m, 1) E [I,...,p]* 22,, the gradient of V/ml is:
dLg 2mpm
@
00 VL,(T,,';;))
Nr mm
(T:
2
mmkk [r']
Omm.
Mmmd
ag mmil
2
D (T(T:
Gradients of kernel parameters dominate the computational cost of gradients. The complexity is of OppI.)) for each kernel parameter, leading to total complexity of Opp2I2) and is independent of the number events Ny. Thus, a trade-off can be made between the precision of the method and its computational efficiency when varying the size of the kernel's support or the discretization.
Optimization The inference is then conducted using gradient descent for the Eg loss Lg. FaDIn thus allows for very general parametric kernels, as exact gradients for each parameter involved in the kernels can be derived efficiently as long as the kernel is differentiable and has a finite support. Gradient-based optimization algorithms can therefor be used without limitation, in contrast with the EM algorithm which requires close form solution to zero the gradient, which is difficult for many kernels. An important remark is that the problem generally non-convex and may converge to local minimum.
2.3 IMPACT OF THE DISCRETIZATION
While discretization allows for efficient computations, it also introduces perturbation in the loss value. In this section, we quantify the impact of this perturbation the parameter estimation when A goes to O. Through this section, consider we observe a process Fr whose intensity function is given by the parametric form X(:; Note that if the process .F 's intensity is not in the parametric family X(=; 0). 6" is defined as the best approximation of its intensity function in the l sense. The goal of the inference process thus to recover the parameters 0".
When working with the discrete process Fy. the events tú of the original process are replaced with projection on a grid i 1, + 5 Here, 6% is uniformly distributed on |-A/2, A/2). We consider
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the discrete FaDIn estimator A defined as 0A arg min (0). We upper-bound the error
incurred by A by the decomposition:
00 0"
0
0-
+
03
0
(5)
where 0-- arg ming (0) is the reference estimator for 0* based on the standard &g estimator for continuous point processes. This decomposition involves the statistical error (*) and the bias error (x*) induced by the discretization. The statistical term measures how far the parameters obtained by minimizing the & continuous loss having access to a finite amount of data are from the true ones. In contrast, the term (**) represents the discretization bias induced by minimizing the discrete loss (4) instead of the continuous one (2). In the following proposition, we focus on the discretization error (**) which is related the computational trade-off offered by our method and not on the statistical error of the continuous lg estimator (**). Our work showcases that this disregarded estimator can be efficiently computed, and we hope it will promote research describe its asymptotic behavior. We now study the perturbation of the loss due to the discretization.
Proposition 1. Let FT and FT be respectively a MHP process and its discretized version on grid with stepsize A. Assume that the intensity function of FT possesses contimuously differentiable finite support kernels on [0, W1. Thus, assuming A < minl M.nn 1t - t//n l. for any E [1 pl. holds:
tym + 0(A2) 0bis (sA Øt
[sss] (ssA))
Σ
8
1=1 th
2 NT C
dos 5tn 0t (t, Mm: 0(A3)
Co) = [()) + (.) +
(8%
The technical proof deferred to Section B.1 the Appendix. The first result is a direct appli- cation of the Taylor expansion of the intensity for the kernels. For the loss, the first perturbation term (f) comes from approximating the integral with a finite Euler sum (Tasaki, 2009) while the second one derives from the perturbation of the intensity. This proposition shows that as the discretization step A goes to O, the perturbed intensity and &g loss are good estimates of their con- tinuous counterpart. We now quantify the discretization error (**) as A goes to O.
Proposition 2. We consider the same assumption as in Proposition 1. Then, the estimators Ō, arg ming L()) and 02 = arg ming (0) are uniquely defined, 02 converges to O. as - O. Moreover, if C is C and its hessian 2² V²)( is positive definite with e > O its smallest eigenvalue, then 00 0ell2 < (g(0s). with (( = O(1).
This proposition shows that asymptotically on A, the estimator 0^ is equivalent to θ.. It also shows that the discrete estimator converges to the continuous one at the same speed A decreases. This confirmed experimentally by results shown in Figure A.7 in the Appendix. Thus, one would need to select A so that the discretization error is small compared to the statistical one.
3 NUMERICAL EXPERIMENTS
We present various synthetic data experiments to support the advantages of the proposed approach. To begin, we investigate the bias induced by the discretization in Section 3.1. Afterwards, the sta- tistical and computational efficiency of FaDIn is highlighted through a benchmark with popular non non-parametric approaches Section 3.2. Due to the space limitation, sensitivity analysis regard- ing the parameter W and additional non-parametric comparisons are provided in Section A.1 and Section A.2, respectively.
3.1 CONSISTENCY OF DISCRETIZATION
In order to study the estimation bias due to discretization, we run two experiments and report the results in Figure 1 (details and further experiments presented in Section A.3 and Section A.4

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in the Appendix). The general paradigm is one-dimensional TPP with intensity parametrized as in (1) with a Truncated Gaussian kernel of mean m E IR and standard deviation o > 0, with fixed support [0,, W| C IR W > O. It corrssonnds to o(-) ax(()) 0 with
<<-) = (: o, W) o
10s-sW
where f (resp. F) is the probability density function (resp. cumulative distribution function) of the standard normal distribution. Hence, the parameters to estimate are fr and rj (a, m, m,
In both experiments, for multiple process length T, the discrete estimates are computed for vary- ing grid stepsize A, from 10-1 to 10-3 The parameter W is set to 1. The 62 norm of the dif- ference between estimates and the true parameter values -the ones used for data simulation- computed and reported. We first computed the parameter estimates with our FaDIn method for T' E {10", 1 10*, 100), }, for 100 simulations each time. Second, since we wish to separate dis- cretization bias from statistical bias, we compute the estimates with an EM algorithm, both con- tinuously and discretely, and that for 50 random data simulations. For the latter, the process is not self-excited, but rather driven by an exogenous homogeneous Poisson Process (Allain al., 2021).
One can observe that the l2 errors between discrete estimates and true parameters tend towards zero T increases. For 7 fixed, one see plateaus starting for stepsize values that are not particularly small, indicating that the discretization bias is limited. The second experiment with the EM algorithm shows that when the plateau mentioned above reached, it corresponds some statistical error. In other words, even for a reasonably coarse stepsize, the bias induced by the discretization slight compared to the statistical error.
T
Method EM
10'
10
10
10
FaDin
Cont. EM
10
39 eerror
10
10
10-1
2
10
10
10
10
A
Figure 1: Median and interquartile error bar of the & norm between true parameters and parameter estimates computed with FaDIn (left) and with EM algorithm (right), continuously and discretely, w.r.t. the stepsize of the grid A.
3.2 STATISTICAL AND COMPUTATIONAL EFFICIENCY OF FADIN
We compare FaDIn with non-parametric methods by assessing approaches' statistical and compu- tational efficiency. To learn the parammeteric kernel, we select various existing methods. The first benchmarked method uses histogram kernels and relies on the EM algorithm, provided Zhou et al. (2013a) and implemented in the ti iick library (Bacry et al., 2017). The kernel is set with one basis function. The three other approaches involve a linear combination of pre-defined raised cosine functions as non- non-parametric kernels. The inference is made either by stochastic gradient descent algorithm (Non-paree non-param SGD; Linderman & Adams, 2014) or by Bayesian approaches such as Gibbs sampling (Gibbs) or Variational Inference (VB) from Linderman & Adams (2015). These algo- rithms are implemented in the pyhawkes library. In the following experiments, we set the number of basis to five for each method. More precisely, we simulate a two-dimensional Hawkes process (repeated ten times) using the tick library with baseline fe [0.1, 0.2) and Raised Cosine kernels:
( = asi
1+ cos
Mi.j π
[.- E [ma.s u. + 20, (1,5) E [1,2]
Tij
1.5
0.1
0.1 0.3
00.3 0.25
with parameters a
0.1
1.5
u
0.3
0.3
and o = 0.3 0.3
Further, we infer
the intensity function of these underlying Hawkes processes using FaDIn and the four previously
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mentioned methods setting A = 0.01 for all these discrete approaches. This experiment is repeated for varying values of T' E {10*, 10*, 10"}. The averaged (over the ten runs) normalized & error the same discrete grid), as well as the associated computation time,
on the intensity (evaluated are reported in Figure 2. From a statistical perspective, we can observe the advantages of FaDIn inference for varying 7 over the benchmarked methods. It is worth noting that this result is expected by parametric approach when the used kernel belongs to the same family as the one with which events have been simulated. From computational perspective, FaDIn is very efficient compared to benchmarked approaches. Indeed, it scales very well with an increasing time T and then with growing number of events. In contrast, other methods depend on the number of events and scale linearly with the time T. For completeness, different kernel shapes are provided in Section 1.2.1.
FaDin
Non-param EM
Non-param SGD
Gibbs
VB
Estimation error
Computation time
A 1 G
109
10a
Time (s.)
10
10
10a
10a
10 T
10
10a
10a T
109
Figure 2: Comparison of the statistical and computational efficiency of FaDIn with four bench- marked methods. The averaged (over ten runs) statistical error on the intensity function (left) and the computational time (right) are computed regarding the time T (and thus the number of events).
4 APPLICATION TO MEG DATA
Electrophysiology signals recorded with M/EEG contain recurring prototypical waveforms that can be related to human behavior (Shin et al., 2017). Convolutional Dictionary Learning (CDL; Jas al. 2017) is an unsupervised method efficiently extract such patterns and study them in quantitative way. With CDL, multivariate neural signals are represented by set of spatio-temporal patterns, called atoms, with their respective onsets, called activations. Here, we make use of the alphacse software for CDL with rank-1 constraint (Dupré la Tour et al., 2018), as it includes physical priors for the patterns to recover, namely that the spatial propagation of the signal from the brain to sensors is linear and instantaneous. The schema in Figure A.14 in the Appendix presents how CDL applies to MEG recordings.
Experiments on magnetoenceehaalggaphy (MEG) data were run on two datasets from the MNE Python package (Gramfort et al., 2013; 2014): the sample dataset and the somatosensory (somato) dataset'. These datasets were selected as they elicit two distinct types of event-related neural ac- tivations: evoked responses which are time-locked to the onsets of the stimulation, and induced responses which exhibit larger random jitters. The sample dataset contains M/EEG recordings of a human subject presented with audio and visual stimuli. This experiment presents checkerboard patterns to the subject in the left and right visual field, interspersed with tones to the left or right ear. The experiment lasts about 4.6 min, and approximately 70 stimuli per type are presented to the subject. For the somato dataset, a human subject is scanned with MEG during 15 min, while 111 stimulations of his left median nerve were made.
For both datasets, raw data first preprocessed as done by Allain et al. (2021), and CDL is then applied: 40 atoms of duration 1 s each are extracted on the sample dataset, and 20 atoms of duration 0.53s for the somato dataset. Finally, each dataset is represented by two sets of Temporal Point Processes: a set of stochastic ones representing atoms' activations, and a set of deterministic ones coding for external stimuli events. The main goal of applying the TPP framework such data
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Indx. htm1
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auditory
visual
Spatial
Temporal
Intensity TG
Intensity RC
Intensity NP
0.4
40
0.095
Atom 2
0.2
20
0.087
0.090
0.0
M~
W
0
0.4
40
0.187
0.185
0.2
Atom
20
0.190
0.0
0.5
0
0.25
0.5
0
0.25
0.5
0.00
0.25
0.50
Time (s)
Time (s)
Time (s)
Time (s)
Figure 3: Spatial and temporal patterns of atoms from sample dataset, and their respective esti- mated intensity functions following stimulus (cue at time = 0 s), for auditory and visual stimuli with non- paraparaettrcc kernel (NP) and two kernel parametrizations: Truncated Gaussian (TG) and Raised Cosine (RC).
to characterize directly when and how each stimulus is responsible for the occurrence of neural responses, especially by estimating the distribution of latencies. We are interested in the paradigm of Driven Point Process (DriPP; Allain et al. 2021) and for every extracted atom, its intensity function related to the corresponding stimuli is estimated using non parametric kernel (NP) and two kernel parametrizations: Truncated Gaussian (TG) and Raised Cosine (RC). Results on the sample (resp. somato) dataset are presented in Figure 3 (resp. Figure A. in the Appendix), where only the kernel related to each type of stimulus is plotted, for the sake of clarity. See Appendix A.5 for more details.
Results show that all three kernels agree on a peak latency around 90 ms for the auditory condition and 190 ms for the visual condition. Due to the limited number of events, one can observe that the non-parametric kernel estimated less smooth, with spurious peaks later the interval. Overall, these results on real MEG data demonstrate that approach with RC kernel parametrization is able to recover correct latency estimates even with the discretization of stepsize 0.02. Furthermore, the usage of RC allows to have sharper peaks in the intensity compared TG, enforcing the link between the external stimulus and the atom's 's activation. This difference mainly comes from the fact that RC does not need to have its support pre-determined. This advantage is even more pronounced in the case of induced responses, such as in the somato dataset (see Figure A.15), where the range of possible latency values is more difficult determine beforehand.
DISCUSSION
This work proposed efficient approach, FaDIn, infer general parametric kernels for Multivari- ate Hawkes processes. Our method makes the use of parametric kernels computationally tractable, beyond exponential kernels. The development of FaDIn based on the three key features: (i) finite- support kernels. (ži) timeline discretization and (iii) precomputations reducing the computational cost of the gradients. These allow for a computationally efficient gradient-based approach, improv- ing state-of-theeart methods while providing flexible use of kernels well-fitted to the considered applications. Moreover, this work shows that the bias induced by the discretization negligible, both theoretically and numerically. By allowing the use of general parametric kernel in Hawkes processes, this contribution opens new possibilities for many applications. This is the case with M/EEG data, where estimating information about the rate and latency of occurrences of brain signal patterns at the core of neuroscience questions. Therefore, FaDIn makes it possible to use a Raised Cosine kernel, allowing for efficient retrieval of these parameters.

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Under review as a conference paper at ICLR 2023
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13


Reviews:
[{"review_id": "wTm_Q2zgNLv", "reviewer": "Reviewer_EZux", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors deal with the problem of performin inference for multivariate hawkes processes with parametric kernels. They propose an efficient discretized inference scheme that aims to optimize the least square objective (rather than max likelihood for instance) which does not assume exact events but aggregated counts, hence the name discretized inference. Their approach covers parametric kernels and the efficiency of the compution relies on pre-computations and on finite supports. Finite supports means that they discard events in the long past as they claim these have low infleunce for the scope of their applications being to quantify the dependence between neural response and external stimuli.", "summary_of_the_review": "Overall this paper targets a known problem, that of computationally expensive inference for multivariate hawkes processes. The authors motivate their work within the context of neuroscience in which hawkes process can find great use and can bring interesting conclusions on neural activity. The assumptions the aithors make to achieve their computationally efficient inference are reasonable but the choice of loss function is not well motivated or justified and needs further support (theoretical guarantees, empirical examples on higher dimensions and comparisons). Additionally several claims supporting the proposal choices are incorrect. Overall although the authors make interesting suggestions for a known problem, the paper as it is does not have enough methodological support for a publication.", "strength_and_weaknesses": "Strengths of the paper are firstly the efficient computation for multivariate hawkes processes which is a non-trivial task. To achieve this the authors make some assumptions and relaxations such as assuming aggregated rather than exact events and finite support. They choose to minimize a least square loss which they achieve efficiently with some pre-computations. The authors motivate well their proposal and the benefit of hawkes process efficient inference is clear especially withing the neuroscience context. The application is both relevant and interesting.\n\nThe paper disregards bayesian inference approaches to the problem - and there are quite a lot both in previous and in the recent literature covering both exact and aggregated approaches. Many authors have proposed different approaches to multivariate hawkes processes that in fact rely on less approximations than those proposed here. For instance when it comes to efficient computation of discretized hawkes processes approaches already exist (G. Mohler 2013). \nAdditionally when it comes to non-parametric kernel approaches, there are way more than the paper suggests (see papers by J. Rousseau). In the context of non-parametric vs parametric approaches, the authors claim throughout that parametric approaches are less prone to over-fitting in comparison to parametric ones and use this as an advantage of their parametric kernels. However this is not true. If the authors wanted to make a statements about the potential advantages of parametric approaches vs nonparametric ones, these could be around other topics such as model mismatch. Other claims that are not true are for instance that Hawkes processes are easy to fit among other temporal point processes. The contrary, considering homogeneous or other inhomogeneous poisson processes often used to explain temporal or spatial patterns, hawkes processes are not striaghtforward to fit.\nThe choice of the loss function is not well motivated nor justified and should be compared theoretically and empirically with max likelihood-based approaches, as well as bayesian ones ideally. The loss function choice should not be arbitrary (or merely serving computational purposes) but should be properly justified in regards to the model being used. \n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "qrJTxn0OQJZ", "reviewer": "Reviewer_h6fy", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes an approximate version of the multivariate Hawkes process. The authors claim that the existing estimation procedure has significant limitations, especially in the computational cost. The authors propose a variant of the binned Hawkes process, where the continuous decay function is replaced by a discrete vector. \n", "summary_of_the_review": "The paper is overall well-written. However, the technical novelty is not clearly justified by the authors not only empirically but also methodologically. ", "strength_and_weaknesses": "**Strength**\n\n- Addresses the important issue of computational issue of the multivariate Hawkes process.\n- Developed a practically feasible procedure. \n- Provided a theoretical analysis on the L2 approach. \n\n**Weakness**\n\n- The theoretical basis is not very clear. Especially the justification of the main loss function is not clearly provided. Chapter 2 of Bompaire (2019) does not explicitly derive Eq.(2). The text looks misleading. \n\n- Introduciton misses a major part of the efforts that have been made so far to address the computational issues. Notably,  an EM-like iterative scheme, first proposed by Veen and Schoenberg, has been known as one of the standard approaches to address the issue. But apparently, little attention is paid to that thread. \n\t> Veen, Alejandro, and Frederic P. Schoenberg. \"Estimation of space–time branching process models in seismology using an em–type algorithm.\" Journal of the American Statistical Association 103.482 (2008): 614-624.\n\n\n\n- Also, I don't think it is true that existing multivariate Hawkes processes fail to estimate the decay parameter. In fact, many recent works such as the below seem to estimate the decay parameter without significant issues with the aid of the Veen-Schoenberg algorithm.\n\t>Idé, T., Kollias, G., Phan, D., & Abe, N. (2021). Cardinality-Regularized Hawkes-Granger Model. Advances in Neural Information Processing Systems, 34, 2682-2694.\n\n\n- Finally, it is not very clear how the proposed method is compared with the existing binned Hawkes process, in particular Kirchner's INAR(p) model. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "pyOsweAVZR", "reviewer": "Reviewer_GU8n", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The submission provided a TPP model estimation method based on a L2 loss by assuming parametric finite-support kernel, discretization and precomputation. The statistical and computational efficiency of the method is compared to a nonparametric method. Finally, the method is applied to the MEG data. ", "summary_of_the_review": "The major concern of the submission is novelty. All key ideas of the proposed method: L2 loss, finite-support, discretization and precomputation, are all existing in previous works. There is nothing new in the methodology in this submission. Another conceren is that the experiment is not convincing. The submission compares the statistical and computational efficiency of the proposed method to a nonparametric method, and shows the proposed method is better w.r.t. accuracy and efficiency. On the one hand, the simulated data is from a parametric model and the proposed method also assumes the same parametric form for the kernel, of course the ground-truth parametric model would get a more accurate estimation with limited data; on the other hand, the efficiency of nonparametric methods are certainly less efficient than parametric methods in general. The experiment section does not provide any meaningful comparision. ", "strength_and_weaknesses": "Strength: The paper is easy to follow and the derivation of inference seems correct but I did not check it carefully.\n\nWeakness: The submission is lacking in novelty and the experiment is not convincing. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "BTORP9mulhO", "reviewer": "Reviewer_mV6v", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes discretizing and limiting kernel support for efficiently learning kernel parameters of a Hawkes process via gradient descent. Additionally, the paper provides theoretical insights measuring the discretization bias. Experimental results on synthetic and electrophysiology datasets show improved estimation of kernel parameters and computation time compared to baselines.", "summary_of_the_review": "The technical novelty is limited and experimental analysis is unconvincing.", "strength_and_weaknesses": "**Strengths**\n- Theoretical analysis of discretization impact may be of interest to some researchers \n- The proposed approach improves computational efficiency \n- The proposed approach seems easy to implement\n\n**Weaknesses**\n\n*Technical limitations*\n- The proposed approach is limited to parametric self-exciting Hawkes kernels which are often violated in practice \n- The proposed approach could be sensitive to the selected discretization step-sizes $\\Delta$ and the length of kernel's support $W$\n- The novelty is quite limited. Discretizing and limiting support of existing parametric kernels seems straightforward\n\n*Lack of clarity*\n- In general, the writing and notation should be improved to increase clarity, e.g., it's not necessary to write kernel convolution with Dirac functions since it's the kernel evaluated at that location. \n\n*Experimental Limitations*\n- Figure 1: For easy comparisons, the paper should plot all methods on the same plot. As is,  the y-axis is on a different scale making it hard to make comparisons.\n- The paper should consider adding additional real-world datasets and recent state-of-the-art  baselines (*e.g.*, neural Hawkes [1] models flexible kernels with neural networks)\n- Figure 3:  Are the TG and RC intensity predictions from the proposed approach?\n- The performance improvements over baselines seem marginal\n- The paper should provide summary statistics of the synthetic and electrophysiology, *e.g.*, number of event types, sequence lengths, number of sequences, *etc.*\n- The paper should provide sensitivity analysis for step-sizes $\\Delta$ and length of kernel's support $W$\n\n**References**\n- [1] Mei et al., \"The neural hawkes process: A neurally self-modulating multivariate point process\", NeurIPS, 2017", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "2WROkFN1BKn", "reviewer": "Reviewer_rbja", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "In this paper, the authors study discrete Hawkes process to model temporal data, along with parametric kernels with fixed support. \nThey propose to use least squares loss to learn the model parameters (compared to the more popular log likelihood estimation). They propose to use gradient descent, where the gradients may be efficiently computed thanks to the discreteness of the process and the finite support of kernels. They also derive the results for the approximated solution to converge to the true solution as the discretization window approaches zero. Numerical results are presented on (a) simulated data to evaluate the kernel learning and to estimate the recovery error as a function of \\Delta (the bin width)  (b) MEG data illustrating the learnt latency. ", "summary_of_the_review": "Overall, the paper has good technical contributions, but may be improved significantly with convincing experimental claims. ", "strength_and_weaknesses": "Strengths\n- the paper is well written and easy to follow. \n- The impact of discretization is studied well, theoretically as well as experimentally\n\nWeakness / Questions / Comments\n- Discrete time Hawkes processes are also considered in some of the earlier works such as https://www.cs.princeton.edu/~rpa/pubs/linderman2015scalable.pdf, https://arxiv.org/pdf/2003.02810.pdf\nIt would be better to compare the methodologies in this submission agianst them, theoretically as well as in experiments. \n\n- The experiments section may be improved with more convincing claims. For instance, in kernel recovery comparisons, only a single kernel is used for generating the samples. For fair comparisons, more kernel types should be compared, assuming that FasDin is not aware of the true kernel. \n\n- In Figure 1, it is not clear how the recovery error compares across the models for all choices of T.\n\n- Section 4 can benefit from better description of the data and the setup. What are the choices of \\Delta tried? What Is the typical value of T  or G for the samples ? While the latency estimates are similar, what is the specific computational advantage in terms of the total time ?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject