Intensity-Free Learning of Temporal Point Processes

1 INTRODUCTION . Visits to hospitals , purchases in e-commerce systems , financial transactions , posts in social media — various forms of human activity can be represented as discrete events happening at irregular intervals . The framework of temporal point processes is a natural choice for modeling such data . By combining temporal point process models with deep learning , we can design algorithms able to learn complex behavior from real-world data . Designing such models , however , usually involves trade-offs along the following dimensions : flexibility ( can the model approximate any distribution ? ) , efficiency ( can the likelihood function be evaluated in closed form ? ) , and ease of use ( is sampling and computing summary statistics easy ? ) . Existing methods ( Du et al. , 2016 ; Mei & Eisner , 2017 ; Omi et al. , 2019 ) that are defined in terms of the conditional intensity function typically fall short in at least one of these categories . Instead of modeling the intensity function , we suggest treating the problem of learning in temporal point processes as an instance of conditional density estimation . By using tools from neural density estimation ( Bishop , 1994 ; Rezende & Mohamed , 2015 ) , we can develop methods that have all of the above properties . To summarize , our contributions are the following : • We connect the fields of temporal point processes and neural density estimation . We show how normalizing flows can be used to define flexible and theoretically sound models for learning in temporal point processes . • We propose a simple mixture model that performs on par with the state-of-the-art methods . Thanks to its simplicity , the model permits closed-form sampling and moment computation . • We show through a wide range of experiments how the proposed models can be used for prediction , conditional generation , sequence embedding and training with missing data . 2 BACKGROUND . Definition . A temporal point process ( TPP ) is a random process whose realizations consist of a sequence of strictly increasing arrival times T = { t1 , ... , tN } . A TPP can equivalently be represented ∗Equal contribution Code and datasets are available under https : //github.com/shchur/ifl-tpp . as a sequence of strictly positive inter-event times τi = ti− ti−1 ∈ R+ . Representations in terms of ti and τi are isomorphic — we will use them interchangeably throughout the paper . The traditional way of specifying the dependency of the next arrival time t on the history Ht = { tj ∈ T : tj < t } is using the conditional intensity function λ∗ ( t ) : = λ ( t|Ht ) . Here , the ∗ symbol reminds us of dependence on Ht . Given the conditional intensity function , we can obtain the conditional probability density function ( PDF ) of the time τi until the next event by integration ( Rasmussen , 2011 ) as p∗ ( τi ) : = p ( τi|Hti ) = λ∗ ( ti−1 + τi ) exp ( − ∫ τi 0 λ∗ ( ti−1 + s ) ds ) . Learning temporal point processes . Conditional intensity functions provide a convenient way to specify point processes with a simple predefined behavior , such as self-exciting ( Hawkes , 1971 ) and self-correcting ( Isham & Westcott , 1979 ) processes . Intensity parametrization is also commonly used when learning a model from the data : Given a parametric intensity function λ∗θ ( t ) and a sequence of observations T , the parameters θ can be estimated by maximizing the log-likelihood : θ∗ = arg maxθ ∑ i log p ∗ θ ( τi ) = arg maxθ [ ∑ i log λ ∗ θ ( ti ) − ∫ tN 0 λ∗θ ( s ) ds ] . The main challenge of such intensity-based approaches lies in choosing a good parametric form for λ∗θ ( t ) . This usually involves the following trade-off : For a ” simple ” intensity function ( Du et al. , 2016 ; Huang et al. , 2019 ) , the integral Λ∗ ( τi ) : = ∫ τi 0 λ∗ ( ti−1+s ) ds has a closed form , which makes the log-likelihood easy to compute . However , such models usually have limited expressiveness . A more sophisticated intensity function ( Mei & Eisner , 2017 ) can better capture the dynamics of the system , but computing log-likelihood will require approximating the integral using Monte Carlo . Recently , Omi et al . ( 2019 ) proposed fully neural network intensity function ( FullyNN ) — a flexible , yet computationally tractable model for TPPs . The key idea of their approach is to model the cumulative conditional intensity function Λ∗ ( τi ) using a neural network , which allows to efficiently compute the log-likelihood . Still , in its current state , the model has downsides : it doesn ’ t define a valid PDF , sampling is expensive , and the expectation can not be computed in closed form1 . This work . We show that the drawbacks of the existing approaches can be remedied by looking at the problem of learning in TPPs from a different angle . Instead of modeling the conditional intensity λ∗ ( t ) , we suggest to directly learn the conditional distribution p∗ ( τ ) . Modeling distributions with neural networks is a well-researched topic , that , surprisingly , is not usually discussed in the context of TPPs . By adopting this alternative point of view , we are able to develop new theoretically sound and effective methods ( Section 3 ) , as well as better understand the existing approaches ( Section 4 ) . 3 MODELS . We develop several approaches for modeling the distribution of inter-event times . First , we assume for simplicity that each inter-event time τi is conditionally independent of the history , given the model parameters ( that is , p∗ ( τi ) = p ( τi ) ) . In Section 3.1 , we show how state-of-the-art neural density estimation methods based on normalizing flows can be used to model p ( τi ) . Then in Section 3.2 , we propose a simple mixture model that can match the performance of the more sophisticated flowbased models , while also addressing some of their shortcomings . Finally , we discuss how to make p ( τi ) depend on the historyHti in Section 3.3 . 1A more detailed discussion of the FullyNN model follows in Section 4 and Appendix C. 3.1 MODELING p ( τ ) WITH NORMALIZING FLOWS The core idea of normalizing flows ( Tabak & Turner , 2013 ; Rezende & Mohamed , 2015 ) is to define a flexible probability distribution by transforming a simple one . Assume that z has a PDF q ( z ) . Let x = g ( z ) for some differentiable invertible transformation g : Z → X ( where Z , X ⊆ R ) 2 . We can obtain the PDF p ( x ) of x using the change of variables formula as p ( x ) = q ( g−1 ( x ) ) ∣∣ ∂ ∂xg −1 ( x ) ∣∣ . By stacking multiple transformations g1 , ... , gM , we obtain an expressive probability distribution p ( x ) . To draw a sample x ∼ p ( x ) , we need to draw z ∼ q ( z ) and compute the forward transformation x = ( gM ◦ · · · ◦ g1 ) ( z ) . To get the density of an arbitrary point x , it is necessary to evaluate the inverse transformation z = ( g−11 ◦ · · · ◦ g −1 M ) ( x ) and compute q ( z ) . Modern normalizing flows architectures parametrize the transformations using extremely flexible functions fθ , such as polynomials ( Jaini et al. , 2019 ) or neural networks ( Krueger et al. , 2018 ) . The flexibility of these functions comes at a cost — while the inverse f−1θ exists , it typically doesn ’ t have a closed form . That is , if we use such a function to define one direction of the transformation in a flow model , the other direction can only be approximated numerically using iterative root-finding methods ( Ho et al. , 2019 ) . In this work , we don ’ t consider invertible normalizing flows based on dimension splitting , such as RealNVP ( Dinh et al. , 2017 ) , since they are not applicable to 1D data . In the context of TPPs , our goal is to model the distribution p ( τ ) of inter-event times . In order to be able to learn the parameters of p ( τ ) using maximum likelihood , we need to be able to evaluate the density at any point τ . For this we need to define the inverse transformation g−1 : = ( g−11 ◦· · ·◦g −1 M ) . First , we set zM = g−1M ( τ ) = log τ to convert a positive τ ∈ R+ into zM ∈ R. Then , we stack multiple layers of parametric functions fθ : R → R that can approximate any transformation . We consider two choices for fθ : deep sigmoidal flow ( DSF ) from Krueger et al . ( 2018 ) and sum-ofsquares ( SOS ) polynomial flow from Jaini et al . ( 2019 ) fDSF ( x ) = σ−1 ( K∑ k=1 wkσ ( x− µk sk ) ) fSOS ( x ) = a0 + K∑ k=1 R∑ p=0 R∑ q=0 ap , kaq , k p+ q + 1 xp+q+1 ( 1 ) where a , w , s , µ are the transformation parameters , K is the number of components , R is the polynomial degree , and σ ( x ) = 1/ ( 1 + e−x ) . We denote the two variants of the model based on fDSF and fSOS building blocks as DSFlow and SOSFlow respectively . Finally , after stacking multiple g−1m = fθm , we apply a sigmoid transformation g −1 1 = σ to convert z2 into z1 ∈ ( 0 , 1 ) . For both models , we can evaluate the inverse transformations ( g−11 ◦ · · · ◦ g −1 M ) , which means the model can be efficiently trained via maximum likelihood . The density p ( τ ) defined by either DSFlow or SOSFlow model is extremely flexible and can approximate any distribution ( Section 3.4 ) . However , for some use cases , this is not sufficient . For example , we may be interested in the expected time until the next event , Ep [ τ ] . In this case , flow-based models are not optimal , since for them Ep [ τ ] does not in general have a closed form . Moreover , the forward transformation ( gM ◦ · · · ◦g1 ) can not be computed in closed form since the functions fDSF and fSOS can not be inverted analytically . Therefore , sampling from p ( τ ) is also problematic and requires iterative root finding . This raises the question : Can we design a model for p ( τ ) that is as expressive as the flow-based models , but in which sampling and computing moments is easy and can be done in closed form ? 3.2 MODELING p ( τ ) WITH MIXTURE DISTRIBUTIONS Model definition . While mixture models are commonly used for clustering , they can also be used for density estimation . Mixtures work especially well in low dimensions ( McLachlan & Peel , 2004 ) , which is the case in TPPs , where we model the distribution of one-dimensional inter-event times τ . Since the inter-event times τ are positive , we choose to use a mixture of log-normal distributions to model p ( τ ) . The PDF of a log-normal mixture is defined as p ( τ |w , µ , s ) = K∑ k=1 wk 1 τsk √ 2π exp ( − ( log τ − µk ) 2 2s2k ) ( 2 ) 2All definitions can be extended to RD for D > 1 . We consider the one-dimensional case since our goal is to model the distribution of inter-event times τ ∈ R+ . z1 zm τ q1 ( z1 ) qm ( zm ) p ( τ ) ... ... g1 gm−1 gM g−11 g−1M Figure 2 : Normalizing flows define a flexible distribution via transformations . where w are the mixture weights , µ are the mixture means , and s are the standard deviations . Because of its simplicity , the log-normal mixture model has a number of attractive properties . Moments . Since each component k has a finite mean , the mean of the entire distribution can be computed as Ep [ τ ] = ∑ k wk exp ( µk+s 2 k/2 ) , i.e. , a weighted average of component means . Higher moments can be computed based on the moments of each component ( Frühwirth-Schnatter , 2006 ) . Sampling . While flow-based models from Section 3.1 require iterative root-finding algorithms to generate samples , sampling from a mixture model can be done in closed form : z ∼ Categorical ( w ) ε ∼ Normal ( 0 , 1 ) τ = exp ( sTz · ε+ µTz ) where z is a one-hot vector of size K. In some applications , such as reinforcement learning ( Upadhyay et al. , 2018 ) , we might be interested in computing gradients of the samples w.r.t . the model parameters . The samples τ drawn using the procedure above are differentiable with respect to the means µ and scales s. By using the Gumbel-softmax trick ( Jang et al. , 2017 ) when sampling z , we can obtain gradients w.r.t . all the model parameters ( Appendix D.6 ) . Such reparametrization gradients have lower variance and are easier to implement than the score function estimators typically used in other works ( Mohamed et al. , 2019 ) . Other flexible models ( such as multi-layer flow models from Section 3.1 ) do not permit sampling through reparametrization , and thus are not well-suited for the above-mentioned scenario . In Section 5.4 , we show how reparametrization sampling can also be used to train with missing data by performing imputation on the fly .

Summary: This paper describes a simple yet effective technique for learning temporal point processes using a mixture of log-normal densities whose parameters are estimated with neural networks that also adds conditional information. The method is shown to perform better than more recent techniques for density estimation such as different versions of normalising flows. Experiments were reported on 6 datasets, comparing the approach against flow models and assessing the benefits of adding extra conditional information, performance with missing data, and benefits of sequence embeddings.