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Title: Informed Temporal Modeling via Logical Specification of Factorial LSTMs

Paper content:
Consider a world in which events occur that involve various entities.
Learning how to predict future events from patterns of past events becomes more difficult as we consider more types of events.
Many of the patterns detected in the dataset by an ordinary LSTM will be spurious since the number of potential pairwise correlations, for example, grows quadratically with the number of events.
We propose a type of factorial LSTM architecture where different blocks of LSTM cells are responsible for capturing different aspects of the world state.
We use Datalog rules to specify how to derive the LSTM structure from a database of facts about the entities in the world.
This is analogous to how a probabilistic relational model (Getoor & Taskar, 2007) specifies a recipe for deriving a graphical model structure from a database.
In both cases, the goal is to obtain useful inductive biases by encoding informed independence assumptions into the model.
We specifically consider the neural Hawkes process, which uses an LSTM to modulate the rate of instantaneous events in continuous time.
In both synthetic and real-world domains, we show that we obtain better generalization by using appropriate factorial designs specified by simple Datalog programs.
Temporal sequence data is abundant in applied machine learning.
A common task is to impute missing events, e.g., to predict the future from the past.
Often this is done by fitting a generative probability model.
For evenly spaced sequences, historically popular models have included hidden Markov models and discrete-time linear dynamical systems, with more recent interest in recurrent neural network models such as LSTMs.
For irregularly spaced sequences, a good starting point is the Hawkes process, a self-exciting temporal point process; many variations and enhancements have been published, including neural variants using LSTMs.
All of these models can be described schematically by Figure 1a .
Events e i , e i+1 , . . . are assumed to be conditionally independent of previous events, given the system state s i (which may or may not be fully known given events e 1 , . . . , e i ).
That is, s i is enough to determine the joint distribution of the i th event and the updated state s i+1 , which is needed to recursively predict all subsequent events.
Figure 1a and its caption show the three types of influence in the model.
The update, affect, and depend arrows are characterized by parameters of the model.
In the case of a recurrent neural network, these are the transition, input, and output matrices.
Our main idea in this paper is to inject structural zeros into these weight matrices.
Structural zeros are weights that are fixed at zero regardless of the model parameters.
In other words, we will remove many connections (synapses) from both the recurrent and non-recurrent portions of the neural network.
Parameter estimation must use the sparse remaining connections to explain the observed data.
Specifically, we partition the neural state s i ∈ R d into a number of node blocks.
Different node blocks are intended to capture different aspects of the world's state at step i.
By zeroing out rectangular blocks of the weight matrix, we will restrict how these node blocks interact with the events and with one another.
An example is depicted in Figures 1b (affect, depend) and 1d (update).
In addition, by reusing nonzero blocks within a weight matrix, we can stipulate (for example) that event e affects node block b in the same way in which event e affects node block b .
Such parameter tying makes it possible to generalize from frequent events to rare events of the same type.
Although our present experiments are small, we are motivated by the challenges of scale.
Real-world domains may have millions of event types, including many rare types.
To model organizational behavior, we might consider a dataset of meetings and emails in a large organization.
To model supply chains, we might consider purchases of goods and services around the world.
In an unrestricted model, anything in the past could potentially influence anything in the future, making estimation extremely difficult.
Structural zeroes and parameter tying, if chosen carefully, should help us avoid overfitting to coincidental patterns in the data.
Analogous architectures have been proposed in the world of graphical models and causal models.
Indeed, to write down such a model is to explicitly allow specific direct interactions and forbid the rest.
For example, the edges of a Gaussian graphical model explicitly indicate which blocks of the inverse covariance matrix are allowed to be nonzero.
Some such models reuse blocks (Hojsgaard & Lauritzen, 2008) .
As another example, a factorial HMM (Ghahramani & Jordan, 1997 )-an HMM whose states are m-tuples-can be regarded as a simple example of our architecture.
The state s i can be represented using m node blocks, each of which is a 1-hot vector that encodes the value of a different tuple element.
The key aspect of a factorial HMM is that the stochastic transition matrix (update in Figure 1d ) is fully block-diagonal.
The affect matrix is 0, since the HMM graphical model does not feed the output back into the next state; the depend matrix is unrestricted.
But how do we know which interactions to allow and which to forbid?
This is a domain-specific modeling question.
In general, we would like to exploit the observation that events are structured objects with participants (which is why the number of possible event types is often large).
For example, a travel event involves both a person and a place.
We might assume that the probability that Alice travels to Chicago depends only on Alice's state, the states of Alice's family members, and even the state of affairs in Chicago.
Given that modeling assumption, parameter estimation cannot try to derive this probability (presumably incorrectly) from the state of the coal market.
These kinds of systematic dependencies can be elegantly written down using Datalog rules, as we will show.
Datalog rules can refer to database facts, such as the fact that Alice is a person and that she is related to other people.
Given these facts, we use Datalog rules to automatically generate the set of possible events and node blocks, and the ways in which they influence one another.
Datalog makes it easy to give structured names to the events and node blocks.
The rules can inspect these structures via pattern-matching.
In short, our contribution is to show how to use a Datalog program to systematically derive a constrained neural architecture from a database.
Datalog is a blend of logic and databases, both of which have previously been used in various formalisms for deriving a graphical model architecture from a database (Getoor & Taskar, 2007) .
There has been extensive research about having inductive biases in the architecture design of a machine learning model.
The epitome of this direction is perhaps the graphical models where edges between variables are usually explicitly allowed or forbidden (Koller & Friedman, 2009 ).
There has also been work in learning such biases from data.
For example, Stepleton et al. (2009) proposed to encourage the block-structured states for Hidden Markov Models (HMM) by enforcing a sparsityinducing prior over the non-parametric Bayesian model.
Duvenaud et al. (2013) and Bratières et al. (2014) attempted to learn structured kernels for Gaussian processes.
Our work is in the direction of injecting inductive biases into a neural temporal model-a class of models that is useful in various domains such as demand forecasting (Seeger et al., 2016) , personalization and recommendation (Jing & Smola, 2017) , event prediction (Du et al., 2016) and knowledge graph modeling (Trivedi et al., 2017) .
Incorporating structural knowledge in the architecture design of such a model has drawn increasing attention over the past few years.
Shelton & Ciardo (2014) introduced a factored state space in continuous-time Markov processes.
Meek (2014) and Bhattacharjya et al. (2018) proposed to consider direct dependencies among events in graphical event models.
Wang et al. (2019) developed a hybrid model that decomposes exchangeable sequences into a global part that is associated with common patterns and a local part that reflects individual characteristics.
However, their approaches are all bounded to the kinds of inductive biases that are easy to specify (e.g. by hand).
Our work enables people to use a Datalog program to conveniently specify the neural architecture based on a deductive database-a much richer class of knowledge than the previous work could handle.
Although logic programming languages and databases have both previously been used to derive a graphical model architecture (Getoor & Taskar, 2007) , we are, to the best of our knowledge, the first to develop such a general interface for a neural event model.
As future work, we hope to develop an extension where events can also trigger assertions and retractions of facts in the Datalog database.
Thanks to the Datalog rules, the model architecture will dynamically change along with the facts.
For example, if Yoyodyne Corp. hires Alice, then the Yoyodyne node block begins to influence Alice's actions, and K expands to include a new (previously impossible) event where Yoyodyne fires Alice.
Moreover, propositions in the database-including those derived via other Datalog rules-can now serve as extra bits of system state that help define the λ k intensity functions in (1).
Then the system's learned neural state s i is usefully augmented by a large, exact set of boolean propositions-a division of labor between learning and expert knowledge.
In this section, we elaborate on the details of the transition function Ψ that is introduced in section 2.1; more details about them may be found in Mei & Eisner (2017) .
where the interval (t i−1 , t i ] has consecutive observations k i−1 @t i−1 and k i @t i as endpoints.
At t i , the continuous-time LSTM reads k i @t i and updates the current (decayed) hidden cells c(t) to new initial values c i+1 , based on the current (decayed) hidden state h(t i ), as follows:
At time t i , the updated state vector is
] is given by (26), which continues to control h(t) except that i has now increased by 1).
On the interval (t i , t i+1 ], c(t) follows an exponential curve that begins at c i+1 (in the sense that lim t→t + i c(t) = c i+1 ) and decays, as time t increases, toward c i+1 (which it would approach as t → ∞, if extrapolated).

assistant: text: Factorize LSTM states and zero-out/tie LSTM weight matrices according to real-world structural biases expressed by Datalog programs.