2Instituto de Investigación en Ciencias de la Computación, CONICET-UBA, Argentina In this work, we present a framework for incorporating declarative logical rules in state-of-the-art neural networks, enabling them to learn how to handle unseen labels without the introduction of any new training data. The rules are integrated into existing networks without modifying their architecture, through an additional term in the network's loss function that penalizes states of the network that do not obey the designed rules. As a case study, the framework is applied to an existing neural-based Dialog State Tracker. Our experiments demonstrate that the inclusion of logical rules allows the prediction of unseen labels, without deteriorating the predictive capacity of the original system.
When deploying machine-learning-based systems, it is common for users to detect problems related to functionalities that do not meet the expected requirements. In particular, in dialog systems, problems arise when for a certain input, the model makes a prediction that is diferent from the user’s inferred decision. This is often due to the model structure and to the inherent characteristics of the dataset used to train the models. In the same direction, new user requirements for the existing system may require outputing unseen labels (not present in training data). In such situations, a typical solution consists of the collection of new annotated data aligned with the expected functionality. However, collecting new data every time such a need arises is an expensive and time-consuming efort, therefore, an alternative approach is desired.
In this work, we propose a solution to the above mentioned problems by incorporating
prescriptive logical rules into learned neural network models. These rules are designed by
*These authors contributed equally to this work. This work was performed while the authors were still at SRI.
† Work done prior to joining Amazon.
nEvelop-O
domain experts and can influence the system output, enabling also the prediction of unseen
labels. Similar to other works [
We apply our logic rules framework to Dialog State Tracking, a challenging and complex task
in the field of Dialog Systems. More specifically, we extend the Multi-Domain Neural Belief
State Tracker (MDNBT), proposed in [
The main contributions of our work are the following: a) we enhance a Neural-based Dialog State Tracker with logic rules, without degrading the performance of the base system, b) we show that the addition of the logic rules allow the predictions of unseen labels which can be very useful in the case of unlabeled or partially labeled data.
Due to the increasing popularity of neural network models for supervised learning, there is a
growing body of material related to the inclusion of structural knowledge as a tool of biasing
certain models decisions and as a way to mitigate the uninterpretability of results. One way to
introduce this knowledge is to integrate logical rules through the use of FOL — a declarative
language that can represent high-level knowledge [1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, inter allia]. In
the majority of prior works, diferent types of posterior regularization terms are implemented
to afect the optimization process. For example, in a seminal work by Hu et al. [
There are also works that use these techniques in semi-supervised or even zero-shot learning
settings. For example, in [
The proposed framework consists of the addition of a plug-in component into an existing computational graph with no extra learning parameters. A rules-dependent loss term is introduced into the system’s loss function as a way of integrating rules to the learning process and as a way of allowing the use of of-the-shelf optimizers. By including rules as part of the training process, we generate extra cost when the network does not satisfy a certain rule.
A Dialog State Tracker (DST) is a key component in task-based spoken dialog systems. It models the user’s intent at any point of an ongoing conversation [15] which is then used by the downstream dialog management component to choose the next system response. DST models estimate the belief state, which is the system’s internal probability distribution over possible dialog states, by taking into account the user goals at every turn as extracted by a Spoken Language Understanding (SLU) component. The dialog states are defined by a domain-specific ontology that lists the slot-value pairs that describe the constraints the users can express (e.g. price range-expensive, price range-cheap, area-west, area-east, etc) [16].
In our work, we use the Multi-domain Neural Belief State Tracker (MDNBT) which jointly
identifies the domain and tracks the belief states corresponding to that domain by utilizing the
semantic similarity between dialog utterances and ontology terms [
For our experiments, we used the MultiWOZ 2.0 dataset [17], following Ramadan et al. [
1. utterance: Hi, can you help me find a place to eat on the northside? {T r u e s t a t e : r e s t a u r a n t - a r e a - n o r t h } system: Yes, I have 15 options, do you have any preferences for the price range? 2. utterance: Yes, I would like an expensive restaurant {T r u e s t a t e : r e s t a u r a n t - a r e a - n o r t h , r e s t a u r a n t - p r i c e r a n g e - e x p e n s i v e } system: There are 2 expensive places, an Italian restaurant and a gastropub.
Rules are defined as formulas in a relaxation of FOL that represent truth values in a continuous
domain (FOL Fuzzy Logic) in which the satisfaction of rules is a diferentiable function that
can be maximized to perform learning [
We formulated two types of rules for this study. The type 1, which triggers when some specific keyword (in this case expensive) is explicitly uttered in a specific domain (in this case hotel), and the type 2 in order to preserve belief-state predictions related to the price-range slots throughout the turns for the cases when the user’s price-range intent does not change 1https://github.com/osmanio2/multi-domain-belief-tracking 2In our experiments, we are not interested in FOL functions and quantifiers, and we leave them out of the discussion. throughout the turns. Examples of the two types of rules for the HOTEL domain are given below: 1: IF the user’s utterance contains a word like EXPENSIVE AND also contains a word like HOTEL THEN the prediction in the domain HOTEL, slot PRICERANGE should be EXPENSIVE 2: IF the previous prediction for the domain HOTEL, slot PRICERANGE was EXPENSIVE AND the user did NOT uttered a word like MODERATE AND did NOT uttered a word like CHEAP THEN the next prediction in the domain HOTEL, slot PRICERANGE should be EXPENSIVE.
In these examples, underlined words represent predicates; bold-uppercase terms represent logical connectives; bold-lowercase terms represent nodes in the computational graph on the network (or concepts that are mapped into embeddings such as in the case of the user’s utterance); ifnally, italic-uppercase words refer to the constants of our system.
As we will discuss in further detail in the next sections, rules are implemented in the model’s computational graph through the addition of new operations that are applied over existing nodes. When the graph is evaluated for a specific instance, each rule produces a number that indicates the truthiness of that rule for the specific instance under evaluation. The loss function will be a function of the truthiness, allowing the network to learn the rules.
Unsatisfied rules generate a cost that the optimizer minimizes in conjunction with the misclassification cost. Our system is trained not only to learn from labels (by minimizing the cross-entropy loss function) but also to learn how to make the set of rules ℛ as true as possible (a concept called best satisfiability
as presented in [18]). For this, we use the simplest posterior
regularization approach, in which the objective function of the rule-based model is the sum of
the loss function of the base MDNBT model (ℒ
) and the rules’ loss function (ℒ
) [
is defined as the sum of the loss of each individual rule: ℒ (; ; ℛ) = ∑ ℒ (; ) ∈ℛ ∈ℛ = ∑ 1 − ℎ (; ) Here, each individual rule loss (ℒ ) is defined in terms of the ℎ , i.e. the degree of truth of the rule. Finally, the optimization problem consists of finding the optimal weights and biases for the network given the composed loss function, ∗ = arg min ℒ (; ; ; ℛ) (1) The backpropagation mechanism computes gradients that update all trainable parameters in the network that are reachable from the loss function. For weights and biases to be reachable, the operations that define the loss function have to be fully diferentiable. Thus the logical operations that define truthiness of the rules need to be diferentiable.
We define two types of formulas: (i) atomic formulas: predicates applied to constants and nodes of the computational graph; and (ii) composed formulas, which are built up from atomic formulas using the Boolean connectives.
Predicates define relations among entities of the neural network. For example, in the case
of the MDNBT model, we can refer to the embedding representation of a word in the input
data; to the actual belief states ( ) — a slot-specific distribution of probabilities estimated by
the MDNBT output layer [
The implementation of a predicate, as opposed to that of a logic connective, strongly depends on the underlying architecture and on the input representations. For example, the predicate contains a word like CHEAP can be expressed as the cosine similarity function over nodes of the computational graph: ℎℎ= (, CHEAP). This function may check if there is any word in the utterance whose cosine similarity with the word embedding for cheap is greater than threshold .
Logic operators (namely ∧, ∨, ¬, → ) are implemented through the following equations based on
Product -norm for conjunction, -norm for disjunction, and the residuum of the -norm for the
implication (see [19] and [
¬ ≝ 1 − , ∧ ≝ ⋅ , ∨ ≝ ¬(¬ ∧ ¬ ) = + − → ≝ ¬( ∧ ¬ ) = 1 − (1 − ) Having defined all the aforementioned components, we can represent rules in terms of logic formulas. For example, 1 is: 1 ≡ ( ℎℎ= (, EXPENSIVE) ∧ ℎℎ= (, HOTEL)) →
( , HOTEL-PRICERANGE-EXPENSIVE) where ( , HOTEL-PRICERANGE-EXPENSIVE) is a predicate that returns the model’s belief state probability in the index corresponding to the HOTEL-PRICERANGE-EXPENSIVE state.
When a gradient-based method is used to solve the optimization problem of Eq. 1, the parameters of the network are updated in the opposite direction to the gradient of ℒ. The rule loss term produces an update associated with it. For example, in the case of a rule with the form = → in which the loss is ℒ = (1 − ) , the update associated to the rule looks like: Δ = − ( (ℒ ) ( )) + (ℒ ) ( )) ) where is the learning rate and the partial derivatives of the implication are (ℒ ) = 1 − and (ℒ ) = − .
For example, in the case where this implication is not satisfied ( = 1 and = 0 ), the partial derivatives are (ℒ ) = 1 and (ℒ ) = −1, and the update is,
Δ = − ( ) + ( ) The network will update in the direction of growth of Y and in the direction of decrease of X. That is, during the learning step the antecedent tends to decrease and the consequent tends to increase simultaneously.
Depending on the rule, one may want the antecedent or the consequent learning to be frozen (i.e. that the learning process occurs only through one of them). This is the case of some if-type rules like 2, in which we are not interested in learning how to make the condition True, but we want to make the then branch True (in case the condition is met). For these types of cases, one can use two diferent alternatives: either implementing predicates using non-derivable functions such as ; or by programmatically stopping the back-propagation in the corresponding subterms.
The main question we address in this section is: can we extend a system with a new slot without any additional data and without degrading the existing system?
To answer this question, we simulated this scenario by removing all existing annotations for the PRICERANGE slot in the training set of the MULTIWOZ dataset. Next, we built a set of twelve rules designed to learn the PRICERANGE slot values. Six of these rules were in the form of 1, and six were in the form of 2 (as described in Section 3.2), addressing the six possible combinations of domains (hotel and restaurant) and price ranges (cheap, moderate and expensive).
We compare our models to a base MDNBT model that does not contain rules and in which we removed the PRICERANGE slot-value pairs from the ontology. In our experiments, we used 100 Bi-LSTM cells and we trained the models from scratch using the ADAM optimizer [20] with batch size 64 for 600 epochs. A dropout rate [21] of 50% was used in all the intermediate representations. Also, all the weights were initialized using normal distribution of zero mean and unit variance and biases were initialized to zero. For the rule-based MDNBT we trained models with four diferent values of the rules’ weight, i.e., = 10, 30, 100, 300 .
The F1 performance of the rules-based MDNBT models for the PRICERANGE slot and for all the remaining slots are depicted in Fig. 1. F1 is measured by considering the correct and incorrect predictions in each slot of each domain. The performance of the base MDNBT model for the remaining slots is also shown (right plot). From the left panel, it is evident that the rule-based MDNBT model shows predictive capability over the PRICERANGE slot values without any training data (zero-shot learning). The increase in the performance measured in the PRICERANGE slots for diferent rules’ weight values afects the overall performance on the rest of the slots in the ontology depending on the rules’ weight value. As can be seen from the right plot, for lower rules’ weight values (e.g. = 10 ) in the rule-based MDNBT there is not an appreciable performance drop compared to the base MDNBT ( 0.7% relative decrease), while for large rules’ weights (e.g. = 300 ) there is a considerable degradation of performance (i.e., 29% relative decrease). A selection of an optimal rules’ weight (e.g. = 30 ) can guarantee the optimal trade-of between the performance on unlabeled data (i.e. PRICERANGE slot) and the performance on labeled data (e.g. the remaining slots in the ontology).
Our results do not show a significant performance degradation in the other slots (right panel) for low rules-weight values ( = 10, 30 ), since it does not show significant diference with the base model (two-sided -test -val > 0.1 in both values). However, using = 100 or = 300 we see a notable decrease in the general performance ( -val = 0.06 and -val= 0.004 respectively with two-sided -tests). That is, the system learns how to identify price ranges at the expense of producing unwanted efects in the performance of the rest of the slots.
From the experimental results, we observe that it is possible to integrate rules into an existing system to allow the prediction of unseen labels without degrading the predictive capabilities over the rest of the labels (as it is the case with = 30 ). However, it is necessary to pay special attention to the trade-of that is generated between learning the rules and the degradation of the system. In particular, it is important to notice that the weights depend on the number of times the rules are actually satisfied, the number of rules, and the design properties of the system.
In this work we show how a set of rules can be incorporated into a Neural Network to predict a new category that did not exist in the training set.
It is worth noting that this rule-based setup completely depends on the coverage of a set of rules established by the domain experts, and also, that the rules/weight/predicate and values/slot names are problem specific.
The presented approach serves specifically for the case in which there are new requirements for a system that can easily be covered – allowing the creation of an updated version of the system without the need of additional annotations.
As we described in section 2, there are other methods to integrate rules in neural networks, and also, there are new state-of-the-art Dialogue State Tracking models that produce better metrics [22]. Since the objective of this work was to show that it is possible to perform ZeroShot Learning by adding rules to an existing neural network, we have chosen a simple and transparent rule integration method, and we have not focused on the selection of the State Tracking model. In the future, we plan to compare diferent methods of integration of rules in state-of-the-art models.
Campagna et al. [23] have used data synthesis techniques for transfering knowledge into new domains (Zero-Shot Transfer Learning). In the future, we plan to adapt this method for our task (i.e. predict slots not seen during training) to compare with our approach. We believe that the integration of rules in neural networks has the benefit of using existing information in the internal states and to predicate, for example, about the prediction of the model during training.
This paper presented how the addition of prescriptive logical rules designed by domain experts can enable neural networks to predict unseen labels without the need for creating new labeled training data. The rules are integrated into an existing neural network without modifying the original architecture. A posterior regularization approach was used to introduce the rules into the learning process, penalizing the objective function when inputs and the internal state of the network do not obey one of the designed rules. Our rules-based framework was applied and tested to an existing neural-based Dialog State Tracker for Dialog Systems where rules were implemented so that the model learns to identify PRICERANGE labels, which were not seen during training. Our experiments showed that the inclusion of logical rules allows the prediction of new labels, without jeopardizing the predictive capacity on the rest of the data. It is finally worth noting that our rules-based solution is independent of the neural network model and thus can be applied to any application (and neural network model) given the formulation of appropriate rules. [13] M. Diligenti, M. Gori, C. Sacca, Semantic-based regularization for learning and inference,
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