GENN : P REDICTING C ORRELATED D RUG - DRUG I NTERACTIONS WITH G RAPH E NERGY N EURAL N ETWORKS
Anonymous authors
Paper under double-blind review
A BSTRACT
Gaining more comprehensive knowledge about drug-drug interactions (DDIs) is one of the most important tasks in drug development and medical practice.
Recently graph neural networks have achieved great success in this task by modeling drugs as nodes and drug-drug interactions as links and casting DDI predictions as link prediction problems.
However, correlations between link labels (e.g., DDI types) were rarely considered in existing works.
We propose the graph energy neural network ( GENN ) to explicitly model link type correlations.
We formulate the DDI prediction task as a structure prediction problem, and introduce a new energy-based model where the energy function is deﬁned by graph neural networks.
Experiments on two real world DDI datasets demonstrated that GENN is superior to many baselines without consideration of link type correlations and achieved 13 .
77% and 5 .
01% PR-AUC improvement on the two datasets, respectively.
We also present a case study in which GENN can better capture meaningful
DDI correlations compared with baseline models.
I NTRODUCTION
The use of drug combinations is common and often necessary for treating patients with complex diseases.
However, it also increases the risk of drug-drug interactions (DDI).
DDIs are pharmacological interactions between drugs that can alter the action of either or both drugs and cause adverse effects.
Overall DDIs result in a large number of fatalities per year and incur ∼ $177 billion DDI associated cost annually (Giacomini et al., 2007).
To mitigate these risks and costs, accurate prediction of DDI becomes a clinically important task.
While DDI knowledge is expensive to gather, several deep learning approaches have been proposed to search the large biomedical data for predicting potential DDIs (Zitnik et al., 2018;
Ma et al., 2018; Ryu et al., 2018).
Among them, graph neural networks (GNN) that consider DDI prediction in the graph setting have obtained great performance.
In DDI graphs, drugs are represented as nodes and embedded into a low-dimensional space.
Further prediction can be made based on casting
DDI prediction as a link prediction problem where DDI types are represented as link types (Kipf &
Welling, 2016; Zitnik et al., 2018).
However, existing works rarely exploit the correlations between these link types (e.g., DDI types) despite of their importance.
For example, in Fig.
1 Warfarin is a drug for treating blood clots.
Its combined use with antibiotic or nonsteroidal anti-inﬂammatory drugs can cause multiple DDI types including nhibition of clotting, gastrointestinal bleeding, and hemorrhage.
These DDI types are correlated since they are all bleeding related DDIs caused by the increase of Warfarin’s effect.
Explicit modeling such correlations can help infer unseen DDI types.
To ﬁll the gap, we propose GENN , a new deep architecture that predicts correlated DDIs based on graph neural networks and energy based models.
Speciﬁcally, we leverage the dependency structures among DDI types and formulate multi-type DDI detection as a structure prediction problem (Lafferty et al., 2001a; LeCun et al., 2006; Belanger & McCallum, 2016b).
Here we use an energy based approach to incorporate such dependency structures.
To summarize, GENN is enabled by the following technical contributions.
Likewise, if Warfarin if concomitant with nonsteroidal anti-inﬂammatory drugs (NSAIDs) such as
Celecoxib and Naproxen, it will likely to cause DDI hemorrhage (i.e., loss of blood from broken blood vessel).
These DDIs are correlated since they are all bleeding related DDIs caused by the increase of Warfarin’s effect
Modeling link type correlations .
GENN bridges graph neural networks and structure prediction to directly capture link type correlations for more accurate link prediction in graphs through minimizing an energy function.
A new graph neural network based energy function .
Inspired by a family of structure prediction model called structured prediction energy networks (SPENs) (Belanger & McCallum 2016a) , we design a new graph energy function based on graph neural networks to capture the global dependencies among link types in DDI graphs. This is more ﬂexible and expressive than previous approaches which used CRF-type energy deﬁnitions Qu et
al.
Efﬁcient semi-supervised training .
We designed one cost-augmented inference network to approximate the output in training and one test inference network to approximate the output in testing.
Different from the previous inference method for SPENs, we propose a new semi-supervised joint optimization procedure to adapt to our problem on graphs. The new inference method optimizes the training-set structured hinge loss together with the test-set energy with respect to the parameters for both inference networks and the energy function.
We evaluated GENN on two real world DDI datasets with both quantitative and qualitative study.
Results demonstrated GENN outperformed basic graph neural networks with 13 .
77% and 5 .
PR-AUC improvement on the two datasets, respectively.
R ELATED W ORK 2. DDI P REDICTION
To predict unseen DDIs based on known ones, drug similarity has been learned via nearest neighbor approaches (Zhang et al., 2017), and using random walk methods including label propagation (Zhang et al., 2015; Wang et al., 2010), and unsupervised methods (Wang et al., 2014; Angione et al., 2016).
Recently, deep graph neural networks have been demonstrated to provide much improved performance in DDI prediction.
Among them, (Ma et al., 2018) integrates different sources of drug-related information with heterogeneous formats into an coherent and informationpreserving representation using attentive multi-view graph auto-encoders (Kipf & Welling, 2016).
Decagon (Zitnik et al., 2018) develops a new graph auto-encoder approach, which allows us to develop an end-to-end trainable model for link prediction on a multi-modal graph.
As the experimental results in Zitnik et al.
(2018) show graph neural networks achieved signiﬁcantly improved performance than shallow models such as tensor decomposition (Nickel et al., 2011; Papalexakis et al., 2017), random walk based methods (Perozzi et al., 2014; Zong et al., 2017), and non-graph neural ﬁngerprinting (Duvenaud et al., 2015; Jaeger et al., 2018), DeepDDI (Ryu et al., 2018).
None of the existing DDI prediction works consider the correlation among multiple DDI types.
2. S TRUCTURE P REDICTION AND E NERGY B ASED M ODELS
Structure prediction is an important problem in various application domains where we want to predict structured outputs instead of independent labels, e.g., structured labels prediction for object detection (Zheng et al., 2015b), semantic structure prediction (Belanger et al., 2017) or part-ofspeeches tagging (Ma & Hovy, 2016).
For structure prediction problems, feed-forward networks is insufﬁcient since they cannot directly model the interaction or constraints of the outputs.
Instead, energy based models (LeCun et al., 2006) provide one way to address the challenge by deﬁning a much more ﬂexible and expressive energy score, and the prediction is conducted by minimizing the energy function.
One of the well-known structure prediction approaches is the conditional random ﬁelds (CRF) (Lafferty et al., 2001b) which show success in application areas such as named entity recognition (Sato et al., 2017) and image segmentation (Zheng et al., 2015a).
But as a structured linear model, CRF has limited representation ability.
To enhance the ﬂexibility of structure prediction models, Belanger &
McCallum (2016b) proposed Structured Prediction Energy Networks (SPENs) which uses arbitrary neural networks to deﬁne the energy function and optimizes the energy over a continuous relaxation of labels.
Thus it can capture high-arity interaction among output labels and approximate the optimization of energy function via gradient descent and repeated “loss-augmented inference”.
Belanger et al.
(2017) further developed an “end-to-end” method that unroll the approximate energy optimization to a different computation graph.
However, after learning the energy function, they still have to use gradient descent for test-time inference.
Later, Tu & Gimpel (2018) replaced the gradient descent with a neural network trained to do inference directly.
However, it separates the training of cost-augmented inference network and the ﬁne-tuning of another inference network for testing, which makes it inefﬁcient.
In addition, how to optimize the SPENs on graphs for semi-supervised learning remains challenging.
In this paper our method improves the algorithm of Tu & Gimpel (2018) to solve this problem.
Very recently, GMNN (Qu et al., 2019) combines the statistical relational learning (SRL) and graph neural networks.
It includes a CRF to model the joint distribution of object labels conditioned on object attributes, and then it is trained with a variational EM algorithm.
The model has been used for node classiﬁcation and link classiﬁcation.
However, it cannot be easily extended for prediction of missing links, since they have to use links as nodes in a “dual graph” for link classiﬁcation, but the missing links generally cannot be modeled as nodes.
P RELIMINARY
In this section, we ﬁrst summarize our task in Section 3.1, then a basic message passing graph neural network (MPNN) is described for embedding the DDI graph in Section 3.2.
3. T ASK F ORMULATION
Deﬁnition 1 (DDI Graph) Given a DDI graph G = {V , E} , V is the node set which contains N drug nodes with node features X ∈ R N × D , and E is the edge set which contains all DDIs between drug pairs.
For a speciﬁc drug pair x i ∈ R D , x j ∈ R D , the DDIs of this pair e i,j can have multiple types, i.e. e i,j ∈ { 0 , 1 } L , where D is the feature dimension and L is the total number of DDI types.
And these DDI vectors e form the edge features (or labels)
Y ∈ { 0 , 1 } |E|× L .
Problem 1 (DDI Prediction)
We cast the DDI prediction into a multi-type link prediction problem.
We assume there are some missing edges Y U in the graph, i.e. Y =
{ Y L , Y U } .
Given the node features X and some known DDI links Y L , the goal is to predict the unknown DDIs Y U .
3. M ESSAGE P ASSING N EURAL N ETWORKS FOR DDI PREDICTION
Recently graph neural networks (GNNs) have been successfully applied to DDI prediction (Zitnik et al., 2018; Ma et al., 2018).
To predict the unknown DDIs in a graph, a GNN ﬁrst learns the node embeddings of all the drug nodes and then make prediction based on these embedding. In this x i x j x k x m e i,j e i,k e i
[Figure] m [Figure] section, we describe a basic message passing based graph neural network ( et al., 2017) for drug node embedding.
Considering that our graph contains different types of edges, we select a schema including edge information in the message passing.
Given the graph G with node features
X , for each node with hidden state h t i and its neighborhood N ( i ) , we deﬁne the message passing layer to update the node hidden state as follows:
[Equation]
where e i,j ∈ R L is the edge (DDI type vector) between nodes x i , x j ∈ R D , W s ∈ R M × M , W 0 ∈
R M × D are both learnable weight, and f ( · ) ∈ R M × M is a neural network, i.e. MLP.
The message passing layer can then be stacked to get the ﬁnal state
h
i for each node, i.e. the node embedding.
Then we can use these ﬁnal states to predict the probability of DDI types for each edge:
[Equation]
where || indicates concatenation, W ∈ R L × 2 M and b ∈ R 2 M are parameters and Sigmoid is the activation function commonly used for multi-label prediction.
The model is trained on the partially known DDI edges Y L by optimizing cross entropy loss
L CE ( ˆY L , Y L ) =
(cid:80) e i,j ∈ Y L L CE ( ˆy i,j , e i,j ) between the predictions and the true edges.
After training, it can be directly used for prediction of unknown DDIs Y U .
T HE GENN M ETHOD
We present our GENN framework in this section (Fig. 2).
We ﬁrst discuss the design of energy based framework for DDI prediction and the design of our graph energy functions based on MPNN.
Then we propose a joint learning strategy with a semi-supervised learning setting in Section 4.3.
Obviously the aforementioned message passing neural network does not explicitly consider the DDI correlation in the prediction.
However, in practice, the labels are often correlated as we stated in the introduction section.
So we re-formulate our problem as a structure prediction problem and infer the labels by optimizing an energy function.
In order to get extremely powerful representation for the energy model, we follow the ideas of SPENs to formulate the energy function as a neural network.
4. E NERGY F UNCTION D ESIGN
The SPENs deﬁne the energy E ( X, Y ) as a neural network that takes both the input features X and labels Y as inputs and returns the energy.
Although the SPENs can almost obtain arbitrarily high expressiveness, in practice it requires a trade-off between using increasingly expressive energy networks and being more vulnerable to overﬁtting (Belanger & McCallum, 2016b), so the energy function need to be properly designed.
Energy function via GNN Recall the message passing neural network we used before, the edge information is also included in the neural networks.
If we aggregate all the nodes’ and edges’ information to get the whole graph’s representation, it is natural to derive an “energy” over the graph.
Motivated by this intuition, we formulate a new and simple energy function from a graph neural network:
[Equation]
Where G 0 is an arbitrary graph neural network (GNN) which accepts node features X and edge features Y which can be initialized by multi-hot encoding storing labels, h i is the node embedding of ﬁnal output of GNN which contains information of neighbor nodes and edges, MLP is a multilayer perceptron network.
Obviously, Eq.
(3) contains both the interaction between each X and Y and the interaction among all labels Y .
When we have an energy function, the inference of the predicted label then simply becomes to minimize the energy function:
[Equation]
4. T RAINING C OST -A UGMENTED I NFERENCE N ETWORK
The training of SPENs usually takes the structured hinge loss (Tsochantaridis et al., 2004; Belanger & McCallum, 2016b) and relaxes the edge labels to be continuous for easier optimization.
Assume the parameters for the energy function Eq.
(3) is Θ , our problem becomes:
[Equation]
where Y L are the ground-truth DDI edges for training, and ˆY L are the predictions on these edges.
(cid:52) is the structured error function which returns a non-negative value to indicate the difference between the ground truth and prediction; and [ · ] + means max(0 , · ) .
In this paper, we use L1 loss as (cid:52) .
This above training loss is expensive to directly optimize due to the cost-augmented inference step.
Following the idea of Tu & Gimpel (2018), we use a cost-augmented inference network (i.e. a graph neural network) G Φ ( X , Y L ) to approximate the output ˆY L in the training phase.
[Equation]
The problem can be seen as minimax game and optimized by alternatively optimizing Θ and Φ .
(1) Fix Θ , we can optimize the parameters for the cost-augmented inference networks
[Equation]
Φ , optimize
[Equation]
4. S EMI - SUPERVISED J OINT T RAINING AND I NFERENCE
The trained cost-augmented inference network cannot be directly used for test inference because of the cost augmentation.
Notice that objective function to infer the unknown labels in the testing phase is Eq.
Due to the additional term in the loss function Eq. (8) during training, the derived training inference network cannot lead to a minimized energy.
Thus we need another test inference network to optimize Eq. (4) in the testing phase. We propose the test inference network ˆY U =
G Ψ ( X , Y L ) to directly approximate the test output.
Since the test edge features are not given(instead, we only know the indices of edges to test), so this test inference network also use X and Y L as its inputs.
In the DDI prediction problem, the training set and test set share the same set of nodes X in a single graph. In this graph, the energy optimization in the test phase should not be regarding only Y U but the all graph including both Y L and the predicted ˆY U , as we showed in Eq. Since this energy function is a graph neural network, that is to say, we use both Y L and the prediction ˆY U as edge attributes for the computation of Eq.
When the parameter Θ for the energy function is given, the objective for this test inference network is as below:
[Equation]
The previous methods (e.g. Tu & Gimpel (2018))
ﬁrst ﬁne-tune the trained inference network on training data again with respect to the original inference objective without cost augmentation arg min Φ E ( X , G Φ ( X , Y L )) .
But the real effect of this ﬁne-tuning is actually doubted (Tu & Gimpel (2018)). Also considering that the deﬁnition of training-set energy and the test-set energy are in the same graph, and the test inference network should not deviate too much from the training cost-augmented inference network, we are proposing a new joint training strategy as follows. We design the test inference network to share some bottom layers with the training cost-augmented inference networks.
Then we do the optimization with the two following steps: (1) Fix Θ , we optimize the parameters for both the cost-augmented inference networks and the test inference network.
[Equation]
To better discriminate the two inference networks, we use G L to indicates the predictions on training edges, and G U are the predictions on and missing edges.
λ 1 is a hyperparameter which is often set as 1 in practice.
Notice that Φ and Ψ are not independent but share some parameters.
To better understand the beneﬁt of the joint training, let us rewrite the parameter Φ and Ψ into { Φ 0 , Φ 1 } and { Φ 0 , Ψ 1 } , where Φ 0 is the shared parameter set between Φ and Ψ for the shared layers of the two inference networks.
Then the above optimization problem becomes:
[Equation]
, Y L , G U Φ 0 , Ψ 1 ( X , Y L )) From this equation we can see the optimization of the second term, i.e. the test-set energy, can also impact the training of the cost-augmented inference network. If we ﬁx all other parameters, the Ψ 1 still gives the test inference network enough ﬂexibility to only optimize the test energy term; meanwhile the shared layers and the shared parameter Φ 0 kept the two inference networks not deviate too much. (2) Fix Φ and Ψ , we do not want the test inference network to impact the energy function (so that the objective for optimizing Θ is still from Eq.
(6), so the min function is unchanged, i.e. still Eq.
E XPERIMENT
We evaluate GENN 1 model to answer the following questions:
Q1 :
Does GENN provide more accurate DDI prediction than feed-forward GNNs?
If so, we can demonstrate the usefulness of the new energy function.
Q2 :
Does GENN improves the supervised inference method (Section 4.2)?
If so, we can demonstrate the effectiveness of our new optimization algorithm.
Q3 :
How does GENN respond to various fraction of edge missingness? 4.
Q4 : Does the model really capture meaningful label correlation?
5. E XPERIMENTAL S ETUP
Data We implemented our experiments on two public datasets (with some modiﬁcation of the setting): DeepDDI and Decagon.
For both datasets, we randomly select 80% of the drug-drug pairs as training data, 10% as validation data and the remaining 10% as test data.
For a given drug-drug pair, we predict its DDI types.
Note that one drug-drug pair can have zero, one or multiple DDI types, so it is a multi-label prediction problem.
DeepDDI Dataset (Ryu et al., 2018) consists of 1861 drugs (nodes) and 222,127 drug-drug pairs (edges) from DrugBank which results in 113 different DDI types as labels.
99.87% drug-drug pairs only have one type of DDI.
The edge exists when two drugs has at least one DDI type.
The input feature for each drug is generated based on structural similarity proﬁle (SSP), then its dimension is reduced to 50 using principal components analysis (PCA) as suggested by DeepDDI.
Moreover,
Chemical-based similarity measure is used to reduce the the effect of redundant drugs on prediction accuracy where redundant similar drugs in training and test dataset will overestimate the predictive performanc as noted in Gottlieb et al.
In particular, we use Tanimoto score (Tanimoto, 1957) and ﬁnd 90% drug pairs has less than 0.4721 similarity score which show that there is no need to tackle the effect of redundant drugs.
BIOSNAP-sub Dataset (Marinka Zitnik & Leskovec, 2018) consists of 645 drugs (nodes) and 46,221 drug-drug pairs (edges) from TWOSIDES dataset which results in 200 different DDI types as labels.
We extract data from Zitnik et al.
and only keep 200 medium commonly-occurring
DDI types ranging from Top-600 to Top-800 that every ddi type has at least 90 drug combinations which contains proper number of edges for fast evaluation.
73.27% drug-drug pairs have more than one type of DDI.
The input feature is 32-dimension vector by transforming one-hot encoding by
Gaussian random projection using Pedregosa et al.
Baselines We compared GENN with the following baselines.
Label Propragation (LP) (Zhu et al., 2003) is a similarity-based semi-supervised method which makes use of unlabeled data to better generalize to new samples.
MLP is the model used in original DeepDDI dataset which accepts pairwise features (e
g. structural similarity proﬁle (SSP)) between two drugs to predict DDI. 3.
DeepWalk learns d-dimensional neural features for nodes based on a biased random walk procedure exploring network neighborhoods of nodes.
For each drug pair, we concatenate the learned DeepWalk feature vectors and its original drug feature representation and use the MLP same as the above method to do classiﬁcation.
GNN is a basic graph neural network based on MPNN framework as introduced in 3.2. 5.
GLENN is
designed to be an energy model with a CRF-style local energy function and the same inference algorithm as GENN .
Notice that each node and all its connected edges form a clique (because these edges are mutual neighbors and they are also connected to each other due to the correlation), so we use E = (cid:80) i
f 1 ( x i + (cid:80) j ∈ N ( j ) f
( e i,j )) .
The difference between Eq. (3) and
Eq. (1) is that it does not include the neighbor node features in the equation.
It is linear and has only one layer.
We call it a graph local energy neural network (GLENN) and use the same optimization strategy as our ﬁnal model.
GENN − is an ablation model without semi-supervised joint training and inference.
GENN is our ﬁnal model that incorporates the power of GNNs and energy models.
The Implementation Details can be found in Appendix B.
Metrics.
To measure the prediction accuracy, we used the following metrics
ROC-AUC : Area under the receiver operating characteristic curve: the area under the plot of the true positive rate against the false positive rate at various thresholds.
PR-AUC : Area under the precision-recall curve: the area under the plot of precision versus recall curve at various thresholds.
P@K : short for Precision at K which is the mean percentage of correct predicted labels among TOP-K over all samples.
Here, we specially choose P@1 and P@5 to validate performence as usually done in multi-label learning setting.
Following Decagon (Zitnik et al., 2018), we calculated the ROC-AUC and PR-AUC for each single label, and then use the average scores as the ﬁnal values.
5. R ESULTS
Performance Comparison For each dataset, we use 3 different random splits and run all the models on that split.
The results are then averaged over 3 runs and we report the mean and std values in test dataset see Table 1.
Note that the results on DeepDDI dataset in Table 1 is based on the 60% randomly sampled data, because it will take more than 2 days for models to converge with limited performance gain on total dataset.
From Table.
1, we can see DeepWalk and the methods based on MPNN framework outperform MLP and LP in both datasets with a large margin.
It indicates the neighbor edges and nodes information
[Table] [Figure] [Figure] ) actually help the representation learning which improve the performance.
Compared with GNN,
GLENN achieves nearly the same performance and gets little better score with respect to ROCAUC.
As for GENN , it outperforms others with respect to all metrics in both datasets which give the positive answer to Q1 that GENN does provide more accurate DDI prediction.
Especially, compared to GNN and GLENN, it demonstrates the power of our new designed graph neural network based global energy function.
It is more expressive and ﬂexible to capture the label correlation.
In addition, for variants of GENN , the supervised learning model GENN − setting performs worse compared with GENN which give the answer of Q2 .
Analysis of Robustness
[Figure]
We design experiments to show that compared with baseline models, GENN achieved empirically more robust performance in the setting of high edge missingness.
We let each method be trained on 5%, 10%, and 30% of edges on DeepDDI dataset, and predict on the rest of them.
From Figure.
3, we can see that GENN is better than all the baselines in every percentage point ( Q3 ).
Correlation Analysis
In correlation analysis experiment, we chose the models trained on 60% DeepDDI dataset same as the models reported in Table.
1 and made DDI type predictions on the rest of the edges (40% left test dataset).
To illustrate that GENN is able to capture correlation between labels, i.e., different
DDI types by incorperating energy function, we further calculate the pearson correlation coefﬁcent between distribution of DDI types.
For each of the 113 DDI types 2 , we deﬁne its distribution by counting the number of occurrences in the prediction on each drug node over all edges in test dataset.
The PairGrid ﬁgure 4 shows the pairwise relation between these randomly chosed four DDI types, numbered as 17, 36, 75, 102.
For instance, DDI type 17 means “Drug a may increase the cardiotoxic activities of Drug b ” and DDI type 36 means “Drug a may decrease the bronchodilatory activities of Drug b ”.
For the ground truth, DDI type 17 has medium correlation with DDI type 36 but MLP and DeepWalk overestimate the coefﬁcient.
Especially for DDI type 102, only GLENN and GENN capture the true distribution same with the ground truth.
On the one hand, it is consistent with the high performance achieved by GENN .
On the other hand, the consistency between Truth and GENN demonstrates that GENN really capture some label correlation which answers Q4 .
C ONCLUSION
In this paper, we proposed GENN to cast DDI detection as a structure prediction problem with a new GNN based energy function.
Experiments on two real DDI datasets demonstrated that GENN is superior to the models without consideration of link type correlations and achieved up to 0 .
53% relative improvement) with respect to PR-AUC.
Future works includes extending the correlated GNNs to heterogeneous networks and incorporating medical domain knowledge with structure information such as drug classiﬁcation ontology in the learning.
B I MPLEMENTATION D ETAILS
Most methods are implemented in PyTorch-v1.1.0 (Paszke et al., 2017) based on Fey & Lenssen (2019) and trained on an Ubuntu 18.04 with 56 CPUs and 64GB memory.
We use the implementation from Pedregosa et al.
(2011) for Label Propagation method with the setting of 0.25 gamma value for rbf kernel and 200 maximum number of iteration.
As for DeepWalk method, the ofﬁcial implementation 3 is used with the setting of embedding dimension as 50 and the length of random walk as
[Table]
We implement training inference network ( G Φ ), test inference network ( G Ψ ) and energy network
E Θ with the same 2-layer message passing neural network (MPNN) encoder and 1-hidden-layer
MLP decoder structure.
Hidden dimension are all set as
[Table]
and to reduce the number of parameters and ensure the two inference networks ( G Φ and G Ψ ) do not deviate too much, we share the message passing layer for them, while make the MLP layer different.
For energy network E Θ , we also use message passing for node encoding, but the layer has a different set of parameters from the inference networks in order to make the minimax training schema more effective.
For MLP decoder, we use batch-normalization technology (Ioffe & Szegedy, 2015) with Relu non-linear activation function.
For training efﬁciency, we ﬁrst train the basic MPNN (i.e. GNN baseline) on the training data, and then initialize all MPNN layers used in inference networks and energy networks with the these pre-trained parameters.
We use the learning rate 0.01 for methods to train and 0.001 to ﬁne tuning on two datasets with early-stop mechanism (i.e., training is stopped when there is no performance improvement on validation dataset with 35 consecutive epochs).
Threshold for prediction is simply set as 0.4.
C C ROSS E NTROPY R EGULARIZATION
From the experience of Tu & Gimpel (2018), adding an local cross entropy loss to the energy function could improve the performance a lot.
It can be seen as a multi-task training with two objectives: energy optimization and reconstruction error minimization.
We add the reconstruction cross-entropy loss of both the training and test inference networks in Eq.
(10) to train better inference network approximations.
[Equation] [Equation]
where L CE Φ
=
L CE ( G U Φ ( X , Y L ) , Y L ) , L CE Ψ =
L CE ( G U Ψ
( X , Y L ) , Y L ) and λ 2 , λ 3 are hyperparameters which can all be set as 1 the same as λ 1 in practice.
Note that these additional regularization terms are independent of Θ , so we do not need to add it when minimizing over Θ (Eq.
(8) is still unchanged).
[Table]
D C ORRELATION A NALYSIS
As shown in table 3, there are three pairs of DDI type which shows large (0.6423), medium (0.3347) and small postive correlation (0.2854) indicated by Truth column.
For the ﬁrst pair of DDI type, decreasing metabolism has large correlation with increasing the bradycardic activities in ground truth but GNN only got medium correlation coefﬁcient much lower than GENN ’.
As for the last two pairs of DDI type, GNN performed bad and gave a nearly zero coefﬁcient for the second and overestimated the coefﬁcient for the last.
In the meantime, GENN corresponds with the ground truth which in some extent gives the evidence that GENN has the power to capture correlation between labels.