Gated Relational Graph Attention Networks

1 INTRODUCTION . In this work , we focus on learning long-range patterns in multi-relational graphs using graph neural networks ( GNN ) , which heavily relies on the ability to train deep networks1 . However , GNNs suffer from decreasing performance when the number of layers is increased . Zhao & Akoglu ( 2020 ) point out that this may be due to ( 1 ) over-fitting , ( 2 ) vanishing gradients , and ( 3 ) over-smoothing ( the phenomenon where node representations become less distinguishable from each other when more layers are used ) . Recently , several works have investigated over-fitting ( Vashishth et al. , 2020 ) , over-smoothing ( Li et al. , 2018 ; Chen et al. , 2019 ; Zhao & Akoglu , 2020 ; Rong et al. , 2019 ; Yang et al. , 2020 ) , over-squashing ( Alon & Yahav , 2020 ) , and possible vanishing gradient ( Hochreiter & Schmidhuber , 1997 ; Pascanu et al. , 2013 ; He et al. , 2016 ) problems in GNNs ( Li et al. , 2019a ; 2020 ; Rahimi et al. , 2018 ) . One simple but effective technique to improve the training of deeper GNNs is using skip connections , for example , implemented by the Gated Recurrent Unit ( GRU ) ( Cho et al . ) in the Gated GNN ( GGNN ) ( Li et al. , 2015 ) . Such connections can improve learning deep GNNs as it avoids vanishing gradients towards lower-layer representations of the same node , and reduces over-smoothing ( Hamilton , 2020 ) . However , such vertical skip connections are not sufficient to enable learning long-range patterns . In addition , in relational GNNs , such as the Relational Graph Convolutional Network ( RGCN ) ( Schlichtkrull et al. , 2018 ) , training difficulties may arise from the methods used for integrating relation information , which may suffer from over-parameterization and impact backpropagation . In this work , we develop a novel GNN architecture for multi-relational graphs that reduces the vanishing gradient and over-parameterization problems that occur with existing methods and improves generalization when learning long-range patterns using deeper networks . Several changes are proposed to the Graph Attention Network ( GAT ) ( Veličković et al. , 2018 ) , including a modified attention mechanism , an alternative GRU-based update function , and a gated relation-aware message function . An extensive experimental study is conducted that ( 1 ) shows that different existing relationaware GNNs fail to learn simple patterns in a simple synthetic sequence-based graph classification task , ( 2 ) presents a comparison and ablation study on a synthetic node classification task , ( 3 ) shows that our architecture is competitive with existing ones on an entity classification task using real-world data from previous work . 1We need at least K GNN layers to capture information that is K hops away . 2 BACKGROUND : GNNS AS MESSAGE PASSING NETWORKS . Many popular GNNs can be formulated within a message passing ( MP ) framework ( Gilmer et al. , 2017 ) . Consider a multi-relational graph G = ( V , E ) , where E consists of edges e which are specified by a triple ( u , v , r ) that define a directed edge from u to v , labeled by edge type ( relation ) r. A GNN maps each node v ∈ V onto representation vectors h1v . . .hKv by repeatedly aggregating the representations of the immediate neighbours of every node and updating node representations in every step of the encoding process , each associated with one of K layers of the GNN . Relational GNNs must also take into account the edge types between nodes in the graph G. In the following sections , we use the message passing framework where a single GNN layer/step is decomposed into a three-step process : h ( k ) v = φ ( h ( k−1 ) v , γ ( { µ ( h ( k−1 ) u , r ) } ( u , v , r ) ∈E ( · , v ) ) ) , ( 1 ) where µ ( · ) computes a “ message ” along a graph edge using the neighbour representation h ( k−1 ) v and the edge type r of an edge from u to v , γ ( · ) aggregates the incoming messages into a single vector , and φ ( · ) computes a new representation for node v. E ( · , v ) denotes the set of all edges in G that end in v. After subsequently applying Eq . 1K times to each node of the graph , the final node representations h ( K ) v can be used for different tasks , such as graph or node classification . One commonly used message function is a relation-specific linear transformation , which is used by RGCNs and GGNNs : µMM ( hu , r ) = Wrhu , where Wr is a RD×D parameter matrix associated with edge type r. Moreover , GGNNs implement the update function φ ( · ) using a GRU : h ( k ) v = GRU ( h ( k−1 ) v , h ( k−1 ) v ) , where the GRU ’ s input argument is h ( k−1 ) v = γ ( { µ ( h ( k−1 ) u , r ) } ( u , v , r ) ∈E ( · , v ) ) , the vector for the aggregated neighbourhood of v . 3 MOTIVATION : BREADTH-WISE BACKPROPAGATION . Many methods have very recently been proposed to mitigate vanishing gradients in GNNs , using techniques such as residual connections ( Li et al. , 2019a ; 2020 ) , “ jumping knowledge ” connections ( Xu et al. , 2018 ) , and DenseNets ( Huang et al. , 2017 ) , which have effects similar to the Gated GNN from Li et al . ( 2015 ) . However , these techniques address only backpropagation in the depth of the network ( i.e. , vertically towards lower-level features of the same node ) . As we will now discuss , this is not sufficient for learning long-range patterns since breadth-wise backpropagation ( i.e. , horizontally towards the neighbours ) in such GNNs may still result in learning problems and vanishing gradients . Consider a simple graph convolutional network ( GCN ) with the general update of the hidden units h ( k ) v = σ ( W ( k ) ( h ( k−1 ) v + ∑ ( u , v ) ∈E h ( k ) u ) ) , where σ is a non-linearity such as a ReLU . Following the classical arguments , functions of this form will suffer from vanishing or exploding gradients because of the stacking of linear transformations ( W ) and non-linearities ( σ ) . The backpropagation path from the top-level features h ( K ) v of a node to its initial features h ( 0 ) v is σ ( W ( K ) ( σ ( W ( K−1 ) . . . ( σ ( W ( 1 ) h ( 0 ) v ) . . . ) ) ) . With many layers ( i.e. , whenK is large ) , the gradient magnitude diminishes at every step because of the multiplication with the derivative of the activation function , which is always < 1 for some choices . Additionally , depending on the values of the weights W , repeated multiplication with the weights may cause the gradients to either vanish or explode . This could become especially problematic when sharing weights between layers , which could be useful to combat over-parameterization in deeper networks , which are necessary to encode multiple hops . These problems can be mitigated by the choice of activation function ( such as using a ReLU ) , using good initializations for the weights , and using normalization layers . LSTMs ( Hochreiter & Schmidhuber , 1997 ) , GRUs ( Cho et al . ) , ResNets ( He et al. , 2016 ) , Highway Networks ( Srivastava et al. , 2015 ) , and DenseNets ( Huang et al. , 2017 ) propose an alternative solution by introducing some form of skip connection around one or more layers , such that every layer contributes an update additively . This ensures that the gradient to deeper layers does not vanish , and also that the gradient w.r.t . features ( or states ) in different layers ( or timesteps ) is similar ( GRUs , Highway Network ) or the same ( ResNets ) . In fact , He et al . ( 2016 ) point out that the improvement they obtain is not due to gradient magnitudes and attribute the effect to the fact that modeling the residual error is an easier task for the network . Depth-wise backpropagation for the simple GCN can be improved by adapting it to a residual GCN ( ResGCN ) ( Li et al. , 2019a ) , such that we get the following update of the hidden units describing the node features : h ( k ) v = h ( k−1 ) v + σ ( W ( k ) ( h ( k−1 ) v + ∑ ( u , v ) ∈E h ( k ) u ) ) . Alternatively , GGNN uses a GRU-based update function that implements a gated skip connection . However , these techniques do not address breadth-wise backpropagation ( towards the neighbouring nodes ) , which is essential for learning long-range patterns . For an illtustration of this problem , consider the hidden update of the ResGCN . We can easily see that every backpropagation path from the top-level features h ( K ) v of a node v to the ( K − L ) -level features h ( K−L ) u of a node u that is L hops away is σ ( W ( K ) ( σ ( W ( K−1 ) . . . ( σ ( W ( 1 ) h ( K−L ) u ) . . . ) ) ) , which contains L linear transformations and non-linearities . And all the backpropagation paths from h ( K ) v to h ( 0 ) u also contain L such applications . A similar argument can be made for the GGNN , for which we provide a derivation of gradients in Appendix A.4 . While proper initialization , the use of normalization layers , and non-saturating nonlinearities may maintain good gradient magnitudes , the training of the network can still be improved by implementing some form of breadth-wise skip connections . In addition , as we will experimentally confirm , the use of only depth-wise skip connections can actually hurt performance when the task requires learning patterns containing distant nodes . Note that the breadth-wise gradient issues are further exacerbated when averaging aggregators are used , which diminish the gradient towards a neighbour by a factor equal to the number of incoming edges . This leads to a gradient magnitude decay that is exponential in the distance between nodes . This drop is also related to the issues of oversmoothing and over-squashing ( Alon & Yahav , 2020 ) . 4 GATED RELATIONAL GRAPH ATTENTION NETWORKS . In this section , we describe a novel class of GNN architectures that addresses the breadth-wise backpropagation issues and prevents exponential breadth-wise gradient decay . We propose several changes which all follow the general principle of using additive ( gated ) feature updates not only depth-wise ( vertically ) but also breadth-wise ( horizontally ) .

Summary:     The authors propose a new gating based recurrent graph attention networks for multi-relational graphs to capture long-range neighbor dependencies. The authors provide an interesting analysis of current gated GNN models (in the appendix + Figure 3) in light of their ability to capture long-range dependencies in graphs. Experimental results are reported for node classification with two synthetic datasets and two real-world datasets.