summaries:     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.   This paper proposes a new GNN model (GR-GAT) for multi-relational graphs. The proposed method has better ability of capturing the long-range information. Essentially, the proposed GR-GAT is modified from GAT so that it can apply to the multi-relational graphs. Since the modifications are common and frequently used techniques, the novelty of this work is not enough. Also, why these modifications can help to capture long-range information is not well explained in this paper. Overall, this work is ok but not good enough for ICLR. This paper presents a graph attention architecture that captures long-range interactions. The novelties in the architectures are (1) vector-based parameterization of edge type in modeling message, (2) slight modification of graph attention (Section 3.2), and (3) GRU-based node update function. The experiments are primarily on synthetic tasks. However, it is unclear if modeling such long-range interaction is useful in real tasks. The paper fails to demonstrate convincing results on the real tasks of entity classification in knowledge graphs.
abstractText: Relational Graph Neural Networks (GNN), like all GNNs, suffer from a drop in performance when training deeper networks, which may be caused by vanishing gradients, over-parameterization, and over-smoothing. Previous works have investigated methods that improve the training of deeper GNNs, which include normalization techniques and various types of skip connection within a node. However, learning long-range patterns in multi-relational graphs using GNNs remains an under-explored topic. In this work, we propose a novel relation-aware GNN architecture based on the Graph Attention Network that uses gated skip connections to improve long-range modeling between nodes and uses a more scalable vector-based approach for parameterizing relations. We perform extensive experimental analysis on synthetic and real data, focusing explicitly on learning long-range patterns. The proposed method significantly outperforms several commonly used GNN variants when used in deeper configurations and stays competitive to existing architectures in a shallow setup.
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sections.heading: 1 INTRODUCTION
sections.text: 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.
sections.heading: 2 BACKGROUND: GNNS AS MESSAGE PASSING NETWORKS
sections.text: 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.
sections.heading: 3 MOTIVATION: BREADTH-WISE BACKPROPAGATION
sections.text: 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).
sections.heading: 4 GATED RELATIONAL GRAPH ATTENTION NETWORKS
sections.text: 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).
sections.heading: 4.1 SYMMETRICALLY GATED RECURRENT UNIT
sections.text: As elaborated above, depth-wise residual update functions and GRUs as used in the GGNN improve depth-wise backpropagation towards lower-level features of the same node. To also improve the breadth-wise backpropagation, we propose to use an adapted version of the GRU. We call the proposed update function, which is described by the following equations, a Symmetrically Gated Recurrent Unit (SGRU):
  r (k) h = σ(Wrhh (k−1) v + Urhh (k−1) v + brh) , r (k) x = σ(Wrxh (k−1) v + Urxh (k−1) v + brx) (2) z(k)x =Wzxh (k−1) v + Uzxh (k−1) v + bzx , z (k) h =Wzhh (k−1) v + Uzhh (k−1) v + bzh (3) z(k)u =Wzuh (k−1) v + Uzuh (k−1) v + bzu , ẑ (k) x,i ,ẑ (k) h,i , ẑ (k) u,i = softmax([z (k) x,i , z (k) h,i , z (k) u,i ]) (4)
  ĥ(k)v = tanh(W (h (k−1) v r(k)x ) + U(h(k−1)v r (k) h )) (5)
  h(k)v = ẑ (k) x h (k−1) v + ẑ (k) h h (k−1) v + ẑ (k) u ĥ(k)v . (6)
  The SGRU differs from the GRU used in the GGNN in (i) introducing an additional reset gate r(k)x that is applied to the aggregated neighbour states h (k−1) v and (ii) computing the output state h (k) v as a three-way mixture between the previous node state h(k−1)v , the aggregated neighbour states h (k−1) v , and the candidate state ĥ (k) v instead of a two-way mixture between h (k−1) v and ĥ (k) v . The interpolation coefficients ẑ are produced by an elementwise softmax between the three z vectors. See also Figure 2 for an illustration.
  The vanilla GRU as applied to sequences has two inputs: the input vector xt at a certain timestep t and the state vector h. While it implements additive gated updates on h, it applies linear transformations
  and nonlinearities on xt before merging it into h. In the SGRU, both inputs are gated similarly, which lets the xt input benefit from the same additive gradient behavior as h and gave rise to the name "symmetrically gated". This is crucial for improving horizontal information flow during the training of GNN since the xt input of the SGRU receives the aggregated neighbourhood vector.
  In synthetic task experiments (see Sections 5.1 and 5.2), we show that the SGRU is superior to the GRU as a GNN update function. However, this modification alone proved insufficient. To actually enable learning of long-range patterns, we also need to address the problem of exponential breadth-wise gradient decay and over-squashing (Alon & Yahav, 2020). This is addressed by the modifications described in the following sections.
sections.heading: 4.2 ATTENTION-BASED NEIGHBOURHOOD AGGREGATION
sections.text: As elaborated at the end of Section 3, averaging aggregators, which are commonly used to stabilize GNN training, lead to decaying gradient magnitudes w.r.t. distant nodes. In addition, a simple, non-weighted sum based aggregator squashes the entire K-hop receptive field of a K-layer GNN into a single vector uniformly, making it very challenging to distinguish features from the exponentially growing number of connected distant nodes (Alon & Yahav, 2020). For this reason, we believe that using an attention mechanism is crucial to enable the learning of long-range patterns in graphs. In the initial stages of training, when the attention weights are more uniform, the network might still suffer from exponentially decaying gradients. However, this problem should be reduced during training if attention learns to focus on a few specific node.
  To implement the aggregation function γ(·), we adapt the scaled multi-head multiplicative attention mechanism (Vaswani et al., 2017). The per-head attention distributions are computed as described for transformers, with two small but important differences: (1) the edge type embedding are used directly in the keys when computing the attention, and (2) the linear transformations normally used to obtain value vectors are omitted. This last change follows the principle of using only additive operations to enable better long-range backpropagation through messages and node features. See Appendix A.1 for a more detailed description.
sections.heading: 4.3 USING EDGE FEATURES
sections.text: Different ways of using edge features have been proposed. RGCN and GGNN use multiplication with a relation-specific matrix (µMM). CompGCN (Vashishth et al., 2020) and VR-GCN (Ye et al., 2019) make use of vector-based relation parameterization, and explore composition functions inspired by graph embedding learning methods (Bordes et al., 2013; Yang et al., 2014; Nickel et al., 2016). However, all these methods implement a linear transformation based update of the neighbour state.
  To preserve the gradient in the message function, we must choose an additive function. A simple possible choice is vector addition: µADD(hu,ar) = hu + ar. However, additive update functions of this form have the problem that under a non-weighted sum aggregator, the relation vectors are interchangeable. It is not clear whether simply using an attention-based aggregator can resolve this problem completely since (subsets of) the attention weights can be close to each other. This leads us to the additional requirement of non-interchangeability of relations: ∀a,b, c,d ∈ R : a 6= c ∧ b 6= d ⇐⇒ µ(a,b) + µ(c,d) 6= µ(a,d) + µ(c,b). The third requirement is vector-based parameterization of relation types to combat over-parameterization in RGCN and GGNN.
  We propose the following two-layer gated message function, which fulfills these requirements and can model complex interactions between node and edge features (unlike some simpler choices that
  also fulfill the requirements for the message function).
  z = CELU(bA +WA[hu;ar])) , m = σ(bM +WMz) , (7) u = bB +WBz , µGCM(hu, r) = m hu + (1−m) u , (8)
  where ar is a vector ∈ RD associated with the edge type from u to v, and WA, WB, and WM are trainable weight matrices (which have their corresponding biases bA, bB, and bM) that are shared between different edge types. See also Figure 1 for an illustration. CELU (Barron, 2017) is used as the activation function. Similarly to highway networks (Srivastava et al., 2015), µGCM(·) implements a gate that interpolates (based on m) between the original node state hu and the relation-aware update u using a two-layer feedforward network. See also Appendix A.3.
sections.heading: 5 EXPERIMENTAL ANALYSIS
sections.text: We wish to answer the following questions in our experiments:
  Q.1 How well can the baselines learn simple rules for a sequence classification task? (5.1) Q.2 How does the proposed model compare to the baselines on a synthetic node classification
  task requiring long-range pattern learning? (5.2) Q.3 Are all the proposed changes necessary to achieve the best performance? (5.2) Q.4 How does the proposed method perform in a shallow setup on entity classification using
  real-world knowledge graph data? (5.3) Q.5 Does the performance degrade on entity classification when deeper networks are used? (5.3)
  The following baselines are used: (1) RGCN (Schlichtkrull et al., 2018), (2) GGNN (Li et al., 2015) and (3) RGAT (Busbridge et al., 2019). We implement our RGCN and GGNN baselines based on the code provided by the DGL framework. For GGNN, the parameters between layers are shared. We found several relational variants of the GAT in the literature (Busbridge et al., 2019; Brockschmidt, 2019; Sinha et al., 2019) and chose to adapt (Busbridge et al., 2019) for the attention-based baseline (RGAT). See Appendix A.2 for an elaborate description of the used RGAT and other details regarding memory-efficiency.
  The following ablations of our method (GR-GAT) are considered in the experiments: (1) “GR-GAT (X), mean agg.” replaces the attention based aggregation function with a simple mean (equivalent to uniform attention), (2) “GR-GAT (X), edge-MM” replaces µGCM with µMM, and (3) “GR-GAT (X), val. transf.” uses a linear layer to transform the value while using attention, like in transformers. The X in GR-GAT(X) is the update function and can be one of the following: Ident (identity function), GRU (Gated Recurrent Unit), or SGRU (see Section 4.1). If X is omitted, the SGRU has been used. Weight are always shared between all layers of GR-GAT.
sections.heading: 5.1 CONDITIONAL RECALL
sections.text: In this experiment, we aim to answer Q.1 and show that the different popular relational GNNs are not able to solve a simple sequence classification task that involves remembering one of the symbols in
  a sequence according to a set of simple rules. Here we deliberately choose the following setup to measure how well the different GNNs can learn over a larger numbers of hops in a very simple graph.
  We define a sequence classification task where given a sequence of characters [x1, . . . , xN ], the model is asked to predict the correct class based on the representation of the last node (corresponding to input xN ). The input sequences consist of strings of letters and digits of a given length (which was varied between different experiments). The class of the sequence is determined by the following rules: (i) if there is a digit in the sequence, the first digit corresponds to the class label; (ii) otherwise, if there is an upper case character, the first upper case character is the class label; (iii) otherwise, the class is given by the first character in the sequence. Some examples are: ”abcdefg“→ ”a“, ”abcDefg“ → ”D“, ”abcd3Fg“→ ”3“, ”abCd3fg“→ ”3“. Twenty examples were generated per output class for a total of 1220 examples and the data was split in 80/10/10 train/validation/test splits. For more details about the experimental setup for this task see Appendix A.5.
  The input sequences are transformed into graphs by (i) creating a node for every character of the sequence and (ii) adding edges labeled using the next type between every adjacent element in the sequence. Given these edges, the GNN has to use at least N layers/steps in order to propagate information from the first node to the last node. The number of layers is always set to N + 1. The readout for prediction is done by taking the representation of the last node in the sequence.
sections.heading: 5.1.1 RESULTS AND DISCUSSION:
sections.text: The results for the different GNNs and varying sequence lengths (L) are shown in Table 1 (Note that all examples have the same, specified length in one experiment). While all baselines reach acceptable accuracy for length 5, their accuracy severely suffers when the length is increased. In contrast, our method solves the task well for all lengths. Figure 4 reports the training behaviour in terms of training and validation set accuracies per epoch for the best runs of all considered GNNs, showing that the proposed GR-GAT model achieves much faster convergence and superior generalization on this task when compared to the baselines.
  Additional experiments are conducted to directly compare SGRU and GRU. Here, the message function as well as the attention mechanism from GR-GAT were disabled, and self-edges were removed. The update function (SGRU or GRU) receives the neighbourhood aggregating vector as the input and the lower-level features of a node as the state. Clearly, the SGRU performs better, confirming our claims. Note that in this task, there is only one real edge type and every node has at most one neighbour so the message and aggregation functions are unnecessary.
  Surprisingly, the GGNN severely underperforms compared to the skip-less RGCN and RGAT. We suspect that, because of the gated skip connections to lower-layer node representations, it is easier for the GGNN to memorize combinations of nearby nodes rather than learning meaningful patterns. Additional experiments on a modified task were conducted to verify the GGNN, for which the results are shown in the last column of Table 1. The modified task follows the same rules but uses short sequences (L=5) and a large number of layers (N=16). Even though many layers are used, the task thus requires looking over at most 4 hops in the graph. As we can see, the GGNN performs better than the other networks without depth-wise skip connections.
  It is important to keep in mind that the computational graph for a GNN applied to this graph is different from an RNN applied to the same sequence. The “update distance” (the number of RNN or GNN updates performed) between the last state hT of an RNN and a token xT−k (from k timesteps before) is k. In contrast, as illustrated in Figure 3, the update distance between the top-level features of any node and the initial features of any node is always equal to the number of layers in the GNN. In the figure, the final vector of the blue node is used for prediction and the red node specifies the desired output. Red arrows are “vertical” self-messages and blue arrows “horizontal” messages to neighbours. In the case of an RGCN, the neighbours and the node itself are treated equally, while the GGNN provides a skip connection to previous node features, enabling easier backpropagation “vertically” within the node. However, this does not improve backpropagation to distant nodes in the graph (see also Appendix A.4).
  We also inspected the gradients w.r.t. the input vectors and experimented with different initializations for the edge type matrices of the baselines. We found that the initialization indeed strongly affects the gradient magnitudes and affects training behaviour. However, even when the weights are initialized correctly and the initial gradients are not small, the baselines are still not able to solve the task well for larger task lengths. We believe that our proposed changes encourage more effective weight sharing across nodes and layers and may benefit from other effects that make it easier for the network to generalize. This is similar to the findings of He et al. (2016).
sections.heading: 5.2 TREE MAX
sections.text: In this experiment, we aim to answer Q.2 and Q.3 using a synthetic node classification task on trees. The input graphs are trees with nodes labeled with random integers between 1 and 100. The expected output labels for the node classification task are defined as the largest value of all the descendants of a node and the node itself. The graphs contain edges from a parent to its children, and from children to their parent, as well as self-edges. We use numbered child edges and child-of edges, for example, :CHILD-1-OF for the edge going from the first child of a node to its parent.
  As an example of this task, consider the following input tree: (1 (2 (3 ) (4 )) (5 (6 ) (7 (8 ) (9 ) (10 )))). The nodes in the tree should be labeled with the value of their highest-valued descendant. Thus, the tree with its output labels will become: (10 (4 (3 ) (4 )) (10 (6 ) (10 (8 ) (9 ) (10 )))). In this case, to correctly predict the output label of the root node (1), the GNN must handle three hops. Please see Figure 6 in Appendix A.6 for an illustration of the graph constructed for this example.
  A total of 800 examples are generated, each containing one tree with a randomly chosen depth between 5 and 17. The largest generated trees contained more than 200 nodes. 17-layer GNNs (and fewer layers for the baselines) are used in the experiments. A 50/25/25% training/validation/test split is used. Instead of training and evaluating with the expected labels of all nodes, we choose a setting where the labels are omitted of node for which fewer hops must be resolved. Of the remaining nodes, only a random 50% selection retain their labels. Unlabeled nodes are not used in loss or metric computation. More details about the data and the experimental setup can be found in Appendix A.7.
sections.heading: 5.2.1 RESULTS AND DISCUSSION:
sections.text: As shown in Table 2, the best node-level and graph-level2 accuracies were obtained using the proposed GR-GAT model. Using the SGRU doesn’t bring benefit compared to having no update function (Identity) in this task. A possible explanation is that the task is retrieval-oriented and can thus easily be solved using just the attention mechanism.
  Concerning the claimed improvements of SGRU vs a normal GRU, we can see that GRU is the worst among the three tested options (SGRU, GRU or Identity) by a large margin. The ablation study further indicates that the gated edge function (Section 4.3) and adapted attention mechanism (Section 4.2) are essential for achieving the best performance since the "edge-MM" and "val. transf." variants reach significantly worse performance.
  Even though using no update function works well in this task, we believe the SGRU can be useful since it provides an alternative and more fine-grained mechanism for focusing on the previous node
  2The graph-level accuracy is 100% for an example only if all labeled nodes in the graph have been classified correctly, and is 0% otherwise
  features (whereas the attention mechanism uses a self-edge relation when attending to the previous node features and the granularity is fixed to the number of heads). This can also be seen in the ablation study, where the SGRU variant performs much better with the matrix multiplication based edge function µMM (“edge-MM” in the table) than the variant without an update function.
sections.heading: 5.3 ENTITY CLASSIFICATION USING REAL-WORLD DATA
sections.text: To answer Q.4 and Q.5, we run experiments on real-world data for the node classification tasks, where a certain property about an entity (which are nodes in the graph) must be predicted. It is not clear whether this dataset could benefit from a GNN that is able to model longer patterns so we do not expect to see improvements using the original data. With these entity classification experiments, we mainly want to verify that our proposed model does not severly underperform compared to previously proposed relation-aware GNNs in a shallow setup and study the performance of our model when more layers are used.
  The experimental setup is the same as in Schlichtkrull et al. (2018), who use previously proposed datasets for entity classification in knowledge graphs. The same preprocessed datasets and train/test splits are used here. However, compared to Schlichtkrull et al. (2018), who fit the entire dataset in memory, we implement subsampled mini-batch training. For more details, see Appendix A.8. Experiments are performed using two layers (similarly to Schlichtkrull et al. (2018)), for which the results are reported in Table 3 for the AM dataset and in the first column of Table 4 for the AIFB dataset. Further experiments are performed using the AIFB dataset to measure how much GR-GAT degrades when more layers are used, which is shown in the rest of Table 4.
sections.heading: 5.3.1 RESULTS AND DISCUSSION
sections.text: The results in Tables 3 and the first column of Table 4 show that our method is competitive with the previously proposed relation-aware GNNs in a two-layer setup (L = 2). The second column of Table 4 shows that GR-GAT degrades less strongly than RGCN when ten layers (L = 10) are used on the same two-hop subgraph (“2 hops” ) as in the first column. The GGNN, which has vertical
  skip-connections and also shares parameters between layers, is similarly more resilient to the increase in layer count. When a sampled subgraph is used that is built by random walks from the labeled node (which goes beyond two hops; see also Appendix A.8), which is the setting reported in the last column of Table 4, we see some small improvement from all methods compared to a 10-layer network applied on the two-hop subgraphs. However, there is still no improvement compared to the 2-layer GNN applied to a 2-hop subgraph. It is unclear whether the dataset could benefit from long-range pattern learning.
sections.heading: 6 RELATED WORK
sections.text: Li et al. (2019a) investigate the GCN equivalents of ResNet (He et al., 2016) and DenseNet (Huang et al., 2017), evaluated on point cloud semantic segmentation, as well as dilated aggregation. ResidualGCNs and Dense-GCNs (Li et al., 2019a), Highway GCN (Rahimi et al., 2018), Column Networks (Pham et al., 2017) and Li et al. (2020) are similar to the GGNN (Li et al., 2015) in using gated, residual or concatenated skip connections to previous nodes states. Xu et al. (2018) propose to add skip connections from all layers straight to the output layer to combat oversmoothing. The neighbourhood dilation proposed by Li et al. (2019a) for point cloud semantic segmentation increases the receptive field of the network without requiring the information to pass through other nodes. However, it is not clear how to apply this technique to multi-relational graphs. Additionally, different normalization techniques (Zhou et al., 2020; Zhao & Akoglu, 2020; Li et al., 2020) have very recently been proposed that improve the training of deep GNNs. AGGCN (Guo et al., 2019) transforms a graph into a fully connected graph and uses self-attention and thus, similarly to transformers (Vaswani et al., 2017), benefits from direct access to distant nodes. However, this leads to quadratic complexity in the number of nodes (due to attention in a fully connected graph). In contrast to previous work, we focus on improving the communication between neighbours in GNNs for multi-relational graphs.
  A family of methods closely related to GNNs are TreeLSTMs (Tai et al., 2015) and GraphLSTMs (Peng et al., 2017; Liang et al., 2016; Song et al., 2018; Bresson & Laurent, 2017). Some versions of these methods have similar beneficial properties regarding backpropagation to distant nodes but are not suited for use in a general graph setup. Song et al. (2018) adapt Graph LSTMs into a message passing network, however, their implementation of the LSTM-based update suffers from the same issues as the GGNN. Compared to our proposed method, GraphLSTMs use independent forget gates for every neighbour instead of attention. Song et al. (2018) uses a global forget gate, which reduces the ability of the model to focus on certain nodes. The mixing gates in our SGRU bear some similarity to those in the Sentence State LSTM (Zhang et al., 2018). However, the original formulation of the Sentence State LSTM is not applicable for graphs in general. Fi-GNN (Li et al., 2019b), similarly to a basic version of our work, uses a GGNN with attention-based aggregation. However, it lacks our modifications and investigation to improve long-range pattern learning.
sections.heading: 7 CONCLUSION
sections.text: In this work, we proposed a novel GNN architecture that can handle long-term dependencies in multi-relational graphs better than commonly used GNN architectures. Experiments on synthetic tasks show that popular GNNs are bad at learning long-range patterns, and are outperformed by the proposed method (GR-GAT). Real-world experiments show that our proposed method is at least competitive with previous work.An ablation study indicates that the proposed changes w.r.t. the GAT and the GGNN are essential. The SGRU, which beats the GRU, is not absolutely necessary to have as the update function because the attention mechanism can mostly compensate for it while avoiding introducing additional parameters. However, we believe that the SGRU could still be useful as an orthogonal mechanism to control node representations and should be explored in applications. In future work, we plan to investigate additional model improvements and training techniques to further improve training of long-range relational GNNs.
sections.heading: A APPENDIX
sections.heading: A.1 ATTENTION-BASED AGGREGATOR
sections.text: The query and key vectors for each attention head h (of in total H heads), q(h)v ∈ RD for node v and k(h)e ∈ RD for an edge e = (u, v, r) ending in v, respectively, are computed using head-specific linear transformations parameterized by Q(h) ∈ RD/H×D and K(h) ∈ RD/H×2D:
  q(h)v = Q (h)hv , k (h) e = K (h)[µ(hu, r);ar] , (9)
  where [·; ·] denotes the (vertical) concatenation of two (column) vectors and ar is the same edge type-specific parameter vector that is used in Eqs. 7-8. Note that Eq. 9 is slightly different from a standard attention mechanism, which would compute key vectors as K(h)µ(hu, r). We expect that including the edge type vector ar directly in the keys will make it easier to learn to place attention.
  The attention score w(h)e for an edge e and head h is computed by a scaled dot product and then normalized to obtain α(h) as usual: w(h)e = q(h)v ·k (h) e√
  D/H and α(h)e = exp(w(h)e )∑ e∗∈E(·,v) exp(w (h) e∗ ) .
  Unlike multi-head attention in transformers and previous work (Veličković et al., 2018; Busbridge et al., 2019), we do not use linear transformations to obtain the value vectors. Instead, the vector µ(hu, r) = µe = [µ (1) e , . . . ,µ (H) e ] is split in H equally sized chunks and the h-th part µ (h) e ∈ RD/H is taken as value vector for the h-th head. The attention scores α(h)e and value vectors µ (h) e are then used to compute the neighbourhood aggregation vector h (h)
  v for head h as a weighted sum. The full neighbourhood aggregation vector hv is then given as a concatenation over all heads h:
  h (h) v = ∑
  e∈E(·,v)
  α(h)e µ (h) e , hv = [h
  (1) v ; . . . ;h (H) v ] . (10)
sections.heading: A.2 BASELINES
sections.heading: A.2.1 A NOTE ON MEMORY-EFFICIENCY OF THE BASELINES
sections.text: When experimenting with the baselines (RGCN, GGNN and RGAT), we quickly ran out of memory for larger hidden dimensions of the GNN and larger numbers of layers. This is caused by the relation matrices, which have O(n2) parameters for every relation. In the implementations we used, the relations were indexed from a tensor and have to be retained in memory for every edge. While more memory-efficient implementations are possible, the existing ones sacrifice execution time, which can increase significantly with a large number of different relations.
  Gradient accumulation significantly slows down training. As a simple fix to maintain reasonable training speed for our experiments, we replace µMM(·) in our RGCN and GGNN with µMM−red(hu) = WBW ∗ rWAhu, where W ∗ r is an edge type specific square matrix of lower dimensionality than Wr and WA and WB are matrices projecting into and out of W ∗r ’s dimensionality that are shared for all edge types. Note that we did this only for the Tree Max experiments.
sections.heading: A.2.2 RGAT BASELINE
sections.text: Like in (Busbridge et al., 2019), we use relation-specific transformation matrices W(r) and relation and head-specific query and key matrices Q(r,h) and K(r,h).
  First, we define relation-dependent representations for a node, which is computed based on its current state z(k−1):
  g (r) i = W (r)z (k−1) i . (11)
  Subsequently, for every head h, we define the relation-specific query, key and value projections as:
  q (h,r) i = Q (h,r)g (r) i (12)
  k (h,r) i = K (h,r)g (r) i (13)
  v (h,r) i = V (h,r)z (k−1) i (14)
  We compute attention between messages µx, where x indexes over all edges in the input graph. First, we compute the attention score for a message as in (Vaswani et al., 2017):
  s(h)µx = q (h,r) i · k (h,r) j , (15)
  where µx is the message sent along an edge j → i labeled by the relation r. Please note that a node i may receive multiple messages from a node j, and that those messages could contain the same relation.
  The scores are normalized over all messages that node i receives:
  α(h)µx = s (h) µx∑
  µ x ′ s
  (h) µ
  x ′
  (16)
  The normalized scores are used to compute a summary:
  z (h,k) i = ∑ µx v (h,r) j α (h) µx , (17)
  where r is the relation of µx and j is the source node id and i is the target node id of message µx.
  The updated representation for node i is then the concatenation over all heads, fed through the activation function σ:
  z (k) i = σ([z (0,k) i , . . . , z (H,k) i ]) . (18)
  We experiment with a ReLU, as well as with a linear σ.
sections.heading: A.3 NOTES ON RELATION-AWARE MESSAGE FUNCTIONS.
sections.text: Many different message function implementations are possible that take into account the relation connecting a node to a neighbour. The default message function used in earlier literature is simply a matrix multiplication: µMM(hu, r) = Wrhu (19) This can easily lead to overparameterization and leads to difficulties implementing the GNNs in a both memory-efficient and computationally efficient way.
  In CompGCN, Vashishth et al. (2020) propose the following function:
  µCompGCN(hu, r) = Wλ(r)φ(hu,ar) , (20)
  where Wλ(r) is one of the three matrices: one for forward relations, one for reverse and one for self-edges, and φ(hu,ar) is a composition function that combines the neighbour state and relation vector. For the composition function, the authors explore functions inspired by knowledge graph embedding literature: subtraction (hu − ar), multiplication or circular-correlation. While this provides an alternative solution to over-parameterizing relations, and enables more efficient implementation, it does not improve gradient behavior since the backpropagation paths to the lower node features as well as to distant nodes still contain linear transformation and non-linearities.
  To improve backpropagation, we can make use of (gated or residual) skip connections. The residual version would have the following general form:
  µRes(hu, r) = µX(hu, r) + hu , (21)
  where µX is some function. Similarly to transformers and ResNets, we choose to use a two-layer MLP, which is more expressive than µMM, and unlike the composition-based message functions in Vashishth et al. (2020), can effortlessly incorporate additional edge features that are not based on the relation embeddings.
  We chose the gated skip connection described in the text over the simple residual one because it enables better control of the additive contribution of the current layer and the residual version did not perform better in our early experiments. We leave a further investigation of residuals to replace the gated skip connections for future work.
sections.heading: A.3.1 A NOTE ON INTERCHANGEABILITY OF RELATIONS
sections.text: Note that the simple choice of adding a relation vector, similarly to CompGCN’s subtraction version, might pose problems since it loses information about which relation was associated with which neighbour:
  (hu + ar) + (hw + ar) = (hw + ar) + (hu + ar) . (22)
  This is undesireable since it limits the expressive power of the network when using a sum aggregator. It is not clear to us how much this would affect models using attention-based aggregation, however, we prefer to avoid this since the attention-based aggregator may behave like a uniform sum, especially throughout the early stages of training.
  This leads to the following condition that we impose on the message function: The message function µ must be chosen such that
  µ(a, b) + µ(c, d) 6= µ(a, d) + µ(c, b) (23)
  iff a 6= c and b 6= d. The gated message function µGCM proposed earlier in our work satisfies this condition.
sections.heading: A.4 BACKPROPAGATION TO DISTANT NODES IN GATED GRAPH NEURAL NETWORKS
sections.text: The GGNN (Li et al., 2015) implements the update function φ(·) based on Gated Recurrent Units (GRUs) (Cho et al.):
  r(k) = σ(Wrh (k−1) v + Urh (k−1) v + br) (24) z(k) = σ(Wzh (k−1) v + Uzh (k−1) v + bz) (25) ĥ(k)v = tanh(Wh (k−1) v + U(h (k−1) v r(k))) (26) h(k)v = (1− z(k)) h(k−1)v + z(k) ĥ(k)v , (27)
  where h (k−1) v = γ({µ(h (k−1) u , r)}(u,v,r)∈E(·,v)) is the vector representing the aggregated neighbourhood of node v.
  Consider a node classification task, where the top-level state h(K)v of node v is fed into a classifier that produces a loss Lv . To compute the gradient ∇wLv = ∂Lv ∂w
  , where w are the parameters of the model, we apply the chain rule as follows (here we ignore other nodes and only take into account only node v’s contribution to the gradient):
  ∇wLv = ∂Lv ∂w = K∑ k=0 ∂Lv ∂h (k) v ∂h (k) v ∂w (28)
  with
  ∂Lv ∂h (k) v = ∂Lv ∂h (K) v K∏ j=k+1 ∂h (k) v ∂h (k−1) v
  (29)
  Given the gated update equations of the GRU, the partial derivatives ∂h
  (k) v
  ∂h (k−1) v
  of the updated node
  representation w.r.t. the previous node representation are:
  ∂h (k) v
  ∂h (k−1) v
  = (1− z) I+ z ∂ĥ (k) v
  ∂h (k−1) v + ∂(−z) ∂h (k−1) v h(k−1)v + ∂z ∂h (k−1) v ĥ(k)v (30)
  The closer z is to a zero vector, the closer the partial derivatives across layers ∂h
  (k) v
  ∂h (k−1) v
  get to the
  identity matrix I and the more the following holds:
  ∂Lv ∂h (k) v ≈ ∂Lv ∂h (K) v , (31)
  which has the effect that all the GGNN layers are skipped. In practice, the gate z will take on different values throughout training, but the additive update still allows to retain some contribution of the original gradient ∂Lv ∂h (K) v deeper into the network.
  On the other hand, consider a (K-1)-layer GGNN applied to a graph representing a sequence of length K. The sequence is represented as a graph according to the same rules as described in Section 5.1: the nodes are connected using only next-edges. Thus, only left-to-right propagation in the graph can be performed, and every node has only one incoming message, except the first, which has none. This is the setup that maximizes the ratio of graph diameter per number of nodes and edges.
  When a (K-1)-layer GGNN is applied to such a graph of K nodes, there is only one backpropagation
  path from node K to node 0: ∂h
  (K) K ∂h (0) 0 , which is also the longest backpropagation path possible in this
  graph:
  ∂h (K) K ∂h (0) 0 = K∏ k=1 ∂h (k) k ∂h (k−1) k−1
  (32)
  Ignoring the used message function µMM and considering every node has only one neighbour, the neighbourhood aggregation vector is simply the neighbour node vector: h (k−1) v = h (k−1) k−1 .
  Plugging this into the GGNN equations (Eqs. 24-27) gives:
  ∂h (k) k
  ∂h (k−1) k−1
  = ∂z(k)tanh(Wh
  (k−1) k−1 )
  ∂h (k−1) k−1
  (33)
  = tanh(Wh (k−1) k−1 )
  ∂z(k)
  ∂h (k−1) k−1
  + z(k) ∂tanh(Wh
  (k−1) k−1 )
  ∂h (k−1) k−1
  (34)
  = tanh(Wh (k−1) k−1 )
  ∂σ(Wzh (k−1) k−1 )
  ∂h (k−1) k−1
  + z(k) ∂tanh(Wh
  (k−1) k−1 )
  ∂h (k−1) k−1
  (35)
  Both terms contain a partial derivative of the form ∂σ(Wh
  (k−1) k−1 )
  ∂h (k−1) k−1
  , which is essentially identical to
  the terms found in the backpropagation equations for a vanilla RNN.
  As in vanilla RNNs (without any gating like in GRUs and LSTMs), these terms can cause vanishing gradients due to repeated multiplication with the derivative of a σ or tanh nonlinearity and vanishing or exploding gradients depending on the values of W (which also change throughout training and thus could potentially reach unstable values).
  In addition to the gradient-based argument outlined above, residual networks (as well as GRUs and LSTMs), benefit from better generalization abilities due to effects which were not formalized or deeply studied in the literature known to us. Intuitively, the argument for residual networks (He et al., 2016) is that the individual residual blocks can easily learn to not contribute if necessary, which enables to learn very deep architectures, and otherwise, they only need to model the function that models the residual error computed from the output of the previous layer. In addition, we believe that adding such skip connections enables more effective weight sharing across layers and nodes.
sections.heading: A.5 CONDITIONAL RECALL EXPERIMENTAL DETAILS:
sections.text: In our experiments for this task, we randomly explore the following hyperparameter values for all compared methods: (1) dimensionality of node feature vectors in [150, 300] (larger values can cause
  memory issues with the baselines), (2) dropout in [0., 0.25, 0.5], (3) we set the dropout of embedding vectors to 0.1 (4) learning rate in [0.001, 0.0005, 0.0001]. For the baselines, we run for at least 200 epochs. For our GR-GAT, we reach high accuracy in 20 epochs and thus run only for a maximum of 50 epochs. Throughout all of the experiments, we use the Adam (Kingma & Ba, 2015) optimizer. We also use label smoothing with a factor 0.1 everywhere. For our experiments with task length 15, we use the best hyperparameters found for length 10 for all methods and also experiment with smaller dropout values.
  For the final reported accuracies, we trained with the best hyperparameters and loaded the model with the best validation error and evaluated on the test set. This was repeated with three different seeds that were shared between the tested methods.
  The initial node states are initialized by embedding the node type using a low-dimensional embedding matrix (dimensionality 20; for a vocabulary of 62 characters) and projecting the low-dimensional embeddings to the node state dimension using a learned transformation.
  An example of a graph used in this task is given in Figure 5. The output in this example should be "8".
sections.heading: A.6 TREE MAX EXAMPLE
sections.text: See Figure 6 on page 17.
sections.heading: A.7 TREE MAX EXPERIMENTAL DETAILS:
sections.text: During data generation, we first randomly pick a tree depth between five and fifteen. Then, we generate a tree, choosing between 0, 2 or 3 children for each node until we reach the chosen tree depth. The largest trees in our generated data contained over 200 nodes. The generated dataset contained 800 examples; we used a 50/25/25 train/validation/test split.
  Not every node in the dataset is annotated. For every example, which consists of one graph, the nodes which are at most N/2 away from their “answer” node (which contains their output label) are ignored (not used in training or testing). Of the remaining nodes, which require to propagate information for at least N/2 to be solved, only 50% is retained for training and testing. Here, N is the maximum depth of the tree. This setup defines a semi-supervised node classification setup. The reason for using the semi-supervised version instead of the previous version, which was fully supervised is that the task can be solved iteratively using immediate neighbour states only. Because of this, all methods reached very high node classification accuracy. In the semi-supervised version, the models have to learn what information to retain in unlabeled nodes, which are in between labeled nodes and their corresponding “answer” nodes.
  For every model tested, we perform a hyperparameter search using all predefined seeds. For all models, we experiment only with dimensionality of 150 to avoid memory issues and ensure a fair comparison regarding representation dimensionality. Then we run the best hyperparameter setting with three different random seeds and report the test results in Table 2. Note that the same seeds are re-used for experiments for all models, and that every seed results in a different dataset being generated. We used early stopping and reloaded the best model based on element-wise accuracy on the validation set. Patience was set to 10 epochs but every experiment was run for a minimum of 50 epochs.
  The models were evaluated using node-wise and graph-wise accuracy. The graph-level accuracy is 100% for an example only if all gold-labeled nodes in the graph have been classified correctly, and is 0% otherwise.
  In our experiments, we randomly explore different combinations of different hyperparameter settings: dropout is selected from {0., 0.1, 0.25, 0.5}, dimensionality of node vectors is fixed to 150 for all methods. Learning rate is chosen from {0.001, 0.000333, 0.0001}. For all baselines, the number of layers was chosen from 10, 17. Note however that 10 layers is insufficient to perfectly solve the task.
sections.heading: A.8 ENTITY CLASSIFICATION EXPERIMENTAL DETAILS
sections.text: We use the datasets as provided at https://github.com/tkipf/relational-gcn, removing some code for more memory efficient storage. The datasets contain graphs, where only a fraction of nodes carries a label that we should predict (we call these nodes labeled nodes). We train on the train portion as provided on the repository and test on the predefined test portion.
  During experiments, we train with two different settings: (1) a fixed two-hop (subsampled) subgraph and (2) a fixed subsampled random walk subgraph.
  In the first setting (two-hop subgraph), we retrieve for every labeled node that will be classified a two-hop, at most 6000 nodes (and 10000 edges) subgraph. During batching, these subgraphs are merged. Note that if the number of nodes (or edges) exceeds the maximum number of nodes (6000), we stop expanding. We experiments with the following hyperparameter ranges: (1) dropout from 0.25, 0.5, (2) embedding dropout from 0.25, 0.5, (3) dimension from 64, 96, (4) learning rate from 0.005, 0.001, 0.00025. We use at least 50 epochs and early stopping with patience set to 10 epochs to terminate validation runs earlier and save computation time. The number of heads in our GR-GAT method is set to four for all tried dimensions. During hyperparameter search, we run a random selection of all possible combinations with three different seeds, using a subset of the training set (80%) for training and another set (20%) for validation. After determining the best hyperparameters, we use them to train a final model on the entire training set and evaluate on the test set. The reported numbers use three different seeds.
  In the second setting (random walk subgraph), we use random walks to generate the subgraph. For every labeled node, we precompute a subgraph of at most 2000 nodes and 3000 edges, using 5% reset probability in the random walk. During batching, these subgraphs are merged. Note that the subgraphs are precomputed once before training and fixed throughout, so the subgraphs do not get resampled at every epoch. We use the same set of seeds for the different methods we test to ensure that the methods have been tested on the same sets of data. We experiments with the following hyperparameter ranges: (1) dropout from 0., 0.1, 0.25, 0.5, (2) embedding dropout from 0., 0.1, 0.25, (3) dimension from 32, 64, (4) learning rate from 0.01, 0.005, 0.001, 0.00025. We use at least 50 epochs and early stopping with patience set to 10 epochs to terminate validation runs earlier and save
  computation time. The number of heads in our GR-GAT method is set to 4 for all tried dimensions. Since GR-GAT (Ident) reached near 100% training accuracy with the chosen hyperparameters after 15 epochs, we ran it for 20 epochs during final tests. During hyperparameter search, we randomly run all possible combinations with three different seeds, using a subset of the training set (80%) for training and another set (20%) for validation. After determining the best hyperparameters, we use them to train a final model on the entire training set and evaluate on the test set.
  In both settings, we erase the initial features (embeddings) of the nodes for which we make predictions by setting them to the same learnable vector that is reused everywhere.