Graph inference learning for semi-supervised classification
Authors: Chunyan Xu
Zhen Cui
Xiaobin Hong
Tong Zhang
Jian Yang
Wei Liu
Venue:  (2020)
Keywords: semi-supervised classification
graph inference learning

Abstract: In this work, we address the semi-supervised classification of graph data, where the categories of those unlabeled nodes are inferred from labeled nodes as well as graph structures. Recent works often solve this problem with the advanced graph convolution in a conventional supervised manner, but the performance could be heavily affected when labeled data is scarce. Here we propose a Graph Inference Learning (GIL) framework to boost the performance of node classification by learning the inference of node labels on graph topology. To bridge the connection of two nodes, we formally define a structure relation by encapsulating node attributes, between-node paths and local topological structures together, which can make inference conveniently deduced from one node to another node. For learning the inference process, we further introduce meta-optimization on structure relations from training nodes to validation nodes, such that the learnt graph inference capability can be better self-adapted into test nodes. Comprehensive evaluations on four benchmark datasets (including Cora, Citeseer, Pubmed and NELL) demonstrate the superiority of our GIL when compared with other state-of-the-art methods in the semi-supervised node classification task.

Published as a conference paper at ICLR 2020
GRAPH INFERENCE LEARNING FOR SEMI-SUPERVISED CLASSIFICATION
Chunyan Xu, Zhen Cui', Xiaobin Hong, Tong Zhang, and Jian Yang School Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China [cyx, zhen. cui, xbhong, tong. zh zhang, es jyang) ên just. mossteduuccm
Wei Liu
Tencent AI Lab, China w1 w122230com co Lumbia. edu
ABSTRACT
In this work, we address semi-supervised classification of graph data, where the categories of those unlabeled nodes inferred from labeled nodes as well graph structures. Recent works often solve this problem via advanced graph convolution in a conventionally supervised manner, but the performance could degrade significantly when labeled data is scarce. To this end, we propose a Graph Inference Learning (GIL) framework to boost the performance of semi- supervised node classification by learning the inference of node labels on graph topology. To bridge the connection between two nodes, we formally define a structure relation by encapsulating node attributes, between-node paths, and local topological structures together, which can make the inference conveniently deduced from one node to another node. For learning the inference process, we further introduce opttaippiiiiztiion on structure relations from training nodes to validation nodes, such that the learnt graph inference capability can be better self-adapted testing nodes. Comprehensive evaluations on four benchmark datasets (including Cora, Citeseer, Pubmed, and NELL) demonstrate the superiority of our proposed GIL when compared against state-of-the-art methods on the semi-supervised node classification task.
INTRODUCTION
Graph, which comprises a set of vertices/nodes together with connected edges, is formal structural representation of non-regular data. Due to the strong representation ability. it accommodates many potential applications, e.g., social network (Orsini et al. 2017), world wide data (Page et al., 1999), knowledge graph (Xu et al. 2017). and protein interaction network (Borgwardt et al. (007)) ). Among these, semi-superviie supervised node classification on graphs is one of the most interesting also popular topics. Given graph in which some nodes are labeled, the aim of superviid classification is to infer the categories of those remaining unlabeled nodes by using various priors of the graph.
While there have been numerous previous works (Brandes al. 2008 Zhou et al. 2004 Zhu jet al.) [2003] Yang et al., 2016, Zhhao et al.) 2019) devoted to semi- supervisd node classification based on explicit graph Laplacian regularizations, is hard efficiently boost the performance of label prediction due to the strict assumption that connected nodes are likely to share the same label information. With the progress of deep learning on grid-shaped images/videos (He et al. 2016).
few of graph convolutional neural networks (CNN) based methods, including spectral (Kipf &
Welling 2017) and spatial methods (Niepert al. 2016 Pan et al., 2018 Yu al.) 2018 have been proposed to learn local convolution filters on graphs in order to extract more discriminative node representations. Although graph CNN based methods have achieved considerable capabilities of graph embedding by optimizing filters, they are limited into conventionally semi-supervised framework and lack an efficient inference mechanism on graphs. Especially, in the case few-shot learning, where small number of training nodes are labeled, this kind of methods would drastically compromise the performance. For example, the Pubmed graph dataset (Sen et al. 2008)) consists
"Corresoonding author: Zhen Cui.

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Figure 1: The illustration of our proposed GIL framework. For the problem of graph node labeling, the category information these unlabeled nodes depends on the similarity computation between query node (e.g.. 12) and these labeled reference nodes (c.g., 2,). We consider the similarity from three points: node attributes, the consistency of local topological structures (i.c., the circle with dashed line), and the between-node path reachability (i.c., the red wave line from 8, to vy). Specifically, the local structures well node attributes encoded as high-level features with graph convolution, while the between-node path reachability is abstracted reachable probabilities of random walks. better make inference generalize test nodes, we introduce a meta-learning strategy to optimize the structure relations learning from training nodes to validation nodes.
of 19,717 nodes and 44,338 edges, but only 0.3% nodes are labeled for the semi-supervised node classification task. These aforementioned works usually boil down to general classification task, where the model is learnt do not put great efforts on how to learn infer from one node to another node on topological graph, especially in the few-shot regime.
training set and selected by checking validation set. However, they
In this paper, we propose a graph inference learning (GIL.) framework to teach the model itself to adaptively infer from reference labeled nodes to those query unlabeled nodes, and finally boost the performance of semi-supervised node classification in the case a few number labeled samples. Given an input graph, GIL attempts to infer the unlabeled nodes from those observed nodes by building between-node relations. The between-node relations are structured as the integration of node attributes, connection paths, and graph topological structures. It means that the similarity between two nodes is decided from three aspects: the consistency of node attributes, the consistency of local topological structures, and the between-node path reachability, shown in Fig. T The local structures anchored around each node as well as the attributes of nodes therein are jointly encoded with graph convolution (Defferrard al., 2016) for the sake of high-level feature extraction. For the between-node path reachability, we adopt the random walk algorithm to obtain the characteristics from labeled reference node 2; to a query unlabeled node 2j in given graph. Based on the computed node representation and between-node reachability, the structure relations can be obtained by computing the similar scores/relationssiip from reference nodes to unlabeled nodes in a graph. Inspired by the recent meta-learning strategy (Finn et al.) 2017), we learn to infer the structure relations from a training set to a validation set, which can benefit the generalization capability of the learned model. In other words, proposed GIL attempts to learn some transferable knowledge underlying in the structure relations from training samples to validation samples, such that the learned structure relations be better self-adapted to the new testing stage.
We summarize the main contributions of this work as three folds:
We propose a novel graph inference learning framework by building structure relations to infer unknown node labels from those labeled nodes in an end-to-end way. The structure relations are well defined by jointly considering node attributes, between-node paths, and graph topological structures.
To make the inference model better generalize to test nodes, we introduce a meta-learning procedure optimize structure relations, which could be the first time for graph node classification to the best of our knowledge.
Comprehensive evaluations on three citation network datasets (including Cora, Citeseer, and Pubmed) and one knowledge graph data (i.e., NELL) demonstrate the superiority of our proposed GIL in contrast with other state-of-the-art methods on the semi-supervised classification task.
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2 RELATED WORK
Graph CNNs: With the rapid development of deep learning methods, various graph convolution neural networks (Kashima et al., 2003) Morris et al., 2017; Shervashidze et al. 2009 Yanardag & Vishwanathan 2015: Tiang et al.; 2019 Zhang et al.) (020) have been exploited to analyze the irregular graph-structure data. For better extending general convolutional neural networks to graph domains, two broad strategies have been proposed, including spectral and spatial convolution methods. Specifically, spectral filtering methods (Henaff et al.) 2015; Kipf & Welling (017) develop convolution-liks operators in the spectral domain, and then perform a series of spectral filters by decomposing the graph Laplacian. Unfortunately, the spectral-based approaches often lead to a high computational complex due to the operation of eigenvalue decomposition, especially for a large number of graph nodes. To alleviate this computation burden, local spectral filtering methods (Defferaard et al. (016) are then proposed by parameterizing the frequency responses as Chebyshev polynomial approximation. Another type of graph CNNs, namely spatial methods (Li et al. 2016 Niepert et al.) 2016), can perform the filtering operation by defining the spatial structures of adjacent vertices. Various approaches can be employed to aggregate or sort neighboring vertices, such as diffusion CNNs (Atwood & Towsley| (2016), GraphS (Hamilton et al., 2017). PSCN (Niepert) jet al. 2016), and NgramCNN (Luo et al. (2017). From the perspective of data distribution, recently, the Gaussian induced convolution model (Jiang et al. 2019) is proposed to disentangle the aggregation process through encoding adjacent regions with Gaussian mixture model.
Semi-supervised node classification: Among various graph-related applications, sumi-supervised node classification has gained increasing attention recently, and various approaches have been proposed to deal with this problem, including explicit graph Laplacian regularization and graph- embedding approaches. Several classic algorithms with graph Laplacian regularization contain the label propagation method using Gaussian random fields (Zhu et al.) 2003). the regularization framework by relying on the local/global consistency (Zhou et al. 2004), and the random walk- based sampling algorithm for acquiring the context information Yang et al. 2016). To further address scalable supervied learning issues (Liu et al.) (2012), the Anchor Graph regularization approach (Liu et al. 2010) is proposed to scale linearly with the number of graph nodes and then applied to saslive-scale graph datasets. Several graph convolution network methods (Abu-El-Haija et al. 2018; Du et al. 2017: Thekumparampi et al., 2018: Velickovic et al. 2018 Zhuang & Ma 2018) then developed obtain discriminative representations of input graphs. For example, Kipf et al. Kipf& & Welling 2017) proposed a scalable graph CNN model, which can scale linearly in the number of graph edges and learn graph representations by encoding both local graph structures and node attributes. Graph attention networks (GAT) (Velickovic et al. 2018) are proposed to compute hidden representations of each node for attending to its neighbors with a self-attention strategy. By jointly considering the local- and global-consistenn networks (Zhuang & Ma. 2018) are presented to deal with supervipe node classification. The critical difference between our proposed GIL and those previous semi-supervised node classification methods is to adopt a graph inference strategy by defining structure relations on graphs and then leverage meta optimization mechanism to learn an inference model, which could be the first time to the best of our knowledge, while the existing graph CNNs take semi- supervisd node classification as general classification task.
information, dual graph convolutional
THE PROPOSED MODEL
3.1 PROBLEM DEFINITION
Formally, we denote an undirected/directed graph as G (V,E, where V [ur / is the finite set of n (or [V)) vertices, & E =x= defines the adjacency relationships (i.e., edges) between vertices representing the topology of G. X' E Ru records the explicit/implicin attributes/signals of vertices, and y E TR. is the vertex labels of C classes. The edge E,j = E(,, myy) = 0 if and only vertices U4 by are not connected, otherwise Enj # 0. The attribute matrix X is attached to the vertex set V, whose i-th row Xz, (or Xx,) represents the attribute of the i-th vertex us It means that i, e V carries vector of d-dimensional signals. Associated with each node U; V, there is a discrete label y: E ,..,,.
We consider the task of semi-supervised node classification over graph data, where only small number of vertices are labeled for the model learning, i.e., |Laberl < [ Generally, we have three node sets: training set Ver. a validation set Voal, and testing set V.e- In the standard protocol
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of prior literatures Yang al. 2016), the three node sets share the same label space. We follow but do not restrict this protocol for our proposed method. Given the training and validation node sets, the aim is to predict the node labels of testing nodes by using node attributes as well as edge connections. A sophisticated machine learning technique used in most existing methods (Kipf & Welling, 2017, 7hou al.) 2004) is to choose the optimal classifier (trained checking the performance on the validation set. However, these methods essentially ignore how to extract transferable knowledge from these known labeled nodes to unlabeled nodes, as the graph structure itself implies node commectivityrrrachaalliy Moreover, due to the scarcity of labeled samples, the performance of such classifier is usually not satisfying. To address these issues, we introduce a meta-learning mechanism (Finn et al., 2017] Ravi & Larochelle, 2017 Sung et al.) 2017) to learn to infer node labels on graphs. Specifically, the graph structure, between-nor -ode path reachability, and node attributes are jointly modeled into the learning process. Our aim is to learn to infer from labeled nodes to unlabeled nodes, so that the learner can perform better on a validation set and thus classify testing set more accurately.
training set) after
3.2 STRUCTURE RELATION
For convenient inference, we specifically build structure relation between two nodes on the topology graph. The labeled vertices (in training set) are viewed the reference nodes, and their information can be propagated into those unlabeled vertices for improving the label prediction accuracy. Formally, given a reference node vi E VLabel, we define the score of a query node Vy similar to o as
$i-jj (L. fe(G ), (Us, E),
(1)
where Gr, and G. may be understood as the centralized subgraphs around 2s and Vy. respectively. Se. Jr. fr are three abstract functions that we explain as follows:
Node representation fG..). - Rdd,, encodes the local representation of the centralized subgraph Gv around node Us and may thus be understood as a local filter function on graphs This function should not only take the signals of nodes therein as input, but also consider the local topological structure of the subgraph for more accurate similarity computation. To this end, we perform the spectral graph convolution on subgraphs to learn discriminative node features, analogous to the pixel-level feature extraction from convolution maps of gridded images. The details of feature extraction Se are described in Section 4
Path reachability fr(U,, 2y. 8) + 4 represents the characteristics of path reachability from U2 to vy. As there usually exist multiple traversal paths between two nodes, we choose the function as reachable probabilities of different lengths of walks from iy to vy. More
details will be introduced in Section 4
Structure relation fr (Rdd, [Rd,, RR4) - IR, is a relational function computing the score of Un similar to ug. This function is not exchangeable for different orders of two nodes, due to the asymmetric reachable relationship fp. If necessary, we may easily revise it as a symmetry function, e.g., summarizing two traversal directions. The score function depends on triple inputs: the local representations extracted from the subgraphs w.r.t. fo (G..) and fe(9.,)) respectively, and the path reachability from f; to Uz--
In semi- cumi--uperviied node classification, we take the training node set Ver as the reference samples, and the validation set Vunl as the query samples during the training stage. Given a query node by Voats we can derive the class similarity score of uy w.r.t. the e-th (c = ..... C) category by weighting the reference samples C. = (ux|y>, c). Formally, we can further revise Eqn. (1) and define the class-to-node relationship function as
Sc, -j = or (Fc.
Σ Wi-s fe(9or), U.EC.
(2)
s.t. Wi-s (Fr((, E),, where the function th maps a reachable vector f "y E) into weight value, and the function di computes the similar score between 2 and the c-th class nodes. The normalization factor Fc..rey of the c-th category w.r.t. By is defined as
(3)
Fc. Ex,ec_ Wa-+j
(4)
For the relation function o, and the weight function Omn we may choose some subnetworks to instantiate them in practice. The detailed implementation of our model can be found in Section 4
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3.3 INFERENCE LEARNING
According to the class-to-node relationship function in Eqn. 2 given a query node ej. we can obtain a score vector Sc-+j = (c,-); &Cc-tgl E RC after computing the relations to all classes The indexed category with the maximum score assumed to be the estimated label. Thus, we can define the loss function based on cross entropy as follows:
L h=
E
Σ T/y o=l
Mi.o log DC.
(5)
where Vsee is binary indicator (i.c., 0 or 1) of class label for node vy. and the softmax operation imposed on 8C.->j, i.e., Iic.-j = exp( Ei= exp($c,-1)) Other enror functions may be chosen as the loss function, e.g., mean square error. In the regime of general classification, the cross entropy loss is a standard one that performs well.
Given training set Ver. expect that the best performance can be obtained on the validation set Veat after optimizing the model on Vrr. Given a trained//retraineed model Q = fefe Pur, Or J. we perform iteratively gradient updates on the training set Ver to obtain the new model, formally,
o' - - (Q),
(6)
where a is the updating rate. Note that, in the computation of class scores, since the reference node and query node can be both from the training set Vr.. we set the computation weight wi-rj 0 if i = j in Eqn. B After several iterates of gradient descent on Vtr. we expect a better performance on the validation set Vvals ie., min Le (Q'). Thus, can perform the gradient update as follows
= θ Vo
where 3 is the learning rate of meta optimization (Finn et al. 2017).
During the training process, we may perform batch sampling from training nodes and validation nodes, instead of taking all one time. In the testing stage, we may take all training nodes and perform the model update according to Eqn. (6) like the training process. The updated model is used as the final model and is then fed into Eqn. (2) to infer the class labels for those query nodes.
MODULES
In this section, we instantiate all modules (i.e., functions) of the aforementioned structure relation. The implementation details can be found in the following.
Node Representation fe(G.,): The local representation at vertex can be extracted by performing the graph convolution operation on subgraph G.,- Similar gridded images/videos, on which local convolution kernels are defined as multiple lattices with various receptive fields, the spectral graph convolution used encode the local representations of an input graph in our work.
Given a graph sample G = (V,E the normalized graph Laplacian matrix is L ==
D-1/2gD-117 = UAU"", with a diagonal matrix of its eigenvalues A. The spectral graph convo- lution can be defined as the multiplication of signal X' with filter go (A) = diag(8) parameterized by 0 in the Fourier domain: conv(X') = go (L) * X' = Uga (A)U" X, where parameter 0 E IR" vector of Fourier coefficients. To reduce the computational complexity and obtain the local information, we use an approximate local filter of the Chebyshev polynomial (Defferrard et al.
2016), go (A) Σ (A),, where parameter 0 E RRK is vector of Chebyshev coefficients and (A( E T the Chebyshev polynomial of order k evaluated at A 2A/Amaa I,, diagonal matrix of scaled eigenvalues. The graph filtering operation (A)) X Σk-1 (L)x where To (L) Raxa is the Chebyshev polynomial of order evaluated at the scaled Laplacian = 3L/Amae In. Further, we can construct multi-scale receptive fields for each vertex based the Laplacian matrix L, where each receptive field records hopping neighborhood relationships around the reference vertex vi, and forms a local centralized subgraph.
then be expressed as
Path Reachability fr By, E): Here we compute the probabilities of paths from vertex i to vertex j by employing random walks on graphs, which refers to traversing the graph from 0; to U, according to the probability matrix P. For the input graph G with n vertices, the random-wall walk transition matrix
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Datasets
Nodes
Edges
Classes
Features
Label Rates
Cora
2,708
5,429
1,433
0.052
Citeseer
3,327
4,732
6
3,703
0.036
Pubmed
19.717
44,338
500
0.003
NELL
65,755
266,144
210
5,414
0.001
Table 1: The properties (especially for label rate) of various graph datasets used for the semi-supervised classification task.
can be defined as P D-1g, where D E R is the diagonal degree matrix with Dis = Ei Esg- That is to say, each element P.j is the probability of going from vertex to vertex in one step.
The sequence of nodes from vertex to vertex j is random walk on the graph, which can be modeled as a classical Markov chain by considering the set of graph vertices. To represent this formulation, we show that Pú is the probability of getting from vertex v; to vertex vy in steps. This fact is easily exhibited by considering t-step path from vertex e; to vertex Uy as first taking single step to some vertex h. and then taking - steps to Vy. The transition probability ps in I steps can be formulated as
Pú P.j Σ
[
if tt=
PMP
t>
(8)
where each matrix entry P denotes the probability of starting at vertex and ending at vertex j in t steps. Finally, the node reachability from U, to B can be written as dp di-demeniional vector:
fr(u; uj, [P.j P....... 1.
where dp refers to the step length of the longest path from e, to Vz-
Class-to-Node Relationship sc, To define the node relationship 8:-3 from U; Uj. we simulta- neously consider the property of path reachability ( Uy E), local representations fe (9..) ). and fo(G.,) of nodes U, Uj. The function (fr(( V/ E) in Eqn. 3.. which is to map the reachable vector fr(U, Vy, E) E Rd into weight value, can be implemented with two 16-dimensional fully connected layers in our experiments. The computed value Wi+jj can be further used to weight the local features at node Ei. (G.. E Rdd For obtaining the similar score between 7/ and the c-th class nodes C. in Eqn. ( we perform a concatenation of two input features, where one refers the weighted features of vertex us. and another is the local features of vertex V/. One fully connected layer (w.rt. Φb) with C-dimensions is finally adopted to obtain the relation regression score.
5 EXPERIMENTS
5.1 EXPERIMENTAL SETTINGS
We evaluate our proposed GII. method on three citation network datasets: Cora, Citeseer, Pubmed (Sen) ctt al. 2008), and one knowledge graph NELL dataset (Carlson et al., 2010). The statistical properties of graph data are summarized in Table T Following the previous protocol in (Kipf & Welling 2017] Zhuang & Ma 2018), we split the graph data into a training set, a validation set, and a testing set. Taking into account the graph convolution and pooling modules, we may alternately stack them into a multi-layer Graph convolutional network. The GIL model consists of two graph convolution layers, each of which followed by mean-pooling layer, class-to-node relationship regression module, and a final softmax layer. We have given the detailed configuration of the relationship regression module in the class-to-node relationship of Section 4 The parameter dp in Eqn. (9) is set to the mean length of between-node reachability paths in the input graph. The channels of the 1-st and 2-nd convolutional layers are set to 128 and 256, respectively. The scale of the respective filed is in both convolutional layers. The dropout rate is set to 0.5 in the convolution and fully connected layers to avoid over-fitting, and the ReLU unit is leveraged as a nonlinear activation function. We pre-train our proposed GIL model for 200 iterations with the training set, where its initial learning rate, decay factor, and momentum are set to 0.05, 0.95, and 0.9, respectively. Here we train the GIL model using the stochastic gradient descent method with the batch size of 100. We further improve the inference learning capability of the GIL model for 1200 iterations with the validation set, where the meta-learning rates a and / are both set to 0.001.

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5.2 COMPARISON WITH STATE-OF- ARTS
We compare the GIL approach with several state-of-the-art methods (Monti et al. 2017 Kipf &
Welling 2017 Zhou et al. 2004, Zhuang & Ma 2018) over four graph datasets, including Cora. Citeseer, Pubmed, and L.LL. The classification accuracies for all methods are reported in Table 2
Our proposed GIL can significantly outperform these graph Laplacian regularized methods on four graph datasets, including Deep walk (Zhou et al. 2004). modularity clustering (Brandes et al. 2008). Gaussian fields (Zhu et al. 2003), and graph embedding (Yang et al. 2016) methods. For example, we can achieve much higher performance than the deepwalk method (Zhou et al.. 2004). e.g., 43.2% vy 74.1% % on the Citeseer dataset, 65.3% vy 83.1% 1% on the Pubmed dataset, and 58. % vs 78.9% on the NELL dataset. We find that the graph embedding method (Yang et al. 2016), which has considered both label information and graph structure during sampling, can obtain lower accuracies than our proposed GIL by 9.4% the Citeseer dataset and 10.5% on the Cora dataset, respectively. This indicates that our proposed GIL can better optimize structure relations and thus improve the network generalization. We further compare proposed GIL. with several existing deep graph embedding methods, including graph attention network (Velickovic et al., 2018), dual graph convolutional networks (Zhuang & Ma 2018). topology adaptive graph convolutional networks (Du al. (0177). Multi-scale graph convolution (Abu-El-Haija et al., 2018). etc. For example, our proposed GIL achieves a very large gain, e-g., 86.2% vs 83.3% (Du et al., 2017) on the Cora dataset, and 78. 78.9% vs 66.0% (Kipf & Welling (017)) on the NELL dataset. We evaluate our proposed GIL method on a large graph dataset with lower label rate, which can significantly outperform existing baselines on the Pubmed dataset: 3. 1% over DGCN (Zhuang & Ma 2018), 4.1% % over classic GCN (Kipf & Welling 2017) and TAGCN (Du et al., 2017). 3.2% over AGNN (Thekumparampil et al.) 2018), and 3.6% over N-GCN (Abu-El-Haijj El---aaaaaa et al., 2018). It demonstrates that our proposed GIL performs very well on various graph datasets by building the graph inference learning process, where the limited label information and graph structures can be well employed in the predicted framework.
Table 2: Performance comparisons of semi-supervisee classification methods.
Methods
Cora
Citeseer
Pubmed
NELL
Clustering (Brandes et al.
2008
59.5
60.1
70.7
21.8
DeepWalk (Zhou et al. 2004)
67.2
43.2
65.3
58.1
Gaussian (Zhu et al. 2003
68.0
45.3
63.0
26.5
G-embedding Yang et al.
2016
75.7
64.7
77.2
61.9
DCNN (Atwood & Towsley
2016
76.8
73.0
GCN (Kipi & Wellin 2017
81.5
70.3
79.0
66.0
MoNet (Monti et al. (2077)
81.7
78.8
N-GCN (Abu-El-Haija
al. 2018)
83.0
72.2
79.5
GAT /Velickovie et al. (0188)
83.0
72.5
79.0
AGNN (Tekumpaaamm et al.
2018
83.1
71.7
79.9
TAGCN (Du et al. 2017)
83.3
72.5
79.0
DGCN (Zhuang & Ma 2018)
83.5
72.6
80.0
74.2
Our GIL
86.2
74.1
83.1
78.9
5.3 ANALYSIS
Meta-optimization: As be seen in Table we report the classification accuracies of ceni-supervised classification with several variants of our proposed GIL and the classical GCN method (Kipf & Welling, 2017) when evaluating them on the Cora dataset. For analyzing the perfor- mance improvement of our proposed GIL with the graph inference learning process, we report the classification accuracies of GCN (Kipf & Welling 2017) and our proposed GIL on the Cora dataset under two different situations, including "only learning with the training set V,r"" and "'with jointly learning on a training set Vi, and validation set Vent" "GCN /w jointly learning on Vir & Venit achieves a better result than "GCN /w learning on V,," by 3.6%, which demonstrates that the network performance can be improved by employing validation samples. When using structure relations, "GIL /w learning can be attributed to the building connection between nodes. The meta-optimization strategy ("GIL /w meta-training training from Ver + Veal" vs "GIL /w learning on Ver"") has a gain of 2.9%, which indicates that a good inference capability can be learnt through meta- mntta-otiiizzttion It is worth noting that, GIL adopts meta-optimization strategy learn the inference model, which is process of migrating
V,," obtains an improvement of 1.9% (over "GCN /w learning on V.."), which
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from training set to validation set. In other words, the validation set is only used to teach the model itself how to transfer to unseen data. In contrast, the conventional methods often employ a validation set to tune parameters of a certain model of interest.
Table Performance comparisons with several GIL variants and the classical GCN method on the Cora dataset.
Methods
Acc. (%)
/w learning on Ver
81.4
GCN (Kipf & Welling 2017
/w jointly learning on Ver Vunt
84.0
/w learning on Ver
83.3
GIL
/w meta-train Ver Vual
86.2
/w conv. layer
84.5
Glln+ pooling
/w conv. layers
86.2
/w 3 conv. layers
85.4
/w max-pooling
85.2
GIL++2 conv. layers
/w mcan pooling
86.2
Network settings: We explore the effectiveness of our proposed GIL with the same mean pooling mechanism, but with different numbers of convolutional layers, i.e., "GIL + mean pooling" with one, two, and three convolutional layers, respectively. As can be seen in Table the proposed GIL with two convolutional layers can obtaín a better performance on the Cora data than the other two network settings (i.e., GIL with one or three convolutional layers). For example, the performance of 'GIL /w I conv. layer + mean pooling" is slightly decreased by 1.7% over "GIL /w 2 conv. layers mean pooling" on the Cora dataset. Furthermore, we report the classification results of our proposed GIL by using mean and max- max-pooling mechanisms, respectively. GIL with 2 conv layers + mean pooling") ") can get a better result than the GIL method with max-pooling (i.e., "GIL /w 2 conv layers + max-pooling"). e.g.. 86.2% 85.2% the Cora graph dataset. The reason may be that the graph network with two convolutional layers and the mean pooling mechanism can obtain the optimal graph embeddings, but when increasing the network layers, more parameters of certain graph model need to be optimized, which may lead to the ever-fitting issue.
pooling (i.e. "GIL /w
Influence of different between-node steps: We compare the classification performance within different between-ner node steps for our proposed GIL and GCN (Kipf' & Welling (017), illustrated in Fig. [(a) The length of between-node steps can be computed with the shortest path between reference nodes and query nodes. When the step between nodes is smaller, both GIL and GCN methods can predict the category information for a small part of unlabeled nodes the testing set. The reason may be that the node category information may be disturbed by its nearest neighboring nodes with different labels and fewer nodes are within 1 or 2 steps in the testing set. The GIL and GCN methods infer the category information for part of unlabeled nodes by adopting node attributes, when two nodes are not connected in the graph (i.e., step=oc). By increasing the length of reachability path, the inference process of the GIL method would become difficult and more graph structure information may be employed in the predicted process. GIL can outperform the classic GCN by analyzing the accuracies within different between-node steps, which indicates that our proposed GIL has a better reference capability than GCN by using the meta-optimization mechanism from training nodes to validation nodes.
85.0%
78.0%
- (jor)))
Label pitess
(a)
Figure 2: (a) Performance comparisons within different between-node steps on the Cora dataset. The accuracy equals the number of correctly classified nodes divided by all testing samples, and is accumulated from step step k. (b) Performance comparisons with different label rates Pubmed dataset.
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Published as a conference paper at ICLR 2020
Influence of different label rates: We also explore the performance comparisons of the GIL method with different label rates, and the detailed results on the Pubmed dataset can be shown in Fig. ((b) When label rates increase by multiplication, the performances of GIL and GCN are improved, but the relative gain becomes narrow. The reason is that, the reachable path lengths between unlaheled nodes and labeled nodes will be reduced with the increase of labeled nodes, which will weaken the effect of inference learning. In the extreme case, labels of unlabeled nodes could be determined by those step reachability. In summary, our proposed GIL method prefers small ratio
neighbors with the labeled nodes on the semi-supervised node classification task.
Inference learning process: Classification errors of different epochs on the validation set of the Cora dataset can be illustrated in Fig. 3 Classification errors are rapidly decreasing as the number of iterations increases from the beginning to 400 iterations, while they are with a slow descent from 400 iterations to 1200 iterations. It demonstrates that the learned knowledge from the training samples can be transferred for inferring node category information from these reference labeled nodes. The performance of semi-supervised classification can be further increased by improving the generalized capability of the Graph CNN model.
Table 4: Performance comparisons with different mod- ules on the Cora dataset, where fe. fr, and f. denote node representation, path reachability, and structure re- lation, respectively.
fe
Jr
fp
A%c
56.0
1he sumbero of tenttoon
81.5
v
85.0
Figure 3: Classification emrors of different itera- tions on the validation set of the Cora dataset.
/ 86.2
Module analysis: We evaluate the effectiveness of different modules within our proposed GIL framework, including node representation fe, path reachability fr, and structure relation fr. Note that the last one fr defines on the former two ones, so we consider the cases in Table by adding modules. When not using all modules, only original attributes of nodes are used to predict labels. The case of only using fo belongs to the GCN method, which can achieve 81.5% on the Cora dataset. The large gain of using the relation module fr (i.e., from 81.5% to 85.0%) may be contributed to the ability of inference learning on attributes as well as local topology structures which are implicitly encoded in fe. The path information fo can further boost the performance by 1.2%, e.g., 86.2% vs 85.0%. It demonstrates that three different modules of our method can improve the graph inference learning capability.
Computational complexity: For the computational complexity of our proposed GIL, the cost is mainly spent on the computations of node representation, node reachability, and class-to- node relationship, which are about O( + nie) * di,, dout). O( nte) e P), and O(ner * n respectively. nhr and Nite refer to the numbers of training and testing nodes, din and dout denote the input and output dimensions of node representation, is about the average degree of graph node, and P is the step length of node reachability. Compared with those classic Graph CNNs (Kipf & Welling 2017). our proposed GIL has a slightly higher cost due an extra inference learning process, but can complete the testing stage with several seconds on these benchmark datasets.
CONCLUSION
In this work, we tackled the supervipe node classification task with a graph inference learning method, which can better predict the categories of these unlabeled nodes in an end-to-end framework. We can build a structure relation for obtaining the connection between any two graph nodes, where node attributes, between-node paths, and graph structure information can be encapsulated together. For better capturing the transferable knowledge, our method further learns to transfer the mined knowledge from the training samples to the validation set, finally boosting the prediction accuracy of the labels of unlabeled nodes in the testing set. The extensive experimental results demonstrate the effectiveness of our proposed GIL for solving the semi-supervised learning problem, even in the few-shot paradigm. In the future, we would extend the graph inference method to handle more graph-related tasks, such graph generation and social network analysis.

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Published as a conference paper at ICLR 2020
ACKNOWLEDGMENT
This work was supported by the National Natural Science Foundation of China (Nos. 61972204, 61906094, U1713208), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20191283 and BK20190019), and Tencent Al Lab Rhino-Bire Focused Research Program (No. JR201922).
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11


Reviews:
[{"review_id": "rkxlbSepYB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper suggests a learning architecture for graph-based semi-supervised learning. The input graph is given with only some labeled, and the goal is to label the rest. The architecture is trained on a validation set to learn features that optimize the accuracy of inference on the unlabeled node.\n\nAll parts of the architecture seem reasonable, and the experiments report advantage over prior work, which counts in favor of the paper. I found the writing to be somewhat dense and lack in clarity. My main concern is that I was unable to find a comparison of the techniques to prior work and an explanation of novelty. Unfortunately I am not very familiar with prior work on using neural networks for graph-based semi-supervised learning, and the lack of discussion made it difficult to put this work in context and assess its contribution. As of now I set my score to a weak reject, though I am willing to raise it if the positioning against prior work is explained and points to a substantial contribution.\n\nA remark on clarity: it would help to use the appropriate commands for citations (citet and citep). The constant interruption of unbracketed references burdens the flow and readability of the text.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "SJxR53RhtS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes to leverage the between-node-path information into the inference of conventional graph neural network methods. Specifically, the proposed method treats the nodes in training set as a reference corpus and, when infering the label of a specific node, makes this node \"attend\" to the reference corpus, where the \"attention\" weights are calculated based on the node representations and the between-node paths. (The paper used different terms about the \"attention\".)\n\nIn general, this paper shares similar insights like those in memory networks, which use a reference corpus to assist the inference on an individual sample. The major technical novelty of this paper seems to lie in the introduction of this idea on graphs and the use of between-node-path information for the \"attention mechanism\".\n\nThe paper is also well-organized and relatively easy to follow. \n\nOne of my major concerns is the unfair experimental comparison. The proposed method uses both the training set and validation set to train the model. And it's well-known that the offical split of the Cora, Citeseer, and Pubmed datasets has a validation set larger than the training set. As most of the baseline performances in Table 1 are obtained by using the training set only, the comparison of Table 1 is unfair and totally meaningless. A valid comparison is Table 2, where the baseline GCN is trained on both training set and validation set. But only the results on Cora is given and results on all other datasets are missing. Actually, I tried a quick run of GCN, without much tuning of hyper-parameters, on Citeseer with both training set and validation set as training data. And it can easily achieve a test accuracy of 0.75+, which is better than the performance of the proposed method reported in the paper. \n\nAnother concern is that, while the proposed method is intuitively sound, it is unclear how this method compares to naively increasing the number of layers of the GNN. Increasing the number of layers of the GNN can also eventually make the test node able to interact with all the training nodes, and the model should be able to implicitly distinguish the distances between nodes. Why incorporating between-node paths, and particularly in this way, helps?\n\nSome minor points:\n\t- Could you clarify how the figure 2 is generated? By grouping the test nodes with differen between-nodes steps? It is confusing why performance on the nodes far away from the training nodes is better?\n\t- Why figure 3 is on PubMed while all other ablation analyses are done on Cora?\n\nIn summary, this paper shows some technical novelty but the motivation behind the proposed method is not strong enough. The experiment design is questionable and especially the main experiment comparison Table 1 is meaningless. Therefore I think this work is not ready for publish yet.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "BJxqaF2WqB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents a semi-supervised classification method for classifying unlabeled nodes in graph data. The authors propose a Graph Inference Learning (GIL) framework to learn node labels on graph topology. The node labeling is based of three aspects: 1) node representation to measure the similarity between the centralized subgraph around the unlabeled node and reference node; 2) structure relation that measures the similarity between node attributes; and 3) the reachability between unlabeled query node and reference node.\n\nThe authors propose to use graph convolution to learn the node representation and random work on graph to evaluate the reachability from query node to reference node.\n\nThe presentation of the paper is clear and easy to follow. The idea of the paper seems straightforward and the experimental results seems promising in semi-supervised classification for nodes in graph data.\n\nJust a few concerns on model performance:\n\n1) The node labeling is based on the comparison with reference nodes. Would such method get biased toward major classes in the data, if the data is imbalanced among different classes?\n\n2) This method adopts three different modules for node labeling. It would be helpful if the authors can add some results to show the contribution of the different modules, i.e., what would be the performance if reachability or consistency of local graph topology structure is not considered in the classification?\n\nOne typo: in Table 1, it should be \"Label rate\".", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors propose a graph inference learning framework to address the issues of sparse labeled data in graphs. The authors use structural information and node attributes to define a structure relation which is then use to infer unknown labels from known labels. The authors demonstrate the effectiveness of their approach on four benchmark datasets.

The approach presented in the paper is sound and the empirical results are convincing. All reviewers have given a positive rating for this paper. Two reviewers had some initial concerns about the paper but after the rebuttal they have acknowledged the answers given by the authors and adjusted their scores. R1 still has concerns about the motivation of the paper and I request the authors to adequately address this in their final version.
Decision: Accept (Poster)