Information Gain Propagation: a New Way to Graph Active Learning with Soft Labels

1 INTRODUCTION . Graph Neural Networks ( GNNs ) have recently achieved remarkable success in various graph-based tasks , ranging from traffic networks , biology , to social networks ( Zhang et al. , 2020 ; Wang et al. , 2019 ; Li et al. , 2019 ; Do et al. , 2019 ) . Despite their effectiveness and popularity , GNNs typically require a large amount of labeled data to achieve satisfactory performance . However , obtaining these labels is a time-consuming , laborious , and costly process . Therefore , how to do this economically and efficiently has attracted great attention both in academia and industry . One of the most popular strategies to tackle this challenge is Active Learning ( AL ) ( Aggarwal et al. , 2014 ) . By combining model training and node selection for labeling , AL significantly reduces labeling cost by selecting the most valuable nodes to label . However , previous AL methods assume the hard label ( namely the exact label , that is , the label specifying the exact class of the node ) can always be provided by the oracle . As illustrated in Fig . 1 , for any selected node , the active learners in the aforementioned methods ask questions like “ which category does it exactly belong to ? ” . Such queries and assumptions exceed the capability of oracle in many labeling tasks requiring domain knowledge . For example , the task in ogbn-papers100M is to leverage the citation graph to infer the labels of the arXiv papers into 172 arXiv subject areas ( a single-label , 172-class classification problem ) . In this example , a specialist/expert in the subject areas of machine learning is incapable of labeling query instances with subject areas of finance ( such as mathematical finance or computational finance ) , which is out of his domain knowledge . In this paper , we propose a novel active learning paradigm for GNNs in which only soft labels are required . Two salient features of our paradigm are as follows : First , we propose relaxed queries where domain expert ( oracle ) only judges the correctness of the predicted labels ( by GNN models ) rather than the exact categorization . Specifically , we propose to select the class label with the highest predicted probability and then ask the oracle to judge whether the class label is correct rather than directly asking for the exact class label . In this way , the multi-class classification task is relaxed into a binary-class classification task for the oracle . Under such queries , if the model prediction is incorrect ( we assume the oracle does not make mistakes ) , we annotate the node with the soft label by re-normalizing the model predicted softmax outputs over the remaining 0.5 0.3 0.2 Model Prediction Oracle Which category does it exactly belong to ? Does it belong to the first category ? Query Oracle 1 0 0 0 0.6 0.4 Yes No Ground truth label Our strategy Previous strategy Figure 1 : An example of our new node labeling strategy . [ 1 , 0 , 0 ] [ 0.5 , 0.3 , 0.2 ] [ 0 , 0.4 , 0.6 ] 0.5 0.4 0.1 [ 0.7 , 0.16 , 0.14 ] IGP ( ) = H ( 0.7 , 0.16 , 0.14 ) – H ( 1/3 , 1/3 , 1/3 ) Figure 2 : An example of IGP . classes . As illustrated in Fig.1 , we allow relaxed queries such as “ Does the sample belong to this class ? ” ( e.g. , the first class in Fig . 1 ) , since this query is specific to the domain of experts , hence more ready to be answered . As far as we are concerned , no previous AL methods can deal with such relaxed queries . Second , the node selection criteria in previous AL methods are specially designed for hard-labeling oracles . As one of the most commonly used AL query strategies , uncertainty sampling queries the labels of the nodes which current model is least certain with , ensuring the largest entropy reduction on a single node when given the hard label from oracle . However , under our relaxed queries , we prove that the uncertainty sampling can not guarantee the largest information gain ( i.e. , entropy reduction ) on a single node . In this paper , we propose a new criterion to explicitly maximize information gain propagation ( IGP ) for active learners with relaxed queries and soft labels on graph . As shown in Fig . 2 , the labeled nodes ( with color ) can propagate their label information with different influence magnitude and then reduce the uncertainty of adjacent unlabeled nodes ( without color ) . Considering the influence propagation in GNNs , we select the nodes that can maximize expected information gain in total under the relaxed queries , i.e. , reducing the aggregated uncertainty of a neighborhood region beyond a single node . To the best of our knowledge , this is the first work to propose the information gain propagation with relaxed queries , and showing that it is highly effective . In summary , the core contributions of this work are the following : First , we provide a new AL paradigm for GNNs methodology with relaxed queries and soft labels . Second , we propose a novel node selection criterion that explicitly maximizes propagation of information gain under the new AL paradigm on GNNs . Third , experimental results show our paradigm significantly outperforms the compared baselines by a large margin in five open graph datasets . 2 PRELIMINARY . 2.1 PROBLEM FORMULATION . Let G = ( V , E ) with |V| = N nodes and |E| = M edges be the graph , and c is the number of classes . X = { x1 , x2 ... , xN } is the node feature matrix , the one-hot vector yi ∈ Rc and and xi ∈ Rd are the ground-truth label and node feature for node vi ∈ V , respectively . We consider a general AL setting in this work . Suppose the full node set V is partitioned into training set Vtrain , validation set Vval , and test set Vtest , and Vtrain can further be partitioned into the labeled set Vl and unlabeled set Vu , respectively . Given the labeling size B , the loss function ` , the goal of graph-based AL is to select a subset Vl ⊂ Vtrain to label , so that the model f trained with the supervision of Vl can get the lowest loss on Vtest : arg min Vl : |Vl|=B Evi∈Vtest [ ` ( yi , ŷi ) ] , ( 1 ) where ŷi is the softmax outputs of node vi given by the model f . Different from the previous setting , B is the labeling cost ( i.e. , money ) rather than the size in our new scenario . As the experts are hard to annotate the node out of their domain , we assume the conventional query cost c− 1 times as much as our relaxed query for each selected node . Correspondingly , the objective in our new scenario is argmin Vl : M ( Vl ) =B Evi∈Vtest [ ` ( yi , ŷi ) ] , ( 2 ) where M ( Vl ) is the labeling cost for annotating Vl , and we focus on f being GNNs due to their state-of-the-art performance in many semi-supervised node classification tasks . 2.2 GRAPH NEURAL NETWORKS . Unlike the images , text or tabular data , where training data are independently and identically distributed , samples are connected by edges in the graph , and each node in a GNN can aggregate the node embedding or feature from its neighbors to enhance their embedding along edges . Let f ( A , X ( k ) ; W ( k ) ) be any GNN , where A is an adjacency matrix , X ( k ) is the feature matrix of k-th layer , and W ( k ) are the learned weights of k-th layer . Taking the widely used Graph Convolution Network ( GCN ) ( Kipf & Welling , 2017a ) as an example , each GCN layer can be formulated as : X ( k+1 ) = f ( A , X ( k ) , W ( k ) ) = δ ( D̃−1ÃX ( k ) W ( k ) ) , ( 3 ) where X ( and X ( 0 ) ) is the original node feature , X ( k ) and X ( k+1 ) are the embeddings of layer k and k+1 respectively . Besides , D̃ is the diagonal node degree matrix for normalization and Ã = A+ IN is the adjacent matrix with self connection , where IN is the identity matrix . Compared with the DNN layer which formulated as X ( k+1 ) = δ ( X ( k ) W ( k ) ) , the feature matrix X ( k ) will be firstly enhanced with the feature propagation operation D̃−1ÃX ( k ) , and then boosts the semi-supervised node classification performance by getting more nodes involved in the model training . 2.3 ACTIVE LEARNING . Common AL . AL can improve labeling efficiency by selecting the most valuable samples to label . Considering the informativeness , Uncertainty Sampling ( Yang et al. , 2015 ; Zhu et al. , 2008 ) selects the nodes which has the most uncertain model prediction . Besides , Query-by-Committee ( Burbidge et al. , 2007 ; Melville & Mooney , 2004 ) trains a committee of models , and selects samples according to the extent to which the models disagree . Furthermore , Density-based ( Zhu et al. , 2008 ; Tang et al. , 2002 ) and Clustering-based ( Du et al. , 2015 ; Nguyen & Smeulders , 2004 ) and Diversity-based ( Wu et al. , 2020 ; Jiang & Gupta , 2021 ) methods can effectively select the most representative samples . All the methods above are proposed for binary or multi-class classification , and some methods ( Qi et al. , 2008 ; Yan & Huang , 2018 ) are proposed for the multi-label classification problem . Besides , considering the class imbalance , some methods ( Aggarwal et al. , 2020 ; Attenberg & Provost , 2010 ) are also proposed to tackle this issue with AL . GNN-based AL . Despite the effectiveness of common AL methods , it is unsuitable to directly apply them to GNNs since the characteristic of influence propagation has not been considered . To tackle this issue , both AGE ( Cai et al. , 2017 ) and ANRMAB ( Gao et al. , 2018 ) introduce the density of node embedding and PageRank centrality into the node selection criterion . Besides , ALG ( Zhang et al. , 2021a ) proposes to maximize the effectiveness of all influenced nodes of GNNs . Recently , Grain ( Zhang et al. , 2021b ) introduces a new diversity influence maximization objective , which takes the diversity of influenced nodes into consideration . Both AGE ( Cai et al. , 2017 ) and ANRMAB ( Gao et al. , 2018 ) are designed for AL on GNNs , which adopt the uncertainty , density , and node degree to select nodes . ANRMAB improves AGE by introducing a multi-armed bandit mechanism for adaptive decision-making . Considering the high training time of GNN , ALG ( Zhang et al. , 2021a ) decouples the GNN model and proposes a new node selection metric that maximizes the effective reception field . Grain ( Zhang et al. , 2021b ) further generalizes the reception field to the number of activated nodes in social influence maximization and introduces the diversity influence maximization for node selection . However , all these AL methods assume the oracle knows the information of all categories , thereby being able to correctly annotate all the selected nodes . This is impractical if the oracle encounters some areas that they are not familiar with , especially when the number of categories is large . IGP is the first work to actively labeling graph data under domain-specific experts .

Summary: The paper proposes an active learning method for GNNs that is based on an information gain maximization where ethe information gain is obtained by querying a data point and looking at the influence of the queried node on the neighborhood relative to their previous information.  They also claim the setting of relaxing the oracle answer to be a binary confirmation of the most probable label. Experiments are presented where some advantage is shown for the method.