Regional based query in graph active learning

1 INTRODUCTION . Relational information is often presented as graphs or multi-graphs , where nodes represent entities and edges represent relations between these entities . Such relations can be used to predict the class of the nodes , using two main principles . The first and most used method is based on node class homophily , where neighboring nodes belong to the same class with a high probability ( Ji & Han , 2012 ; Berberidis & Giannakis , 2018 ; Zhu et al. , 2003a ; b ; Sindhwani et al. , 2005 ; Belkin & Niyogi , 2004 ) . This has been used in many propagation-based algorithms where the class of a node is predicted using the class of neighboring nodes . The second approach presumes a correlation between the topological attributes ( e.g . degree , centrality , clustering coefficient ... ) of nodes and their class ( Shi & Malik , 2000 ; Yang et al. , 2013 ; Rosen & Louzoun , 2015 ; Naaman et al. , 2018 ; Cannistraci et al. , 2013 ) . These two principles are combined in Graph Convolutional Networks ( GCN ) . Such networks received much interest over the last decade , and especially following the works of Kipf & Welling ( 2016 ) , where they have produced higher accuracies than other label propagation methods . The main formalism proposed in such networks is the weighted combination of the input from previous layers in neighboring nodes : Xk+1 = σ ( Ã×Xk ×Wk ) , ( 1 ) with WK being the weights of the k layer , Xk the input to this layer and Ã a matrix derived from the adjacency matrix ( e.g . Ã = D−1/2 [ A+ I ] D−1/2 for undirected graphs ) . X0 is usually external information about the nodes . In the absence of such information , the identity matrix ( Schlichtkrull et al. , 2018 ) , topological features of nodes or the frequency of neighbors belonging to each class in the training set ( Benami et al. , 2019 ) have been proposed . In the absence of a predefined set of classified nodes , and when the nodes composing the training set can be chosen , active learning can be used to query the class of nodes that would produce the highest precision ( as defined through the prediction accuracy or any other measure on the entire dataset ) using the minimal number of classified samples . Many Active Learning ( AL ) methods have been proposed ( Lewis & Catlett , 1994 ; Culotta & McCallum , 2005 ; Settles & Craven , 2008 ) . The most frequently used approaches are uncertainty sampling and representative sampling ( Settles , 2009 ) . Uncertainty sampling is a general framework for measuring informativeness ( Lewis & Catlett , 1994 ) , where a learner queries the instance whose class is the most uncertain.Culotta & McCallum ( 2005 ) employed a simple uncertainty-based strategy for sequence models called least confidence ( LC ) : φLC ( x ) = 1 − P ( y∗|x ; Θ ) . Here , y∗ is the most likely label . This approach queries the instance for which the current model has the least confidence in its most probable label . Scheffer et al . ( 2001 ) proposed another uncertainty strategy , which queries the instance with the smallest margin between the posteriors for its two most probable labels : φM ( x ) = P ( y∗1 |x ; Θ ) −P ( y∗2 |x ; Θ ) , where y∗1 and y ∗ 2 are the first and second best labels , respectively . Another uncertainty-based measure of informativeness is entropy ( see Shannon ( 1948 ) ) . For a discrete random variable Y , the entropy is given by H ( Y ) = − ∑ ( P ( yi ) log ( P ( yi ) ) ) . A different approach to uncertainty involves several independent models and looks for disagreement among them ( Seung et al. , 1992 ) . In representative sampling , one assumes that informative instances are “ representative ” of the underlying distribution , and the query is based on the properties of the nodes in contrast with the uncertainty sampling , where the predicted scores for each label are used and not the samples themselves . Measures of the distribution include the KullbackLeibler ( KL ) divergence similarity ( McCallum & Nigam ) or clustering ( Xu et al. , 2007 ) , where the goal is to obtain representative labeled data samples . Fujii et al . ( 1998 ) considered a query strategy for nearest-neighbor methods that select queries that are ( i ) least similar to the labeled instances , and ( ii ) most similar to the unlabeled instances . Nguyen & Smeulders ( 2004 ) proposed a density-based approach that first clusters instances and then avoids querying outliers by propagating label information to instances in the same cluster . Settles and Craven ( 2008 ) suggested a general density-weighting technique combining both uncertainty and representative sampling . They query instances as follows : arg maxX φA ( X ) × ( 1 U ∑ U sim ( X , XU ) ) β where φA ( X ) represents the informativeness of x according to some “ base ” query strategy A , and U are the unlabeled samples . The second term weights the informativeness of x by its average similarity to all other instances in the input distribution ( as approximated by U ) , subject to a parameter β that controls the relative importance of the density term ( Settles & Craven , 2008 ) . Zhu et al . ( 2009 ) also proposed sampling by a combination of uncertainty and density to solve the outliers problem emerging in some uncertainty techniques . Another frequently used measure is the influence of the unlabeled samples on the model , using varying methods , such as length of gradients ( Settles et al. , 2008 ) , expected change or Fisher information ratio ( Cohn et al. , 1996 ) . Finally , there are many hybrid methods that combine different criteria ( Settles & Craven , 2008 ) . When applied to graphs , uncertainty methods were based on the properties of the classifier scores and did not explicitly use the graph information to select nodes . However , since neighboring nodes share classes more often than non-neighboring nodes , the graph itself can be used not only to predict the class of nodes but also to predict the diffusion of uncertainty . Similarly , representative sampling takes advantage of the graph but ignores the information on the nodes class . Assume , for example , two nodes : 1 ) One node that is a distinct connectivity component , where the current classifier gives the same score for both classes in binary classification . Checking the class of such a node , would probably not improve the accuracy of other nodes . 2 ) Similarly , a node with a predicted probability of 99 % to the first class would also not be of interest , even if this node has a very high degree , and checking its class would be of limited use since we already know it with a high probability . The interesting nodes to classify would be nodes that combine uncertainty and connection to many other nodes . We here propose two methods to combine uncertainty with graph properties and show that combining the graph within the AL leads to significantly higher accuracies than all current AL methods on standard datasets . 2 RELATED WORK . Graphs have been extensively used for machine learning , especially in the context of GCN ( Kipf & Welling , 2016 ; Schlichtkrull et al. , 2018 ; Berg et al. , 2017 ) and GNN ( Grover & Leskovec , 2016 ; Rosen & Louzoun , 2015 ) . GCNs were also used in combination with AL . However , as mentioned , in most such models , the graph is only used for the ML part , while the AL is performed ignoring the graph structure . Recently a few works have proposed to use the graphs themselves for AL . Three main types methods have been proposed : modularity , centrality , and label propagation : In modularity approaches , nodes are divided into communities . Macskassy ( 2009 ) proposed to reveal the most central node in each community and sample it . Then each community is divided into sub-communities and the most central node in each sub-community is sampled and so on . Mackassy further suggested a hybrid method combining communities , centrality , and uncertainty with the Empirical Risk Minimization ( ERM ) framework . Ping et al . ( 2017 ) proposed combining community structure to perform batch-mode AL . They used communities to consider the overlap in information content among the “ best ” instances . Centrality based approaches focus on “ central ” nodes using some method ( e.g . high degree ) . The assumption is that the central nodes will have a major impact on the unknown labels . Macskassy ( 2009 ) in the ERM algorithm , showed that betweenness centrality is a good measure for centrality . Cai et al . ( 2017 ) proposed to calculate a node representativeness score based on graph centrality . They tested several centrality measures : degree centrality , betweenness centrality , harmonic centrality , closeness centrality , and page-rank centrality . They concluded that the Page-Rank centrality is superior , and suggested using it when the prediction model is not informative enough . In label propagation approaches , the implicit assumption is of label smoothness over the graph or over the projection of the graph into some manifold in RN . Ming Ji proposed to select the samples such that the total variance of the Gaussian field over unlabeled examples , as well as the expected prediction error of the harmonic Gaussian field classifier , is minimized . An efficient computation scheme was then proposed to solve the corresponding optimization problem with no additional parameter ( Ji & Han , 2012 ) .Ma et al . ( 2013 ) extended sub-modularity guarantees from V-optimality to Σ -optimality using properties specific to Gaussian Markov Random Field ( GRMF ) s. Finally , Dimitris Berberidis proposed to sample the nodes with the highest expected change of the unknown labels . Thus , in contrast with the expected error reduction and entropy minimization approaches that actively sample with the goal of increasing the confidence on the model , Berberidis et al . focus on maximally perturbing the model with each node sampled . The intuition behind this approach is that by sampling nodes with the largest impact , one may take faster steps towards an increasingly accurate model ( Berberidis & Giannakis , 2018 ) . 3 MAIN CONTRIBUTIONS OF THE CURRENT WORK . We here propose to combine uncertainty and representative sampling methods . Specifically , we propose to sample within regions of uncertainty in the graph ( in contrast with sampling uncertain nodes ) . While the sampling of nodes with high uncertainty was proposed in a vast array of applications ( Cai et al. , 2017 ; Zhou & Sun , 2014 ; Macskassy , 2009 ) , sampling using regional uncertainty ( and in the cases of directed graphs , regional directed uncertainty , as will be further explained ) has never been successfully applied . Chen et al . recently proposed as one of their models , a regional uncertainty algorithm , but obtained very low accuracies ( Chen et al. , 2019 ) . Specifically , three novel claims are proposed here : • We propose that replacing the node entropy by the regional entropy leads to a higher accuracy in GCN based AL methods . • We propose an extension of the PageRank algorithm to detect nodes with limited diffusion of information from labeled nodes in directed graphs , and show that an AL method based on this Adaptive PageRank ( APR ) outperforms other methods in some datasets . • We hypothesize that information can be gained from the graph , only when the sampling rate is low enough so that most unlabeled nodes have no labeled networks . At this stage , we only exemplify this hypothesis but do not bring a clear proof for it .

Summary: The paper studies active learning in graph representations. To decide which nodes in a graph to label during the active labeling, the paper proposes two approaches. First it argues that one should consider region-based measures rather than single nodes. Second, it proposes to adapt the page rank algorithm (APR) to determine which nodes are far away from labeled nodes.