RIM: Reliable Influence-based Active Learning on Graphs

1 Introduction . Graphs are ubiquitous in the real world , such as social , academic , recommendation , and biological networks [ 38 , 30 , 31 , 10 , 28 ] . Unlike the independent and identically distributed ( i.i.d ) data , nodes are connected by edges in the graph . Due to the ability to capture the graph information , message passing is the core of many existing graph models assuming that labels and features vary smoothly over the edges of the graph . Particularly in Graph Convolutional Neural Network ( GCN ) [ 16 ] , the feature of each node is propagated along edges and transformed through neural networks . In Label Propagation ( LP ) [ 25 ] , node labels are propagated and aggregated along edges in the graph . The message passing typically requires a large amount of labeled data to achieve satisfactory performance . However , labeling data , be it by specialists or crowd-sourcing , often consumes too much time and money . The process is also tedious and error-prone . As a result , it is desirable to achieve good classification results with labeled data that is both few and unreliable . Active Learning ( AL ) [ 1 ] is a promising strategy to tackle this challenge , which minimizes the labeling cost by prioritizing the selection of data in order to improve the model performance as much as possible . Unfortunately , conventional AL methods [ 3 , 8 , 20 , 41 , 21 , 27 ] treat message passing and AL independently without explicitly considering their impact on each other . In this paper , we advocate that a better AL method should unify node selection and message passing towards minimizing labeling cost , and we make two contributions towards that end . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . The first contribution is that we quantify node influence by how much the initial feature/label of label node v influences the output feature/label of node u in GCN/LP , and then connect AL with influence maximization , e.g. , the problem of finding a small set of seed nodes in a network that maximizes the spread of influence . To demonstrate this idea , we randomly select different sets of |Vl| = 20 labeled nodes under the clean label and train a 2-layer GCN with a different labeled set on the Cora dataset [ 16 ] . For LP , we select two sets with the average node degree of 2 and 10 . As shown in Figure 1 , the model performance in both GCN/LP tends to increase along with the receptive field/node degree under the clean label , implying the potential gain of increasing the node influence . Note that in real life , both humans and automated systems are prone to mistakes . To examine the impact of label noise , we set the label accuracy to 50 % , and Figure 1 ( a ) and Figure 1 ( b ) show that the test accuracy could even drop with the increase of node influence under the noisy label . This is because the noise of labels will also be widely propagated with node influence increasing , thus diminishing the benefit of influence maximization . Therefore , our second contribution is that we further propose to maximize the reliable influence spread when label noise is taken into consideration . Specifically , each node is associated with a new parameter called the quality factor , indicating the probability that the label given by the oracle is correct . We recursively infer the quality of newly selected nodes based on the smoothing features/labels of previously selected nodes across the graph ’ s edges , i.e. , nodes that share similar features or graph structure are likely to have the same label . Based on the above insights , we propose a fundamentally new AL selection criterion for GCN and LP–reliable influence maximization ( RIM ) –by considering both quantity and quality of influence simultaneously . Under a high-quality factor , we enforce the influence of selected label nodes for large overall reaches , while under a low-quality factor , we make mistake penalization to limit the selected node influence . RIM also maintains some nice properties such as submodularity , which allows a greedy approximation algorithm for maximizing reliable influence to reach an approximation ratio of 1− 1e compared with the optimum . Empirical studies on public datasets demonstrate that RIM significantly outperforms the state-of-the-art methods GPA [ 13 ] by 2.2 % -5.1 % in terms of predictive accuracy when the labeling accuracy is 70 % , even if it is enhanced with the anti-noise mechanism PTA [ 6 ] . Furthermore , in terms of efficiency , RIM achieves up to 4670× and 18652× end-to-end runtime speedups compared to GPA in GPU and CPU , respectively . In summary , the core contributions of this paper are 1 ) We open up a novel perspective for efficient and effective AL for GCN and LP by enforcing the feature/label influence with a connection to social influence maximization [ 18 , 9 , 4 ] ; 2 ) To the best of our knowledge , we are the first to consider the influence quality in graph-based AL , and we propose a new method to estimate the influence quantity based on the feature/label smoothing ; 3 ) We combine the influence quality and quantity in a unified RIM framework . The empirical study on both GCNs and LP demonstrates that RIM significantly outperforms the compared baselines in performance and efficiency . 2 Preliminary . 2.1 Active Learning . Problem Formulation . Let c be the number of label classes and the ground-truth label for a node vi be a one-hot vector yi ∈ Rc . Suppose that the entire node set V is partitioned into training set Vtrain ( including both the labeled set Vl and unlabeled set Vu ) , validation set Vval and test set Vtest . Given a graph G = ( V , E ) with |V| = N nodes and |E| = M edges , feature matrix X = { x1 , x2 ... , xN } in which xi ∈ Rd , a labeling budget B , and a loss function ` , the goal of an AL algorithm is to select a subset of nodes Vl ⊂ Vtrain to label from a noisy oracle with the labeling accuracy α , so that it can produce a model f with the lowest loss on the test set : arg min Vl : |Vl|=B Evi∈Vtest [ ` ( yi , P ( ŷi|f ) ) ] , ( 1 ) where P ( ŷi|f ) is the predictive label distribution of node vi . To measure the influence of feature or label propagation , we focus on f being GCN or LP . AL on graphs . Both GCN and LP are representative semi-supervised graph models which can utilize additional unlabeled data to enhance the model learning [ 40 ] , and lots of AL methods are specially designed for these two models . For example , both AGE [ 3 ] and ANRMAB [ 8 ] adopt uncertainty , density , and node degree when composing query strategies . LP-ME chooses the node that maximizes entropy for itself , while LP-MRE chooses the node such that the acquisition of its label reduces entropy most for the total graph [ 20 ] . 2.2 Graph Models . LP . Based on the intuitive assumption that locally connected nodes are likely to have the same label , LP iteratively propagates the label influence to distant nodes along the edges as follows : Y ( k+1 ) = D̃−1ÃY ( k ) , Y ( k+1 ) u = Y ( k+1 ) , Y ( k+1 ) l = Y ( 0 ) , ( 2 ) where Y ( 0 ) = { y1 , y2 ... , yl } is the initial label matrix consisting of one-hot label indicator vectors , Y ( k ) u and Y ( k ) l denote the soft label matrix of the labeled nodes set Vl and unlabeled nodes set Vu in iteration k respectively . According to [ 39 ] , we iteratively set the nodes in Vl back to their initial label Y ( 0 ) since their labels are correct and should not be influenced by the unlabeled nodes . GCN . Each node in GCN iteratively propagates its feature influence to the adjacent nodes when predicting a label . Especially , each layer updates the node feature embedding in the graph by aggregating the features of neighboring nodes : X ( k+1 ) = δ ( D̃−1ÃX ( k ) W ( k ) ) , ( 3 ) where X ( k ) and X ( k+1 ) are the embeddings of layer k and k + 1 respectively . Specifically , X ( and X ( 0 ) ) is the original node feature . Ã = A + IN is used to aggregate feature vectors of adjacent nodes , where A is the adjacent matrix of the graph and IN is the identity matrix . D̃ is the diagonal node degree matrix used to normalize Ã , W ( k ) is a layer-specific trainable weight matrix and δ ( · ) is the activation function . 2.3 Influence Maximization . The influence maximization ( IM ) problem in social networks aims to select B nodes so that the number of nodes activated ( or influenced ) in the social networks is maximized [ 15 ] . That being said , given a graph G = ( V , E ) , the formulation is as follows : max S |σ ( S ) | , s.t . S ⊆ V , |S| = B , ( 4 ) where σ ( S ) is the set of nodes activated by the seed set S under certain influence propagation models , such as Linear Threshold ( LT ) and Independent Cascade ( IC ) models [ 15 ] . The maximization of σ ( S ) is NP-hard . However , if σ ( S ) is nondecreasing and submodular with respect to S , a greedy algorithm can provide an approximation guarantee of ( 1− 1e ) [ 23 ] . RIM is the first to connect social influence maximization with graph-based AL under a noisy oracle by defining reliable influence . 3 Reliable Influence Maximization . This section presents RIM , the first graph-based AL framework that considers both the influence quality and influence quantity . At each batch of node selection , RIM first measures the proposed reliable influence quantity and selects a batch of nodes that can maximize the number of activated nodes , and then it updates the influence quality for the next iteration . The above process is repeated until the labeling budget B runs out . We will introduce each component of RIM in detail below . 3.1 Influence Propagation . We measure the feature/label influence of a node vi on vj by how much change in the input feature/label of vi affects the aggregated feature/label of vj after k iterations propagation [ 26 , 33 ] . Definition 3.1 ( Feature Influence ) . The feature influence score of node vi on node vj after k-step propagation is the L1-norm of the expected Jacobian matrix Îf ( vj , vi , k ) = ∥∥∥E [ ∂X ( k ) j /∂X ( 0 ) i ] ∥∥∥ 1 . The normalized influence score is defined as If ( vj , vi , k ) = Îf ( vj , vi , k ) ∑ vw∈V I ( vj , vw , k ) . ( 5 ) Definition 3.2 ( Label Influence ) . The label influence score of node vi on node vj after k-step propagation is the gradient of y ( k ) j with respect to yi : Il ( vj , vi , k ) = ∥∥∥E [ ∂y ( k ) j /∂yi ] ∥∥∥ 1 . ( 6 ) Given the k-step feature/label propagation mechanism , the feature/label influence score captures the sum over probabilities of all possible paths of length k from vi to vj , which is the probability that a random walk ( or variants ) starting at vj ends at vi after taking k steps [ 33 ] . Thus , we could take them as the probability that vi propagates its feature/label to vj via random walks from the influence propagation perspective .

Summary: This proposes a novel graph-based active learning framework by reliable social influence maximization. The proposed model aims to select a subset of nodes to be labeled, so as to maximize the number of nodes influenced. Whether a node is influenced is determined by both the influence quantity and quality (assuming label noise). A greedy algorithm is adopted for optimization, whose approximation precision is guaranteed theoretically. SeveraleExperiments are conducted to validate the effectiveness and efficiency of RIM.