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 paper proposes the AL method which combines node selection and message passing of the graph models for minimizing labeling cost. The first technical contribution is to quantify the influence of labeling in GCN/LP and relate AL to the problem of finding seed nodes for maximizing the influence. In addition, this paper shows how to maximize the spread of influence by considering the noise and reliability of labels.