LINGUINE: LearnIng to pruNe on subGraph convolUtIon NEtworks

1 INTRODUCTION . Graph Representation Learning has attracted much attention from the research communities in recent years , with emerging new work every year . Graph Convolution Neural Networks ( GCNs ) were proposed as the extension of Convolutional Neural Networks ( CNNs ) ( LeCun et al. , 1995 ) on geometric data . The first spectral-based GCN was designed on Spectral Graph Theory ( Bruna et al. , 2013 ) and was extended by many following works ( Henaff et al. , 2015 ; Defferrard et al. , 2016 ) . Over recent years , the spatial-based counterpart ( Kipf & Welling , 2016a ) gained more attention and had facilitated many machine learning tasks ( Wu et al. , 2020 ; Cai et al. , 2018 ) including semisupervised node classification ( Hamilton et al. , 2017b ) , link prediction ( Kipf & Welling , 2016b ; Berg et al. , 2017 ) and knowledge graphs ( Schlichtkrull et al. , 2018 ) . In this work , we primarily focused on large-scale spatial-based GCNs ( Hamilton et al. , 2017a ; Chen et al. , 2018b ; Gao et al. , 2018 ; Huang et al. , 2018 ; Zeng et al. , 2019 ; Zou et al. , 2019 ; Chiang et al. , 2019 ) , where a given node aggregates hidden states from its neighbors in the previous layer , followed by a non-linear activation to obtain the topological representation . However , as the graph gets larger , GNN models suffer from the challenges imposed by limited physical memory and exponentially growing computation overhead . Recent work adopted sampling methods to handle the large volume of data and facilitate batch training . The majority of them could be classified as 3 types , layer-wise sampling ( Hamilton et al. , 2017a ; Gao et al. , 2018 ; Huang et al. , 2018 ; Zou et al. , 2019 ) , node-wise sampling ( Chen et al. , 2018b ) and subgraph sampling ( Chiang et al. , 2019 ; Zeng et al. , 2019 ) . In layer-wise sampling , we take samples from the neighbors of a given node in each layer . The number of nodes is growing exponentially as the GCNs gets deeper , which resulted in ’ neighbor explosion ’ . In node-wise sampling , the nodes in each layer are sampled independently to form the structure of GCNs , which did avoid ’ neighbor explosion ’ . But the GCN ’ s structure is unstable and resulted in inferior convergence . In subgraph sampling , the GCNs are trained on a subgraph sampled on the original graph . The message was passed within the subgraph during training . This approach resolved the problem of neighbor explosion and can be applied to training deep GCNs . However , the subgraph ’ s structure and connectivity had a great impact in the training phase . It might result in suboptimal performance and slow convergence if the subgraph is overly sparse ( Chen et al. , 2018b ) . Different sampling methods can make a huge difference in the final accuracy and the convergence speed as shown in ( Zeng et al. , 2019 ) . In the context of large-scale GCN training , the limitations in GPU memory makes the maximum batch size restricted . Research communities are actively seeking efficient sampling methods to deal with the challenges in scalability , accuracy , and computation complexity on large-scale GCNs . The initiative of subgraph GCN is a stochastic approach to approximate their full-graph counterparts . However , experiments showed that GCN trained with partial information can achieve even less bias . ( Zou et al. , 2019 ) This was even more so when variance reduction is applied in GCN ’ s training , which overcomes the negative effect induced by subgraph GCNs and improves its convergence speed as well as inference performance . ( Hamilton et al. , 2017a ; Chen et al. , 2018a ; Zeng et al. , 2019 ) However , the inner mechanism behind this random sample has yet to be studied . GCNs are also made to be deeper and more complicated with architecture design . The model overcomes the drawbacks of the gradient vanishing problem via applying Deep CNN ’ s residual/dense connection and dilated convolution , achieving state of art performance on open graph benchmarks . ( Li et al. , 2019 ; 2020 ; Weihua Hu , 2020 ) However , as their model is significantly larger ( amounting to more than 100 layers in some occasions ) than previous approaches , the model is suffered from high computation overhead and memory cost . With the model taking up much space , the batch sizes are also strictly limited . We aim to provide an easier solution to train complex models while maintaining a relatively large receptive field in the graph , and keeping the training quality . We propose Linguine framework . In each forward pass , we ’ smartly prune ’ inferior nodes to extract a concentrated smaller subgraph from the large subgraph randomly sampled previously . Therefore , we reduce the batch size and the memory requirement in training the model . We are also able to train the complex GCNs with larger receptive field and achieve better performance with the same budget . We parameterize the decision function in smart pruning with a light-weight meta-model , which is fed with the meta-information we obtained from training a light-weight proxy model . This keeps the extra cost of algorithm under control . Our framework is built upon existing subgraph sampling methods and utilize joint training to learn the meta-model . Our meta-model improved the quality of the subgraphs in training via actively dropping redundant nodes from its receptive field and concentrate the information . We summarize the contributions of this work as follows : 1 . We designed a new training framework Linguine which aims to train high-performance GCNs on large graphs via model-parameterized smart pruning techniques . 2 . Linguine provides a joint-training algorithm called bootstrapping , originally designed to train the meta-model in smart pruning , but also has a favorable impact in augmenting existing models and converge to better solutions with scalability and lower bias . It is also an effective algorithm inspired by the idea in the real-world learning process . 3 . Empirical study justified that Linguine framework worked well on different scaled public benchmarks and compares favorably to previous methods . 4 . We did an analysis on the mechanism behind smart-pruning via graph visualization . 2 RELATED WORK . Our framework was inspired and built upon two popular branches in Machine Learning . Graph Neural Networks Many GNNs have emerged over the recent years . Spatial-based GCNs are the most popular approaches among them and have gained broad interest from research communities ( Atwood & Towsley , 2016 ; Niepert et al. , 2016 ; Gilmer et al. , 2017 ) . They stack multiple graph convolutional layers to extract high-level representations . In each layer , every node in the graph aggregates the hidden states from its neighbors on the previous layer . The final output is the embedding of each node in the graph . Existing work on GCNs utilized sampling technique to perform efficient minibatch training . Common approaches can be categorized as Layer-wise Sampling , Node-wise Sampling and Subgraph Sampling . In Layer-wise sampling and Node-wise Sampling , the layers are sampled recursively from top to bottom to form mini-batches . The major difference between Node-wise Sampling and Layer-wise Sampling lied in the sampling mechanism . Node-wise Sampling sampled from the whole graph dataset whereas Layer-wise Sampling only focused on the neighbors from the upper layer . In Subgraph Sampling we only sample once and propagate through the subgraph to formalize batch training . Among work in Layer-wise Sampling , GraphSAGE ( Hamilton et al. , 2017a ) performed uniform Node-wise Sampling on the neighbors of nodes in the previous layer . ( Ying et al. , 2018 ) introduced the importance score to each neighbor and performed weighted aggregation to enhance the model . The potential drawback of Layer-wise Sampling is Neighbor Explosion where we created an overly-large receptive field in the bottom layer and resulted in memory overflow . In Node-wise Sampling , FastGCN ( Chen et al. , 2018b ) performs sampling independently across all layers and perform importance sampling on the node level to reduce the variance . This approach was further extended by LADIES ( Zou et al. , 2019 ) which performs importance sampling adaptively from top to bottom . Recently , Subgraph Sampling emerged as a low complexity solution for large-scale GCN training . ClusterGCN ( Chiang et al. , 2019 ) performs clustering on the graph . This work partitions the graph into densely connected clusters . Random sampling is performed on the cluster level to form minibatches in training . GraphSAINT ( Zeng et al. , 2019 ) directly sampled the mini-batches at node-level . This approach also performed normalization at the node level to reduce the sampling variance . It achieves the new state of the art performance on multiple public datasets . The main difference between Subgraph Sampling and other approaches is that GCN is trained on the subgraph instead of a massage passing flow containing different nodes on different layers . Sampling subgraph has been an important method leveraging limited memory to achieve better performance . However , the variance introduced by sampling negatively impacts the training . Recent approaches adopted Variance Reduction ( Hamilton et al. , 2017a ; Chen et al. , 2018a ; Zeng et al. , 2019 ) to stochastically approximate the activations of sampled nodes . which was shown to boost the final model performance and convergence speed ( Zeng et al. , 2019 ) . Besides , empirical study found that different sampling methods can also create a great gap in the final performance of GCNs and varied convergence speed . The advance of deepening GCNs has drawn extensive attention in the research community , leveraging the computation power of high-performance GPU devices . ( Li et al. , 2019 ) adapted residual connections and dilated convolutions into GCNs . ( Li et al. , 2020 ) proposed a general aggregation function , which further improves the capacity of deep GCNs . Meta-Learning Our work is inspired by recent meta-learning and model-teaching approaches , which is also a trendy topic in recent years . Learning to Learn , or Meta-Learning ( Schmidhuber , 1987 ; Thrun & Pratt , 2012 ) was proposed as an exploration of using automatic learning via transferring knowledge learned from meta tasks . It was designed as a two-level structure , in which the meta-level model evolves slowly and the task model progresses quickly . Recently , meta-learning has been utilized in different machine learning scenarios . There is work trying to design optimizer and neural network architectures . ( Andrychowicz et al. , 2016 ; Li & Malik , 2016 ; Zoph & Le , 2016 ) . Other approaches similar to Meta-Learning is Teaching ( Anderson et al. , 1985 ; Goldman & Kearns , 1992 ) , which can be categorized as machineteaching and hardness based methods . Machine teaching ( Zhu , 2013 ) is to construct a minimal training set for the student model to learn a target model . However , due to the strong assumption of the oracle ’ s existence in machine teaching . Hardness-based models are proposed on the assumption that a data order from easy to hard can benefit model training . There are many different works in the hardness-based method . Curriculum Learning ( CL ) ( Bengio et al. , 2009 ; Spitkovsky et al. , 2010 ; Graves et al. , 2017 ) utilizes a hardness measure based on a heuristic understanding of data . Self-paced learning ( SPL ) ( Kumar et al. , 2010 ) heuristically define the scheduling of training data in an easy-to-hard manner . Learning to Teach ( Fan et al. , 2018 ; Wu et al. , 2018 ) designed a new teacher-student model and perform transferable model teaching . Our meta-model incorporates graph topological information and node-wise performance information into meta information . We used a learned model instead of heuristic and fixed rules in the model to explore the optimal policy .

Summary: This paper proposes an approach to compute GNNs on pruned subgraphs. The authors use a "meta-model" to learn a good node pruning strategy during training. Then the meta-model is used to generate pruned subgraphs during inference. The proposed pruning algorithm can be applied to various graph samplers. The authors evaluate such pruning on two simple samplers, random node and random walk samplers.  Some theoretical analysis is performed. However, the analysis is not quite convincing. Evaluation on several graph benchmarks show that the proposed LINGUINE framework achieves good accuracy using pruned subgraphs.