Boost then Convolve: Gradient Boosting Meets Graph Neural Networks

1 INTRODUCTION . Graph neural networks ( GNNs ) have shown great success in learning on graph-structured data with various applications in molecular design ( Stokes et al. , 2020 ) , computer vision ( Casas et al. , 2019 ) , combinatorial optimization ( Mazyavkina et al. , 2020 ) , and recommender systems ( Sun et al. , 2020 ) . The main driving force for progress is the existence of canonical GNN architecture that efficiently encodes the original input data into expressive representations , thereby achieving high-quality results on new datasets and tasks . Recent research has mostly focused on GNNs with sparse data representing either homogeneous node embeddings ( e.g. , one-hot encoded graph statistics ) or bag-of-words representations . Yet tabular data with detailed information and rich semantics among nodes in the graph are more natural for many situations and abundant in real-world AI ( Xiao et al. , 2019 ) . For example , in a social network , each person has socio-demographic characteristics ( e.g. , age , gender , date of graduation ) , which largely vary in data type , scale , and missing values . GNNs for graphs with tabular data remain unexplored , with gradient boosted decision trees ( GBDTs ) largely dominating in applications with such heterogeneous data ( Bentéjac et al. , 2020 ) . GBDTs are so successful for tabular data because they possess certain properties : ( i ) they efficiently learn decision space with hyperplane-like boundaries that are common in tabular data ; ( ii ) they are well-suited for working with variables of high cardinality , features with missing values , and of different scale ; ( iii ) they provide qualitative interpretation for decision trees ( e.g. , by computing decrease in node impurity for every feature ) or for ensembles via post-hoc analysis stage ( Kaur et al. , 2020 ) ; ( iv ) in practical applications , they mostly converge faster even for large amounts of data . In contrast , a crucial feature of GNNs is that they take into account both the neighborhood information of the nodes and the node features to make a prediction , unlike GBDTs that require additional preprocessing analysis to provide the algorithm with graph summary ( e.g. , through unsupervised graph embeddings ( Hu et al. , 2020a ) ) . Moreover , it has been shown theoretically that message-passing GNNs can compute any function on its graph input that is computable by a Turing machine , i.e. , GNN is known to be the only learning architecture that possesses universality properties on graphs ( approximation ( Keriven & Peyré , 2019 ; Maron et al. , 2019 ) and computability ( Loukas , 2020 ) ) . Furthermore , gradient-based learning of neural networks can have numerous advantages over the treebased approach : ( i ) relational inductive bias imposed in GNNs alleviates the need to manually engineer features that capture the topology of the network ( Battaglia et al. , 2018 ) ; ( ii ) the end-to-end nature of training neural networks allows multi-stage ( Fey et al. , 2019 ) or multi-component ( Wang et al. , 2020 ) integration of GNNs in application-dependent solutions ; ( iii ) pretraining representations with graph networks enriches transfer learning for many valuable tasks such as unsupervised domain adaptation ( Wu et al. , 2020 ) , self-supervised learning ( Hu et al. , 2020b ) , and active learning regimes ( Satorras & Estrach , 2018 ) . Undoubtedly , there are major benefits in both GBDT and GNN methods . Is it possible to get advantages of both worlds ? All previous approaches ( Arik & Pfister , 2020 ; Popov et al. , 2019 ; Badirli et al. , 2020 ) that attempt to combine gradient boosting and neural networks are computationally heavy , do not consider graph-structured data , and suffer from the lack of relational bias imposed in GNN architectures , see Appendix A for a more detailed comparison with related literature . To the best of our knowledge , the current work is the first to explore using GBDT models for graph-structured data . In this paper , we propose a novel learning architecture for graphs with tabular data , BGNN , that combines GBDT ’ s learning on tabular node features with GNN that refines the predictions utilizing the graph ’ s topology . This allows BGNN to inherit the advantages of gradient boosting methods ( heterogeneous learning and interpretability ) and graph networks ( representation learning and end-toend training ) . Overall , our contributions are the following : ( 1 ) We design a novel generic architecture that combines GBDT and GNN into a unique pipeline . To the best of our knowledge , this is the first work that systematically studies the application of GBDT to graph-structured data . ( 2 ) We overcome the challenge of end-to-end training of GBDT by iteratively adding new trees that fit the gradient updates of GNN . This allows us to backpropagate the error signal from the topology of the network to GBDT . ( 3 ) We perform an extensive evaluation of our approach against strong baselines in node prediction tasks . Our results consistently demonstrate significant performance improvements on heterogeneous node regression and node classification tasks over a variety of real-world graphs with tabular data . ( 4 ) We show that our approach is also more efficient than the state-of-the-art GNN models due to much faster loss convergence during training . Furthermore , learned representations exhibit discernible structure in the latent space , which further demonstrates the expressivity of our approach . 2 BACKGROUND . Let G = ( V , E ) be a graph with nodes having features and target labels . In node prediction tasks ( classification or regression ) , some target labels are known , and the goal is to predict the remaining ones . Throughout the text , by lowercase variables xv ( v ∈ V ) or x we denote features of individual nodes , and X represents the matrix of all features for v ∈ V . Individual target labels are denoted by yv , while Y is the vector of known labels . Graph Neural Networks ( GNNs ) use both the network ’ s connectivity and the node features to learn latent representations for all nodes v ∈ V . Many popular GNNs use a neighborhood aggregation approach , also called the message-passing mechanism , where the representation of a node v is updated by applying a non-linear aggregation function of v ’ s neighbors representation ( Fey & Lenssen , 2019 ) . Formally , GNN is a differentiable , permutation-invariant function gθ : ( G , X ) 7→ Ŷ , where Ŷ is the vector of predicted labels . Similar to traditional neural networks , GNNs are composed of multiple layers , each representing a non-linear message-passing function : xtv = COMBINE t ( xt−1v , AGGREGATE t ( { ( xt−1w , x t−1 v ) : ( w , v ) ∈ E } ) ) , ( 1 ) where xtv is the representation of node v at layer t , and COMBINE t and AGGREGATEt are ( parametric ) functions that aggregate representations from the local neighborhood of a node . Then , the GNN mapping gθ includes multiple layers of aggregation ( 1 ) . Parameters of GNN model θ are optimized with gradient descent by minimizing an empirical loss function LGNN ( Y , gθ ( G , X ) ) . Gradient Boosted Decision Trees ( GBDT ) is a well-known and widely used algorithm that is defined on non-graph tabular data ( Friedman , 2001 ) and is particularly successful for tasks containing heterogeneous features and noisy data . The core idea of gradient boosting is to construct a strong model by iteratively adding weak ones ( usually decision trees ) . Formally , at each iteration t of the gradient boosting algorithm , the model f ( x ) is updated in an additive manner : f t ( x ) = f t−1 ( x ) + ht ( x ) , ( 2 ) where f t−1 is a model constructed at the previous iteration , ht is a weak learner that is chosen from some family of functions H , and is a learning rate . The weak learner ht ∈ H is chosen to approximate the negative gradient of a loss function L w.r.t . the current model ’ s predictions : ht = argmin h∈H ∑ i ( −∂L ( f t−1 ( xi ) , yi ) ∂f t−1 ( xi ) − h ( xi ) ) 2 . ( 3 ) The gradient w.r.t . the current predictions indicates how one should change these predictions to improve the loss function . Informally , gradient boosting can be thought of as performing gradient descent in functional space . The set of weak learnersH is usually formed by shallow decision trees . Decision trees are built by a recursive partition of the feature space into disjoint regions called leaves . This partition is usually constructed greedily to minimize the loss function ( 3 ) . Each leaf Rj of the tree is assigned to a value aj , which estimates the response y in the corresponding region . In our case , aj is equal to the average negative gradient value in the leaf Rj . To sum up , we can write h ( x ) = ∑ j aj1 { x∈Rj } . 3 GBDT MEETS GNN . GBDT : 𝑓 Node features 𝑿 GNN : 𝑔𝜃 𝑓 ( 𝑿 ) Pass to GNN 𝐺 , 𝑿′ Loss 𝐿 𝑔𝜃 𝐺 , 𝑿 ′ , 𝑌 New tree : 𝑓𝑖 𝜕𝐿 𝑔𝜃 𝐺 , 𝑿 ′ , 𝑌 𝜕𝑿′ Compute update 1 2 3 4 Gradient boosting approach is successful for learning on tabular data ; however , there are challenges of applying GBDT on graph-structured data : ( i ) how to propagate relational signal , in addition to node features , to otherwise inherently tabular model ; and ( ii ) how to train it together with GNN in an end-to-end fashion . Indeed , optimizations of GBDT and GNN follow different approaches : the parameters of GNN are optimized via gradient descent , while GBDT is constructed iteratively , and the decision trees remain fixed after being built ( decision trees are based on hard splits of the feature space , which makes them non-differentiable ) . A straightforward approach would be to train the GBDT model only on the node features and then use the obtained predictions , jointly with the original input , as new node features for GNN . In this case , the graph-insensitive predictions of GBDT will further be refined by a graph neural network . This approach ( which we call Res-GNN ) can already boost the performance of GNN for some tasks . However , in this case , the GBDT model completely ignores the graph structure and may miss descriptive features of the graph , providing inaccurate input data to GNN . In contrast , we propose end-to-end training of GBDT and GNN called BGNN ( for Boost-GNN ) . As before , we first apply GBDT and then GNN , but now we optimize both of them , taking into account the quality of final predictions . The training of BGNN is shown in Figure 1 . Recall that one can not tune already built decision trees due to their discrete structure , so we iteratively update the GBDT model by adding new trees that approximate the GNN loss function . In Algorithm 1 , we present the training of BGNN that combines GBDT and GNN for any node-level prediction problem such as semi-supervised node regression or classification . In the first iteration , we build a GBDT model f1 ( x ) with k decision trees by minimizing the loss function LGBDT ( f1 ( x ) , y ) ( e.g. , RMSE for regression or cross-entropy for classification ) averaged over the train nodes , following the equations ( 2 ) - ( 3 ) . Using all predictions f1 ( X ) , we update the node features to X′ that we pass to GNN . Possible update functions that we experiment with include concatenation with the original node features and their replacement by f1 ( X ) . Next , we train a graph neural network gθ on a graph G with node features X′ by minimizing LGNN ( gθ ( G , X′ ) , Y ) with l steps of gradient descent.1 Importantly , we optimize both the parameters θ of GNN and the node features X′ . Then , we use the difference between the optimized node features X′new and the input node features X ′ = f1 ( X ) as the target for the next decision trees built by GBDT . If l = 1 , the difference X′new −X′ exactly equals the negative gradient of the loss function w.r.t . the input features X′ multiplied by the learning rate η : X′new = X ′ − η ∂LGNN ( gθ ( G , X ′ ) , Y ) ∂X′ . In the second iteration , we train a new GBDT model f2 with the original input features X but new target labels : X′new −X′ . Intuitively , f2 fits the direction that would improve GNN prediction based on the first predictions f1 ( X ) . In other words , GBDT approximates the gradient steps made by GNN for the node features X′ . This is a regression problem , so here LGBDT is the RMSE loss . After f2 is trained , we combine the predictions f ( X ) = f1 ( X ) + f2 ( X ) and pass the obtained values X′ to GNN as node features . GNN model gθ again does l steps of backpropagation and passes the new difference X′new −X′ as a target to the next iteration of GBDT . In total , the model is trained for N epochs and outputs a GBDT model f : X 7→ Y and GNN model gθ : ( G , X ) 7→ Y , which can be used for downstream tasks . Intuitively , BGNN model consists of two consecutive blocks , GBDT and GNN , which are trained end-to-end , and therefore can be interpreted from two angles : GBDT as an embedding layer for GNN or GNN as a parametric loss function for GBDT . In the former case , GBDT transforms the original input features X to new node features X′ , which are then passed to GNN . In the latter case , one can see BGNN as a standard gradient boosted training where GNN acts as a complex loss function that depends on the graph topology .

Summary: Review: This paper proposes a fusion of GBDT and graph neural network that works on graphs with heterogeneous tabular features. Previous approaches are computationally heavy and do not consider graph-structured data and suffer from lack of relational bias imposed in GNNs. The proposed method is a new ensemble tree method which alternates between functional gradient step in GBDT (which train on the current latent features) and SGD training of graph neural network (to generate the latent features which are fed into the subsequent trees).