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1. How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks

   Xu, K. ; Zhang, M ; Li, J ; Du, S. ; Kawarabayashi, K ; Jegelka, S. ( January 2021 , International Conference on Learning Representations (ICLR))

   We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while feedforward neural networks, a.k.a. multilayer perceptrons (MLPs), do not extrapolate well in certain simple tasks, Graph Neural Networks (GNNs) – structured networks with MLP modules – have shown some success in more complex tasks. Working towards a theoretical explanation, we identify conditions under which MLPs and GNNs extrapolate well. First, we quantify the observation that ReLU MLPs quickly converge to linear functions along any direction from the origin, which implies that ReLU MLPs do not extrapolate most nonlinear functions. But, they can provably learn a linear target function when the training distribution is sufficiently “diverse”. Second, in connection to analyzing the successes and limitations of GNNs, these results suggest a hypothesis for which we provide theoretical and empirical evidence: the success of GNNs in extrapolating algorithmic tasks to new data (e.g., larger graphs or edge weights) relies on encoding task-specific non-linearities in the architecture or features. Our theoretical analysis builds on a connection of over-parameterized networks to the neural tangent kernel. Empirically, our theory holds across different training settings. 

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2. Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depth

   Xu, K. ; Zhang, M. ; Jegelka, S. ; Kawaguchi, K. ( January 2021 , International Conference on Machine Learning (ICML))

   Graph Neural Networks (GNNs) have been studied through the lens of expressive power and generalization. However, their optimization proper- ties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real- world graphs. Second, we study what may affect the GNNs’ training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice. 

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3. Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depth

   Xu, Keyulu ; Zhang, Mozhi ; Jegelka, Stefanie ; Kawaguchi, Kenji ( January 2021 , Proceedings of the 38th International Conference on Machine Learning, PMLR)

   null (Ed.)

   Graph Neural Networks (GNNs) have been studied through the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs’ training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice. 

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4. Subgraph Neural Networks

   Alsentzer, Emily ; Finlayson, Samuel ; Li, Michelle ; Zitnik, Marinka ( January 2020 , Advances in neural information processing systems)

   null (Ed.)

   Deep learning methods for graphs achieve remarkable performance on many node-level and graph-level prediction tasks. However, despite the proliferation of the methods and their success, prevailing Graph Neural Networks (GNNs) neglect subgraphs, rendering subgraph prediction tasks challenging to tackle in many impactful applications. Further, subgraph prediction tasks present several unique challenges: subgraphs can have non-trivial internal topology, but also carry a notion of position and external connectivity information relative to the underlying graph in which they exist. Here, we introduce SubGNN, a subgraph neural network to learn disentangled subgraph representations. We propose a novel subgraph routing mechanism that propagates neural messages between the subgraph's components and randomly sampled anchor patches from the underlying graph, yielding highly accurate subgraph representations. SubGNN specifies three channels, each designed to capture a distinct aspect of subgraph topology, and we provide empirical evidence that the channels encode their intended properties. We design a series of new synthetic and real-world subgraph datasets. Empirical results for subgraph classification on eight datasets show that SubGNN achieves considerable performance gains, outperforming strong baseline methods, including node-level and graph-level GNNs, by 19.8% over the strongest baseline. SubGNN performs exceptionally well on challenging biomedical datasets where subgraphs have complex topology and even comprise multiple disconnected components. 

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5. Wide and deep neural networks achieve consistency for classification

   https://doi.org/10.1073/pnas.2208779120  

   Radhakrishnan, Adityanarayanan ; Belkin, Mikhail ; Uhler, Caroline ( April 2023 , Proceedings of the National Academy of Sciences)

   While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are consistent for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that are consistent. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and neural tangent kernels, we provide explicit activation functions that can be used to construct networks that achieve consistency. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: 1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); 2) majority vote (model predictions are given by the label of the class with the greatest representation in the training set); or 3) singular kernel classifiers (a set of classifiers containing those that achieve consistency). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful. 

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