Graph Neural Networks (GNNs), neural network architectures targeted to learning representations of graphs, have become a popular learning model for prediction tasks on nodes, graphs and configurations of points, with wide success in practice. This article summarizes a selection of the emerging theoretical results on approximation and learning properties of widely used message passing GNNs and higher-order GNNs, focusing on representation, generalization and extrapolation. Along the way, it summarizes mathematical connections.
Many machine learning tasks involve processing eigenvectors derived from data. Especially valuable are Laplacian eigenvectors, which capture useful structural information about graphs and other geometric objects. However, ambiguities arise when computing eigenvectors: for each eigenvector $v$, the sign flipped $-v$ is also an eigenvector. More generally, higher dimensional eigenspaces contain infinitely many choices of basis eigenvectors. These ambiguities make it a challenge to process eigenvectors and eigenspaces in a consistent way. In this work we introduce SignNet and BasisNet -- new neural architectures that are invariant to all requisite symmetries and hence process collections of eigenspaces in a principled manner. Our networks are universal, i.e., they can approximate any continuous function of eigenvectors with the proper invariances. They are also theoretically strong for graph representation learning -- they can approximate any spectral graph convolution, can compute spectral invariants that go beyond message passing neural networks, and can provably simulate previously proposed graph positional encodings. Experiments show the strength of our networks for molecular graph regression, learning expressive graph representations, and learning implicit neural representations on triangle meshes. Our code is available at https://github.com/cptq/SignNet-BasisNet .
The implicit bias induced by the training of neural networks has become a topic of rigorous study. In the limit of gradient flow and gradient descent with appropriate step size, it has been shown that when one trains a deep linear network with logistic or exponential loss on linearly separable data, the weights converge to rank-$1$ matrices. In this paper, we extend this theoretical result to the much wider class of nonlinear ReLU-activated feedforward networks containing fully-connected layers and skip connections. To the best of our knowledge, this is the first time a low-rank phenomenon is proven rigorously for these architectures, and it reflects empirical results in the literature. The proof relies on specific local training invariances, sometimes referred to as alignment, which we show to hold for a wide set of ReLU architectures. Our proof relies on a specific decomposition of the network into a multilinear function and another ReLU network whose weights are constant under a certain parameter directional convergence.
Contrastive learning relies on an assumption that positive pairs contain related views, e.g., patches of an image or co-occurring multimodal signals of a video, that share certain underlying information about an instance. But what if this assumption is violated? The literature suggests that contrastive learning produces suboptimal representations in the presence of noisy views, e.g., false positive pairs with no apparent shared information. In this work, we propose a new contrastive loss function that is robust against noisy views. We provide rigorous theoretical justifications by showing connections to robust symmetric losses for noisy binary classification and by establishing a new contrastive bound for mutual information maximization based on the Wasserstein distance measure. The proposed loss is completely modality-agnostic and a simple drop-in replacement for the InfoNCE loss, which makes it easy to apply to existing contrastive frameworks. We show that our approach provides consistent improvements over the state-of-the-art on image, video, and graph contrastive learning benchmarks that exhibit a variety of real-world noise patterns.
Modeling the time evolution of discrete sets of items (e.g., genetic mutations) is a fundamental problem in many biomedical applications. We approach this problem through the lens of continuous-time Markov chains, and show that the resulting learning task is generally underspecified in the usual setting of cross-sectional data. We explore a perhaps surprising remedy: including a number of additional independent items can help determine time order, and hence resolve underspecification. This is in sharp contrast to the common practice of limiting the analysis to a small subset of relevant items, which is followed largely due to poor scaling of existing methods. To put our theoretical insight into practice, we develop an approximate likelihood maximization method for learning continuous-time Markov chains, which can scale to hundreds of items and is orders of magnitude faster than previous methods. We demonstrate the effectiveness of our approach on synthetic and real cancer data.
The generalization of representations learned via contrastive learning depends crucially on what features of the data are extracted. However, we observe that the contrastive loss does not always sufficiently guide which features are extracted, a behavior that can negatively impact the performance on downstream tasks via "shortcuts", i.e., by inadvertently suppressing important predictive features. We find that feature extraction is influenced by the difficulty of the so-called instance discrimination task (i.e., the task of discriminating pairs of similar points from pairs of dissimilar ones). Although harder pairs improve the representation of some features, the improvement comes at the cost of suppressing previously well represented features. In response, we propose implicit feature modification (IFM), a method for altering positive and negative samples in order to guide contrastive models towards capturing a wider variety of predictive features. Empirically, we observe that IFM reduces feature suppression, and as a result improves performance on vision and medical imaging tasks. The code is available at: \url{https://github.com/joshr17/IFM}.
This work explores the hypothesis that the complexity of the function a deep neural network (NN) is learning can be deduced by how fast its weights change during training. Our analysis provides evidence for this supposition by relating the network's distribution of Lipschitz constants (i.e., the norm of the gradient at different regions of the input space) during different training intervals with the behavior of the stochastic training procedure. We first observe that the average Lipschitz constant close to the training data affects various aspects of the parameter trajectory, with more complex networks having a longer trajectory, bigger variance, and often veering further from their initialization. We then show that NNs whose biases are trained more steadily have bounded complexity even in regions of the input space that are far from any training point. Finally, we find that steady training with Dropout implies a training- and data-dependent generalization bound that grows poly-logarithmically with the number of parameters. Overall, our results support the hypothesis that good training behavior can be a useful bias towards good generalization.
Understanding the generalization of deep neural networks is one of the most important tasks in deep learning. Although much progress has been made, theoretical error bounds still often behave disparately from empirical observations. In this work, we develop margin-based generalization bounds, where the margins are normalized with optimal transport costs between independent random subsets sampled from the training distribution. In particular, the optimal transport cost can be interpreted as a generalization of variance which captures the structural properties of the learned feature space. Our bounds robustly predict the generalization error, given training data and network parameters, on large scale datasets. Theoretically, we demonstrate that the concentration and separation of features play crucial roles in generalization, supporting empirical results in the literature. The code is available at \url{https://github.com/chingyaoc/kV-Margin}.
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.