Levi Rauchwerger

Levi Rauchwerger, Stefanie Jegelka, Ron Levie

We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on the weight values, we show that dense neural networks do not possess this universality. Our argument is based on a model compression approach, combining the weak regularity lemma with an interpretation of feedforward networks as message passing graph neural networks. We consider ReLU neural networks subject to natural constraints on weights and input and output dimensions, which model a notion of dense connectivity. Within this setting, we demonstrate the existence of Lipschitz continuous functions that cannot be approximated by such networks. This highlights intrinsic limitations of neural networks with dense layers and motivates the use of sparse connectivity as a necessary ingredient for achieving true universality.

Levi Rauchwerger, Stefanie Jegelka, Ron Levie

We analyze the universality and generalization of graph neural networks (GNNs) on attributed graphs, i.e., with node attributes. To this end, we propose pseudometrics over the space of all attributed graphs that describe the fine-grained expressivity of GNNs. Namely, GNNs are both Lipschitz continuous with respect to our pseudometrics and can separate attributed graphs that are distant in the metric. Moreover, we prove that the space of all attributed graphs is relatively compact with respect to our metrics. Based on these properties, we prove a universal approximation theorem for GNNs and generalization bounds for GNNs on any data distribution of attributed graphs. The proposed metrics compute the similarity between the structures of attributed graphs via a hierarchical optimal transport between computation trees. Our work extends and unites previous approaches which either derived theory only for graphs with no attributes, derived compact metrics under which GNNs are continuous but without separation power, or derived metrics under which GNNs are continuous and separate points but the space of graphs is not relatively compact, which prevents universal approximation and generalization analysis.

Levi Rauchwerger, Ron Levie

A recent paper, ``A Graphon-Signal Analysis of Graph Neural Networks'', by Levie, analyzed message passing graph neural networks (MPNNs) by embedding the input space of MPNNs, i.e., attributed graphs (graph-signals), to a space of attributed graphons (graphon-signals). Based on extensions of standard results in graphon analysis to graphon-signals, the paper proved a generalization bound and a sampling lemma for MPNNs. However, there are some missing ingredients in that paper, limiting its applicability in practical settings of graph machine learning. In the current paper, we introduce several refinements and extensions to existing results that address these shortcomings. In detail, 1) we extend the main results in the paper to graphon-signals with multidimensional signals (rather than 1D signals), 2) we extend the Lipschitz continuity to MPNNs with readout with respect to cut distance (rather than MPNNs without readout with respect to cut metric), 3) we improve the generalization bound by utilizing robustness-type generalization bounds, and 4) we extend the analysis to non-symmetric graphons and kernels.