Weisfeiler and Leman Go Gambling: Why Expressive Lottery Tickets Win

TL;DR

This paper provides a theoretical analysis of the lottery ticket hypothesis for graph neural networks, emphasizing the importance of expressivity in sparse subnetworks for graph distinction and model performance.

Contribution

It establishes conditions under which sparse GNNs match the expressivity of full networks, introducing a Strong Expressive Lottery Ticket Hypothesis with theoretical proofs.

Findings

Expressivity of sparse subnetworks can match full networks under certain conditions.

Increased initialization expressivity accelerates convergence and enhances generalization.

Theoretical foundations for GNN lottery tickets and their expressivity are established.

Abstract

The lottery ticket hypothesis (LTH) is well-studied for convolutional neural networks but has been validated only empirically for graph neural networks (GNNs), for which theoretical findings are largely lacking. In this paper, we identify the expressivity of sparse subnetworks, i.e. their ability to distinguish non-isomorphic graphs, as crucial for finding winning tickets that preserve the predictive performance. We establish conditions under which the expressivity of a sparsely initialized GNN matches that of the full network, particularly when compared to the Weisfeiler-Leman test, and in that context put forward and prove a Strong Expressive Lottery Ticket Hypothesis. We subsequently show that an increased expressivity in the initialization potentially accelerates model convergence and improves generalization. Our findings establish novel theoretical foundations for both LTH and GNN research, highlighting the importance of maintaining expressivity in sparsely initialized GNNs. We illustrate our results using examples from drug discovery.