Asymmetric Learning for Spectral Graph Neural Networks

Optimizing spectral graph neural networks (GNNs) remains a critical challenge in the field, yet the underlying processes are not well understood. In this paper, we investigate the inherent differences between graph convolution parameters and feature transformation parameters in spectral GNNs and their impact on the optimization landscape. Our analysis reveals that these differences contribute to a poorly conditioned problem, resulting in suboptimal performance. To address this issue, we introduce the concept of the block condition number of the Hessian matrix, which characterizes the difficulty of poorly conditioned problems in spectral GNN optimization. We then propose an asymmetric learning approach, dynamically preconditioning gradients during training to alleviate poorly conditioned problems. Theoretically, we demonstrate that asymmetric learning can reduce block condition numbers, facilitating easier optimization. Extensive experiments on eighteen benchmark datasets show that asymmetric learning consistently improves the performance of spectral GNNs for both heterophilic and homophilic graphs. This improvement is especially notable for heterophilic graphs, where the optimization process is generally more complex than for homophilic graphs. Code is available at https://github.com/Mia-321/asym-opt.git.