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.

翻译：优化谱图神经网络（GNNs）仍是该领域的一个关键挑战，但其内在过程尚未得到充分理解。本文研究了谱图神经网络中图卷积参数与特征变换参数之间的固有差异及其对优化地形的影响。我们的分析表明，这些差异导致了病态问题，从而造成次优性能。为解决此问题，我们引入了Hessian矩阵的块条件数概念，用以刻画谱图神经网络优化中病态问题的困难程度。随后，我们提出了一种非对称学习方法，通过在训练过程中动态预处理梯度来缓解病态问题。理论上，我们证明了非对称学习能够降低块条件数，从而简化优化过程。在十八个基准数据集上的大量实验表明，非对称学习方法能持续提升谱图神经网络在同配图和异配图上的性能。这一改进在异配图上尤为显著，因为异配图的优化过程通常比同配图更为复杂。代码可在 https://github.com/Mia-321/asym-opt.git 获取。