  Polynomial graph filters have been widely used as guiding principles in the
design of Graph Neural Networks (GNNs). Recently, the adaptive learning of the
polynomial graph filters has demonstrated promising performance for modeling
graph signals on both homophilic and heterophilic graphs, owning to their
flexibility and expressiveness. In this work, we conduct a novel preliminary
study to explore the potential and limitations of polynomial graph filter
learning approaches, revealing a severe overfitting issue. To improve the
effectiveness of polynomial graph filters, we propose Auto-Polynomial, a novel
and general automated polynomial graph filter learning framework that
efficiently learns better filters capable of adapting to various complex graph
signals. Comprehensive experiments and ablation studies demonstrate significant
and consistent performance improvements on both homophilic and heterophilic
graphs across multiple learning settings considering various labeling ratios,
which unleashes the potential of polynomial filter learning.
