Graph-based machine learning, encompassing Graph Edit Distances (GEDs), Graph Kernels, and Graph Neural Networks (GNNs), offers extensive capabilities and exciting potential. While each model possesses unique strengths for graph challenges, interrelations between their underlying spaces remain under-explored. In this paper, we introduce a novel framework for bridging these distinct spaces via GED cost learning. A supervised metric learning approach serves as an instance of this framework, enabling space alignment through pairwise distances and the optimization of edit costs. Experiments reveal the framework's potential for enhancing varied tasks, including regression, classification, and graph generation, heralding new possibilities in these fields.