Graph Edit Distance (GED) is a commonly used method for measuring graph similarity and is often applied in graph search and retrieval tasks, like identifying the chemical compound most similar to a target one. However, exact GED computation is NP-complete. Traditional A* algorithms face scalability challenges due to their exhaustive nature, while learning-based methods cannot recover the edit path. To address this, we present UGED, an Unsupervised Learning (UL) framework that integrates the interpretability of conventional search techniques for edit path generation. The Graph Neural Networks (GNNs) produce a probability matrix that shows the likelihood of each node pair being part of the optimal matching. Following this, we conduct a local search based on the probability matrix to generate the final prediction. Experimental results across various graph datasets demonstrate that our approach can considerably simplify the local search process with minimal impact on accuracy.