To determine the similarity of labeled graphs, the graph edit distance (GED) is widely used due to its metric properties on the graph space and its interpretability. It is defined as the minimal cost of a sequence of edit operations transforming one graph into another one, with the cost of each edit operation being a parameter of the distance. Although calculating GED is NP-hard, various heuristics exist which, in practice, typically yield tight upper or lower bounds. Since determining appropriate edit operation costs for a given dataset or application can be challenging, it is attractive to learn these costs from the data, e.g., using metric learning architectures. However, for this approach to be feasible, a differentiable algorithm to approximate the GED is required. In this work, we present such an algorithm and show via an empirical evaluation on three datasets that the obtained distances closely match the distances computed by a state-of-the-art combinatorial GED heuristic.