As an effective way to learn a distance metric between pairs of samples, deep metric learning (DML) has drawn significant attention in recent years. The key idea of DML is to learn a set of hierarchical nonlinear mappings using deep neural networks, and then project the data samples into a new feature space for comparing or matching. Although DML has achieved practical success in many applications, there is no existing work that theoretically analyzes the generalization error bound for DML, which can measure how good a learned DML model is able to perform on unseen data. In this paper, we try to fill up this research gap and derive the generalization error bound for DML. Additionally, based on the derived generalization bound, we propose a novel DML method (called ADroDML), which can adaptively learn the retention rates for the DML models with dropout in a theoretically justified way. Compared with existing DML works that require predefined retention rates, ADroDML can learn the retention rates in an optimal way and achieve better performance. We also conduct experiments on real-world datasets to verify the findings derived from the generalization error bound and demonstrate the effectiveness of the proposed adaptive DML method.