Supervision for metric learning has long been given in the form of equivalence between human-labeled classes. Although this type of supervision has been a basis of metric learning for decades, we argue that it hinders further advances in the field. In this regard, we propose a new regularization method, dubbed HIER, to discover the latent semantic hierarchy of training data, and to deploy the hierarchy to provide richer and more fine-grained supervision than inter-class separability induced by common metric learning losses. HIER achieves this goal with no annotation for the semantic hierarchy but by learning hierarchical proxies in hyperbolic spaces. The hierarchical proxies are learnable parameters, and each of them is trained to serve as an ancestor of a group of data or other proxies to approximate the semantic hierarchy among them. HIER deals with the proxies along with data in hyperbolic space since the geometric properties of the space are well-suited to represent their hierarchical structure. The efficacy of HIER is evaluated on four standard benchmarks, where it consistently improved the performance of conventional methods when integrated with them, and consequently achieved the best records, surpassing even the existing hyperbolic metric learning technique, in almost all settings.