Despite the success of deep-learning models in many tasks, there have been concerns about such models learning shortcuts, and their lack of robustness to irrelevant confounders. When it comes to models directly trained on human faces, a sensitive confounder is that of human identities. Many face-related tasks should ideally be identity-independent, and perform uniformly across different individuals (i.e. be fair). One way to measure and enforce such robustness and performance uniformity is through enforcing it during training, assuming identity-related information is available at scale. However, due to privacy concerns and also the cost of collecting such information, this is often not the case, and most face datasets simply contain input images and their corresponding task-related labels. Thus, improving identity-related robustness without the need for such annotations is of great importance. Here, we explore using face-recognition embedding vectors, as proxies for identities, to enforce such robustness. We propose to use the structure in the face-recognition embedding space, to implicitly emphasize rare samples within each class. We do so by weighting samples according to their conditional inverse density (CID) in the proxy embedding space. Our experiments suggest that such a simple sample weighting scheme, not only improves the training robustness, it often improves the overall performance as a result of such robustness. We also show that employing such constraints during training results in models that are significantly less sensitive to different levels of bias in the dataset.