\section{LIMITATIONS and BROADER IMPACTS}
The results of this paper are primarily theoretical and ideally a suggestion towards what practical guarantees can and cannot be inferred from the output of metric learning algorithms. When attempting to apply guarantees to real world settings, special care must be taken to ensure that the assumptions of the theory match the practical situation to which they are being applied. Indeed, any assumptions taken in this work may be reasonably considered a limitation depending on the practitioner’s desired application, though we hope that this work serves more as a guide for research and that its assumptions can be altered and its results can be reshown in new settings of practical interest. 

Finally, we remark on the broader impacts of this work. While this paper is primarily theoretical, it is worth considering the impacts of the problems to which we are providing theory. Metric learning has recently been adopted as a model for facial recognition which may lead to negative sociopolitical externalities. Preference learning allows for algorithms to more adeptly specialize to users’ preferences in recommendation and this may cause adverse effects owing to what content is shown to users.
