The authors address an important area of study, aiming to ensure fairness beyond the training data by optimizing a worst case fairness loss across any weighted combination of the training set. They show that such fairness robustness comes at the cost of lower accuracy. Please add the material from the rebuttal and incorporate the reviewers' detailed comments. This includes: Add all experimental and hyperparameter details Improve clarity of writing and notation Add brief explanations of methods and proofs in the main body