This paper is about structures output predictions analysis under 'fairness' constraint. This paper shows that constraints relative to fairness can help to increases accuracies. Fairness is one of the notion whose importance is rising in our community, and this paper give interesting insights about it. One of the main issue raised by one of the reviewer that pleads for non acceptance is the "vagueness" of the definition of fairness here. I personally think that this issue should not be taken to much into account here, there is still in our community some "vagueness" according to what the good definition should be. Another point that has been raised by one of the reviewers is "The idea that fairness improves the bounds is not counter intuitive AS CLAIMED IN THE INTRODUCTION: it is clear that the fairness assumption which applies to both the ground truth and the function search space reduces the complexity of the problem. I agree with the reviewer that adding constraint reduces complexity, but it is nevertheless an important result to see how it can be captured by a bound. So overall, a nice paper on an important ML issue, the fairness ... clear acceptance.