Title: On Learning with Fairness Trade-Offs. Abstract: Previous literature has shown that bias mitigating algorithms were sometimes prone to overfitting and had poor out-of-sample generalisation. This paper is first and foremost concerned with establishing a mathematical framework to tackle the specific issue of generalisation. Throughout this work, we consider fairness trade-offs and objectives mixing statistical loss over the whole sample and fairness penalties on categories (which could stem from different values of protected attributes), encompassing partial de-biasing.  We do so by adopting two different but complementary viewpoints: first, we consider a PAC-type setup and derive probabilistic upper bounds involving sample-only information; second, we leverage an asymptotic framework to derive a closed-form limiting distribution for the difference between the empirical trade-off and the true trade-off. While these results provide guarantees for learning fairness metrics across categories, they also point out to the key (but asymmetric) role played by class imbalance. To summarise, learning fairness without having access to enough category-level samples is hard, and a simple numerical experiment shows that it can lead to spurious results. 