Supplementary Material for " A Unified Approach to Fair Online Learning via Blackwell Approachability " by Evgenii Chzhen, Christophe Giraud, Gilles Stoltz

This supplementary material contains all the proofs omitted from the main body. We now move to the proof of Theorem 1, which we restate below. Assume that the condition in Eq. (2) is not satisfied, then (q Then, for any strategy of the player, it holds by martingale convergence (e.g., by the Hoeffding-Azuma Note that in this part we did not use that the target set C was convex, only that it was a closed set. We substitute the bound from Eq. A.1 Two lemmas used in the proof of Theorem 1 We use the less famous one, for non-negative super-martingales. We first detail the two counter-examples alluded at in the proof of Proposition 1, relative to Example 2 on group-wise no-regret.