123650dd0560587918b3d771cf0c0171-Supplemental.pdf

This is clarified in the following lemma, extracted from Vialard (2019), Proposition19. We write(ft,gt)t the sequence of iterates. Given a pair of optimal potentials(f?,g?), we writeut, ft f?,vt, gt g?,uTt, Tα(ft) g? andvTt, Tα(gt) f?. It is upper bounded by 1 e Ldiam(X),thankstoAssumption1andLemma1. We writeˆβ = ˆβn to make explicit the dependency of the quantities on the batchsizen.