's weights define an exponential family.

The fair-ranking problem, which asks to rank a given set of items to maximize utility subject togroup fairness constraints, has received attention inthe fairness, information retrieval, and machine learning literature.

The definition of Sinkhorn divergences usually starts from that oftheground cost onobservations.

Step decay step-size schedules (constant and then cut) are widely used in practice because of their excellent convergence and generalization qualities, but their theoretical properties are not yet well understood. Weprovide convergence results for step decay in the non-convexregime, ensuring that the gradient norm vanishes at an O(lnT/ T)rate.

In the sequel, we empirically show the effect of different numbers of local updates on the fixed point. We consider cases withK = 1, K = 10, K = 20, K = 50. From Assumption 1, it is obvious thatgi(x,y) is convex-concave. Then, we conclude that there exists someη1 > 0 such that h(η) > 0, 0 < η < η1.

These algorithms iterate with first-order oracle and achieve linear convergence.

Infact, bysubstitutingp =1n into Equation (8), weobtain: Proposition 5.For q = ( c,..., c, 0) Rn withc = 2 Δ logn, theexpectederrorE [ E ( MPF, q)] ofpermute-and-flipisatleastΔ2 log ( n).