On the Linear Convergence of SISTA for Entropic Martingale Optimal Transport

Abstract

We prove a linear convergence rate of a Sinkhorn-type algorithm for entropic martingale optimal transport in arbitrary dimension. The algorithm follows a block-coordinate ascent scheme, alternating exact Sinkhorn updates for the marginal potentials with an explicit gradient step for the potential associated with the martingale constraint. Under irreducibility and non-degeneracy assumptions on the compactly supported marginals, we establish linear convergence of the dual objective values and iterates. Our analysis relies on a pointwise coercivity estimate involving the gradient of the dual functional with respect to the potential for the martingale constraint. This property allows us to establish uniform $L^{\infty}$ bounds on the iterates. Combining these bounds with smoothness and strong convexity modulo affine invariances yields a uniform Polyak--{\L}ojasiewicz inequality along the iterations and, consequently, a linear convergence rate. We complement our theoretical results with numerical experiments illustrating the convergence behavior of the algorithm.