Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

Abstract

Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert's metric originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.

Numerical methods for model-free pricing in finance, optimal transport, and cyber risk management

“…These methods typically involve solving an infinite-dimensional optimization problem parametrized by deep neural networks; see, e.g., , Eckstein and Kupper, 2021, De Gennaro Aquino and Bernard, 2020, De Gennaro Aquino and Eckstein, 2020, Henry-Labordère, 2019. See also [Cuturi, 2013, Nutz and Wiesel, 2021, Eckstein and Nutz, 2022 for the theoretical properties of entropic regularization and the Sinkhorn algorithm. One downside of these methods is the challenge posed by the non-convexity of the objective function when training these neural networks, and there is hence no theoretical guarantee on the quality of the approximate solutions represented by trained neural networks.…”

“…Most notably, Cuturi [2013] proposed to use entropic regularization and the Sinkhorn algorithm for solving the optimal transport problem (i.e., when 𝑁 = 2). See also [Nutz and Wiesel, 2021] and [Eckstein and Nutz, 2022] for the theoretical properties of entropic regularization and the Sinkhorn algorithm. While most regularization-based approaches deal with discrete marginals, see, e.g., [Benamou et al, 2019, Peyré and Cuturi, 2019, Tupitsa et al, 2020, there are also regularization-based approaches for solving MMOT problems with non-discrete marginals.…”

Statistical analysis of the optimal transport problem

“…Generalizations and extensions for discrete measures have been proved by ; . A growing body of work investigates the properties of the entropy regularized optimal transport problem from the perspective of probability and analysis, including its asymptotic properties as ϵ → 0 Nutz and Wiesel (2021); Eckstein and Nutz (2021); Ghosal et al (2021); Nutz and Wiesel (2022); Altschuler et al (2022); Berman (2020); , opening the door to further statistical applications of entropy regularised transport.…”