Several interesting and emerging problems in statistics, machine learning and optimal transport can be cast as minimisation of (entropy-regularised) objective functions defined on an appropriate space of probability distributions. Numerical methods have historically focused on linear objective functions, a setting in which one has access to an unnormalised density for the distributional target. For nonlinear objectives, numerical methods are relatively under-developed; for example, mean-field Langevin dynamics is considered state-of-the-art. In the nonlinear setting even basic questions, such as how to tell whether or not a sequence of numerical approximations has practically converged, remain unanswered. Our main contribution is to present the first computable measure of sub-optimality for optimisation in this context.
Joint work with Clémentine Chazal, Heishiro Kanagawa, Zheyang Shen and Anna Korba.