Convergence of McKean-Vlasov processes and Markov Chain Monte Carlo methods for mean-field models

Abstract

In recent years, analysing the long-time behaviour of stochastic processes has received increasing interest. Firstly, efficient sampling of a given probability measure is an important task that arises in various fields such as Bayesian statistics or computational physics. Markov Chain Monte Carlo (MCMC) algorithms form a powerful class of sampling methods for which guarantees for fast mixing are of particular interest, especially for intractable target measures. Secondly, one would like to better understand the convergence behaviour of stochastic processes which have their origin in modelling phenomena in physics and are used in deep learning, among others. <br /> In this thesis, we focus on specific high-dimensional problems. We are interested in sampling target measures of mean-field particle type consisting of a unary potential that is in general not strongly convex and of a pairwise interaction potential. Correspondingly, we consider a system of many particles moving according to an external confining force and a pairwise interaction force. Further, we address the connection between processes of mean-field particle type and their corresponding McKean-Vlasov process, where only one particle is considered and whose moves are determined by a nonlinear stochastic differential equation (SDE) with an external force and a distribution-dependent interaction force. We are interested in quantitative estimates between the laws of these two types of processes. <br /> The thesis covers three projects. In the first part, we analyse the behaviour of the unadjusted Hamiltonian Monte Carlo (uHMC) algorithm which forms an MCMC method that samples approximately a given target measure. For a target measure of mean-field type, contraction in Wasserstein distance with dimension-free rates is established under certain conditions on the unary part and the interaction part of the mean-field potential. Furthermore, error estimates between the target measure and the measure sampled by uHMC are provided. <br /> In the second part, we investigate nonlinear stochastic differential equations without confinement and their corresponding mean-field particle systems. To show contraction in Wasserstein distance, the so-called sticky coupling is established for nonlinear SDEs and a novel class of nonlinear one-dimensional SDEs with a sticky boundary behaviour at zero is introduced. For these equations, existence and uniqueness of a weak solution are proven and a phase transition from a unique to several invariant probability measures is analysed. Provided a unique invariant probability measure exists and contraction towards this measure holds, we deduce contraction in Wasserstein distance for the nonlinear SDE without confinement. Further, we establish uniform in time propagation of chaos estimates for the corresponding particle system. <br /> In the final part, we study the long-time behaviour of diffusions given by the second-order Langevin dynamics with distribution-dependent forces. Global contraction in Wasserstein distance with dimension-free rates is shown via a coupling approach and a carefully constructed distance function. In addition, we analyse the optimal order of the contraction rates for the classical second-order Langevin dynamics with a strongly convex potential. Finally, we provide uniform in time propagation of chaos bounds for the corresponding mean-field particle system.