A Study Of Gradient Flow Induced Particle-based Variational Inference

Recently developed Particle-based Variational Inference(Par VI)methods have draw much attention in distribution approximation literature due to their impressive success in practice,which are commonly adopted to solve many machine learning tasks,such as Bayesian inference and Wasserstein barycenter.These Par VIs evolve a set of fixed-weight particles by simulating the Wasserstein gradient flow of certain dissimilarity functional,which defines the direction of motion of each particle and evolve particles to iteratively approximate the target distribution.However,existing Par VIs have a common fixed-weight restriction,that they all keep the particles’ weights fixed during the whole procedure and only update the positions of particles.This fixed-weight restriction severely confines the approximation ability of particles,and may even induce erroneous results in facing complex multi-modal distributions,which is known as the particle-collapsing phenomenon in Par VI literature.To address the aforementioned challenge,this thesis investigates dynamic-weight particlebased variational inference methods that are able to evolve particles’ positions and weights simultaneously,thereby improving the approximation accuracy of existing fixed-weight Par VIs and avoiding the problem of incorrect approximation results induced by particle-collapsing phenomenon.This thesis analyzes the limitation of the finite-particle simulation of Wasserstein gradient flow in formally adjusting particles’ weights,and then introduce the Fisher-Rao gradient flow into existing Par VI system to design a composite flow that are able to update particles’ positions and weights simultaneously.Based on the designed composite flow,this thesis proposes a Dynamic-weight Particle-based Variational Inference(DPVI)framework,which provides a general guidance for existing fixed-weight Par VIs to formally adjust particles’ weights.This framework improves the flexibility of particles under the condition of little extra computational cost,avoids the particle-collapsing problem of existing fixed-weight Par VIs in the case of complex multi-modal distribution,and effectively reduces the approximation error under the same time complexity.Moreover,this thesis proofs that the mean-field limit of the proposed composite flow is actually the gradient flow of functional F in the Wasserstein-Fisher-Rao space,which leads to a faster decrease of F compared with the Wasserstein gradient flow.And this proof is the first theoretical analysis of dynamic-weight particle-based variational inference methods.Based on the general DPVI framework,this thesis designs five efficient DPVI algorithms in terms of different functionals.Besides,five corresponding duplicate/kill variants are derived with a special probability discretization of weight adjustment.Finally,this thesis designs a variety of machine learning tasks to verify the effectiveness of the proposed methods.