Gradient Estimation Based On Probability Flow

Variational inference is a machine learning technique aiming to approximate the posterior distribution in probabilistic models.Reparameterization trick is currently the most widely used gradient estimation technique for variational inference;however,this optimization trick can only be applied to a small class of parametric distribution,such as the location-scale probability distribution family.Although there are lines of research trying to extend the reparameterization trick to a larger class of variational distributions such as Gamma or Dirichlet distribution family,these methods cannot generalize to multivariate distributions trivially.This thesis mainly studies the variational optimization problem of multivariate variational distribution,and focuses on the connection between the multivariate variational distribution and fluid motion.Here are the three main contributions:(1)Proposes the probability flow gradient model by comparing the variational distribution to the fluid motion,analyses the reparameterization trick with the proposed model,and discovers that the reparameterization trick is not dependent on the standardization transformation since all the standardization transformations lead to the same velocity field in the view of probability flow.This thesis proves that the velocity field obtained by the reparameterization trick is a special solution to the continuity equation;(2)Proposes three different expressions of the variational gradient,that is the zero-flux gradient,modified total gradient,and flow increment gradient,and analyses the connection between the norm of flow increment or velocity field and the gradient estimation variance.This thesis proves that the gradient estimation variance increases as the L2 norm of the flow increment or velocity field grows,finally to the infinitely great;(3)Proposes a polynomial-based probability flow gradient estimator based on the zero-flux gradient and justifies this estimator for distributions that admit a factorization.By experiments with the proposed gradient estimator on synthetic and real datasets,this theis shows that the gradient estimation based on the probability flow is effective and feasible.The proposed probability flow gradient model would contribute to the research of variational optimization and support the application of variational inference technique.