Functional Gradient Flows for Constrained Sampling

Shiyue Zhang, Longlin Yu, Ziheng Cheng, Cheng Zhang

Keywords: particle-based variational inference, constrained sampling, functional gradient flow, boundary integral

TL;DR: A new functional gradient particle-based variational inference method for sampling on constrained domains.

Abstract: Recently, through a unified gradient flow perspective of Markov chain Monte Carlo (MCMC) and variational inference (VI), particle-based variational inference methods (ParVIs) have been proposed that tend to combine the best of both worlds. While typical ParVIs such as Stein Variational Gradient Descent (SVGD) approximate the gradient flow within a reproducing kernel Hilbert space (RKHS), many attempts have been made recently to replace RKHS with more expressive function spaces, such as neural networks. While successful, these methods are mainly designed for sampling from unconstrained domains. In this paper, we offer a general solution to constrained sampling by introducing a boundary condition for the gradient flow which would confine the particles within the specific domain. This allows us to propose a new functional gradient ParVI method for constrained sampling, called *constrained functional gradient flow* (CFG), with provable continuous-time convergence in total variation (TV). We also present novel numerical strategies to handle the boundary integral term arising from the domain constraints. Our theory and experiments demonstrate the effectiveness of the proposed framework.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 6116