Functional Gradient Flows for Constrained Sampling

TL;DR

This paper introduces a new constrained functional gradient flow method for sampling within specific domains, extending particle-based variational inference to handle domain constraints with proven convergence.

Contribution

It proposes a novel constrained functional gradient flow (CFG) method for sampling, incorporating boundary conditions and boundary integral handling, with theoretical convergence guarantees.

Findings

The CFG method effectively samples within constrained domains.

The approach demonstrates convergence in total variation.

Experimental results validate the method's effectiveness.

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.