Particle-based Variational Inference with Preconditioned Functional Gradient Flow

Particle-based variational inference (VI) minimizes the KL divergence between model samples and the target posterior with gradient flow estimates. With the popularity of Stein variational gradient descent (SVGD), the focus of particle-based VI algorithms has been on the properties of functions in Reproducing Kernel Hilbert Space (RKHS) to approximate the gradient flow. However, the requirement of RKHS restricts the function class and algorithmic flexibility. This paper remedies the problem by proposing a general framework to obtain tractable functional gradient flow estimates. The functional gradient flow in our framework can be defined by a general functional regularization term that includes the RKHS norm as a special case. We use our framework to propose a new particle-based VI algorithm: preconditioned functional gradient flow (PFG). Compared with SVGD, the proposed method has several advantages: larger function class; greater scalability in large particle-size scenarios; better adaptation to ill-conditioned distributions; provable continuous-time convergence in KL divergence. Non-linear function classes such as neural networks can be incorporated to estimate the gradient flow. Both theory and experiments have shown the effectiveness of our framework.