Bridging the Gap Between Variational Inference and Wasserstein Gradient Flows

Variational inference is a technique that approximates a target distribution by optimizing within the parameter space of variational families. On the other hand, Wasserstein gradient flows describe optimization within the space of probability measures where they do not necessarily admit a parametric density function. In this paper, we bridge the gap between these two methods. We demonstrate that, under certain conditions, the Bures-Wasserstein gradient flow can be recast as the Euclidean gradient flow where its forward Euler scheme is the standard black-box variational inference algorithm. Specifically, the vector field of the gradient flow is generated via the path-derivative gradient estimator. We also offer an alternative perspective on the path-derivative gradient, framing it as a distillation procedure to the Wasserstein gradient flow. Distillations can be extended to encompass $f$-divergences and non-Gaussian variational families. This extension yields a new gradient estimator for $f$-divergences, readily implementable using contemporary machine learning libraries like PyTorch or TensorFlow

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.Comment: 34 pages, 8 figure

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.Comment: 26 pages. significant revision from the previous versio