Stein Variational Gradient Descent: many-particle and long-time asymptotics
This work provides theoretical insights into SVGD, a method for Bayesian inference, which is incremental as it clarifies existing relationships rather than introducing new algorithms.
The paper analyzes Stein Variational Gradient Descent (SVGD) by linking its deterministic and stochastic variants through gradient flow structures and large-deviation principles, revealing that the Stein-Fisher information governs the long-time and many-particle asymptotic behavior of the method.
Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the Stein-Fisher information (or kernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $Γ$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.