WIAS Preprint No. 2819, (2021)

Stein variational gradient descent: Many-particle and long-time asymptotics


  • Nüsken, Nikolas
    ORCID: 0000-0001-5415-5284
  • Renger, D. R. Michiel
    ORCID: 0000-0003-3557-3485

2020 Mathematics Subject Classification

  • 68T09 68T09 60F10


  • Stein variational gradient descent, gradient flows, large deviations




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: emphvariational inference and emphMarkov 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 emphStein-Fisher information (or emphkernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $Gamma$-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.

Appeared in

  • Foundations of Data Science, 5:3 (2023), pp. 286--320.

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