On the geometry of Stein variational gradient descent

TY - JOUR

T1 - On the geometry of Stein variational gradient descent

AU - Duncan, Andrew

AU - Szpruch, Lukasz

AU - Nusken, Nikolas

PY - 2023/1/1

Y1 - 2023/1/1

N2 - Bayesian inference problems require sampling or approximating high-dimensional probability dis- tributions. The focus of this paper is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm. This construction leads to interacting particle systems, the mean-field limit of which is a gradient flow on the space of probability distributions equipped with a certain geometrical structure. We leverage this viewpoint to shed some light on the convergence properties of the algorithm, in particular addressing the problem of choosing a suitable positive definite kernel function. Our analysis leads us to considering certain nondifferentiable kernels with adjusted tails. We demonstrate significant performance gains of these in various numerical experiments.

AB - Bayesian inference problems require sampling or approximating high-dimensional probability dis- tributions. The focus of this paper is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm. This construction leads to interacting particle systems, the mean-field limit of which is a gradient flow on the space of probability distributions equipped with a certain geometrical structure. We leverage this viewpoint to shed some light on the convergence properties of the algorithm, in particular addressing the problem of choosing a suitable positive definite kernel function. Our analysis leads us to considering certain nondifferentiable kernels with adjusted tails. We demonstrate significant performance gains of these in various numerical experiments.

M3 - Article

SN - 1532-4435

VL - 24

SP - 1

EP - 39

JO - JOURNAL OF MACHINE LEARNING RESEARCH

JF - JOURNAL OF MACHINE LEARNING RESEARCH

IS - 56

ER -