Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice models

Multilevel Stein variational gradient descent is a method for particle-based variational inference that leverages hierarchies of approximations of target distributions with varying costs and fidelity to computationally speed up inference. This work provides a cost complexity analysis of multilevel Stein variational gradient descent that applies under milder conditions than previous results, especially in discrete-in-time regimes and beyond the limited settings where Stein variational gradient descent achieves exponentially fast convergence. The analysis shows that the convergence rate of Stein variational gradient descent enters only as a constant factor for the cost complexity of the multilevel version, which means that the costs of the multilevel version scale independently of the convergence rate of Stein variational gradient descent on a single level. Numerical experiments with Bayesian inverse problems of inferring discretized basal sliding coefficient fields of the Arolla glacier ice demonstrate that multilevel Stein variational gradient descent achieves orders of magnitude speedups compared to its single-level version.