## Abstract

This work presents a multilevel variant of Stein variational gradient descent to more efficiently sample from target distributions. The key ingredient is a sequence of distributions with growing fidelity and costs that converges to the target distribution of interest. For example, such a sequence of distributions is given by a hierarchy of ever finer discretization levels of the forward model in Bayesian inverse problems. The proposed multilevel Stein variational gradient descent moves most of the iterations to lower, cheaper levels with the aim of requiring only a few iterations on the higher, more expensive levels when compared to the traditional, single-level Stein variational gradient descent variant that uses the highest-level distribution only. Under certain assumptions, in the mean-field limit, the error of the proposed multilevel Stein method decays by a log factor faster than the error of the single-level counterpart with respect to computational costs. Numerical experiments with Bayesian inverse problems show speedups of more than one order of magnitude of the proposed multilevel Stein method compared to the single-level variant that uses the highest level only.

Original language | English (US) |
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Pages (from-to) | 93-117 |

Number of pages | 25 |

Journal | Proceedings of Machine Learning Research |

Volume | 145 |

State | Published - 2021 |

Event | 2nd Mathematical and Scientific Machine Learning Conference, MSML 2021 - Virtual, Online Duration: Aug 16 2021 → Aug 19 2021 |

## Keywords

- Bayesian inference
- Monte Carlo
- multilevel and multifidelity
- particle methods

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability