Multilevel Stein variational gradient descent with applications to Bayesian inverse problems

Terrence Alsup, Luca Venturi, Benjamin Peherstorfer
Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, PMLR 145:93-117, 2022.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v145-alsup22a, title = {Multilevel Stein variational gradient descent with applications to Bayesian inverse problems}, author = {Alsup, Terrence and Venturi, Luca and Peherstorfer, Benjamin}, booktitle = {Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference}, pages = {93--117}, year = {2022}, editor = {Bruna, Joan and Hesthaven, Jan and Zdeborova, Lenka}, volume = {145}, series = {Proceedings of Machine Learning Research}, month = {16--19 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v145/alsup22a/alsup22a.pdf}, url = {https://proceedings.mlr.press/v145/alsup22a.html}, 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.} }
Endnote
%0 Conference Paper %T Multilevel Stein variational gradient descent with applications to Bayesian inverse problems %A Terrence Alsup %A Luca Venturi %A Benjamin Peherstorfer %B Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference %C Proceedings of Machine Learning Research %D 2022 %E Joan Bruna %E Jan Hesthaven %E Lenka Zdeborova %F pmlr-v145-alsup22a %I PMLR %P 93--117 %U https://proceedings.mlr.press/v145/alsup22a.html %V 145 %X 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.
APA
Alsup, T., Venturi, L. & Peherstorfer, B.. (2022). Multilevel Stein variational gradient descent with applications to Bayesian inverse problems. Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, in Proceedings of Machine Learning Research 145:93-117 Available from https://proceedings.mlr.press/v145/alsup22a.html.

Related Material