On Sampling with Approximate Transport Maps

Louis Grenioux, Alain Oliviero Durmus, Eric Moulines, Marylou Gabrié
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:11698-11733, 2023.

Abstract

Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-grenioux23a, title = {On Sampling with Approximate Transport Maps}, author = {Grenioux, Louis and Oliviero Durmus, Alain and Moulines, Eric and Gabri\'{e}, Marylou}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {11698--11733}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/grenioux23a/grenioux23a.pdf}, url = {https://proceedings.mlr.press/v202/grenioux23a.html}, abstract = {Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.} }
Endnote
%0 Conference Paper %T On Sampling with Approximate Transport Maps %A Louis Grenioux %A Alain Oliviero Durmus %A Eric Moulines %A Marylou Gabrié %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-grenioux23a %I PMLR %P 11698--11733 %U https://proceedings.mlr.press/v202/grenioux23a.html %V 202 %X Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.
APA
Grenioux, L., Oliviero Durmus, A., Moulines, E. & Gabrié, M.. (2023). On Sampling with Approximate Transport Maps. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:11698-11733 Available from https://proceedings.mlr.press/v202/grenioux23a.html.

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