Sampling in Unit Time with Kernel Fisher-Rao Flow

Aimee Maurais, Youssef Marzouk
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:35138-35162, 2024.

Abstract

We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, $\pi_0^{1-t} \pi_1^t$, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Ampère equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-maurais24a, title = {Sampling in Unit Time with Kernel {F}isher-Rao Flow}, author = {Maurais, Aimee and Marzouk, Youssef}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {35138--35162}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/maurais24a/maurais24a.pdf}, url = {https://proceedings.mlr.press/v235/maurais24a.html}, abstract = {We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, $\pi_0^{1-t} \pi_1^t$, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Ampère equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.} }
Endnote
%0 Conference Paper %T Sampling in Unit Time with Kernel Fisher-Rao Flow %A Aimee Maurais %A Youssef Marzouk %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-maurais24a %I PMLR %P 35138--35162 %U https://proceedings.mlr.press/v235/maurais24a.html %V 235 %X We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, $\pi_0^{1-t} \pi_1^t$, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Ampère equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.
APA
Maurais, A. & Marzouk, Y.. (2024). Sampling in Unit Time with Kernel Fisher-Rao Flow. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:35138-35162 Available from https://proceedings.mlr.press/v235/maurais24a.html.

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