Stochastic Localization via Iterative Posterior Sampling
Louis Grenioux Maxence Noble Marylou Gabrié Alain Oliviero Durmus
Building upon score-based learning, new interest in stochastic localization
techniques has recently emerged. In these models, one seeks to noise a sample
from the data distribution through a stochastic process, called observation
process, and progressively learns a denoiser associated to this dynamics. Apart
from specific applications, the use of stochastic localization for the problem
of sampling from an unnormalized target density has not been explored
extensively. This work contributes to fill this gap. We consider a general
stochastic localization framework and introduce an explicit class of
observation processes, associated with flexible denoising schedules. We provide
a complete methodology, $\textit{Stochastic Localization via Iterative
Posterior Sampling}$ (SLIPS), to obtain approximate samples of this dynamics,
and as a by-product, samples from the target distribution. Our scheme is based
on a Markov chain Monte Carlo estimation of the denoiser and comes with
detailed practical guidelines. We illustrate the benefits and applicability of
SLIPS on several benchmarks, including Gaussian mixtures in increasing
dimensions, Bayesian logistic regression and a high-dimensional field system
from statistical-mechanics.