Log-Concave Sampling on Compact Supports: A Versatile Proximal Framework

Lu Yu

Keywords: Markov Chain Monte Carlo, Kinetic Langevin, Langevin algorithm, Midpoint method, Mixing rate, Proximal method

Abstract: In this paper, we investigate the theoretical aspects of sampling from strongly log-concave distributions defined on convex and compact supports. We propose a general proximal framework that involves projecting onto the constrained set, which is highly flexible and supports various projection options. Specifically, we consider the cases of Euclidean and Gauge projections, with the latter having the advantage of being performed efficiently using a membership oracle. This framework can be seamlessly integrated with multiple sampling methods. Our analysis focuses on Langevin-type sampling algorithms within the context of constrained sampling. We provide nonasymptotic upper bounds on the $W_1$ and $W_2$ errors, offering a detailed comparison of the performance of these methods in constrained sampling.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Submission Number: 627