A unified complexity bound for logconcave sampling
This work offers a unified analysis that nearly matches known lower bounds for logconcave sampling, benefiting the broader sampling and optimization community.
The paper provides a nearly tight bound for sampling from arbitrary logconcave distributions using the In-and-Out algorithm with exponential lifting, achieving near-optimal convergence rates for both constrained and well-conditioned settings.
We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).