A unified complexity bound for logconcave sampling

2026-06-10

AI Summary

This work proposes a unified framework for sampling from arbitrary log-concave distributions by integrating the In-and-Out algorithm with exponential boosting techniques. By refining the upper bound on the Poincaré constant of the boosted distribution, the method achieves near-optimal convergence rates—matching theoretical lower bounds—for two canonical settings: constrained distributions (e.g., Gaussian restricted to a convex body) and well-conditioned densities (e.g., strongly log-concave and smooth). Combining Poincaré inequality analysis with Markov chain Monte Carlo (MCMC) methodology, the approach efficiently samples from a warm-start initial distribution and establishes nearly tight, unified complexity upper bounds for log-concave sampling.

Abstract

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).

Problem

• logconcave sampling
• complexity bound
• Poincaré constant
• In-and-Out algorithm
• exponential lifting

Innovation

• logconcave sampling
• Poincaré constant
• exponential lifting
• In-and-Out algorithm
• convergence rate