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).

翻译：我们给出一个简单、统一且近乎紧的界限，用于通过In-and-Out算法结合指数提升从热启动中对任意对数凹分布进行采样。分析中的主要新要素是对提升分布庞加莱常数的改进界限。作为结果，所得收敛率在约束设置（例如，限制在凸体上的高斯分布）和良条件设置（例如，强对数凹且光滑的密度）下均近乎紧。